Transcriber's note:

 A few typographical errors have been corrected: they are listed at the end of the text. 

 In mathematical formulae the carat (^) and underscore (_) introduce superscripts or subscripts respectively, of one character or a group enclosed in curly braces ({xyz}). Elsewhere underscores delimit italics in the text, and braces enclose the original page numbers thus {123}. 

BY AUGUSTUS DE MORGAN

A BUDGET OFPARADOXES

REPRINTED WITH THE AUTHOR'S ADDITIONS FROM THE ATHENAEUM

SECOND EDITION EDITED BY DAVID EUGENE SMITH

WITH A NEW INTRODUCTION BY ERNEST NAGEL

PROFESSOR OF PHILOSOPHY, COLUMBIA UNIVERSITY

UNABRIDGED EDITION--TWO VOLUMES BOUND AS ONE

Volume II

DOVER PUBLICATIONS, INC. , NEW YORK

 * * * * *

_This new Dover Edition, published in 1954, is an unabridged republication of the Second Editionof 1915, with a new introduction by Professor Ernest Nagel. _

_Copyright 1954 by Dover Publications, Inc. Manufactured in the United States of America_

 * * * * *

{1}

A BUDGET OF PARADOXES. 

VOLUME II. 

ON SOME PHILOSOPHICAL ATHEISTS. 

With the general run of the philosophical atheists of the last century thenotion of a God was an hypothesis. There was left an admitted possibilitythat the vague somewhat which went by more names than one, might bepersonal, intelligent, and superintendent. In the works of Laplace, [1] whois sometimes called an atheist from his writings, there is nothing fromwhich such an inference can be drawn: unless indeed a Reverend Fellow ofthe Royal Society may be held to be the fool who said in his heart, etc. , etc. , if his contributions to the _Philosophical Transactions_ go no higherthan _nature_. The following anecdote is well known in Paris, but has neverbeen printed entire. Laplace once went in form to present some edition ofhis "Système du Monde" to the First Consul, or Emperor. Napoleon, whom somewags had told that this book contained no mention of the name of God, andwho was fond of putting embarrassing questions, received it with--"M. Laplace, they tell me you have written this large book on the system of theuniverse, and have never even mentioned its Creator. " Laplace, who, thoughthe most supple of politicians, was as stiff as a martyr on every point ofhis philosophy or religion (e. G. , even under Charles X he never concealedhis dislike of the priests), drew himself up and answered {2} bluntly, "Jen'avais pas besoin de cette hypothèse-là. "[2] Napoleon, greatly amused, told this reply to Lagrange, who exclaimed, "Ah! c'est une belle hypothèse;ça explique beaucoup de choses. "[3]

It is commonly said that the last words of Laplace were, "Ce que nousconnaissons est peu de chose; ce que nous ignorons est immense. "[4] Thislooks like a parody on Newton's pebbles:[5] the following is the trueaccount; it comes to me through one remove from Poisson. [6] After thepublication (in 1825) of the fifth volume of the _Mécanique Céleste_, Laplace became gradually weaker, and with it musing and abstracted. Hethought much on the great problems of existence, and often muttered tohimself, _Qu'est ce que c'est que tout cela!_[7] After many alternations, he appeared at last so permanently prostrated that his family applied tohis favorite pupil, M. Poisson, to try to get a word from him. Poisson paida visit, and after a few words of salutation, said, "J'ai une bonnenouvelle à vous annoncer: on a reçu au Bureau des Longitudes une lettred'Allemagne annonçant que M. Bessel a vérifié par l'observation vosdécouvertes théoriques sur les satellites de Jupiter. "[8] Laplace openedhis eyes and answered with deep {3} gravity, "_L'homme ne poursuit que deschimères_. "[9] He never spoke again. His death took place March 5, 1827. 

The language used by the two great geometers illustrates what I have said:a supreme and guiding intelligence--apart from a blind rule called _natureof things_--was an _hypothesis_. The absolute denial of such a ruling powerwas not in the plan of the higher philosophers: it was left for the smallerfry. A round assertion of the non-existence of anything which stands in theway is the refuge of a certain class of minds: but it succeeds only withthings subjective; the objective offers resistance. A philosopher of theappropriative class tried it upon the constable who appropriated _him_: Ideny your existence, said he; Come along all the same, said theunpsychological policeman. 

Euler[10] was a believer in God, downright and straightforward. Thefollowing story is told by Thiébault, [11] in his _Souvenirs de vingt ans deséjour à Berlin_, [12] published in his old age, about 1804. This volume wasfully received as trustworthy; and Marshall Mollendorff[13] told the Duc deBassano[14] in 1807 that it was the most veracious of books written by themost honest of men. Thiébault says that he has no personal knowledge of thetruth of the story, but {4} that it was believed throughout the whole ofthe north of Europe. Diderot[15] paid a visit to the Russian Court at theinvitation of the Empress. He conversed very freely, and gave the youngermembers of the Court circle a good deal of lively atheism. The Empress wasmuch amused, but some of her councillors suggested that it might bedesirable to check these expositions of doctrine. The Empress did not liketo put a direct muzzle on her guest's tongue, so the following plot wascontrived. Diderot was informed that a learned mathematician was inpossession of an algebraical demonstration of the existence of God, andwould give it him before all the Court, if he desired to hear it. Diderotgladly consented: though the name of the mathematician is not given, it wasEuler. He advanced towards Diderot, and said gravely, and in a tone ofperfect conviction: _Monsieur, _ (a + b^{n}) / n = x, _donc Dieu existe;répondez!_[16] Diderot, to whom algebra was Hebrew, was embarrassed anddisconcerted; while peals of laughter rose on all sides. He askedpermission to return to France at once, which was granted. 

ROTATION OF THE MOON. 

 An examination of the Astronomical doctrine of the Moon's rotation. By J. L. [17] Edinburgh, 1847, 8vo. 

A systematic attack of the character afterwards made with less skill andmore notice by Mr. Jellinger Symons. 

July 1866, J. L. Appears as Mr. James Laurie, with a new pamphlet "TheAstronomical doctrines of the Moon's rotation ... " Edinburgh. Of all theworks I have seen on the question, this is the most confident, and thesorest. {5} A writer on astronomy said of Mr. Jellinger Symons, [18] "Ofcourse he convinced no one who knew anything of the subject. " This"ungenerous slur" on the speculator's memory appears to have been keenlyfelt; but its truth is admitted. Those who knew anything of the subject are"the so-called men of science, " whose three P's were assailed; prestige, pride, and prejudice: this the author tries to effect for himself withthree Q's; quibble, quirk, and quiddity. He explains that the Scribes andPharisees would not hear Jesus, and that the lordly bishop of Rome will notcast his tiara and keys at the feet of the "humble presbyter" who now playsthe part of pope in Scotland. I do not know whom he means: but perhaps thefriends of the presbyter-pope may consider this an ungenerous slur. Thebest proof of the astronomer is just such "as might have been expected fromthe merest of blockheads"; but as the giver is of course not a blockhead, this circumstance shows how deeply blinded by prejudice he must be. 

Of course the paradoxers do not persuade any persons who know theirsubjects: and so these Scribes and Pharisees reject the Messiah. We mustsuppose that the makers of this comparison are Christians: for if theythought the Messiah an enthusiast or an impostor, they would be absurd incomparing those who reject what they take for truth with others who oncerejected what they take for falsehood. And if Christians, they are bothirreverent and blind to all analogy. The Messiah, with His Divine missionproved by miracles which all might see who chose to look, is degraded intoa prototype of James Laurie, ingeniously astronomizing upon ignorantgeometry and false logic, and comparing to blockheads those who expose hisnonsense. Their comparison is as foolish as--supposing {6} themChristians--it is profane: but, like errors in general, its other endpoints to truth. There were Pseudochrists and Antichrists; and aConcordance would find the real forerunners of all the paradoxers. But theyare not so clever as the old false prophets: there are none of whom weshould be inclined to say that, if it were possible, they would deceive thevery educated. Not an Egyptian among them all can make uproar enough tocollect four thousand men that are murderers--of common sense--to lead outinto the wilderness. Nothing, says the motto of this work, is so difficultto destroy as the errors and false facts propagated by illustrious menwhose words have authority. I deny it altogether. There are things muchmore difficult to destroy: it is much more difficult to destroy the truthsand real facts supported by such men. And again, it is much more difficultto prevent men of no authority from setting up false pretensions; and it ismuch more difficult to destroy assertions of fancy speculation. Many anerror of thought and learning has fallen before a gradual growth ofthoughtful and learned opposition. But such things as the quadrature of thecircle, etc. , are never put down. And why? Because thought can influencethought, but thought cannot influence self-conceit: learning can annihilatelearning: but learning cannot annihilate ignorance. A sword may cut throughan iron bar; and the severed ends will not reunite: let it go through theair, and the yielding substance is whole again in a moment. 

 Miracles _versus_ Nature: being an application of certain propositions in the theory of chances to the Christian miracles. By Protimalethes. [19] Cambridge, 1847, 8vo. 

The theory, as may be supposed, is carried further than most students ofthe subject would hold defensible. 

{7}

 An astronomical Lecture. By the Rev. R. Wilson. [20] Greenock, 1847, 12mo. 

Against the moon's rotation on her axis. 

 [Handed about in the streets in 1847: I quote the whole:] Important discovery in astronomy, communicated to the Astronomer Royal, December 21st, 1846. That the Sun revolve round the Planets in 25748-2/5 years, in consequence of the combined attraction of the planets and their satellites, and that the Earth revolve round the Moon in 18 years and 228 days. D. T. GLAZIER [altered with a pen into GLAZION. ] Price one penny. 

1847. In the _United Service Magazine_ for September, 1847, Mrs. Borron, [21] of Shrewsbury, published some remarks tending to impeach thefact that Neptune, the planet found by Galle, [22] really was the planetwhich Le Verrier and Adams[23] had a right to claim. This was followed(September 14) by two pages, separately circulated, of "FurtherObservations upon the Planets Neptune and Uranus, with a Theory ofPerturbations"; and (October 19, 1848) by three pages of "A Review of M. Leverrier's Exposition. " Several persons, when the remarkable discovery wasmade, contended that the planet actually discovered was an intruder; andthe future histories of the discovery must contain some account of thislittle afterpiece. Tim Linkinwater's theory that there is no place likeLondon for coincidences, would have been utterly overthrown in favor ofwhat they used to call the celestial spaces, if there had been a planetwhich by chance was put {8} near the place assigned to Neptune at the timewhen the discovery was made. 

EARLY IDEAS OF AVIATION. 

 Aerial Navigation; containing a description of a proposed flying machine, on a new principle. By Dædalus Britannicus. London, 1847, 8vo. 

In 1842-43 a Mr. Henson[24] had proposed what he called an aeronautsteam-engine, and a Bill was brought in to incorporate an "Aerial TransitCompany. " The present plan is altogether different, the moving power beingthe explosion of mixed hydrogen and air. Nothing came of it--not even aBill. What the final destiny of the balloon may be no one knows: it mayreasonably be suspected that difficulties will at last be overcome. Darwin, [25] in his "Botanic Garden" (1781), has the following prophecy:

 "Soon shall thy arm, unconquered Steam! afar Drag the slow barge, or drive the rapid car; Or, on wide-waving wings expanded, bear The flying chariot through the fields of air. "

Darwin's contemporaries, no doubt, smiled pity on the poor man. It is worthnote that the two true prophecies have been fulfilled in a sense differentfrom that of the predictions. Darwin was thinking of the suggestion ofJonathan Hulls, [26] when he spoke of dragging the slow barge: it is onlyvery recently that the steam-tug has been employed on the canals. The carwas to be driven, not drawn, and on the common roads. Perhaps, the flyingchariot will {9} be something of a character which we cannot imagine, evenwith the two prophecies and their fulfilments to help us. [27]

THE SECRET OF THE UNIVERSE DIVULGED. 

 A book for the public. New Discovery. The causes of the circulation of the blood; and the true nature of the planetary system. London, 1848, 8vo. 

Light is the sustainer of motion both in the earth and in the blood. Thenatural standard, the pulse of a person in health, four beats to onerespiration, gives the natural second, which is the measure of the earth'sprogress in its daily revolution. The Greek fable of the Titans is anelaborate exposition of the atomic theory: but any attempt to convincelearned classics would only meet their derision; so much does long-fosteredprejudice stand in the way of truth. The author complains bitterly that menof science will not attend to him and others like him: he observes, that"in the time occupied in declining, a man of science might test themerits. " This is, alas! too true; so well do applicants of this kind knowhow to stick on. But every rule has its exception: I have heard of one. Thelate Lord Spencer[28]--the Lord Althorp of the House of Commons--told methat a speculator once got access to him at the Home Office, and wasproceeding to unfold his way of serving the public. "I do not understandthese things, " said Lord Althorp, "but I happen to have ---- (naming aneminent engineer) upstairs; suppose you talk to him on the subject. " Thediscoverer went up, and in half-an-hour returned, and said, "I am very muchobliged to your Lordship for introducing me to Mr. ----; he has convincedme {10} that I am quite wrong. " I supposed, when I heard the story--but itwould not have been seemly to say it--that Lord A. Exhaled candor andsense, which infected those who came within reach: he would have done so, if anybody. 

THE TRISECTION AND QUADRATURE AGAIN. 

 A method to trisect a series of angles having relation to each other; also another to trisect any given angle. By James Sabben. 1848 (two quarto pages). 

"The consequence of years of intense thought": very likely, and very sad. 

1848. The following was sent to me in manuscript. I give the whole of it:

"_Quadrature of the Circle_. --A quadrant is a curvilinear angle traversinground and at an equal distance from a given point, called a center, no twopoints in the curve being at the same angle, but irreptitiously graduatingfrom 90 to 60. It is therefore a mean angle of 90 and 60, which is 75, because it is more than 60, and less than 90, approximately from 60 to 90, and from 90 to 60, with equal generation in each irreptitiousapproximation, therefore meeting in 75, and which is the mean angle of thequadrant. 

"Or suppose a line drawn from a given point at 90, and from the same pointat 60. Let each of these lines revolve on this point toward each other atan equal ratio. They will become one line at 75, and bisect the curve, which is one-sixth of the entire circle. The result, taking 16 as adiameter, gives an area of 201. 072400, and a circumference of 50. 2681. 

"The original conception, its natural harmony, and the result, to my ownmind is a demonstrative truth, which I presume it right to make known, though perhaps at the hazard of unpleasant if not uncourteous remarks. "

I have added punctuation: the handwriting and spelling {11} are those of aneducated person; the word _irreptitious_ is indubitable. The whole is anatural curiosity. 

 The quadrature and exact area of the circle demonstrated. By Wm. Peters. 8vo. _n. D. _ (circa 1848). [29]

 Suggestions as to the necessity for a revolution in philosophy; and prospectus for the establishment of a new quarterly, to be called the _Physical Philosopher and Heterodox Review_. By Q. E. D. 8vo. 1848. 

These works are by one author, who also published, as appears byadvertisement, "Newton rescued from the precipitancy of his followers through a centuryand a half, "[30] and "Dangers along a coast by correcting (as it is called)a ship's reckoning by bearings of the land at night fall, or in a fog, nearly out of print. Subscriptions are requested for a new edition. "

The area of a circle is made four-fifths of the circumscribed square:proved on an assumption which it is purposed to explain in a longeressay. [31] The author, as Q. E. D. , was in controversy with the _Athenæum_journal, and criticised a correspondent, D. , who wrote against a certainclass of discoverers. He believed the common theories of hydrostatics to bewrong, and one of his questions was:

"Have you ever taken into account anent gravity and gravitation the factthat a five grain cube of cork will of itself half sink in the water, whilst it will take 20 grains of brass, which will sink of itself, to pullunder the other half? Fit this if you can, friend D. , to your notions ofgravity and specific gravity, as applied to the construction of a universallaw of gravitation. "

This the _Athenæum_ published--but without some Italics, for which theeditor was sharply reproved, as a sufficient {12} specimen of the _quoderat_ D. _monstrandum_: on which the author remarks--"D, --Wherefore the ecaret? is it D apostrophe? D', D'M, D'Mo, D'Monstrandum; we cannot find the_wit_ of it. " This I conjecture to contain an illusion to the name of thesupposed author; but whether De Mocritus, De Mosthenes, or De Moivre wasintended, I am not willing to decide. 

 The Scriptural Calendar and Chronological Reformer, for the statute year 1849. Including a review of recent publications on the Sabbath question. London, 1849, 12mo. [32]

This is the almanac of a sect of Christians who keep the Jewish Sabbath, having a chapel at Mill Yard, Goodman's Fields. They wrote controversialworks, and perhaps do so still; but I never chanced to see one. 

 Geometry _versus_ Algebra; or the trisection of an angle geometrically solved. By W. Upton, B. A. [33] Bath (circa 1849). 8vo. 

The author published two tracts under this title, containing differentalleged proofs: but neither gives any notice of the change. Both containthe same preface, complaining of the British Association for refusing toexamine the production. I suppose that the author, finding his first proofwrong, invented the second, of which the Association never had the offer;and, feeling sure that they would have equally refused to examine thesecond, thought it justifiable to {13} present that second as the one whichthey had refused. Mr. Upton has discovered that the common way of findingthe circumference is wrong, would set it right if he had leisure, and, inthe mean time, has solved the problem of the duplication of the cube. 

_The trisector of an angle, if he demand attention from any mathematician, is bound to produce, from his construction, an expression for the sine orcosine of the third part of any angle, in terms of the sine or cosine ofthe angle itself, obtained by help of no higher than the square root. _ Themathematician knows that such a thing cannot be; but the trisectorvirtually says it can be, and is bound to produce it, to save time. This isthe misfortune of most of the solvers of the celebrated problems, that theyhave not knowledge enough to present those consequences of their results bywhich they can be easily judged. Sometimes they have the knowledge andquibble out of the use of it. In many cases a person makes an honestbeginning and presents what he is sure is a solution. By conference withothers he at last feels uneasy, fears the light, and puts self-love in theway of it. Dishonesty sometimes follows. The speculators are, as a class, very apt to imagine that the mathematicians are in fraudulent confederacyagainst them: I ought rather to say that each one of them consents to themode in which the rest are treated, and fancies conspiracy against himself. The mania of conspiracy is a very curious subject. I do not mean theseremarks to apply to the author before me. 

One of Mr. Upton's trisections, if true, would prove the truth of thefollowing equation:

 3 cos ([theta] / 3) = 1 + [root](4 - sin^2[theta])

which is certainly false. [34]

{14}

In 1852 I examined a terrific construction, at the request of the late Dr. Wallich, [35] who was anxious to persuade a poor countryman of his, thattrisection of the angle was waste of time. One of the principles was, that"magnitude and direction determine each other. " The construction wasequivalent to the assertion that, [theta] being any angle, the cosine ofits third part is

 sin 3[theta] . Cos(5[theta]/2) + sin^2 [theta] sin (5[theta]/2)

divided by the square root of

 sin^2 3[theta] . Cos^2 (5[theta]/2) + sin^4 [theta] + sin 3[theta] . Sin 5[theta] . Sin^2 [theta]. 

This is from my rough notes, and I believe it is correct. [36] It is sonearly true, unless the angle be very obtuse, that common drawing, appliedto the construction, will not detect the error. There are many formulae ofthis kind: and I have several times found a speculator who has discoveredthe corresponding construction, has seen the approximate success of hisdrawing--often as great as absolute truth could give in graphicalpractice, --and has then set about his demonstration, in which he alwayssucceeds to his own content. 

There is a trisection of which I have lost both cutting and reference: Ithink it is in the _United Service Journal_. I could not detect any errorin it, though certain there must {15} be one. At least I discovered thattwo parts of the diagram were incompatible unless a certain point lay inline with two others, by which the angle to be trisected--and which wastrisected--was bound to be either 0° or 180°. 

Aug. 22, 1866. Mr. Upton sticks to his subject. He has just published "TheUptonian Trisection. Respectfully dedicated to the schoolmasters of theUnited Kingdom. " It seems to be a new attempt. He takes no notice of thesentence I have put in italics: nor does he mention my notice of him, unless he means to include me among those by whom he has been "ridiculedand sneered at" or "branded as a brainless heretic. " I did neither one northe other: I thought Mr. Upton a paradoxer to whom it was likely to beworth while to propound the definite assertion now in italics; and Mr. Upton does not find it convenient to take issue on the point. He prefersgeneral assertions about algebra. So long as he cannot meet algebra on theabove question, he may issue as many "respectful challenges" to themathematicians as he can find paper to write: he will meet with noattention. 

There is one trisection which is of more importance than that of the angle. It is easy to get half the paper on which you write for margin; or aquarter; but very troublesome to get a third. Show us how, easily andcertainly, to fold the paper into three, and you will be a real benefactorto society. 

Early in the century there was a Turkish trisector of the angle, HusseinEffendi, who published two methods. He was the father of Ameen Bey, who waswell known in England thirty years ago as a most amiable and cultivatedgentleman and an excellent mathematician. He was then a student atCambridge; and he died, years ago, in command of the army in Syria. HusseinEffendi was instructed in mathematics by Ingliz Selim Effendi, whotranslated a work {16} of Bonnycastle[37] into Turkish. [38] This Englishmanwas Richard Baily, brother of Francis Baily[39] the astronomer, whoemigrated to Turkey in his youth, and adopted the manners of the Turks, butwhether their religion also I never heard, though I should suppose he did. 

I now give the letters from the agricultural laborer and his friend, described on page 12, Vol. I. They are curiosities; and the history of thequadrature can never be well written without some specimens of this kind:

"Doctor Morgan, Sir. Permit me to address you

"Brute Creation may perhaps enjoy the faculty of beholding visible thingswith a more penitrating eye than ourselves. But Spiritual objects are asfar out of their reach as though they had no being

"Nearest therefore to the brute Creation are those men who Supposethemselves to be so far governed by external objects as to believe nothingbut what they See and feel And Can accomedate to their Shallowunderstanding and Imaginations

"My Dear Sir Let us all Consult ourselves by the wise proverb. 

"I believe that evry man^s merit & ability aught to be appreciated andvalued In proportion to its worth & utility

"In whatever State or Circumstances they may fortunately or unfortunatelybe placed

"And happy it is for evry man to know his worth and place

"When a Gentleman of your Standing in Society Clad with those honors Cannot understand or Solve a problem That is explicitly explained by words andLetters and {17} mathematically operated by figuers He had best consult thewise proverd

"Do that which thou Canst understand and Comprehend for thy good. 

"I would recommend that Such Gentleman Change his business

"And appropriate his time and attention to a Sunday School to Learn what heCould and keep the Litle Children form durting their Close

"With Sincere feelings of Gratitude for your weakness and Inability I am

 "Sir your Superior in Mathematics ----"

"1849 June th29. "

"Dor Morgin Sir

"I wrote and Sent my work to Professor ---- of ---- State of ---- UnitedStates

"I am now in the possession of the facts that he highly approves of mywork. And Says he will Insure me Reward in the States

"I write this that you may understand that I have knowledge of the unfairway that I am treated In my own nati County

"I am told and have reasons to believe that it is the Clergy that treat meso unjust. 

"I am not Desirous of heaping Disonors upon my own nation. But if I have toLeave this kingdom without my Just dues. The world Shall know how I am andhave been treated. 

 "I am Sir Desirous of my "Just dues ----"

"1849 July 3. "

"July 7th, 1849. 

"Sir, I have been given to understand that a friend of mine one whom Ishall never be ashamed to acknowledge as {18} such tho' lowly his origine;nay not only not ashamed but proud of doing so for I am one of those whoesteem and respect a man according to his ability and probity, deeming withDr. Watts 'that the mind is the standard of the man, '[40] has laid beforeyou and asked your opinion of his extraordinary performance, viz. Thequadrature of the circle, he did this with the firmest belief that youwould not only treat the matter in a straightforward manner but with theconviction that from your known or supposed knowledge of mathematicks wouldhave given an upright and honorable decision upon the subject; but thequestion is have you done so? Could I say yes I would with the greatest ofpleasure and have congratulated you upon your decision whatever it mighthave been but I am sorry to say that I cannot your letter is a paltryevasion, you say 'that it is a great pity that you (Mr. ----) should haveattempted this (the quadrature of the circle) for your mathematicalknowledge is not sufficient to make you know in what the problem consists, 'you don't say in what it does consist _according to your ideas_, oh! nonothing of the sort, you enter into no disquisition upon the subject inorder to show where you think Mr. ---- is wrong and why you have not issimply--_because you cannot_--you know that he has done it and what is if Iam not wrongly informed _you have been heard to say so_. He has done whatyou nor any other mathematician as those who call themselves such havedone. And what is the reason that you will not candidly acknowledge to himas you have to others that he has squared the circle shall I tell you? itis because he has performed the feat to obtain the glory of whichmathematicians have battled from time immemorial that they might encircletheir brows with a wreath of laurels far more glorious than ever conquerorwon it is simply this that it is a poor man a {19} humble artisan who hasgained that victory that you don't like to acknowledge it you don't like tobe beaten and worse to acknowledge that you have miscalculated, you have inshort too small a soul to acknowledge that he is right. 

"I was asked my opinion and _I_ gave it unhesitatingly in the affirmativeand I am backed in my opinion not only by Mr. ---- a mathematician andwatchmaker residing in the boro of Southwark but by no less an authoritythan the Professor of mathematics of ---- College ---- ---- United StatesMr. ---- and I presume that he at least is your equal as an authority andMr. ---- says that the government of the U. S. Will recompense M. D. For thediscovery he has made if so what a reflection upon Old england the boastedland of freedom the nursery of arts and sciences that her sons are obligedto go to a foreign country to obtain that recompense to which they arejustly entitled

"In conclusion I had to contradict an assertion you made to the effect that'there is not nor ever was any reward offered by the government of thiscountry for the discovery of the quadrature of the circle. ' I beg to informyou that there _was_ but that it having been deemed an impossibility thegovernment has withdrawn it. I do this upon no less an authority than theMarquis of Northampton. [41]

 "I am, sir, yours ----"

"Dr. Morgan. "

THE MOON'S ROTATION. 

 Notes on the Kinematic Effects of Revolution and Rotation, with reference to the Motions of the Moon and of the earth. By Henry Perigal, Jun. Esq. London, 1846-1849, 8vo. 

 On the misuse of technical terms. Ambiguity of the terms _Rotation_ and _Revolution_, owing to the double meaning improperly {20} attributed to each of the words. (No date nor place, but by Mr. Perigal, [42] I have no doubt, and containing letters of 1849 and 1850. )

 The moon controversy. Facts _v. _ Definitions. By H. P. , Jun. London, 1856, 8vo. (pp. 4. )

Mr. Henry Perigal helped me twenty years ago with the diagrams, direct fromthe lathe to the wood, for the article "Trochoidal Curves, " in the _PennyCyclopædia_: these cuts add very greatly to the value of the article, which, indeed, could not have been made intelligible without them. He hashad many years' experience, as an amateur turner, in combination of doubleand triple circular motions, and has published valuable diagrams inprofusion. A person to whom the double circular motion is familiar in thelathe naturally looks upon one circle moving upon another as in _simple_motion, if the second circle be fixed to the revolving radius, so that oneand the same point of the moving circle travels upon the fixed circle. Mr. Perigal commenced his attack upon the moon for moving about her axis, inthe first of the tracts above, ten years before Mr. Jellinger Symons;[43]but he did not think it necessary to make it a subject for the _Times_newspaper. His familiarity with combined motions enabled him to handle hisarguments much better than Mr. J. Symons could do: in fact, he is theclearest assailant of the lot which turned out with Mr. J. Symons. But heis as wrong as the rest. The assault is now, I suppose, abandoned, until itbecomes epidemic again. This it will do: it is one of those fallacies whichare very tempting. There was a dispute on the subject in 1748, betweenJames Ferguson[44] and an anonymous opponent; and I think there have beenothers. 

{21}

A poet appears in the field (July 19, 1863) who calls himself Cyclops, andwrites four octavo pages. He makes a distinction between _rotation_ and_revolution_; and his doctrines and phrases are so like those of Mr. Perigal that he is a follower at least. One of his arguments has so oftenbeen used that it is worth while to cite it:

 "Would Mathematicals--forsooth-- If true, have failed to prove its truth? Would not they--if they could--submit Some overwhelming proofs of it? But still it totters _proofless_! Hence There's strong presumptive evidence None do--or can--such proof profound Because _the dogma is unsound_. For, were there means of doing so, They would have proved it long ago. "

This is only one of the alternatives. Proof requires a person who can giveand a person who can receive. I feel inspired to add the following:

 "A blind man said, As to the Sun, I'll take my Bible oath there's none; For if there had been one to show They would have shown it long ago. How came he such a goose to be? Did he not know he couldn't see? Not he!"

The absurdity of the verses is in the argument. The writer was not soignorant or so dishonest as to affirm that nothing had been offered by theother side as proof; accordingly, his syllogism amounts to this: If yourproposition were true, you could have given proof satisfactory to _me_; butthis you have not done, therefore, your proposition is not true. 

The echoes of the moon-controversy reached Benares in 1857, in which yearwas there published a pamphlet "Does the Moon Rotate?" in Sanskrit andEnglish. The {22} arguments are much the same as those of the discussion athome. 

ON THE NAMES OF RELIGIOUS BODIES. 

We see that there are paradoxers in argument as well as in assertion offact: my plan does not bring me much into contact with these; but anotherinstance may be useful. Sects, whether religious or political, givethemselves names which, in meaning, are claimed also by their opponents;loyal, liberal, conservative (of good), etc. Have been severallyappropriated by parties. _Whig_ and _Tory_ are unobjectionable names: thefirst--which occurs in English ballad as well as in Scotland--is sourmilk;[45] the second is a robber. In theology, the Greek Church is_Orthodox_, the Roman is _Catholic_, the modern Puritan is _Evangelical_, etc. 

The word _Christian_ (Vol. I, p. 248[46]) is an instance. When words begin, they carry their meanings. The Jews, who had their Messiah to come, and thefollowers of Jesus of Nazareth, who took _Him_ for their Messiah, were both_Christians_ (which means _Messianites_): the Jews would never haveinvented the term to signify _Jesuans_, nor would the disciples haveinvented such an ambiguous term for themselves; had they done so, the Jewswould have disputed it, as they would have done in later times if they hadhad fair play. The Jews of our day, I see by their newspapers, speak ofJesus Christ as the _Rabbi Joshua_. But the {23} heathens, who knew littleor nothing about the Jewish hope, would naturally apply the term_Christians_ to the only followers of a _Messiah_ of whom they had heard. For the _Jesuans_ invaded them in a missionary way; while the Jews did notattempt, at least openly, to make proselytes. 

All such words as Catholic, etc. , are well enough as mere nomenclature; andthe world falls for the most part, into any names which parties choose togive themselves. Silly people found inferences on this concession; and, asusually happens, they can cite some of their betters. St. Augustine, [47] afreakish arguer, or, to put it in the way of an old writer, _lectorem nemultiloquii tædio fastidiat, Punicis quibusdam argutiis recrearesolet_, [48] asks, with triumph, to what chapel a stranger would bedirected, if he inquired the way to the _Catholic assembly_. But the bestexhibition of this kind in our own century is that made by the excellentDr. John Milner, [49] in a work (first published in 1801 or 1802) which Isuppose still circulates, "The End of Religious Controversy": a startlingtitle which, so far as its truth is concerned, might as well have been "Thefloor of the bottomless pit. " This writer, whom every one of his readerswill swear to have been a worthy soul, though many, even of his own sect, will not admire some of his logic, speaks as follows:

"Letter xxv. _On the true Church being Catholic. _ In treating of this thirdmark of the true Church, as expressed in our common creed, I feel myspirits sink within me, and I am almost tempted to throw away my pen indespair. For what chance is there of opening the eyes of candid Protestantsto the other marks of the Church, if they are capable of keeping them shutto this? Every time they address the {24} God of Truth, either in solemnworship or in private devotion [stretch of rhetoric], they are forced, eachof them, to repeat: _I believe in_ THE CATHOLIC _Church_, and yet if I askany of them the question: _Are you a_ CATHOLIC? he is sure to answer me, _No, I am a_ PROTESTANT! Was there ever a more glaring instance ofinconsistency and self-condemnation among rational beings!"

 "John Milner, honest and true, Did what honest people still may do, If they write for the many and not for the few, But what by and bye they must eschew. "

He _shortened his clause_; and for a reason. If he had used the wholeepithet which he knew so well, any one might have given his argument ahalf-turn. Had he written, as he ought, "the _Holy_ Catholic Church" andthen argued as above, some sly Protestant would have parodied him with "andyet if I ask any of them the question: _Are you_ HOLY? he is sure to answerme _No, I am a_ SINNER. " To take the adjective from the Church, and applyit to the individual partisan, is recognized slipslop, but not ground ofargument. If Dr. M. Had asked his Protestant whether he belonged to theCatholic _Church_, the answer would have been Yes, but not to the Romanbranch. When he put his question as he did, he was rightly answered and inhis own division. This leaving out words is a common practice, especiallywhen the omitter is in authority, and cannot be exposed. A year or two agoa bishop wrote a snubbing letter to a poor parson, who had complained thathe was obliged, in burial, to send the worst of sinners to everlastinghappiness. The bishop sternly said, "_hope_[50] is not _assurance_. " {25}Could the clergyman have dared to answer, he would have said, "No, my Lord!but '_sure and certain_ hope' is as like assurance as a _minikin_ man islike a dwarf. " Sad to say, a theologian must be illogical: I feel sure thatif you took the clearest headed writer on logic that ever lived, and made abishop of him, he would be shamed by his own books in a twelvemonth. 

Milner's sophism is glaring: but why should Dr. Milner be wiser than St. Augustine, one of his teachers? I am tempted to let out the true derivationof the word _Catholic, as exclusively applied to the Church of Rome_. Allcan find it who have access to the _Rituale_ of Bonaventura Piscator[51](lib. I. C. 12, _de nomine Sacræ Ecclesiæ_, p. 87 of the Venice {26} folioof 1537). I am told that there is a _Rituale_ in the Index Expurgatorius, but I have not thought it worth while to examine whether this be the one: Iam rather inclined to think, as I have heard elsewhere, that the book washeld too dangerous for the faithful to know of it, even by a prohibition:it would not surprise me at all if Roman Christians should deny itsexistence. [52]

It amuses me to give, at a great distance of time, a small Rowland for asmall Oliver, [53] which I received, _de par l'Eglise_, [54] so far as lay inthe Oliver-carrier more than twenty years ago. The following contributionof mine to _Notes and Queries_ (3d Ser. Vi. P. 175, Aug. 27, 1864) willexplain what I say. There had been a complaint that a contributor had usedthe term _Papist_, which a very excellent dignitary of the Papal systempronounced an offensive term:

PAPIST. 

The term _papist_ should be stripped of all except its etymologicalmeaning, and applied to those who give the higher and final authority tothe declaration _ex cathedrâ_[55] of the Pope. See Dr. Wiseman's[56]article, _Catholic Church_, in the _Penny Cyclopædia_. 

What is one to do about these names? First, it is clear that offenceshould, when possible, be avoided: secondly, no one must be required togive a name which favors _any_ assumption made by those to whom it isgiven, and not {27} granted by those who give it. Thus the subdivisionwhich calls itself distinctly _Evangelical_ has no right to expect othersto concede the title. Now the word _Catholic_, of course, falls under thisrule; and even _Roman Catholic_ may be refused to those who would restrictthe word _Catholic_ to themselves. _Roman Christian_ is unobjectionable, since the Roman Church does not deny the name of Christian to those whomshe calls heretics. No one is bound in this matter by Acts of Parliament. In many cases, no doubt, names which have offensive association are usedmerely by habit, sometimes by hereditary transmission. Boswell records ofJohnson that he always used the words "dissenting teacher, " refusing_minister_ and _clergyman_ to all but the recipients of episcopalordination. 

This distinctive phrase has been widely adopted: it occurs in the Index of3d S. Iv. [_Notes and Queries_]. Here we find "Platts (Rev. John), Unitarian teacher, 412;" the article indexed has "Unitarian minister. "

This, of course is habit: an intentional refusal of the word _minister_would never occur in an index. I remember that, when I first read about SamJohnson's little bit of exclusiveness, I said to myself: "Teacher? Teacher?surely I remember One who is often called _teacher_, but never _minister_or _clergyman_: have not the dissenters got the best of it?"

When I said that the Roman Church concedes the epithet Christians toProtestants, I did not mean that all its adherents do the same. There is, or was, a Roman newspaper, the _Tablet_, which, seven or eight years ago, was one of the most virulent of the party journals. In it I read, referringto some complaint of grievance about mixed marriages, that if _Christians_would marry _Protestants_ they must take the consequences. My memory notesthis well; because I recollected, when I saw it, that there was in thestable a horse fit to run in the curricle with this one. About seventeenyears ago an Oxford M. A. , who hated {28} mathematics like a genuineOxonian of the last century, was writing on education, and was compelled togive some countenance to the nasty subject. He got out cleverly; for hegave as his reason for the permission, that man is an arithmetical, geometrical, and mechanical _animal_, as well as a rational _soul_. 

The _Tablet_ was founded by an old pupil of mine, Mr. Frederic Lucas, [57]who availed himself of his knowledge of me to write some severearticles--even abusive, I was told, but I never saw them--against me, forcontributing to the _Dublin Review_, and poking my heretic nose intoorthodox places. Dr. Wiseman, the editor, came in for his share, and oughtto have got all. Who ever blamed the pig for intruding himself into thecabin when the door was left open? When Mr. Lucas was my pupil, he was ofthe Society of Friends--in any article but this I should say _Quaker_--andwas quiet and gentlemanly, as members of that Church--in any article butthis I should, from mere habit, say _sect_--usually are. This is due to hismemory; for, by all I heard, when he changed his religion he ceased to beLucas couchant, and became Lucas rampant, fanged and langued gules. (Ilooked into Guillim[58] to see if my terms were right: I could not findthem; but to prove I have been there, I notice that he calls a violin a_violent_. How comes the word to take this form?) I met with several RomanChristians, born and bred, who were very much annoyed at Mr. Lucas and hisdoings; and said some severe things about new converts needingkicking-straps. 

{29}

The mention of Dr. Wiseman reminds me of another word, appropriated byChristians to themselves: _fides_;[59] the Roman faith is _fides_, andnothing else; and the adherents are _fideles_. [60] Hereby hangs a retort. When Dr. Wiseman was first in England, he gave a course of lectures indefence of his creed, which were thought very convincing by those who werealready convinced. They determined to give him a medal, and there was avery serious discussion about the legend. Dr. Wiseman told me himself thathe had answered to his subscribers that he would not have the medal at allunless--(naming some Italian authority, whom I forget) approved of thelegend. At last _pro fide vindicata_[61] was chosen: this may be readeither in a Popish or heretical sense. The feminine substantive _fides_means confidence, trust, (it is made to mean _belief_), but _fidis_, withthe same ablative, _fide_, and also feminine, is a _fiddle-string_. [62] Ifa Latin writer had had to make a legend signifying "For the defence of thefiddle-string, " he could not have done it otherwise, in the terseness of alegend, than by writing _pro fide vindicata_. Accordingly, when a RomanChristian talks to you of the _faith_, as a thing which is his and notyours, you may say _fiddle_. I have searched Bonaventura Piscator in vainfor notice of this ambiguity. But the Greeks said fiddle; according toSuidas, [63] [Greek: skindapsos][64]--a word meaning a four stringedinstrument played with a quill--was an exclamation of contemptuous dissent. How the wits of different races jump!

{30}

I am reminded of a case of _fides vindicata_, which, being in a publicletter, responding to a public invitation, was not meant to beconfidential. Some of the pupils of University College, in which allsubdivisions of religion are (1866; _were_, 1867) on a level, have ofcourse changed their views in after life, and become adherents of varioushigh churches. On the occasion of a dinner of old students of the College, convened by circular, one of these students, whether then Roman orTractarian Christian I do not remember, not content with simply givingnegative answer, or none at all, concocted a jorum of theological rebuke, and sent it to the Dinner Committee. Heyday! said one of them, this man gotout of bed backwards! How is that? said the rest. Why, read his namebackwards, and you will see. As thus read it was--_No grub_![65]

THE WORD CHURCH. 

To return to _Notes and Queries_. The substitution in the (editorial) indexof "Unitarian teacher, " for the contributor's "Unitarian minister, " struckme very much. I have seldom found such things unmeaning. But as the journalhad always been free from editorial sectarianisms, --and very apt to checkthe contributorial, --I could not be sure in this case. True it was, thatthe editor and publisher had been changed more than a year before; but thiswas not of much force. Though one swallow does not make a summer, I havegenerally found it show that summer is coming. However, thought I tomyself, if this be Little Shibboleth, we shall have Big Shibbolethby-and-bye. At last it came. About a twelvemonth afterwards, (3d S. Vii. P. 36) the following was the _editorial_ answer to the question when theestablishment was first called the "Church of England and Ireland":

{31}

"That unmeaning clause, 'The United Church of England and Ireland, ' whichoccurs on the title-page of _The Book of Common Prayer_, was first used atthe commencement of the present century. The authority for this phrase isthe fifth article of the Union of 1800: 'That the Churches of England andIreland be united into one _Protestant_ (!) episcopal Church, to be called"The United Church of England and Ireland. "' Of course, churchmen are notresponsible for the theology of Acts of Parliament, especially those passedduring the dark ages of the Georgian era. "

That is to say, the journal gives its adhesion to the party which--underthe assumed title of _the_ Church of England--claims for the endowedcorporation for the support of religion rights which Parliament cannotcontrol, and makes it, in fact, a power above the State. The State hasgiven an inch: it calls this corporation by the name of the "United_Church_ of England and Ireland, " as if neither England nor Ireland had anyother Church. The corporation, accordingly aspires to an ell. But this thenation will only give with the aspiration prefixed. To illustrate myallusion in a delicate way to polite ears, I will relate what happened in aJohnian lecture-room at Cambridge, some fifty years ago, my informant beingpresent. A youth of undue aspirations was giving a proposition, and at lastsaid, "Let E F be produced to 'L':" "Not quite so far, Mr. ----, " said thelecturer, quietly, to the great amusement of the class, and the utterastonishment of the aspirant, who knew no more than a Tractarian thetendency of his construction. 

This word _Church_ is made to have a very mystical meaning. The followingdialogue between Ecclesiastes and Hæreticus, which I cannot vouch for, hasoften taken place in spirit, if not in letter: E. The word _Church_([Greek: ekklêsia])[66] is never used in the New Testament except generallyor locally for that holy and mystical body to which the sacraments and theordinances of Christianity are entrusted. {32} H. Indeed! E. It is beyond adoubt (here he quoted half a dozen texts in support). H. Do you mean thatany doctrine or ordinance which was solemnly practised by the [Greek:ekklêsia] is binding upon you and me? E. Certainly, unless we should be cutoff from the congregation of the faithful. H. Have you a couple of hours tospare? E. What for? H. If you have, I propose we spend them in crying, Great is Diana of the Ephesians! E. What do you mean? H. You ought to knowthe solemn service of the [Greek: ekklêsia] (Acts xix. 32, 41), at Ephesus;which any one might take to be true Church, by the more part not knowingwherefore they were come together, and which was dismissed, after one ofthe most sensible sermons ever preached, by the Recorder. E. I see yourmeaning: it is true, there is that one exception! H. Why, the Recorder'ssermon itself contains another, the [Greek: ennomos ekklêsia], [67]legislative assembly. E. Ah! the New Testament can only be interpreted bythe Church! H. I see! the Church interprets itself into existence out ofthe New Testament, and then interprets the New Testament out of existenceinto itself!

I look upon all the Churches as fair game which declare of me that _absquedubio in æternum peribo_;[68] not for their presumption towards God, butfor their personal insolence towards myself. I find that their sectariesstare when I say this. Why! they do not speak of you in _particular_! Thesepoor reasoners seem to think that there could be no meaning, as against me, unless it should be propounded that "without doubt he shall perisheverlastingly, especially A. De Morgan. " But I hold, with the schoolmen, that "_Omnis_ homo est animal" in conjunction with "Sortes est homo"amounts to "_Sortes_ est animal. "[69] But they do not mean it _personally_!Every universal proposition is {33} personal to every instance of thesubject. If this be not conceded, then I retort, in their own sense andmanner, "Whosoever would serve God, before all things he must not pronounceGod's decision upon his neighbor. Which decision, except every one leave toGod himself, without doubt he is a bigoted noodle. "

The reasoning habit of the educated community, in four cases out of five, permits universal propositions to be stated at one time, and denied, _prore nata_, [70] at another. "Before we proceed to consider any questioninvolving physical principles, we should set out with _clear ideas_ of thenaturally possible and impossible. " The eminent man who said this, whenwanting it for his views of mental education (!) never meant it for morethan what was in hand, never assumed it in the researches which will givehim to posterity! I have heard half-a-dozen defences of his having saidthis, not one of which affirmed the truth of what was said. A worthyclergyman wrote that if A. B. Had said a certain thing the point inquestion would have been established. It was shown to him that A. B. _had_said it, to which the reply was a refusal to admit the point because A. B. Said it in a second pamphlet and in answer to objections. And I might givefifty such instances with very little search. Always assume more than youwant; because you cannot tell how much you may want: put what is over intothe didn't-mean-that basket, or the extreme case what-not. 

PROTESTANT AND PAPAL CHRISTENDOM. 

Something near forty years of examination of the theologies on andoff--more years very much on than quite off--have given me a good title--tomyself, I ask no one else for leave--to make the following remarks: Aconclusion has _premises_, facts or doctrines from proof or authority, and_mode of inference_. There may be invention or {34} falsehood of premise, with good logic; and there may be tenable premise, followed by bad logic;and there may be both false premise _and_ bad logic. The Roman system hassuch a powerful manufactory of premises, that bad logic is little wanted;there is comparatively little of it. The doctrine-forge of the Roman Churchis one glorious compound of everything that could make Heraclitus[71] soband Democritus[72] snigger. But not the only one. The Protestants, intearing away from the Church of Rome, took with them a fair quantity of theresults of the Roman forge, which they could not bring themselves to giveup. They had more in them of Martin than of Jack. But they would have nopremises, except from the New Testament; though some eked out with a fewgeneral Councils. The consequence is that they have been obliged to findsuch a logic as would bring the conclusions they require out of thecanonical books. And a queer logic it is; nothing but the Roman forge canbe compared with the Protestant loom. The picking, the patching, thepiecing, which goes to the Protestant _termini ad quem_, [73] would be asremarkable to the general eye, as the Roman manufacture of _termini aquo_, [74] if it were not that the world at large seizes the character of anasserted fact better than that of a mode of inference. A grand step towardsthe deification of a lady, made by alleged revelation 1800 years after herdeath, is of glaring evidence: two or three additional shiffle-shufflestowards defence of saying the Athanasian curse in church and unsaying itout of church, are hardly noticed. Swift has bungled his satire where hemakes Peter a party to finding out what he wants, _totidem syllabis_ and_totidem literis_, {35} when he cannot find it _totidem verbis_[75] This isProtestant method: the Roman plan is _viam faciam_; the Protestant plan is_viam inveniam_. [76] The public at large begins to be conversant with theways of _wriggling out_, as shown in the interpretations of the damnatoryparts of the Athanasian Creed, the phrases of the Burial Service, etc. Thetime will come when the same public will begin to see the ways of_wriggling in_. But one thing at a time: neither Papal Rome nor ProtestantRome was built--nor will be pulled down--in a day. 

The distinction above drawn between the two great antitheses of Christendommay be illustrated as follows. Two sets of little general dealers livedopposite to one another: all sold milk. Each vaunted its own produce: oneset said that the stuff on the other side the way was only chalk and water;the other said that the opposites sold all sorts of filth, of which calves'brain was the least nasty. Now the fact was that both sets sold milk, andfrom the same dairy: but adulterated with different sorts of dirty water:and both honestly believed that the mixture was what they were meant tosell and ought to sell. The great difference between them, about which theapprentices fought each other like Trojans, was that the calves' brain menpoured milk into the water, and the chalk men poured water into the milk. The Greek and Roman sects on one side, the Protestant sects on the other, must all have _churches_: the Greek and Roman sects pour the New Testamentinto their churches; the Protestant sects pour their churches into the NewTestament. The Greek and Roman insist upon the New Testament being no morethan part and parcel of their churches: the Protestant insist upon theirchurches being as much part and parcel of the New Testament. All dwellvehemently upon the doctrine that there must be milk {36} somewhere; andeach says--I have it. The doctrine is true: and can be verified by any onewho can and will go to the dairy for himself. Him will the several tradersdeclare to have no milk at all. They will bring their own wares, andchallenge a trial: they want nothing but to name the judges. To vary themetaphor, those who have looked at Christianity in open day, know that allwho see it through painted windows shut out much of the light of heaven andcolor the rest; it matters nothing that the stains are shaped into what aremeant for saints and angels. 

But there is another side to the question. To decompose any substance, itmust be placed between the poles of the battery. Now theology is but onepole; philosophy is the other. No one can make out the combinations of ourday unless he read the writings both of the priest and the philosopher: andif any one should hold the first word offensive, I tell him that I mean_both_ words to be _significant_. In reading these writings, he will needto bring both wires together to find out what it is all about. Time waswhen most priests were very explicit about the fate of philosophers, andmost philosophers were very candid about their opinion of priests. Butthough some extremes of the old sorts still remain, there is now, in themiddle, such a fusion of the two pursuits that a plain man is wofullypuzzled. The theologian writes a philosophy which seems to tell us that theNew Testament is a system of psychology; and the philosopher writes aChristianity which is utterly unintelligible as to the question whether theResurrection be a fact or a transcendental allegory. What between thetheologian who assents to the Athanasian denunciation in what seems thesense of no denunciation, and the philosopher who parades a Christianitywhich looks like no revelation, there is a maze which threatens to have theonly possible clue in the theory that everything is something else, andnothing is anything at all. But this is a paradox far beyond my handling:it is a Budget of itself. {37}

RELIGION AND PHILOSOPHY. 

Religion and Philosophy, the two best gifts of Heaven, set up in oppositionto each other at the revival of letters; and never did competing tradesmenmore grossly misbehave. Bad wishes and bad names flew about like swarms ofwasps. The Athanasian curses were intended against philosophers; who, hadthey been a corporation, with state powers to protect them, would haveformulized a _per contra_. But the tradesmen are beginning to combine: theyare civil to each other; too civil by half. I speak especially of GreatBritain. Old theology has run off to ritualism, much lamenting, with nocomfort except the discovery that the cloak Paul left at Troas was achasuble. Philosophy, which always had a little sense sewed up in itsgarments--to pay for its funeral?--has expended a trifle in accommodatingitself to the new system. But the two are poles of a battery; and aquestion arises. 

 If Peter Piper picked a peck of pepper, Where is the peck of pepper Peter Piper picked?

If Religion and Philosophy be the two poles of a battery, whose is thebattery Religion and Philosophy have been made the poles of? Is the changein the relation of the wires any presumption of a removal of the managers?We know pretty well who handled the instrument: has he resigned or been[77]turned out? Has he been put under {38} restriction? A fool may ask morequestions than twenty sages can answer: but there is hope; for twenty sagescannot ask more questions than one reviewer can answer. I should like tosee the opposite sides employed upon the question, What are the _commoda_, and what the _pericula_, [78] of the current approximation of Religion andPhilosophy?

All this is very profane and irreverent! It has always been so held bythose whose position demands such holding. To describe the Church as it ispasses for assailing the Church as it ought to be with all who cannot dowithout it. In Bedlam[79] a poor creature who fancied he was St. Paul, wastold by another patient that he was an impostor; the first maniac lodged acomplaint against the second for calling St. Paul an impostor, which, heargued, with much appearance of sanity, ought not to be permitted in a wellregulated madhouse. Nothing could persuade him that he had missed thequestion, which was whether _he_ was St. Paul. The same thing takes placein the world _at large_. And especially must be noted the refusal to permitto the _profane_ the millionth part of the licence assumed by the _sacred_. I give a sound churchman the epitaph of St. John Long; the usualpronunciation of whose name must be noted--

 "Behold! ye quacks, the vengeance strong On deeds like yours impinging: For here below lies St. John Long[80] Who now must be _long singeing_. "

How shameful to pronounce this of the poor man! What, Mr. Orthodox! may Inot do in joke to one pretender what {39} you do in earnest--unless youquibble--to all the millions of the Greek Church, and a great many others. Enough of you and your reasoning! Go and square the circle!

The few years which end with 1867 have shown, not merely the intermediatefusion of Theology and Philosophy of which I have spoken, but muchconcentration of the two extremes, which looks like a gathering of forcesfor some very hard fought Armageddon. Extreme theology has been aiming at ahigh Church in England, which is to show a new front to all heresy: andextreme philosophy is contriving a physical organization which is to_think_, and to show that mind is a consequence of matter, or thought arecreation of brain. The physical speculators begin with a possiblehypothesis, in which they aim at explanation: and so the bold aspirationsof the author of the _Vestiges_ find standing-ground in the variation ofspecies by "natural selection. " Some relics--so supposed--of extremelyancient men are brought to help the general cause. Only distant hints aregiven that by possibility it may end in the formation of all livingorganisms from a very few, if not from one. The better heads abovementioned know that their theory, if true, does not bear upon morals. Theformation of solar systems from a nebular hypothesis, followed byorganizations gradually emerging from some curious play of particles, nay, the very evolution of mind and thought from such an apparatus, are all asconsistent with a Personal creative power to whom homage and obedience aredue, and who has declared himself, as with a blind Nature of Things. A purematerialist, as to all things visible, may be even a bigotted Christian:this is not frequent, but it is possible. There is a proverb which says, Apig may fly, but it isn't a likely bird. But when the psychologicalspeculator comes in, he often undertakes to draw inferences from thephysical conclusions, by joining on his tremendous apparatus of _a priori_knowledge. He deduces that he can _do without_ a God: he can deduce allthings {40} without any such necessity. With Occam[81] and Newton he willhave no more causes than are necessary to explain phenomena _to him_: andif by pure head-work combined with results of physical observation he canconstruct his universe, he must be a very _unphilosophical_ man who wouldencumber himself with a useless Creator! There is something tangible aboutmy method, says he; yours is vague. He requires it to be granted that hissystem is _positive_ and that yours is _impositive_. So reasoned the stagecoachman when the railroads began to depose him--"If you're upset in astage-coach, why, there you are! but if you're upset on the railroad, whereare you?" The answer lies in another question, Which is most positiveknowledge, God deduced from man and his history, or the postulates of thefew who think they can reason _a priori_ on the tacit assumption ofunlimited command of data?

We are not yet come to the existence of a school of philosophers whoexplicitly deny a Creator: but we are on the way, though common sense mayinterpose. There are always straws which show the direction of the wind. Ihave before me the printed letter of a medical man--to whose professionalability I have good testimony--who finds the vital principle in highlyrarefied oxygen. With the usual logic of such thinkers, he dismisses the"eternal personal identity" because "If soul, spirit, mind, which aremerely modes of sensation, be the attribute or function of nerve-tissue, itcannot possibly have any existence apart from its material organism!" Howdoes he know this _impossibility_? If all the mind _we_ know be fromnerve-tissue, how does it appear that mind in other planets may not beanother thing? Nay, when we come to _possibilities_, does not his ownsystem give a queer one? If highly rarefied oxygen be vital power, morehighly rarefied oxygen {41} may be more vital and more powerful. Where isthis to stop? Is it _impossible_ that a finite quantity, rarefied _adinfinitum_, may be an Omnipotent? Perhaps the true Genesis, when written, will open with "In the beginning was an imperial quart of oxygen at 60° ofFahrenheit, and the pressure of the atmosphere; and this oxygen wasinfinitely rarefied; and this oxygen became God. " For myself, myaspirations as to this system are Manichæan. The quart of oxygen is theOrmuzd, or good principle: another quart, of hydrogen, is the Ahriman, orevil principle! My author says that his system explains Freewill andImmortality so obviously that it is difficult to read previous speculationswith becoming gravity. My deduction explains the conflict of good and evilwith such clearness that no one can henceforward read the New Testamentwith becoming reverence. The surgeon whom I have described is an early budwhich will probably be nipped by the frost and wither on the ground: butthere is a good crop coming. Material pneuma is destined to high functions;and man is to read by gas-light. 

THE SUN AN ELECTRIC SPACE. 

 The solar system truly solved; demonstrating by the perfect harmony of the planets, founded on the four universal laws, the Sun to be an electric space; and a source of every natural production displayed throughout the solar system. By James Hopkins. [82] London, 1849, 8vo. 

The author says:

"I am satisfied that I have given the true _laws_ constituting the _Sun_ tobe _space_; and I call upon those disposed to maintain the contrary, togive true _laws_ showing him to be a body: until such can be satisfactorilyestablished, I have an undoubted claim to the credit of my theory, --Thatthe Sun is an _Electric Space_, fed and governed by the {42} planets, whichhave the property of attracting heat from it; and the means of supplyingthe necessary _pabulum_ by their degenerated air driven off towards thecentral space--the wonderful alembic in which it becomes transmuted to therevivifying necessities of continuous action; and the central space or Sunbeing perfectly electric, has the counter property of repulsing the bodiesthat attract it. How wonderful a conception! How beautiful, how magnificentan arrangement!

"O Centre! O Space! O Electric Space!"

JOSEPH ADY. 

1849. _Joseph Ady_[83] is entitled to a place in this list of discoverers:his great fault, like that of some others, lay in pushing his method toofar. He began by detecting unclaimed dividends, and disclosing them totheir right owners, exacting his fee before he made his communication. Hethen generalized into trying to get fees from all of the _name_ belongingto a dividend; and he gave mysterious hints of danger impending. Forinstance, he would write to a clergyman that a legal penalty was hangingover him; and when the alarmed divine forwarded the sum required fordisclosure, he was favored with an extract from some old statute or canon, never repealed, forbidding a clergyman to be a member of a corporation, andwas reminded that he had insured his life in the ---- Office, which had aroyal charter. He was facetious, was Joseph: he described himself in hiscirculars as "personally known to Sir Peter Laurie[84] and all otheraldermen"; which was nearly true, {43} as he had been before most of themon charges of false pretence; but I believe he was nearly always within thelaw. Sir James Duke, when Lord Mayor, having particularly displeased him bya decision, his circulars of 1849 contain the following:

"Should you have cause to complain of any party, Sir J. Duke has contriveda new law of evidence, viz. , write to him, he will consider your lettersufficient proof, and make the parties complained of pay without judge orjury, and will frank you from every expense. "

I strongly suspect that Joseph Ady believed in himself. 

He sometimes issued a second warning, of a Sibylline character:

"Should you find cause to complain of anybody, my voluntary referee, theRt. Hon. Sir Peter Laurie, Kt. , perpetual Deputy Lord Mayor, will seejustice done you without any charge whatever: he and his toady, -- --------. The accursed of Moses can hang any man: thus, by catching him aloneand swearing Naboth spake evil against God and the King. Therefore (!) Iadmit no strangers to a personal conference without a prepayment of 20s. Each. Had you attended to my former notice you would have received twice asmuch: neglect this and you will lose all. "

ON MODERN ASTROLOGY. 

 Zadkiel's Almanac for 1849. Nineteenth number. 

 Raphael's Prophetic Almanac for 1849. Twenty-ninth number. 

 Reasons for belief in judicial astrology, and remarks on the dangerous character of popish priestcraft. London, 1849, 12mo. 

 Astronomy in a nutshell: or the leading problems of the solar system solved by simple proportion only, on the theory of magnetic attraction. By Lieut. Morrison, [85] N. N. London (_s. A. _) 12mo. 

{44}

Lieut. Morrison is Zadkiel Tao Sze, and declares himself in real earnest anastrologer. There are a great many books on astrology, but I have not feltinterest enough to preserve many of them which have come in my way. Ifanything ever had a fair trial, it was astrology. The idea itself isnatural enough. A human being, set down on this earth, without anytradition, would probably suspect that the heavenly bodies had something todo with the guidance of affairs. I think that any one who tries willascertain that the planets do not prophesy: but if he should find to thecontrary, he will of course go on asking. A great many persons classtogether belief in astrology and belief in apparitions: the two thingsdiffer in precisely the way in which a science of observation differs froma science of experiment. Many make the mistake which M. Le Marquis madewhen he came too late, and hoped M. Cassini[86] would do the eclipse overagain for his ladies. The apparition chooses its own time, and comes asseldom or as often as it pleases, be it departed spirit, nervousderangement, or imposition. Consequently it can only be observed, and notexperimented upon. But the heavens, if astrology be true, are prophesyingaway day and night all the year round, and about every body. Experimentscan be made, then, except only on rare phenomena, such as eclipses: anybodymay choose his time and his question. This is the great difference: andexperiments were made, century after century. If astrology had been true, it must have lasted in an ever-improving state. If it be true, it is atruth, and a useful truth, which had experience and prejudice both in itsfavor, and yet lost ground as soon as astronomy, its working tool, began toimprove. 

1850. A letter in the handwriting of an educated man, dated from a streetin which it must be taken that educated persons live, is addressed to theSecretary of the {45} Astronomical Society about a matter on which thewriter says "his professional pursuit will enable him to give asatisfactory reply. " In a question before a court of law it is sworn on oneside that the moon was shining at a certain hour of a certain night on acertain spot in London; on the other side it is affirmed that she wasclouded. The Secretary is requested to decide. This is curious, as thequestion is not astrological. Persons still send to Greenwich, now andthen, to have their fortunes told. In one case, not very many years ago, ayoung gentleman begged to know who his wife was to be, and what fee he wasto remit. 

Sometimes the astronomer turns conjurer for fun, and his prophesies arefulfilled. It is related of Flamsteed[87] that an old woman came to knowthe whereabouts of a bundle of linen which had strayed. Flamsteed drew acircle, put a square into it, and gravely pointed out a ditch, near hercottage, in which he said it would be found. He meant to have given thewoman a little good advice when she came back: but she came back in greatdelight, with the bundle in her hand, found in the very place. The lateBaron Zach[88] received a letter from Pons, [89] a successful finder ofcomets, complaining that for a certain period he had found no comets, though he had searched diligently. Zach, a man of much sly humor, told himthat no spots had been seen on the sun for about the same time--which wastrue, --and assured him that when the spots came back, the comets would comewith them. Some time after he got a letter {46} from Pons, who informed himwith great satisfaction that he was quite right, that very large spots hadappeared on the sun, and that he had found a fine comet shortly after. I donot vouch for the first story, but I have the second in Zach's handwriting. It would mend the joke exceedingly if some day a real relation should beestablished between comets and solar spots: of late years good reason hasbeen shown for advancing a connection between these spots and the earth'smagnetism. [90] If the two things had been put to Zach, he would probablyhave chosen the comets. Here is a hint for a paradox: the solar spots arethe dead comets, which have parted with their light and heat to feed thesun, as was once suggested. I should not wonder if I were too late, and thething had been actually maintained. My list does not contain the twentiethpart of the possible whole. 

The mention of coincidences suggests an everlasting source of explanations, applicable to all that is extraordinary. The great paradox of coincidenceis that of Leibnitz, known as the _pre-established harmony_, or _law ofcoincidences_, by which, separately and independently, the body receivesimpressions, and the mind proceeds as if it had perceived them fromwithout. Every sensation, and the consequent state of the soul, areindependent things coincident in time by the pre-established law. Thephilosopher could not otherwise _account for_ the connection of mind andmatter; and he never goes by so vulgar a rule as _Whatever is, is_; to himthat which is not clear as to how, is not at all. Philosophers in general, who tolerate each other's theories much better than Christians do eachother's failings, seldom revive Leibnitz's fantasy: they seem to act uponthe maxim quoted by Father Eustace[91] from the {47} Decretals, _Facinoraostendi dum puniuntur, flagitia autem abscondi debent_. [92]

The great _ghost-paradox_, and its theory of _coincidences_, will rise tothe surface in the mind of every one. But the use of the word _coincidence_is here at variance with its common meaning. When A is constantlyhappening, and also B, the occurrence of A and B at the same moment is themere coincidence which may be casualty. But the case before us is that A isconstantly happening, while B, when it does happen, almost always happenswith A, and very rarely without it. That is to say, such is the phenomenonasserted: and all who rationally refer it to casualty, affirm that B ishappening very often as well as A, but that it is not thought worthy ofbeing recorded except when A is simultaneous. Of course A is here a death, and B the spectral appearance of the person who dies. In talking of thissubject it is necessary to put out of the question all who play fast andloose with their secret convictions: these had better give us a reason, when they feel internal pressure for explanation, that there is noweathercock at Kilve; this would do for all cases. But persons of realinquiry will see that first, experience does not bear out the assertedfrequency of the spectre, without the alleged coincidence of death: andsecondly, that if the crowd of purely casual spectres were so great that itis no wonder that, now and then the person should have died at or near themoment, we ought to expect a much larger proportion of cases in which thespectre should come at the moment of the death of one or another of all thecluster who are closely connected with the original of the spectre. Butthis, we know, is almost without example. It remains then, for all, whospeculate at all, to look upon the asserted phenomenon, think what they mayof it, the thing which is to be explained, as a _connection_ in time of thedeath, and the {48} simultaneous appearance of the dead. Any person theleast used to the theory of probabilities will see that purely casualcoincidence, the _wrong spectre_ being comparatively so rare that it may besaid never to occur, is not within the rational field of possibility. 

The purely casual coincidence, from which there is no escape except theactual doctrine of special providences, carried down to a very low point ofspecial intention, requires a junction of the things the like of each ofwhich is always happening. I will give three instances which have occurredto myself within the last few years: I solemnly vouch for the literal truthof every part of all three:

In August 1861, M. Senarmont, [93] of the French Institute, wrote to me tothe effect that Fresnel[94] had sent to England, in or shortly after 1824, a paper for translation and insertion in the _European Review_, whichshortly afterwards expired. The question was what had become of that paper. I examined the _Review_ at the Museum, found no trace of the paper, andwrote back to that effect at the Museum, adding that everything nowdepended on ascertaining the name of the editor, and tracing his papers: ofthis I thought there was no chance. I posted this letter on my way home, ata Post Office in the Hampstead Road at the junction with Edward Street, onthe opposite side of which is a bookstall. Lounging for a moment over theexposed books, _sicut meus est mos_, [95] I saw, within a few minutes of theposting of the letter, a little catch-penny book of anecdotes of Macaulay, which I bought, and ran over for a minute. My eye was soon caught by thissentence: "One of the young fellows immediately wrote to the editor (Mr. Walker) {49} of the _European Review_. " I thus got the clue by which Iascertained that there was no chance of recovering Fresnel's paper. Of themention of current reviews, not one in a thousand names the editor. 

In the summer of 1865 I made my first acquaintance with the tales ofNathaniel Hawthorne, and the first I read was about the siege of Boston inthe War of Independence. I could not make it out: everybody seemed to havegot into somebody else's place. I was beginning the second tale, when aparcel arrived: it was a lot of old pamphlets and other rubbish, as hecalled it, sent by a friend who had lately sold his books, had not thoughtit worth while to send these things for sale, but thought I might like tolook at them and possibly keep some. The first thing I looked at was asheet which, being opened, displayed "A plan of Boston and its environs, shewing the true situation of his Majesty's army and also that of therebels, drawn by an engineer, at Boston Oct. 1775. " Such detailed plans ofcurrent sieges being then uncommon, it is explained that "The principalpart of this plan was surveyed by Richard Williams, Lieutenant at Boston;and sent over by the son of a nobleman to his father in town, by whosepermission it was published. " I immediately saw that my confusion arosefrom my supposing that the king's troops were besieging the rebels, when itwas just the other way. 

April 1, 1853, while engaged in making some notes on a logical point, anidea occurred which was perfectly new to me, on the mode of conciliatingthe notions _omnipresence_ and _indivisibility into parts_. What it was isno matter here: suffice it that, since it was published elsewhere (in apaper on _Infinity_, _Camb. Phil. Trans. _ vol. Xi. P. 1) I have not had itproduced to me. I had just finished a paragraph on the subject, when aparcel came in from a bookseller containing Heywood's[96] _Analysis ofKant's Critick_, 1844. 

{50} On turning over the leaves I found (p. 109) the identical thoughtwhich up to this day, I only know as in my own paper, or in Kant. I feelsure I had not seen it before, for it is in Kant's first edition, which wasnever translated to my knowledge; and it does not appear in the latereditions. Mr. Heywood gives some account of the first edition. 

In the broadsheet which gave account of the dying scene of Charles II, itis said that the Roman Catholic priest was introduced by P. M. A. C. F. Thechain was this: the Duchess of Portsmouth[97] applied to the Duke of York, who may have consulted his Cordelier confessor, Mansuete, about procuring apriest, and the priest was smuggled into the king's room by the Duchess andChiffinch. [98] Now the letters are a verbal acrostic of _Père Mansuete aCordelier Friar_, and a syllabic acrostic of _PortsMouth and ChifFinch_. This is a singular coincidence. Macaulay adopted the first interpretation, preferring it to the second, which I brought before him as the conjectureof a near relative of my own. But Mansuete is not mentioned in hisnarrative: it may well be doubted whether the writer of a broadside forEnglish readers would use _Père_ instead of _Father_. And the person whoreally "reminded" the Duke of "the duty he owed to his brother, " was theDuchess and not Mansuete. But my affair is only with the coincidence. 

But there are coincidences which are really connected without theconnection being known to those who find in them matter of astonishment. Presentiments furnish marked cases: sometimes there is no mystery to thosewho have the clue. In the _Gentleman's Magazine_ (vol. 80, part 2, p. 33)we read, the subject being presentiment of death, as follows: "In 1778, tocome nearer the recollection of {51} survivors, at the taking ofPondicherry, Captain John Fletcher, Captain De Morgan, and LieutenantBosanquet, each distinctly foretold his own death on the morning of hisfate. " I have no doubt of all three; and I knew it of my grandfather longbefore I read the above passage. He saw that the battery he commanded wasunduly exposed: I think by the sap running through the fort when produced. He represented this to the engineer officers, and to thecommander-in-chief; the engineers denied the truth of the statement, thecommander believed them, my grandfather quietly observed that he must makehis will, and the French fulfilled his prediction. His will bore date theday of his death; and I always thought it more remarkable than thefulfilment of the prophecy that a soldier should not consider any dangershort of one like the above, sufficient reason to make his will. I supposethe other officers were similarly posted. I am told that military men veryoften defer making their wills until just before an action: but to face theordinary risks intestate, and to wait until speedy death must be the allbut certain consequence of a stupid mistake, is carrying the principle veryfar. In the matter of coincidences there are, as in other cases, twowonderful extremes with every intermediate degree. At one end we have theconfident people who can attribute anything to casual coincidence; whoallow Zadok Imposture and Nathan Coincidence to anoint Solomon Selfconceitking. At the other end we have those who see something _very curious_ inany coincidence you please, and whose minds yearn for a deep reason. Aspeculator of this class happened to find that Matthew viii. 28-33 and Lukeviii. 26-33 contain the same account, that of the demons entering into theswine. Very odd! chapters tallying, and verses so nearly: is theversification rightly managed? Examination is sure to show that there aremonstrous inconsistencies in the mode of division, which being corrected, the verses tally as well as the chapters. And then how comes it? I cannotgo on, {52} for I have no gift at torturing a coincidence, but I would laytwopence, if I could make a bet--which I never did in all my life--thatsome one or more of my readers will try it. Some people say that the studyof chances tends to awaken a spirit of gambling: I suspect the contrary. Atany rate, I myself, the writer of a mathematical book and a comparativelypopular book, have never laid a bet nor played for a stake, however small:not one single time. 

It is useful to record such instances as I have given, with precision andon the solemn word of the recorder. When such a story as that of Flamsteedis told, _a priori_ assures us that it could not have been: the story mayhave been a _ben trovato_, [99] but not the bundle. It is also useful toestablish some of the good jokes which all take for inventions. My friendMr. J. Bellingham Inglis, [100] before 1800, saw the tobacconist's carriagewith a sample of tobacco in a shield, and the motto _Quid rides_[101] (_N. _& _Q. _, 3d S. I. 245). His father was able to tell him all about it. Thetobacconist was Jacob Brandon, well known to the elder Mr. Inglis, and theperson who started the motto, the instant he was asked for such a thing, was Harry Calender of Lloyd's, a scholar and a wit. My friend Mr. H. CrabbRobinson[102] remembers the King's Counsel (Samuel Marryat) who took themotto _Causes produce effects_, when his success enabled him to start acarriage. 

The coincidences of errata are sometimes very remarkable: it may be thatthe misprint has a sting. The death of Sir W. Hamilton[103] of Edinburghwas known in London on a Thursday, and the editor of the _Athenæum_ wroteto {53} me in the afternoon for a short obituary notice to appear onSaturday. I dashed off the few lines which appeared without a moment tothink: and those of my readers who might perhaps think me capable ofcontriving errata with meaning will, I am sure, allow the hurry, theoccasion, and my own peculiar relation to the departed, as sufficientreasons for believing in my entire innocence. Of course I could not see aproof: and two errata occurred. The words "addition to Stewart"[104]require "_for_ addition to _read_ edition of. " This represents what hadbeen insisted on by the Edinburgh publisher, who, frightened by the editionof Reid, [105] had stipulated for a simple reprint without notes. Again"principles of logic and mathematics" required "_for_ mathematics _read_metaphysics. " No four words could be put together which would have so gooda title to be Hamilton's motto. 

April 1850, found in the letter-box, three loose leaves, well printed andover punctuated, being

 Chapter VI. Brethren, lo I come, holding forth the word of life, for so I am commanded.... Chapter VII. Hear my prayer, O generations! and walk by the way, to drink the waters of the river.... Chapter VIII. Hearken o earth, earth, earth, and the kings of the earth, and their armies.... 

A very large collection might be made of such apostolic writings. They goon well enough in a misty--meant for mystical--imitation of St. Paul or theprophets, until at last some prodigious want of keeping shows the educationof the writer. For example, after half a page which might {54} pass forIrving's[106] preaching--though a person to whom it was presented as suchwould say that most likely the head and tail would make something more likehead and tail of it--we are astounded by a declaration from the _HolySpirit_, speaking of himself, that he is "not ashamed of the Gospel ofChrist. " It would be long before we should find in _educated_ rhapsody--ofwhich there are specimens enough--such a thing as a person of the Trinitytaking merit for moral courage enough to stand where St. Peter fell. Thefollowing declaration comes next--"I will judge between cattle and cattle, that use their tongues. "

THE FIGURE OF THE EARTH. 

 The figure of the earth. By J. L. Murphy, [107] of Birmingham. (London and Birmingham, 4 pages, 12mo. ) (1850?)

Mr. Murphy invites attention and objection to some assertions, as that theearth is prolate, not oblate. "If the philosopher's conclusion be right, then the pole is the center of a valley (!) thirteen miles deep. " Hence itwould be very warm. It is answer enough to ask--Who knows that it is not?

 *** A paragraph in the MS. Appears to have been inserted in this place by mistake. It will be found in the Appendix at the end of this volume. --S. E. De M. 

PERPETUAL MOTION. 

1851. The following letter was written by one of a class of persons whom, after much experience of them, I {55} do _not_ pronounce insane. But inthis case the second sentence gives a suspicion of actual delusion of thesenses; the third looks like that eye for the main chance which passes forsanity on the Stock Exchange and elsewhere:

15th Sept. 1851. 

"Gentlemen, --I pray you take steps to make known that yesterday I completedmy invention which will give motion to every country on the Earth;--to moveMachinery!--the long sought in vain 'Perpetual Motion'!!--I was supportedat the time by the Queen and H. R. H. Prince Albert. If, Gentlemen, you canadvise me how to proceed to claim the reward, if any is offered by theGovernment, or how to secure the PATENT for the machine, or in any wayassist me by advice in this great work, I shall most graciously acknowledgeyour consideration. 

These are my convictions that my SEVERAL discoveries will be realized: andthis great one can be at once acted upon: although at this moment it onlyexists in my mind, from my knowledge of certain fixed principles innature:--the Machine I have not made, as I only completed the discoveryYESTERDAY, Sunday!

 I have, etc. ---- ----"

 To the Directors of the London University, Gower Street. 

ON SPIRITUALISM. 

 The Divine Drama of History and Civilisation. By the Rev. James Smith, M. A. [108] London, 1854, 8vo. 

I have several books on that great paradox of our day, _Spiritualism_, butI shall exclude all but three. The bibliography of this subject is now verylarge. The question is one both of evidence and speculation;--Are the facts{56} true? Are they caused by spirits? These I shall not enter upon: Ishall merely recommend this work as that of a spiritualist who does notenter on the subject, which he takes for granted, but applies his derivedviews to the history of mankind with learning and thought. Mr. Smith was aman of a very peculiar turn of thinking. He was, when alive, the editor, or_an_ editor, of the _Family Herald_: I say when alive, to speak accordingto knowledge; for, if his own views be true, he may have a hand in itstill. The answers to correspondents, in his time, were piquant andoriginal above any I ever saw. I think a very readable book might be madeout of them, resembling "Guesses at Truth:" the turn given to an inquiryabout morals, religion, or socials, is often of the highest degree of_unexpectedness_; the poor querist would find himself right in a mostunpalatable way. 

Answers to correspondents, in newspapers, are very often the fag ends ofliterature. I shall never forget the following. A person was invited toname a rule without exception, if he could: he answered "A man _must_ bepresent when he is shaved. " A lady--what right have ladies to decidequestions about shaving?--said this was not properly a rule; and the oraclewas consulted. The editor agreed with the lady; he said that "a man _must_be present when he is shaved" is not a _rule_, but a _fact_. 

[Among my anonymous communicants is one who states that I have doneinjustice to the Rev. James Smith in "referring to him as a spiritualist, "and placing his "Divine Drama" among paradoxes: "it is no paradox, nor do_spiritualistic_ views mar or weaken the execution of the design. " Quitetrue: for the design is to produce and enforce "spiritualistic views"; andleather does not mar nor weaken a shoemaker's plan. I knew Mr. Smith well, and have often talked to him on the subject: but more testimony from me isunnecessary; his book will speak for itself. {57} His peculiar style willjustify a little more quotation than is just necessary to prove the point. Looking at the "battle of opinion" now in progress, we see that Mr. Smithwas a prescient:

(P. 588. ) "From the general review of parties in England, it is evidentthat no country in the world is better prepared for the great Battle ofOpinion. Where else can the battle be fought but where the armies arearrayed? And here they all are, Greek, Roman, Anglican, Scotch, Lutheran, Calvinist, Established and Territorial, with Baronial Bishops, andNonestablished of every grade--churches with living prophets and apostles, and churches with dead prophets and apostles, and apostolical churcheswithout apostles, and philosophies without either prophets or apostles, andonly wanting one more, 'the Christian Church, ' like Aaron's rod, to swallowup and digest them all, and then bud and flourish. As if to prepare ourminds for this desirable and inevitable consummation, different partieshave been favored with a revival of that very spirit of revelation by whichthe Church itself was originally founded. There is a complete series ofspiritual revelations in England and the United States, besides mesmericphenomena that bear a resemblance to revelation, and thus gradually openthe mind of the philosophical and infidel classes, as well as the professedbelievers of that old revelation which they never witnessed in livingaction, to a better understanding of that Law of Nature (for it is a Law ofNature) in which all revelation originates and by which its spiritualcommunications are regulated. "

Mr. Smith proceeds to say that there are _only_ thirty-five incorporatedchurches in England, all formed from the New Testament except five, to eachof which five he concedes a revelation of its own. The five are theQuakers, the Swedenborgians, the Southcottians, the Irvingites, and theMormonites. Of Joanna Southcott he speaks as follows: {58}

(P. 592. ) "Joanna Southcott[109] is not very gallantly treated by thegentlemen of the Press, who, we believe, without knowing anything abouther, merely pick up their idea of her character from the rabble. We onceentertained the same rabble idea of her; but having read her works--for wereally have read them--we now regard her with great respect. However, thereis a great abundance of chaff and straw to her grain; but the grain isgood, and as we do not eat either the chaff or straw if we can avoid it, nor even the raw grain, but thrash it and winnow it, and grind it and bakeit, we find it, after undergoing this process, not only very palatable, buta special dainty of its kind. But the husk is an insurmountable obstacle tothose learned and educated gentlemen who judge of books entirely by thestyle and the grammar, or those who eat grain as it grows, like the cattle. Such men would reject all prological revelation; for there never was andprobably never will be a revelation by voice and vision communicated inclassical manner. It would be an invasion of the rights and prerogatives ofHumanity, and as contrary to the Divine and Established order of mundanegovernment, as a field of quartern loaves or hot French rolls. "

Mr. Smith's book is spiritualism from beginning to end; and my anonymousgainsayer, honest of course, is either ignorant of the work he thinks hehas read, or has a most remarkable development of the organ ofimperception. ]

A CONDENSED HISTORY OF MATHEMATICS. 

I cut the following from a Sunday paper in 1849:

"X. Y. --The Chaldeans began the mathematics, in which the Egyptiansexcelled. Then crossing the sea, by means {59} of Thales, [110] theMilesian, they came into Greece, where they were improved very much byPythagoras, [111] Anaxagoras, [112] and Anopides[113] of Chios. These werefollowed by Briso, [114] Antipho, [two circle-squarers; where is Euclid?]and Hippocrates, [115] but the excellence of the algebraic art was begun byGeber, [116] an Arabian astronomer, and was carried on by Cardanus, [117]Tartaglia, [118] Clavius, [119], Stevinus, [120] Ghetaldus, [121]Herig_e_nius, [122] Fran. Van Schooten [meaning Francis Van Schooten[123]], Florida de Beau_m_e, [124] etc. "

Bryso was a mistaken man. Antipho had the disadvantage of being in advanceof his age. He had the notion of which the modern geometry has made somuch, that of {60} a circle being the polygon of an infinitely great numberof sides. He could make no use of it, but the notion itself made him asophist in the eyes of Aristotle, Eutocius, [125] etc. Geber, an Arabastronomer, and a reputed conjurer in Europe, seems to have given his nameto unintelligible language in the word _gibberish_. At one time _algebra_was traced to him; but very absurdly, though I have heard it suggested that_algebra_ and _gibberish_ must have had one inventor. 

Any person who meddles with the circle may find himself the crane who wasnetted among the geese: as Antipho for one, and Olivier de Serres[126] foranother. This last gentleman ascertained, by weighing, that the area of thecircle is very nearly that of the square on the side of the inscribedequilateral triangle: which it is, as near as 3. 162 ... To 3. 141.... He didnot pretend to more than approximation; but Montucla and othersmisunderstood him, and, still worse, misunderstood their ownmisunderstanding, and made him say the circle was exactly double of theequilateral triangle. He was let out of limbo by Lacroix, in a note to hisedition of Montucla's _History of Quadrature_. 

ST. VITUS, PATRON OF CYCLOMETERS. 

 Quadratura del cerchio, trisezione dell' angulo, et duplicazione del cubo, problemi geometricamente risolute e dimostrate dal Reverendo Arciprete di San Vito D. Domenico Angherà, [127] Malta, 1854, 8vo. 

 {61}

 Equazioni geometriche, estratte dalla lettera del Rev. Arciprete ... Al Professore Pullicino[128] sulla quadratura del cerchio. Milan, 1855 or 1856, 8vo. 

 Il Mediterraneo gazetta di Malta, 26 Decembre 1855, No. 909: also 911, 912, 913, 914, 936, 939. 

 The Malta Times, Tuesday, 9th June 1857. 

 Misura esatta del cerchio, dal Rev. D. Angherà. Malta, 1857, 12mo. 

 Quadrature of the circle ... By the Rev. D. Angherà, Archpriest of St. Vito. Malta, 1858, 12mo. 

I have looked for St. Vitus in catalogues of saints, but never found hislegend, though he figures as a day-mark in the oldest almanacs. He must beproperly accredited, since he was an archpriest. And I pronounce andordain, by right accruing from the trouble I have taken in this subject, that he, St. Vitus, who leads his votaries a never-ending and unmeaningdance, shall henceforth be held and taken to be the patron saint of thecircle-squarer. His day is the 15th of June, which is also that of St. Modestus, [129] with whom the said circle-squarer often has nothing to do. And he must not put himself under the first saint with a slantendicularreference to the other, as is much to be feared was done by the Cardinalwho came to govern England with a title containing St. Pudentiana, [130] whoshares a day with _St. Dunstan_. The Archpriest of St. Vitus will have itthat the square inscribed in a semicircle is half of the semicircle, or thecircumference 3-1/5 diameters. He is active and able, with {62} nothingwrong about him except his paradoxes. In the second tract named he hasgiven the testimonials of crowned heads and ministers, etc. As follows. Louis-Napoleon gives thanks. The minister at Turin refers it to the Academyof Sciences, and hopes so much labor will be judged _degna di pregio_. [131]The Vice-Chancellor of Oxford--a blunt Englishman--begs to say that theUniversity has never proposed the problem, as some affirm. The PrinceRegent of Baden has received the work with lively interest. The Academy ofVienna is not in a position to enter into the question. The Academy ofTurin offers the most _distinct_ thanks. The Academy della Crusca attendsonly to literature, but gives thanks. The Queen of Spain has received thework with the highest appreciation. The University of Salamanca givesinfinite thanks, and feels true satisfaction in having the book. LordPalmerston gives thanks, by the hand of "William San. " The Viceroy ofEgypt, not being yet up in Italian, will spend his first moments of leisurein studying the book, when it shall have been translated into French: inthe mean time he congratulates the author upon his victory over a problemso long held insoluble. All this is seriously published as a rate in aid ofdemonstration. If these royal compliments cannot make the circumference ofa circle about 2 per cent. Larger than geometry will have it --which is allthat is wanted--no wonder that thrones are shaky. 

I am informed that the legend of St. Vitus is given by Ribadeneira[132] inhis lives of Saints, and that Baronius, [133] in {63} his _MartyrologiumRomanum_, refers to several authors who have written concerning him. Thereis an account in Mrs. Jameson's[134] _History of Sacred and Legendary Art_(ed. Of 1863, p. 544). But it seems that St. Vitus is the patron saint of_all_ dances; so that I was not so far wrong in making him the protector ofthe cyclometers. Why he is represented with a cock is a disputed point, which is now made clear: next after _gallus gallinaceus_[135] himself, there is no crower like the circle-squarer. 

CELEBRATED APPROXIMATIONS OF [pi]. 

The following is an extract from the _English Cyclopædia_, Art. TABLES:

"1853. William Shanks, [136] _Contributions to Mathematics, comprisingchiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853. (QUADRATURE OF THE CIRCLE. ) Here is a _table_, because it tabulatesthe results of the subordinate steps of this enormous calculation as far as527 decimals: the remainder being added as results only during theprinting. For instance, one step is the calculation of the reciprocal of601. 5^{601}; and the result is given. The number of pages required todescribe these results is 87. Mr. Shanks has also thrown off, as chips orsplinters, the values of the base of Napier's logarithms, and of itslogarithms of 2, 3, 5, 10, to 137 decimals; and the value of the modulus. 4342 ... To 136 decimals: with the 13th, 25th, 37th ... Up to the 721stpowers of 2. These tremendous stretches of calculation--at least we so callthem in our day--are useful in several respects; they prove more than {64}the capacity of this or that computer for labor and accuracy; they showthat there is in the community an increase of skill and courage. We say inthe community: we fully believe that the unequalled turnip which every nowand then appears in the newspapers is a sufficient presumption that theaverage turnip is growing bigger, and the whole crop heavier. All who knowthe history of the quadrature are aware that the several increases ofnumbers of decimals to which [pi] has been carried have been indications ofa general increase in the power to calculate, and in courage to face thelabor. Here is a comparison of two different times. In the day ofCocker, [137] the pupil was directed to perform a common subtraction with avoice-accompaniment of this kind: '7 from 4 I cannot, but add 10, 7 from 14remains 7, set down 7 and carry 1; 8 and 1 which I carry is 9, 9 from 2 Icannot, etc. ' We have before us the announcement of the following _table_, undated, as open to inspection at the Crystal Palace, Sydenham, in twodiagrams of 7 ft. 2 in, by 6 ft. 6 in. : 'The figure 9 involved into the912th power, and antecedent powers or involutions, containing upwards of73, 000 figures. Also, the proofs of the above, containing upwards of146, 000 figures. By Samuel Fancourt, of Mincing Lane, London, and completedby him in the year 1837, at the age of sixteen. N. B. The whole operationperformed by simple arithmetic. ' The young operator calculated bysuccessive squaring the 2d, 4th, 8th, etc. , powers up to the 512th, withproof by division. But 511 multiplications by 9, in the short (or 10-1)way, would have been much easier. The 2d, 32d, 64th, 128th, 256th, and512th powers are given at the back of the announcement. The powers of 2have been calculated for many purposes. In Vol. II of his _MagiaUniversalis Naturæ et Artis_, Herbipoli, 1658, 4to, the Jesuit GasparSchott[138] having discovered, on some grounds of theological {65} magic, that the degrees of grace of the Virgin Mary were in number the 256th powerof 2, calculated that number. Whether or no his number correctlyrepresented the result he announced, he certainly calculated it rightly, aswe find by comparison with Mr. Shanks. "

There is a point about Mr. Shanks's 608 figures of the value of [pi] whichattracts attention, perhaps without deserving it. It might be expectedthat, in so many figures, the nine digits and the cipher would occur eachabout the same number of times; that is, each about 61 times. But the factstands thus: 3 occurs 68 times; 9 and 2 occur 67 times each; 4 occurs 64times; 1 and 6 occur 62 times each; 0 occurs 60 times; 8 occurs 58 times; 5occurs 56 times; and 7 occurs only 44 times. Now, if all the digits wereequally likely, and 608 drawings were made, it is 45 to 1 against thenumber of sevens being as distant from the probable average (say 61) as 44on one side or 78 on the other. There must be some reason why the number 7is thus deprived of its fair share in the structure. Here is a field ofspeculation in which two branches of inquirers might unite. There is butone number which is treated with an unfairness which is incredible as anaccident; and that number is the mystic number _seven_! If the cyclometersand the apocalyptics would lay their heads together until they come to aunanimous verdict on this phenomenon, and would publish nothing until theyare of one mind, they would earn the gratitude of their race. --I was wrong:it is the Pyramid-speculator who should have been appealed to. Acorrespondent of my friend Prof. Piazzi Smyth[139] notices that 3 is thenumber of most frequency, and that 3-1/7 is the nearest approximation to itin simple digits. Professor Smyth himself, whose word on Egypt is paradoxof a very high order, backed by a great quantity of useful labor, theresults which will be made available by those who do not receive {66} theparadoxes, is inclined to see confirmation for some of his theory in thesephenomena. 

CURIOUS CALCULATIONS. 

These paradoxes of calculation sometimes appear as illustrations of thevalue of a new method. In 1863, Mr. G. Suffield, [140] M. A. , and Mr. J. R. Lunn, [141] M. A. , of Clare College and of St. John's College, Cambridge, published the whole quotient of 10000 ... Divided by 7699, throughout thewhole of one of the recurring periods, having 7698 digits. This was done inillustration of Mr. Suffield's method of _Synthetic division_. 

Another instance of computation carried to paradoxical length, in order toillustrate a method, is the solution of x^3 - 2x = 5, the example given ofNewton's method, on which all improvements have been tested. In 1831, Fourier's[142] posthumous work on equations showed 33 figures of solution, got with enormous labor. Thinking this a good opportunity to illustrate thesuperiority of the method of W. G. Horner, [143] not yet known in France, and not much known in {67} England, I proposed to one of my classes, in1841, to beat Fourier on this point, as a Christmas exercise. I receivedseveral answers, agreeing with each other, to 50 places of decimals. In1848, I repeated the proposal, requesting that 50 places might be exceeded:I obtained answers of 75, 65, 63, 58, 57, and 52 places. But one answer, byMr. W. Harris Johnston, [144] of Dundalk, and of the Excise Office, went to101 decimal places. To test the accuracy of this, I requested Mr. Johnstonto undertake another equation, connected with the former one in a way whichI did not explain. His solution verified the former one, but he was unableto see the connection, even when his result was obtained. My reader may beas much at a loss: the two solutions are:

 2. 0945514815423265... 9. 0544851845767340... 

The results are published in the _Mathematician_, Vol. III, p. 290. In1851, another pupil of mine, Mr. J. Power Hicks, [145] carried the result to152 decimal places, without knowing what Mr. Johnston had done. The resultis in the _English Cyclopædia_, article INVOLUTION AND EVOLUTION. 

I remark that when I write the initial of a Christian name, the most usualname of that initial is understood. I never saw the name of W. G. Hornerwritten at length, until I applied to a relative of his, who told me thathe was, as I supposed, Wm. _George_, but that he was named after a relativeof that _surname_. 

The square root of 2, to 110 decimal places, was given {68} me in 1852 bymy pupil, Mr. William Henry Colvill, now (1867) Civil Surgeon at Baghdad. It was

 1. 4142135623730950488016887242096980785696 7187537694807317667973799073247846210703 885038753432764157273501384623

Mr. James Steel[146] of Birkenhead verified this by actual multiplication, and produced

 2 - 2580413 / 10^{117}

as the square. 

 Calcolo decidozzinale del Barone Silvio Ferrari. Turin, 1854, 4to. 

This is a serious proposal to alter our numeral system and to count bytwelves. Thus 10 would be twelve, 11 thirteen, etc. , two new symbols beinginvented for ten and eleven. The names of numbers must of course bechanged. There are persons who think such changes practicable. I thoughtthis proposal absurd when I first saw it, and I think so still:[147] butthe one I shall presently describe beats it so completely in that point, that I have not a smile left for this one. 

ON COMETS. 

 The successful and therefore probably true theory of Comets. London, 1854. (4pp. Duodecimo. )

The author is the late Mr. Peter Legh, [148] of Norbury Booths Hall, Knutsford, who published for eight or ten {69} years the _OmbrologicalAlmanac_, a work of asserted discovery in meteorology. The theory of cometsis that the joint attraction of the new moon and several planets in thedirection of the sun, draws off the gases from the earth, and forms thesecometic meteors. But how these meteors come to describe orbits round thesun, and to become capable of having their returns predicted, is notexplained. 

A NEW PHASE OF MORMONISM. 

 The Mormon, New York, Saturday, Oct. 27, 1855. 

A newspaper headed by a grand picture of starred and striped banners, beehive, and eagle surmounting it. A scroll on each side: on the left, "Mormon creed. Mind your own business. Brigham Young;"[149] on the right, "Given by inspiration of God. Joseph Smith. "[150] A leading article on thediscoveries of Prof. Orson Pratt[151] says, "Mormonism has long taken thelead in religion: it will soon be in the van both in science and politics. "At the beginning of the paper is Professor Pratt's "Law of PlanetaryRotation. " The cube roots of the densities of the planets are as the squareroots of their periods of rotation. The squares of the cube roots of themasses divided by the squares of the diameters are as the periods ofrotation. Arithmetical verification attempted, and the whole very modestlystated {70} and commented on. Dated G. S. L. City, Utah Ter. , Aug. 1, 1855. If the creed, as above, be correctly given, no wonder the Mormonites are insuch bad odor. 

MATHEMATICAL ILLUSTRATIONS OF DOCTRINE. 

 The two estates; or both worlds mathematically considered. London, 1855, small (pp. 16). 

The author has published mathematical works with his name. The presenttract is intended to illustrate mathematically a point which may be guessedfrom the title. But the symbols do very little in the way of illustration:thus, x being the _present value_ of the future estate (eternal happiness), and a of all that this world can give, the author impresses it on themathematician that, x being infinitely greater than a, x + a = x, so that aneed not be considered. This will not act much more powerfully on amathematician by virtue of the symbols than if those same symbols had beendispensed with: even though, as the author adds, "It was this method ofneglecting infinitely small quantities that Sir Isaac Newton was indebtedto for his greatest discoveries. "

There has been a moderate quantity of well-meant attempt to enforce, sometimes motive, sometimes doctrine, by arguments drawn from mathematics, the proponents being persons unskilled in that science for the most part. The ground is very dangerous: for the illustration often turns the otherway with greater power, in a manner which requires only a little moreknowledge to see. I have, in my life, heard from the pulpit or read, atleast a dozen times, that all sin is infinitely great, proved as follows. The greater the being, the greater the sin of any offence against him:therefore the offence committed against an infinite being is infinitelygreat. Now the mathematician, of which the proposers of this argument arenot aware, is perfectly familiar with quantities which increase together, and never cease increasing, but so that one of them remains finite when{71} the other becomes infinite. In fact, the argument is a perfect _nonsequitur_. [152] Those who propose it have in their minds, though in acloudy and indefinite form, the idea of the increase of guilt being_proportionate_ to the increase of greatness in the being offended. Butthis it would never do to state: for by such statement not only would theargument lose all that it has of the picturesque, but the asserted premisewould have no strong air of exact truth. How could any one undertake toappeal to conscience to declare that an offence against a being 4-7/10times as great as another is exactly, no more and no less, 4-7/10 times asgreat an offence against the other?

The infinite character of the offence against an infinite being is laiddown in Dryden's _Religio Laici_, [153] and is, no doubt, an old argument:

 "For, granting we have sinned, and that th' offence Of man is made against Omnipotence, Some price that bears proportion must be paid, And infinite with infinite be weighed. See then the Deist lost; remorse for vice Not paid; or, paid, inadequate in price. "

Dryden, in the words "bears proportion" is in verse more accurate than mostof the recent repeaters in prose. And this is not the only case of the kindin his argumentative poetry. 

My old friend, the late Dr. Olinthus Gregory, [154] who was a sound andlearned mathematician, adopted this dangerous kind of illustration in his_Letters on the Christian Religion_. {72} He argued, by parallel, from whathe supposed to be the necessarily mysterious nature of the _impossible_quantity of algebra to the necessarily mysterious nature of certaindoctrines of his system of Christianity. But all the difficulty and mysteryof the impossible quantity is now cleared away by the advance ofalgebraical thought: and yet Dr. Gregory's book continues to be sold, andno doubt the illustration is still accepted as appropriate. 

The mode of argument used by the author of the tract above named has astriking defect. He talks of reducing this world and the next to "presentvalue, " as an actuary does with successive lives or next presentations. Does value make interest? and if not, why? And if it do, then the presentvalue of an eternity is _not_ infinitely great. Who is ignorant that aperpetual annuity at five per cent is worth only twenty years' purchase?This point ought to be discussed by a person who treats heaven as adeferred perpetual annuity. I do not ask him to do so, and would rather hedid not; but if he _will_ do it, he must either deal with the question ofdiscount, or be asked the reason why. 

When a very young man, I was frequently exhorted to one or another view ofreligion by pastors and others who thought that a mathematical argumentwould be irresistible. And I heard the following more than once, and havesince seen it in print, I forget where. Since eternal happiness belonged tothe particular views in question, a benefit infinitely great, then, even ifthe probability of their arguments were small, or even infinitely small, yet the product of the chance and benefit, according to the usual rule, might give a result which no one ought in prudence to pass over. They didnot see that this applied to all systems as well as their own. I take thisargument to be the most perverse of all the perversions I have heard orread on the subject: there is some high authority for it, whom I forget. 

The moral of all this is, that such things as the preceding should be keptout of the way of those who are not {73} mathematicians, because they donot understand the argument; and of those who are, because they do. 

[The high authority referred to above is Pascal, an early cultivator ofmathematical probability, and obviously too much enamoured of his newpursuit. But he conceives himself bound to wager on one side or the other. To the argument (_Pensées_, ch. 7)[155] that "le juste est de ne pointparier, " he answers, "Oui: mais il faut parier: vous êtes embarqué; et neparier point que Dieu est, c'est parier qu'il n'est pas. "[156] LeavingPascal's argument to make its way with a person who, _being a sceptic_, isyet positive that the issue is salvation or perdition, if a God therebe, --for the case as put by Pascal requires this, --I shall merely observethat a person who elects to believe in God, as the best chance of gain, isnot one who, according to Pascal's creed, or any other worth naming, willreally secure that gain. I wonder whether Pascal's curious imagination everpresented to him in sleep his convert, in the future state, shaken out of ared-hot dice-box upon a red-hot hazard-table, as perhaps he might havebeen, if Dante had been the later of the two. The original idea is due tothe elder Arnobius, [157] who, as cited by Bayle, [158] speaks thus:

"Sed et ipse [Christus] quæ pollicetur, non probat. Ita est. Nulla enim, utdixi, futurorum potest existere comprobatio. Cum ergo hæc sit conditiofuturorum, ut teneri et comprehendi nullius possint anticipationis attactu;nonne {74} purior ratio est, ex duobus incertis, et in ambigua expectationependentibus, id potius credere, quod aliquas spes ferat, quam omnino quodnullas? In illo enim periculi nihil est, si quod dicitur imminere, cassumfiat et vacuum: in hoc damnum est maximum, id est salutis amissio, si cumtempus advenerit aperiatur non fuisse mendacium. "[159]

Really Arnobius seems to have got as much out of the notion, in the thirdcentury, as if he had been fourteen centuries later, with the arithmetic ofchances to help him. ]

NOVUM ORGANUM MORALIUM. 

 The Sentinel, vol. Ix. No. 27. London, Saturday, May 26, 1855. 

This is the first London number of an Irish paper, Protestant in politics. It opens with "Suggestions on the subject of a _Novum Organum Moralium_, "which is the application of algebra and the differential calculus tomorals, socials, and politics. There is also a leading article on thesubject, and some applications in notes to other articles. A separatepublication was afterwards made, with the addition of a long Preface; theauthor being a clergyman who I presume must have been the editor of the_Sentinel_. 

 Suggestions as to the employment of a _Novum Organum Moralium_. Or, thoughts on the nature of the Differential Calculus, and on the application of its principles to metaphysics, with a view to the attainment of demonstration and certainty in moral, {75} political and ecclesiastical affairs. By Tresham Dames Gregg, [160] Chaplain of St. Mary's, within the church of St. Nicholas intra muros, Dublin. London, 1859, 8vo. (pp. Xl + 32). 

I have a personal interest in this system, as will appear from thefollowing extract from the newspaper:

 "We were subsequently referred to De Morgan's _Formal Logic_ and Boole's _Laws of Thought_[161] both very elaborate works, and greatly in the direction taken by ourselves. That the writers amazingly surpass us in learning we most willingly admit, but we venture to pronounce of both their learned treatises, that they deal with the subject in a mode that is scholastic to an excess.... That their works have been for a considerable space of time before the world and effected nothing, would argue that they have overlooked the vital nature of the theme.... On the whole, the writings of De Morgan and Boole go to the full justification of our principle without in any wise so trenching upon our ground as to render us open to reproach in claiming our Calculus as a great discovery.... But we renounce any paltry jealousy as to a matter so vast. If De Morgan and Boole have had a priority in the case, to them we cheerfully shall resign the glory and honor. If such be the truth, they have neither done justice to the discovery, nor to themselves [quite true]. They have, under the circumstances, acted like 'the foolish man, who roasteth not that which he taketh {76} in hunting.... It will be sufficient for us, however, to be the Columbus of these great Americi, and popularize what they found, _if_ they found it. We, as from the mountain top, will then become _their_ trumpeters, and cry glory to De Morgan and glory to Boole, under Him who is the source of all glory, the only good and wise, to Whom be glory for ever! _If_ they be our predecessors in this matter, they have, under Him, taken moral questions out of the category of probabilities, and rendered them perfectly certain. In that case, let their books be read by those who may doubt the principles this day laid before the world as a great discovery, by our newspaper. Our cry shall be [Greek: eurêkasi]![162] Let us hope that they will join us, and henceforth keep their light [_sic_] from under their bushel. "

For myself, and for my old friend Mr. Boole, who I am sure would join me, Idisclaim both priority, simultaneity, and posteriority, and request thatnothing may be trumpeted from the mountain top except our abjuration of allcommunity of thought or operation with this _Novum Organum_. 

To such community we can make no more claim than Americus could make tobeing the forerunner of Columbus who popularized his discoveries. We do notwish for any [Greek: eurêkasi] and not even for [Greek: heurêkasi]. Forself and Boole, I point out what would have convinced either of us thatthis house is divided against itself. 

[Alpha] being an apostolic element, [delta] the doctrinal element, and[Chi] the body of the faithful, the church is [Alpha] [delta] [Chi], we aretold. Also, that if [Alpha] become negative, or the Apostolicity becomeDiabolicity [my words]; or if [delta] become negative, and doctrine becomeheresy; or if [Chi] become negative, that is, if the faithful becomeunfaithful; the church becomes negative, "the very opposite to what itought to be. " For self and Boole, I admit this. But--which is notnoticed--if [Alpha] and [delta] should _both_ become negative, diabolicalorigin {77} and heretical doctrine, then the church, [Alpha] [delta] [Chi], is still positive, what it ought to be, unless [Chi] be also negative, orthe people unfaithful to it, in which case it is a bad church. Now, selfand Boole--though I admit I have not asked my partner--are of opinion thata diabolical church with false doctrine does harm when the people arefaithful, and can do good only when the people are unfaithful. We may bewrong, but this is what we _do_ think. Accordingly, we have caught nothing, and can therefore roast nothing of our own: I content myself with roastinga joint of Mr. Gregg's larder. 

These mathematical vagaries have uses which will justify a large amount ofquotation: and in a score of years this may perhaps be the only attainablerecord. I therefore proceed. 

After observing that by this calculus juries (heaven help them! say I) cancalculate damages "almost to a nicety, " and further that it is madeabundantly evident that c e x is "the general expression for anindividual, " it is noted that the number of the Beast is not given in theRevelation in words at length, but as [Greek: chxw']. [163] On this thefollowing remark is made:

"Can it be possible that we have in this case a specimen given to us of thearithmetic of heaven, and an expression revealed, which indicates by itsfunction of addibility, the name of the church in question, and of eachmember of it; and by its function of multiplicability the doctrine, themission, and the members of the great Synagogue of Apostacy? We merelypropound these questions;--we do not pretend to solve them. "

After a translation in blank verse--a very pretty one--of the 18th Psalm, the author proceeds as follows, to render it into differential calculus:

{78}

"And the whole tells us just this, that David did what he could. Heaugmented those elements of his constitution which were (_exceptisexcipiendis_)[164] subject to himself, and the Almighty then augmented hispersonal qualities, and his vocational _status_. Otherwise, to throw thematter into the expression of our notation, the variable e was augmented, and c x rose proportionally. The law of the variation, according to ourtheory, would be thus expressed. The resultant was David the king c e x [c= r?] (who had been David the shepherd boy), and from the conditions of thetheorem we have

 du/de = ce(dx/de) + ex(dc/de)x + cx

which, in the terms of ordinary language, just means, the increase ofDavid's educational excellence or qualities--his piety, his prayerfulness, his humility, obedience, etc. --was so great, that when multiplied by hisoriginal talent and position, it produced a product so great as to be equalin its amount to royalty, honor, wealth, and power, etc. : in short, to allthe attributes of majesty. "[165]

The "solution of the family problem" is of high interest. It is todetermine the effect on the family in general from a change [of conduct] inone of them. The person chosen is one of the maid-servants. 

"Let c e x be the father; c_1e_1x_1 the mother, etc. The family thenconsists of the maid's master, her mistress, her young master, her youngmistress, and fellow servant. Now the master's calling (or c) is toexercise his share of control over this servant, and mind the rest of hisbusiness: call this remainder a, and let his calling generally, or all hisaffairs, be to his maid-servant as m : y, i. E. , y = (mz/c); ... {79} andthis expression will represent his relation to the servant. Consequently, c e x = (a + mz/c)e x; otherwise (a + mz/c)e x

is the expression for the father when viewed as the girl's master. "

I have no objection to repeat so far; but I will not give the formula forthe maid's relation to her young master; for I am not quite sure that allyoung masters are to be trusted with it. Suffice it that the son will beaffected directly as his influence over her, and inversely as hisvocational power: if then he should have some influence and no vocationalpower, the effect on him would be infinite. This is dismal to think of. Further, the formula brings out that if one servant improve, the other mustdeteriorate, and _vice versa_. This is not the experience of most families:and the author remarks as follows:

"That is, we should venture to say, a very beautiful result, and we may sayit yielded us no little astonishment. What our calculation might lead to wenever dreamt of; that it should educe a conclusion so recondite that ourunassisted power never could have attained to, and which, if we could haveconjectured it, would have been at best the most distant probability, thatconclusion being itself, as it would appear, the quintessence of truth, afforded us a measure of satisfaction that was not slight. "

That the writings of Mr. Boole and myself "go to the full justification of"this "principle, " is only true in the sense in which the Scotch use, or diduse, the word _justification_. 

A TRIBUTE TO BOOLE. 

[The last number of this Budget had stood in type for months, waiting untilthere should be a little cessation of correspondence more connected withthe things of the day. {80} I had quite forgotten what it was to contain;and little thought, when I read the proof, that my allusions to my friendMr. Boole, then in life and health, would not be printed till many weeksafter his death. Had I remembered what my last number contained, I shouldhave added my expression of regret and admiration to the numerous obituarytestimonials, which this great loss to science has called forth. 

The system of logic alluded to in the last number of this series is but oneof many proofs of genius and patience combined. I might legitimately haveentered it among my _paradoxes_, or things counter to general opinion: butit is a paradox which, like that of Copernicus, excited admiration from itsfirst appearance. That the symbolic processes of algebra, invented as toolsof numerical calculation, should be competent to express every act ofthought, and to furnish the grammar and dictionary of an all-containingsystem of logic, would not have been believed until it was proved. WhenHobbes, [166] in the time of the Commonwealth, published his _Computation orLogique_, he had a remote glimpse of some of the points which are placed inthe light of day by Mr. Boole. The unity of the forms of thought in all theapplications of reason, however remotely separated, will one day be matterof notoriety and common wonder: and Boole's name will be remembered inconnection with one of the most important steps towards the attainment ofthis knowledge. ]

DECIMALS RUN RIOT. 

 The Decimal System as a whole. By Dover Statter. [167] London and Liverpool, 1856, 8vo. 

{81}

The proposition is to make everything decimal. The day, now 24 hours, is tobe made 10 hours. The year is to have ten months, Unusber, Duober, etc. Fortunately there are ten commandments, so there will be neither additionto, nor deduction from, the moral law. But the twelve apostles! Evenrejecting Judas, there is a whole apostle of difficulty. These points theauthor does not touch. 

ON PHONETIC SPELLING. 

 The first book of Phonetic Reading. London, Fred. Pitman, [168] Phonetic Depot, 20, Paternoster Row, 1856, 12mo. 

 The Phonetic Journal. Devoted to the propagation of phonetic reading, phonetic longhand, phonetic shorthand, and phonetic printing. No. 46. Saturday, 15 November 1856. Vol. 15. 

I write the titles of a couple out of several tracts which I have by me. But the number of publications issued by the promoters of this spiritedattempt is very large indeed. [169] The attempt itself has had no successwith the mass of the public. This I do not regret. Had the world found thatthe change was useful, I should have gone contentedly with the stream; butnot without regretting our old language. I admit the difficulties which ourunpronounceable spelling puts in the way of learning to read: and I have nodoubt that, as affirmed, it is easier to teach children phonetically, andafterwards to introduce them to our common system, than to proceed in theusual way. But by the usual way I mean proceeding by letters from the verybeginning. If, which I am sure is a better plan, children be taught at thecommencement very much by _complete words_, as if they were learningChinese, and be gradually accustomed to {82} resolve the known words intoletters, a fraction, perhaps a considerable one, of the advantage of thephonetic system is destroyed. It must be remembered that a phonetic systemcan only be an approximation. The differences of pronunciation existingamong educated persons are so great, that, on the phonetic system, different persons ought to spell differently. 

But the phonetic party have produced something which will immortalize theirplan: I mean their _shorthand_, which has had a fraction of the success itdeserves. All who know anything of shorthand must see that nothing but aphonetic system can be worthy of the name: and the system promulgated isskilfully done. Were I a young man I should apply myself to itsystematically. I believe this is the only system in which books were everpublished. I wish some one would contribute to a public journal a briefaccount of the dates and circumstances of the phonetic movement, notforgetting a list of the books published in shorthand. 

A child beginning to read by himself may owe terrible dreams and wakingimages of horror to our spelling, as I did when six years old. In one ofthe common poetry-books there is an admonition against confining littlebirds in cages, and the child is asked what if a great giant, amazinglystrong, were to take you away, shut you up, And feed you with vic-tu-als you ne-ver could bear. 

The book was hyphened for the beginner's use; and I had not the least ideathat _vic-tu-als_ were _vittles_: by the sound of the word I judged theymust be of iron; and it entered into my soul. 

The worst of the phonetic shorthand book is that they nowhere, so far as Ihave seen, give _all_ the symbols, in every stage of advancement, together, in one or following pages. It is symbols and talk, more symbols and moretalk, etc. A universal view of the signs ought to begin the works. {83}

A HANDFUL OF LITTLE PARADOXERS. 

 Ombrological Almanac. Seventeenth year. An essay on Anemology and Ombrology. By Peter Legh, [170] Esq. London, 1856, 12mo. 

Mr. Legh, already mentioned, was an intelligent country gentleman, and alegitimate speculator. But the clue was not reserved for him. 

 The proof that the three angles of a triangle are equal to two right angles looked for in the inflation of the circle. By Gen. Perronet Thompson. London, 1856, 8vo. (pp. 4. )

Another attempt, the third, at this old difficulty, which cannot be putinto few words of explanation. [171]

 Comets considered as volcanoes, and the cause of their velocity and other phenomena thereby explained. London (_circa_ 1856), 8vo. 

The title explains the book better than the book explains the title. 

1856. A stranger applied to me to know what the ideas of a friend of hiswere worth upon the magnitude of the earth. The matter being one involvingpoints of antiquity, I mentioned various persons whose speculations heseemed to have ignored; among others, Thales. The reply was, "I aminstructed by the author to inform you that he is perfectly acquainted withthe works of Thales, Euclid, Archimedes, ... " I had some thought of askingwhether he had used the Elzevir edition of Thales, [172] which is known tobe very incomplete, or that of Professor Niemand with the lections, Nirgend, 1824, 2 vols. Folio; just to see whether the {84} last would nothave been the very edition he had read. But I refrained, in mercy. 

 The moon is the image of the Earth, and is not a solid body. By T^{he} Longitude. [173] (Private Circulation. ) In five parts. London, 1856, 1857, 1857; Calcutta, 1858, 1858, 8vo. 

The earth is "brought to a focus"; it describes a "looped orbit round thesun. " The eclipse of the sun is thus explained: "At the time of eclipses, the image is more or less so directly before or behind the earth that, inthe case of new moon, bright rays of the sun fall and bear upon the spotwhere the figure of the earth is brought to a focus, that is, bear upon theimage of the earth, when a darkness beyond is produced reaching to theearth, and the sun becomes more or less eclipsed. " How the earth is"brought to a focus" we do not find stated. Writers of this kind alwayshave the argument that some things which have been ridiculed at first havebeen finally established. Those who put into the lottery had the same kindof argument; but were always answered by being reminded how many blanksthere were to one prize. I am loath to pronounce against anything: but itdoes force itself upon me that the author of these tracts has drawn ablank. 

LUNAR MOTION AGAIN. 

 _Times_, April 6 or 7, 1856. The moon has no rotary motion. 

A letter from Mr. Jellinger Symons, [174] inspector of schools, whichcommenced a controversy of many letters and pamphlets. This dispute comeson at intervals, and will continue to do so. It sometimes arises frominability to understand the character of simple rotation, geometrically;sometimes from not understanding the mechanical doctrine of rotation. 

{85}

 Lunar Motion. The whole argument stated, and illustrated by diagrams; with letters from the Astronomer Royal. By Jellinger C. Symons. London, 1856, 8vo. 

The Astronomer Royal endeavored to disentangle Mr. J. C. Symons, butfailed. Mr. Airy[175] can correct the error of a ship's compasses, becausehe can put her head which way he pleases: but this he cannot do with aspeculator. 

Mr. Symons, in this tract, insinuated that the rotation of the moon is oneof the silver shrines of the craftsmen. To see a thing so clearly as to besatisfied that all who say they do not see it are telling wilful falsehood, is the nature of man. Many of all sects find much comfort in it, when theythink of the others; many unbelievers solace themselves with it againstbelievers; priests of old time founded the right of persecution upon it, and of our time, in some cases, the right of slander: many of theparadoxers make it an argument against students of science. But I must sayfor men of science, for the whole body, that they are fully persuaded ofthe honesty of the paradoxers. The simple truth is, that all those I havementioned, believers, unbelievers, priests, paradoxers, are not so surethey are right in their points of difference that they can safely allowthemselves to be persuaded of the honesty of opponents. Those who knowdemonstration are differently situated. I suspect a train might be laid forthe formation of a better habit in this way. We know that Suvaroff[176]taught his Russians at Ismail not to fear the Turks by accustoming them tocharge bundles of faggots dressed in turbans, etc. 

 At which your wise men sneered in phrases witty, He made no answer--but he took the city!

Would it not be a good thing to exercise boys, in pairs, in the followingdialogue:--Sir, you are quite wrong!--Sir, {86} I am sure you honestlythink so! This was suggested by what used to take place at Cambridge in myday. By statute, every B. A. Was obliged to perform a certain number ofdisputations, and the _father_ of the college had to affirm that it hadbeen done. Some were performed in earnest: the rest were huddled over asfollows. Two candidates occupied the places of the respondent and theopponent: _Recte statuit Newtonus_, said the respondent: _Recte non statuitNewtonus_, [177] said the opponent. This was repeated the requisite numberof times, and counted for as many _acts_ and _opponencies_. The partiesthen changed places, and each unsaid what he had said on the other side ofthe house: I remember thinking that it was capital drill for the House ofCommons, if any of us should ever get there. The process was repeated withevery pair of candidates. 

The real disputations were very severe exercises. I was badgered for twohours with arguments given and answered in Latin, --or what we calledLatin--against Newton's first section, Lagrange's[178] derived functions, and Locke[179] on innate principles. And though I _took off_ everything, and was pronounced by the moderator to have disputed _magno honore_, [180] Inever had such a strain of thought in my life. For the inferior opponentswere made as sharp as their betters by their tutors, who kept lists ofqueer objections, drawn from all quarters. The opponents used to meet theday before to compare their arguments, that the same might not come twiceover. But, after I left Cambridge, it became the fashion to invite therespondent to be present, who therefore learnt all that was to be broughtagainst him. This made the whole thing a farce: and the disputations wereabolished. 

{87}

 The Doctrine of the Moon's Rotation, considered in a letter to the Astronomical Censor of the _Athenæum_. By Jones L. MacElshender. [181] Edinburgh, 1856, 8vo. 

This is an appeal to those cultivated persons who will read it "to overrulethe _dicta_ of judges who would sacrifice truth and justice to professionalrule, or personal pique, pride, or prejudice"; meaning, the great mass ofthose who have studied the subject. But how? Suppose the "cultivatedpersons" were to side with the author, would those who have conclusions todraw and applications to make consent to be wrong because the "general bodyof intelligent men, " who make no special study of the subject, are againstthem? They would do no such thing: they would request the general body ofintelligent men to find their own astronomy, and welcome. But the truth is, that this intelligent body knows better: and no persons know better thatthey know better than the speculators themselves. 

But suppose the general body were to combine, in opposition to those whohave studied. Of course all my list must be admitted to their trial; andthen arises the question whether both sides are to be heard. If so, thegeneral body of the intelligent must hear all the established side have tosay: that is, they must become just as much of students as the inculpatedorthodox themselves. And will they not then get into _professional rule_, pique, pride, and prejudice, as the others did? But if, which I suspect, they are intended to judge as they are, they will be in a rare difficulty. All the paradoxers are of like pretensions: they cannot, as a class, beright, for each one contradicts a great many of the rest. There will be thepuzzle which silenced the crew of the cutter in Marryat's novel of the DogFiend. [182] "A tog is a tog, " said Jansen. --"Yes, " replied another, "we allknow a dog is a dog; but the question is--Is _this_ dog {88} a dog?" Andthis question would arise upon every dog of them all. 

ZETETIC ASTRONOMY. 

 Zetetic Astronomy: Earth not a globe. 1857 (Broadsheet). 

Though only a traveling lecturer's advertisement, there are so manyarguments and quotations that it is a little pamphlet. The lecturer gainedgreat praise from provincial newspapers for his ingenuity in proving thatthe earth is a flat, surrounded by ice. Some of the journals rather inclineto the view: but the _Leicester Advertiser_ thinks that the statements"would seem very seriously to invalidate some of the most importantconclusions of modern astronomy, " while the _Norfolk Herald_ is clear that"there must be a great error on one side or the other. " This broadsheet isprinted at Aylesbury in 1857, and the lecturer calls himself _Parallax_:but at Trowbridge, in 1849, he was S. Goulden. [183] In this lastadvertisement is the following announcement: "A paper on the above subjectswas read before the Council and Members of the Royal Astronomical Society, Somerset House, Strand, London (Sir John F. W. Herschel, [184] President), Friday, Dec. 8, 1848. " No account of such a paper appears in the _Notice_for that month: I suspect that the above is Mr. S. Goulden's way ofrepresenting the following occurrence: Dec. 8, 1848, the Secretary of theAstronomical Society (De Morgan by name) said, at the close of theproceedings, --"Now, gentlemen, if you will promise not to tell the Council, I will read something for your amusement": and he then read a few of thearguments which had been transmitted by the lecturer. The fact is worthnoting that from 1849 to 1857, arguments on the roundness or flatness ofthe earth did itinerate. I have {89} no doubt they did much good: for veryfew persons have any distinct idea of the evidence for the rotundity of theearth. The _Blackburn Standard_ and _Preston Guardian_ (Dec. 12 and 16, 1849) unite in stating that the lecturer ran away from his second lectureat Burnley, having been rather too hard pressed at the end of his firstlecture to explain why the large hull of a ship disappeared before thesails. The persons present and waiting for the second lecture assuagedtheir disappointment by concluding that the lecturer had slipped off theicy edge of his flat disk, and that he would not be seen again till hepeeped up on the opposite side. 

But, strange as it may appear, the opposer of the earth's roundness hasmore of a case--or less of a want of case--than the arithmetical squarer ofthe circle. The evidence that the earth is round is but cumulative andcircumstantial: scores of phenomena ask, separately and independently, whatother explanation can be imagined except the sphericity of the earth. Theevidence for the earth's figure is tremendously powerful of its kind; butthe proof that the circumference is 3. 14159265... Times the diameter is ofa higher kind, being absolute mathematical demonstration. 

The Zetetic system still lives in lectures and books; as it ought to do, for there is no way of teaching a truth comparable to opposition. The lastI heard of it was in lectures at Plymouth, in October, 1864. Since thistime a prospectus has been issued of a work entitled "The Earth not aGlobe"; but whether it has been published I do not know. The contents areas follows:

"The Earth a Plane--How circumnavigated. --How time is lost or gained. --Whya ship's hull disappears (when outward bound) before the mast head. --Whythe Polar Star sets when we proceed Southward, etc. --Why a pendulumvibrates with less velocity at the Equator than {90} at the Pole. --Theallowance for rotundity _supposed_ to be made by surveyors, not made inpractice. --Measurement of Arcs of the Meridian unsatisfactory. --Degrees ofLongitude North and South of the Equator considered. --Eclipses and Earth'sform considered. --The Earth no motion on axis or in orbit. --How the Sunmoves above the Earth's surface concentric with the North Pole. --Cause ofDay and Night, Winter and Summer; the long alternation of light anddarkness at the Pole. --Cause of the Sun rising and setting. --Distance ofthe Sun from London, 4, 028 miles--How measured. --_Challenge toMathematicians. _--Cause of Tides. --Moon self-luminous, NOT areflector. --Cause of Solar and Lunar eclipses. --Stars _not worlds_; theirdistance. --Earth, the _only material_ world; its true position in theuniverse; its condition and ultimate destruction by fire (2 Peter iii. ), etc. "

I wish there were geoplatylogical lectures in every town; in England(_platylogical_, in composition, need not mean _babbling_). The late Mr. Henry Archer[185] would, if alive, be very much obliged to me for recordinghis vehement denial of the roundness of the earth: he was excited if heheard any one call it a globe. I cannot produce his proof from thePyramids, and from some caves in Arabia. He had other curious notions, ofcourse: I should no more believe that a flat earth was a man's onlyparadox, than I should that Dutens, [186] the editor of Leibnitz, waseccentric only in supplying a tooth which he had lost by one which he foundin an Italian tomb, and fully believed that it had once belonged to ScipioAfricanus, whose family vault was discovered, it is supposed, in 1780. Mr. Archer is of note as {91} the suggester of the perforated border of thepostage-stamps, and, I think, of the way of doing it; for this he got4000l. Reward. He was a civil engineer. 

(_August 28, 1865. _) The _Zetetic Astronomy_ has come into my hands. When, in 1851, I went to see the Great Exhibition, I heard an organ played by aperformer who seemed very desirous to exhibit one particular stop. "What doyou think of that stop?" I was asked. --"That depends on the name of it, "said I. --"Oh! what can the name have to do with the sound? 'that which wecall a rose, ' etc. "--"The name has everything to do with it: if it be aflute-stop, I think it very harsh; but if it be a railway-whistle-stop, Ithink it very sweet. " So as to this book: if it be childish, it is clever;if it be mannish, it is unusually foolish. The flat earth, floatingtremulously on the sea; the sun moving always over the flat, giving daywhen near enough, and night when too far off; the self-luminous moon, witha semi-transparent invisible moon, created to give her an eclipse now andthen; the new law of perspective, by which the vanishing of the hull beforethe masts, usually thought to prove the earth globular, really proves itflat;--all these and other things are well fitted to form exercises for aperson who is learning the elements of astronomy. The manner in which thesun dips into the sea, especially in tropical climates, upsets the whole. Mungo Park, [187] I think, gives an African hypothesis which explainsphenomena better than this. The sun dips into the western ocean, and thepeople there cut him in pieces, fry him in a pan, and then join himtogether again, take him round the underway, and set him up in the east. Ihope this book will be read, and that many will be puzzled by it: for thereare many whose notions of astronomy deserve no better fate. There is nosubject on which there is so little {92} accurate conception as that of themotions of the heavenly bodies. The author, though confident in theextreme, neither impeaches the honesty of those whose opinions he assails, nor allots them any future inconvenience: in these points he is worthy tolive on a globe, and to revolve in twenty-four hours. 

(_October, 1866. _) A follower appears, in a work dedicated to the precedingauthor: it is _Theoretical Astronomy examined and exposed by Common Sense_. The author has 128 well-stuffed octavo pages. I hope he will not be thelast. He prints the newspaper accounts of his work: the _Church Times_says--not seeing how the satire might be retorted--"We never began todespair of Scripture until we discovered that 'Common Sense' had taken upthe cudgels in its defence. " This paper considers our author as the type ofa _Protestant_. The author himself, who gives a summary of his arguments inverse, has one couplet which is worth quoting:

 "How is't that sailors, bound to sea, with _a 'globe'_ would never start, But in its place will always take _Mercator's_[188] LEVEL _chart_!"

To which I answer:

 Why, really Mr. Common Sense, you've never got so far As to think Mercator's planisphere shows countries as they are; It won't do to measure distances; it points out how to steer, But this distortion's not for you; another is, I fear. The earth must be a cylinder, if seaman's charts be true, Or else the boundaries, right and left, are one as well as two; They contradict the notion that we dwell upon a plain, For straight away, without a turn, will bring you home again. There are various plane projections; and each one has its use: I wish a milder word would rhyme--but really you're a goose!

The great wish of persons who expose themselves as above, is to be arguedwith, and to be treated as reputable {93} and refutable opponents. "CommonSense" reminds us that no amount of "blatant ridicule" will turn right intowrong. He is perfectly correct: but then no amount of bad argument willturn wrong into right. These two things balance; and we are just where wewere: but you should answer our arguments, for whom, I ask? Would reasonconvince this kind of reasoner? The issue is a short and a clear one. Ifthese parties be what I contend they are, then ridicule is made for them:if not, for what or for whom? If they be right, they are only passingthrough the appointed trial of all good things. Appeal is made to thefuture: and my Budget is intended to show samples of the long line ofheroes who have fallen without victory, each of whom had his day ofconfidence and his prophecy of success. Let the future decide: they sayroundly that the earth is flat; I say flatly that it is round. 

The paradoxers all want reason, and not ridicule: they are all accessible, and would yield to conviction. Well then, let them reason with one another!They divide into squads, each with a subject, and as many differentopinions as persons in each squad. If they be really what they say theyare, the true man of each set can put down all the rest, and can comecrowned with glory and girdled with scalps, to the attack on the orthodoxmisbelievers. But they know, to a man, that the rest are not fit to bereasoned with: they pay the regulars the compliment of believing that theonly chance lies with them. They think in their hearts, each one forhimself, that ridicule is of fit appliance to the rest. 

 Miranda. A book divided into three parts, entitled Souls, Numbers, Stars, on the Neo-Christian Religion ... Vol. I. London, 1858, 1859, 1860. 8vo. 

The name of the author is Filopanti. [189] He announces himself as the 49thand last Emanuel: his immediate {94} predecessors were Emanuel Washington, Emanuel Newton, and Emanuel Galileo. He is to collect nations into onefamily. He knows the transmigrations of the whole human race. ThusDescartes became William III of England: Roger Bacon became Boccaccio. ButCharles IX, [190] in retribution for the massacre of St. Bartholomew, washanged in London under the name of Barthélemy for the murder of Collard:and many of the Protestants whom he killed as King of France were shoutingat his death before the Old Bailey. 

THE SABBATH--THE GREAT PYRAMID

 A Letter to the members of the Anglo-Biblical Institute, dated Sept. 7, 1858, and signed 'Herman Heinfetter. '[191] (Broadsheet. )

This gentleman is well known to the readers of the _Athenæum_, in which, for nearly twenty years, he has inserted, as advertisements, long argumentsin favor of Christians keeping the Jewish Sabbath, beginning on FridayEvening. The present letter maintains that, by the force of the definitearticle, the _days_ of creation may not be consecutive, but may have anytime--millions of years--between them. This ingenious way of reconcilingthe author of Genesis and the indications of geology is worthy to be addedto the list, already pretty numerous. Mr. Heinfetter has taken such painsto make himself a public agitator, that {95} I do not feel it to be anyinvasion of private life if I state that I have heard he is a largecorn-dealer. No doubt he is a member of the congregation whose almanac hasalready been described. 

 The great Pyramid. Why was it built? And who built it? By John Taylor, 1859, [192] 12mo. 

This work is very learned, and may be referred to for the history ofprevious speculations. It professes to connect the dimensions of thePyramid with a system of metrology which is supposed to have left strongtraces in the systems of modern times; showing the Egyptians to have hadgood approximate knowledge of the dimensions of the earth, and of thequadrature of the circle. These are points on which coincidence is hard todistinguish from intention. Sir John Herschel[193] noticed this work, andgave several coincidences, in the _Athenæum_, Nos. 1696 and 1697, April 28and May 5, 1860: and there are some remarks by Mr. Taylor in No. 1701, June2, 1860. 

Mr. Taylor's most recent publication is--

 The battle of the Standards: the ancient, of four thousand years, against the modern, of the last fifty years--the less perfect of the two. London, 1864, 12mo. 

This is intended as an appendix to the work on the Pyramid. Mr. Taylordistinctly attributes the original system to revelation, of which he saysthe Great Pyramid is the record. We are advancing, he remarks, towards theend of the Christian dispensation, and he adds that it is satisfactory tosee that we retain the standards which were given by unwritten revelation700 years before Moses. This is lighting the candle at both ends; formyself, I shall not undertake to deny or affirm either what is said aboutthe dark past or what is hinted about the dark future. 

{96}

My old friend Mr. Taylor is well known as the author of the argument whichhas convinced many, even most, that Sir Philip Francis[194] was Junius:pamphlet, 1813; supplement, 1817; second edition "The Identity of Juniuswith a distinguished living character established, " London, 1818, 8vo. Hetold me that Sir Philip Francis, in a short conversation with him, madeonly this remark, "You may depend upon it you are quite mistaken:" thephrase appears to me remarkable; it has an air of criticism on the book, free from all personal denial. He also mentioned that a hearer told himthat Sir Philip said, speaking of writers on the question, --"Those fellows, for half-a-crown, would prove that Jesus Christ was Junius. "

Mr. Taylor implies, I think, that he is the first who started thesuggestion that Sir Philip Francis was Junius, which I have no means eitherof confirming or refuting. If it be so [and I now know that Mr. Taylorhimself never heard of any predecessor], the circumstance is veryremarkable: it is seldom indeed that the first proposer of any solution ofa great and vexed question is the person who so nearly establishes hispoint in general opinion as Mr. Taylor has done. 

As to the Junius question in general, there is a little bit of thephilosophy of horse-racing which may be usefully applied. A man who is soconfident of his horse that he places him far above any other, maynevertheless, and does, refuse to give odds against all in the field: formany small adverse chances united make a big chance for one or other of theopponents. I suspect Mr. Taylor has made it at least 20 to 1 for Francisagainst any one competitor who has been named: but what the odds may beagainst the {97} whole field is more difficult to settle. What if the realJunius should be some person not yet named?

Mr. Jopling, _Leisure Hour_, May 23, 1863, relies on the porphyry coffer ofthe Great Pyramid, in which he finds "the most ancient and accuratestandard of measure in existence. "

I am shocked at being obliged to place a thoughtful and learned writer, andan old friend, before such a successor as he here meets with. Butchronological arrangement defies all other arrangement. 

(I had hoped that the preceding account would have met Mr. Taylor's eye inprint: but he died during the last summer. For a man of a very thoughtfuland quiet temperament, he had a curious turn for vexed questions. But hereflected very long and very patiently before he published: and all hisworks are valuable for their accurate learning, whichever side the readermay take. )

MRS. ELIZABETH COTTLE. 

1859. _The Cottle Church. _--For more than twenty years printed papers havebeen sent about in the name of Elizabeth Cottle. [195] It is not soremarkable that such papers should be concocted as that they shouldcirculate for such a length of time without attracting public attention. Eighty years ago Mrs. Cottle might have rivalled Lieut. Brothers or JoannaSouthcott. [196] Long hence, when the now current volumes of our journalsare well-ransacked works of reference, those who look into them will beglad to see this {98} feature of our time: I therefore make a few extracts, faithfully copied as to type. The Italic is from the New Testament; theRoman is the requisite interpretation:

"Robert Cottle '_was numbered_ (5196) _with the transgressors_' at the backof the Church in Norwood Cemetery, May 12, 1858--Isa. Liii. 12. The Rev. J. G. Collinson, Minister of St. James's Church, Chapham, the then districtchurch, before All Saints was built, read the funeral service _over theSepulchre wherein never before man was laid_. 

"_Hewn on the stone_, 'at the mouth of the Sepulchre, ' is his name, --RobertCottle, born at Bristol, June 2, 1774; died at Kirkstall Lodge, ClaphamPark, May 6, 1858. _And that day_ (May 12, 1858) _was the preparation_ (dayand year for 'the PREPARED place for you'--Cottleites---by the widowedmother of the Father's house, at Kirkstall Lodge--John xiv. 2, 3). _And theSabbath_ (Christmas Day, Dec. 25, 1859) _drew on_ (for the resurrection ofthe Christian body on 'the third [Protestant Sun]-day'--1 Cor. Xv. 35). _Why seek ye the living_ (God of the New Jerusalem--Heb. Xii. 22; Rev. Iii. 12) _among the dead_ (men): _he_ (the God of Jesus) _is not here_ (in thegrave), _but is risen_ (in the person of the Holy Ghost, from the supper of'the dead in the second death' of Paganism). _Remember how he spake untoyou_ (in the church of the Rev. George Clayton, [197] April 14, 1839). _Iwill not drink henceforth_ (at this last Cottle supper) _of the fruit ofthis_ (Trinity) _vine, until that day_ (Christmas Day, 1859), _when I_(Elizabeth Cottle) _drink it new with you_ (Cottleites) _in my Father'skingdom_--John xv. _If this_ (Trinitarian) _cup may not pass away from me_(Elizabeth Cottle, April 14, 1839), _except I drink it_ ('new with youCottleites, in my Father's Kingdom'), _thy will be done_--Matt. Xxvi. 29, 42, 64. 'Our Father which art (God) in Heaven, ' _hallowed be thy name, thy_(Cottle) _kingdom_ {99} _come, thy will be done in earth, as it is_ (done)_in_ (the new) _Heaven_ (and new earth of the new name of Cottle--Rev. Xxi. 1; iii. 12). 

"... Queen Elizabeth, from A. D. 1558 to 1566. _And this_ WORD _yet oncemore_ (by a second Elizabeth--the WORD of his oath) _signifieth_ (at JohnScott's baptism of the Holy Ghost) _the removing of those things_ (thoseGods and those doctrines) _that are made_ (according to the Creeds andCommandments of men) _that those things_ (in the moral law of God) _whichcannot be shaken_ (as a rule of faith and practice) _may remain, whereforewe receiving_ (from Elizabeth) _a kingdom_ (of God, ) _which cannot bemoved_ (by Satan) _let us have grace_ (in his Grace of Canterbury) _wherebywe may serve God acceptably_ (with the acceptable sacrifice of Elizabeth'sbody and blood of the communion of the Holy Ghost) _with reverence_ (fortruth) _and godly fear_ (of the unpardonable sin of blasphemy against theHoly Ghost) _for our God_ (the Holy Ghost) _is a consuming fire_ (to thenation that will not serve him in the Cottle Church). We cannot defendourselves against the Almighty, and if He is our defence, no nation caninvade us. 

"In verse 4 the Church of St. Peter is _in prison between four quaternionsof soldiers_--the Holy Alliance of 1815. Rev. Vii. I. Elizabeth, _the Angelof the Lord_ Jesus _appears_ to the Jewish and Christian body with _thevision_ of prophecy to the Rev. Geo. Clayton and his clerical brethren, April 8th, 1839. _Rhoda_ was the name of her maid at Putney Terrace whoused _to open the door to her Peter_, the Rev. Robert Ashton, [198] thePastor of 'the little flock' 'of 120 names together, assembled in an upper(school) room' at Putney Chapel, to which little flock she gave therevelation (Acts. I. 13, 15) _of Jesus the same_ King of the Jews_yesterday_ at the prayer meeting, Dec. 31, 1841, _and to-day_, {100} Jan. 1, 1842, _and for ever_. See book of Life, page 24. Matt. Xviii. 19, xxi. 13-16. In verse 6 the Italian body of St. Peter _is sleeping_ 'in thesecond death' _between the two_ Imperial _soldiers_ of France and Austria. The Emperor of France from Jan. 1, to July 11, 1859, causes the Italian_chains of St. Peter to fall off from his_ Imperial _hands_. 

"_I say unto thee_, Robert Ashton, _thou art Peter_, a stone, _and uponthis rock_, of truth, _will I_ Elizabeth, the angel of Jesus, _build my_Cottle _Church, and the gates of hell_, the doors of St. Peter, at Rome, shall not prevail against it--Matt. Xvi. 18. Rev. Iii. 7-12. "

This will be enough for the purpose. When any one who pleases can circulatenew revelations of this kind, uninterrupted and unattended to, newrevelations will cease to be a good investment of excentricity. I take itfor granted that the gentlemen whose names are mentioned have nothing to dowith the circulars or their doctrines. Any lady who may happen to beintrusted with a revelation may nominate her own pastor, or any otherclergyman, one of her apostles; and it is difficult to say to what courtthe nominees can appeal to get the commission abrogated. 

_March 16, 1865. _ During the last two years the circulars have continued. It is hinted that funds are low: and two gentlemen who are represented asgone "to Bethlehem asylum in despair" say that Mrs. Cottle "will spend allthat she hath, while Her Majesty's Ministers are flourishing on the wagesof sin. " The following is perhaps one of the most remarkable passages inthe whole:

"_Extol and magnify Him_ (Jehovah, the Everlasting God, see the Magnificatand Luke i. 45, 46--68--73--79), _that rideth_ (by rail and steam over landand sea, from his holy habitation at Kirkstall Lodge, Psa. Lxxvii. 19, 20), _upon the_ (Cottle) _heavens, as it were_ (Sept. 9, 1864, see pages 21, 170), _upon an_ (exercising, Psa. Cxxxi. 1), _horse_-(chair, bought of Mr. John Ward, Leicester-square). " {101}

I have pretty good evidence that there is a clergyman who thinks Mrs. Cottle a very sensible woman. 

[_The Cottle Church. _ Had I chanced to light upon it at the time ofwriting, I should certainly have given the following. A printed letter tothe _Western Times_, by Mr. Robert Cottle, was accompanied by a manuscriptletter from Mrs. Cottle, apparently a circular. The date was Nov^{r}. 1853, and the subject was the procedure against Mr. Maurice[199] at King'sCollege for doubting that God would punish human sins by an existence oftorture lasting through years numbered by millions of millions of millionsof millions (repeat the word _millions_ without end, ) etc. The memory ofMr. Cottle has, I think, a right to the quotation: he seems to have been noparticipator in the notions of his wife:

"The clergy of the Established Church, taken at the round number of 20, 000, may, in their first estate, be likened to 20, 000 gold blanks, destined tobecome sovereigns, in succession, --they are placed between the matrix ofthe Mint, when, by the pressure of the screw, they receive the impress thatfits them to become part of the current coin of the realm. In a waysomewhat analogous this great body of the clergy have each passed throughthe crucibles of Oxford and Cambridge, --have been assayed by the Bishop'schaplain, touching the health of their souls, and the validity of theircall by the Divine Spirit, and then the gentle pressure of a prelate's handupon their heads; and the words--'Receive the Holy Ghost, ' have, in a briefspace of time, wrought a {102} change in them, much akin to the miracle oftransubstantiation--the priests are completed, and they become the currentecclesiastical coin of our country. The whole body of clergy, here spokenof, have undergone the preliminary induction of baptism and confirmation;and all have been duly ordained, _professing_ to hold one faith, and tobelieve in the selfsame doctrines! In short, to be as identical as the20, 000 sovereigns, if compared one with the other. But mind is notmalleable and ductile, like gold; and all the preparations of tests, creeds, and catechisms will not insure uniformity of belief. No stamp oforthodoxy will produce the same impress on the minds of different men. Variety is manifest, and patent, upon everything mental and material. TheAlmighty has not created, nor man fashioned, two things alike! How futile, then, is the attempt to shape and mould man's apprehension of divine truthby one fallible standard of man's invention! If proof of this be required, an appeal might be made to history and the experience of eighteen hundredyears. "

This is an argument of force against the reasonableness of expecting tensof thousands of educated readers of the New Testament to find the doctrineabove described in it. The lady's argument against the doctrine itself isvery striking. Speaking of an outcry on this matter among the Dissentersagainst one of their body, who was the son of "the White Stone (Rev. Ii. 17), or the Roman cement-maker, " she says--

"If the doctrine for which they so wickedly fight were true, what wouldbecome of the black gentlemen for whose redemption I have been sacrificedfrom April 8 1839. "

There are certainly very curious points about this revelation. There havebeen many surmises about the final restoration of the infernal spirits, from the earliest ages of Christianity until our own day: a collection ofthem would be worth making. On reading this in proof, I see a possibilitythat by "black gentlemen" may be meant the clergy: {103} I suppose my firstinterpretation must have been suggested by context: I leave the point tothe reader's sagacity. ]

JAMES SMITH, ARCH-PARADOXER. 

 The Problem of squaring the circle solved; or, the circumference and area of the circle discovered. By James Smith. [200] London, 1859, 8vo. 

 On the relations of a square inscribed in a circle. Read at the British Association, Sept. 1859, published in the Liverpool Courier, Oct. 8, 1859, and reprinted in broadsheet. 

 The question: Are there any commensurable relations between a circle and other Geometrical figures? Answered by a member of the British Association ... London, 1860, 8vo. --[This has been translated into French by M. Armand Grange, Bordeaux, 1863, 8vo. ]

 The Quadrature of the Circle. Correspondence between an eminent mathematician and James Smith, Esq. (Member of the Mersey Docks and Harbour Board), London, 1861, 8vo. (pp. 200). 

 Letter to the ... British Association ... By James Smith, Esq. Liverpool, 1861, 8vo. 

 Letter to the ... British Association ... By James Smith, Esq. Liverpool, 1862, 8vo. --[These letters the author promised to continue. ]

 A Nut to crack for the readers of Professor De Morgan's 'Budget of Paradoxes. ' By James Smith, Esq. Liverpool, 1863, 8vo. 

 Paper read at the Liverpool Literary and Philosophical Society, reported in the Liverpool Daily Courier, Jan. 26, 1864. Reprinted as a pamphlet. 

 The Quadrature of the circle, or the true ratio between the diameter and circumference geometrically and mathematically demonstrated. By James Smith, Esq. Liverpool, 1865, 8vo. 

 {104}

 [On the relations between the dimensions and distances of the Sun, Moon, and Earth; a paper read before the Literary and Philosophical Society of Liverpool, Jan. 25, 1864. By James Smith, Esq. 

 The British Association in Jeopardy, and Dr. Whewell, the Master of Trinity, in the stocks without hope of escape. Printed for the authors (J. S. Confessed, and also hidden under _Nauticus_). (No date, 1865). 

 The British Association in Jeopardy, and Professor De Morgan in the Pillory without hope of escape. London, 1866, 8vo. ]

When my work appeared in numbers, I had not anything like an adequate ideaof Mr. James Smith's superiority to the rest of the world in the points inwhich he is superior. He is beyond a doubt the ablest head at unreasoning, and the greatest hand at writing it, of all who have tried in our day toattach their names to an error. Common cyclometers sink into puny orthodoxyby his side. 

The behavior of this singular character induces me to pay him thecompliment which Achilles paid Hector, to drag him round the walls againand again. He was treated with unusual notice and in the most gentlemanner. The unnamed mathematician, E. M. Bestowed a volume of mildcorrespondence upon him; Rowan Hamilton[201] quietly proved him wrong in away accessible to an ordinary schoolboy; Whewell, [202] as we shall see, gave him the means of seeing himself wrong, even more easily than byHamilton's method. Nothing would do; it was small kick and silly fling atall; and he exposed his conceit by alleging that he, James Smith, hadplaced Whewell in the stocks. He will therefore be universally pronounced aproper object of the severest literary punishment: but the opinion of allwho can put two propositions together will be that of the many strokes Ihave given, the hardest and most telling are my republications of his ownattempts to reason. 

He will come out of my hands in the position he ought {105} to hold, theSupreme Pontiff of cyclometers, the vicegerent of St. Vitus upon earth, theMamamouchi of burlesque on inference. I begin with a review of him whichappeared in the _Athenæum_ of May 11, 1861. Mr. Smith says I wrote it: thisI neither affirm nor deny; to do either would be a sin against theeditorial system elsewhere described. Many persons tell me they know me bymy style; let them form a guess: I can only say that many have declared asabove while fastening on me something which I had never seen nor heard of. 

 The Quadrature of the Circle: Correspondence between an Eminent Mathematician and James Smith, Esq. (Edinburgh, Oliver & Boyd; London, Simpkin, Marshall & Co. )

"A few weeks ago we were in perpetual motion. We did not then suppose thatanything would tempt us on a circle-squaring expedition: but thecircumstances of the book above named have a peculiarity which induces usto give it a few words. 

"Mr. James Smith, a gentleman residing near Liverpool, was some years agoseized with the _morbus cyclometricus_. [203] The symptoms soon took adefined form: his circumference shrank into exactly 3-1/8 times hisdiameter, instead of close to 3-16/113, which the mathematician knows to beso near to truth that the error is hardly at the rate of a foot in 2, 000miles. This shrinking of the circumference remained until it becameabsolutely necessary that it should be examined by the British Association. This body, which as Mr. James Smith found to his sorrow, has some interestin 'jealously guarding the mysteries of their profession, ' refused at firstto entertain the question. On this Mr. Smith changed his 'tactics' and thename of his paper, and smuggled in the subject under the form of 'TheRelations of a Circle inscribed in a Square'! The paper was thus forcedupon the Association, for Mr. Smith informs us that he {106} 'gave theSection to understand that he was not the man that would permit even theBritish Association to trifle with him. ' In other words, the Associationbore with and were bored with the paper, as the shortest way out of thematter. Mr. Smith also circulated a pamphlet. Some kind-hearted man, whodid not know the disorder as well as we do, and who appears in Mr. Smith'shandsome octavo as E. M. --the initials of 'eminent mathematician'--wrote tohim and offered to show him in a page that he was all wrong. Mr. Smiththereupon opened a correspondence, which is the bulk of the volume. Whenthe correspondence was far advanced, Mr. Smith announced his intention topublish. His benevolent instructor--we mean in intention--protested againstthe publication, saying 'I do not wish to be gibbeted to the world ashaving been foolish enough to enter upon what I feel now to have been aridiculous enterprise. '

"For this Mr. Smith cared nothing: he persisted in the publication, and thebook is before us. Mr. Smith has had so much grace as to conceal his kindadviser's name under E. M. , that is to say, he has divided the wrong amongall who may be suspected of having attempted so hopeless a task as that ofputting a little sense into his head. He has violated the decencies ofprivate life. Against the will of the kind-hearted man who undertook hiscase, he has published letters which were intended for no other purposethan to clear his poor head of a hopeless delusion. He deserves theseverest castigation; and he will get it: his abuse of confidence willstick by him all his days. Not that he has done his benefactor--inintention, again--any harm. The patience with which E. M. Put the blundersinto intelligible form, and the perseverance with which he tried to find acranny-hole for common reasoning to get in at, are more than respectable:they are admirable. It is, we can assure E. M. , a good thing that thenature of the circle-squarer should be so completely exposed as in thisvolume. The benefit which he intended Mr. James Smith may be {107}conferred upon others. And we should very much like to know his name, andif agreeable to him, to publish it. As to Mr. James Smith, we can only saythis: he is not mad. Madmen reason rightly upon wrong premises: Mr. Smithreasons wrongly upon no premises at all. 

"E. M. Very soon found out that, to all appearance, Mr. Smith got a circleof 3-1/8 times the diameter by making it the supposition to set out withthat there was such a circle; and then finding certain consequences which, so it happened, were not inconsistent with the supposition on which theywere made. Error is sometimes self-consistent. However, E. M. , to be quitesure of his ground, wrote a short letter, stating what he took to be Mr. Smith's hypothesis, containing the following: 'On AC as diameter, describethe circle D, which by hypothesis shall be equal to three and one-eighthtimes the length of AC.... I beg, before proceeding further, to ask whetherI have rightly stated your argument. ' To which Mr. Smith replied: 'You havestated my argument with perfect accuracy. ' Still E. M. Went on, and wecould not help, after the above, taking these letters as the initials ofEverlasting Mercy. At last, however, when Mr. Smith flatly denied that thearea of the circle lies between those of the inscribed and circumscribedpolygons, E. M. Was fairly beaten, and gave up the task. Mr. Smith was leftto write his preface, to talk about the certain victory of truth--which, oddly enough, is the consolation of all hopelessly mistaken men; to comparehimself with Galileo; and to expose to the world the perverse behavior ofthe Astronomer Royal, on whom he wanted to fasten a conversation, and whoreplied, 'It would be a waste of time, Sir, to listen to anything you couldhave to say on such a subject. '

"Having thus disposed of Mr. James Smith, we proceed to a few remarks onthe subject: it is one which a journal would never originate, but which isrendered necessary from time to time by the attempts of the autopseustic tobecome {108} heteropseustic. To the mathematician we have nothing to say:the question is, what kind of assurance can be given to the world at largethat the wicked mathematicians are not acting in concert to keep down theirsuperior, Mr. James Smith, the current Galileo of the quadrature of thecircle. 

"Let us first observe that this question does not stand alone:independently of the millions of similar problems which exist in highermathematics, the finding of the diagonal of a square has just the samedifficulty, namely, the entrance of a pair of lines of which one cannot bedefinitely expressed by means of the other. We will show the reader who isup to the multiplication-table how he may go on, on, on, ever nearer, neverthere, in finding the diagonal of a square from the side. 

"Write down the following rows of figures, and more, if you like, in theway described:

 1 2 5 12 29 70 169 408 985 1 3 7 17 41 99 239 577 1393

After the second, each number is made up of double the last increased bythe last but one: thus, 5 is 1 more than twice 2, 12 is 2 more than twice5, 239 is 41 more than twice 99. Now, take out two adjacent numbers fromthe upper line, and the one below the first from the lower: as

 70 169 99. 

Multiply together 99 and 169, giving 16, 731. If, then, you will say that 70diagonals are exactly equal to 99 sides, you are in error about thediagonal, but an error the amount of which is not so great as the 16, 731stpart of the diagonal. Similarly, to say that five diagonals make exactlyseven sides does not involve an error of the 84th part of the diagonal. 

"Now, why has not the question of _crossing the square_ been as celebratedas that of _squaring the circle_? Merely because Euclid demonstrated theimpossibility of the first {109} question, while that of the second was notdemonstrated, completely, until the last century. 

"The mathematicians have many methods, totally different from each other, of arriving at one and the same result, their celebrated approximation tothe circumference of the circle. An intrepid calculator has, in our owntime, carried his approximation to what they call 607 decimal places: thishas been done by Mr. Shanks, [204] of Houghton-le-Spring, and Dr. Rutherford[205] has verified 441 of these places. But though 607 lookslarge, the general public will form but a hazy notion of the extent ofaccuracy acquired. We have seen, in Charles Knight's[206] _EnglishCyclopædia_, an account of the matter which may illustrate theunimaginable, though rationally conceivable, extent of accuracy obtained. 

"Say that the blood-globule of one of our animalcules is a millionth of aninch in diameter. Fashion in thought a globe like our own, but so muchlarger that our globe is but a blood-globule in one of its animalcules:never mind the microscope which shows the creature being rather a bulkyinstrument. Call this the first globe _above_ us. Let the first globe aboveus be but a blood-globule, as to size, in the animalcule of a still largerglobe, which call the second globe above us. Go on in this way to thetwentieth globe above us. Now go down just as far on the other side. Letthe blood-globule with which we started be a globe peopled with animalslike ours, but rather smaller: {110} and call this the first globe belowus. Take a blood-globule out of this globe, people it, and call it thesecond globe below us: and so on to the twentieth globe below us. This is afine stretch of progression both ways. Now give the giant of the twentiethglobe _above_ us the 607 decimal places, and, when he has measured thediameter of his globe with accuracy worthy of his size, let him calculatethe circumference of his equator from the 607 places. Bring the littlephilosopher from the twentieth globe _below_ us with his very bestmicroscope, and set him to see the small error which the giant must make. He will not succeed, unless his microscopes be much better for his sizethan ours are for ours. 

"Now it must be remembered by any one who would laugh at the closeness ofthe approximation, that the mathematician generally goes _nearer_; in facthis theorems have usually no error at all. The very person who isbewildered by the preceding description may easily forget that if therewere _no error at all_, the Lilliputian of the millionth globe below uscould not find a flaw in the Brobdingnagian of the millionth globe above. The three angles of a triangle, of perfect accuracy of form, are_absolutely_ equal to two right angles; no stretch of progression willdetect _any_ error. 

"Now think of Mr. Lacomme's mathematical adviser (_ante_, Vol. I, p. 46)making a difficulty of advising a stonemason about the quantity of pavementin a circular floor!

"We will now, for our non-calculating reader, put the matter in anotherway. We see that a circle-squarer can advance, with the utmost confidence, the assertion that when the diameter is 1, 000, the circumference isaccurately 3, 125: the mathematician declaring that it is a trifle more than3, 141½. If the squarer be right, the mathematician has erred by about a200th part of the whole: or has not kept his accounts right by about 10s. In every 100l. Of course, if he set out with such an error he willaccumulate blunder upon blunder. Now, if there be a process in which {111}close knowledge of the circle is requisite, it is in the prediction of themoon's place--say, as to the time of passing the meridian at Greenwich--ona given day. We cannot give the least idea of the complication of details:but common sense will tell us that if a mathematician cannot find his wayround the circle without a relative error four times as big as astockbroker's commission, he must needs be dreadfully out in his attempt topredict the time of passage of the moon. Now, what is the fact? His erroris less than a second of time, and the moon takes 27 days odd to revolve. That is to say, setting out with 10s. In 100l. Of error in hiscircumference, he gets within the fifth part of a farthing in 100l. Inpredicting the moon's transit. Now we cannot think that the respect inwhich mathematical science is held is great enough--though we find it notsmall--to make this go down. That respect is founded upon a notion thatright ends are got by right means: it will hardly be credited that thetruth can be got to farthings out of data which are wrong by shillings. Even the celebrated Hamilton[207] of Edinburgh, who held that inmathematics there was no way of going wrong, was fully impressed with thebelief that this was because error was avoided from the beginning. He neverwent so far as to say that a mathematician who begins wrong must end rightsomehow. 

"There is always a difficulty about the mode in which the thinking man ofcommon life is to deal with subjects he has not studied to a professionalextent. He must form opinions on matters theological, political, legal, medical, and social. If he can make up his mind to choose a guide, thereis, of course, no perplexity: but on all the subjects mentioned thedirection-posts point different ways. Now why should he not form hisopinion upon an abstract mathematical question? Why not conclude that, asto the circle, it is possible Mr. James Smith may be the man, just {112} asAdam Smith[208] was the man of things then to come, or Luther, or Galileo?It is true that there is an unanimity among mathematicians which prevailsin no other class: but this makes the chance of their all being wrong onlydifferent in degree. And more than this, is it not generally thought amongus that priests and physicians were never so much wrong as when there wasmost appearance of unanimity among them? To the preceding questions we seeno answer except this, that the individual inquirer may as rationallydecide a mathematical question for himself as a theological or a medicalquestion, so soon as he can put himself into a position in mathematics, level with that in which he stands in theology or medicine. The every-daythought and reading of common life have a certain resemblance to thethought and reading demanded by the learned faculties. The research, thebalance of evidence, the estimation of probabilities, which are used in aquestion of medicine, are closely akin in character, however different thematter of application, to those which serve a merchant to draw hisconclusions about the markets. But the mathematicians have methods of theirown, to which nothing in common life bears close analogy, as to the natureof the results or the character of the conclusions. The logic ofmathematics is certainly that of common life: but the data are of adifferent species; they do not admit of doubt. An expert arithmetician, such as is Mr. J. Smith, may fancy that calculation, merely as such, ismathematics: but the value of his book, and in this point of view it is notsmall, is the full manner in which it shows that a practised arithmetician, venturing into the field of mathematical demonstration, may show himselfutterly destitute of all that distinguishes the reasoning geometricalinvestigator from the calculator. 

{113}

"And further, it should be remembered that in mathematics the power ofverifying results far exceeds that which is found in anything else: andalso the variety of distinct methods by which they can be attained. Itfollows from all this that a person who desires to be as near the truth ashe can will not judge the results of mathematical demonstration to be opento his criticism, in the same degree as results of other kinds. Should hefeel compelled to decide, there is no harm done: his circle may be 3-1/8times its diameter, if it please him. But we must warn him that, in orderto get this circle, he must, as Mr. James Smith has done, _make it athome_: the laws of space and thought beg leave respectfully to decline theorder. "

I will insert now at length, from the _Athenæum_ of June 8, 1861, the easyrefutation given by my deceased friend, with the remarks which precede. 

"Mr. James Smith, of whose performance in the way of squaring the circle wespoke some weeks ago in terms short of entire acquiescence, has advertisedhimself in our columns, as our readers will have seen. He has alsoforwarded his letter to the Liverpool _Albion_, with an additionalstatement, which he did not make in _our_ journal. He denies that he hasviolated the decencies of private life, since his correspondent revised theproofs of his own letters, and his 'protest had respect only to making hisname public. ' This statement Mr. James Smith precedes by saying that wehave treated as true what we well knew to be false: and he follows bysaying that we have not read his work, or we should have known the abovefacts to be true. Mr. Smith's pretext is as follows. His correspondentE. M. Says, 'My letters were not intended for publication, and I protestagainst their being published, ' and he subjoins 'Therefore I must desirethat my name may not be used. ' The obvious meaning is that E. M. Protestedagainst the publication altogether, but, judging that Mr. Smith was {114}determined to publish, desired that his name should not be used. That heafterwards corrected the proofs merely means that he thought it wiser tolet them pass under his own eyes than to leave them entirely to Mr. Smith. 

"We have received from Sir W. Rowan Hamilton[209] a proof that thecircumference is more than 3-1/8 diameters, requiring nothing but aknowledge of four books of Euclid. We give it in brief as an exercise forour juvenile readers to fill up. It reminds us of the old days when realgeometers used to think it worth while seriously to demolish pretenders. Mr. Smith's fame is now assured: Sir W. R. Hamilton's brief and easyexposure will procure him notice in connection with this celebratedproblem. 

"It is to be shown that the perimeter of a regular polygon of 20 sides isgreater than 3-1/8 diameters of the circle, and still more, of course, isthe circumference of the circle greater than 3-1/8 diameters. 

"1. It follows from the 4th Book of Euclid, that the rectangle under theside of a regular decagon inscribed in a circle, and that side increased bythe radius, is equal to the square of the radius. But the product 791 (791+ 1280) is less than 1280 × 1280; if then the radius be 1280 the side ofthe decagon is greater than 791. 

"2. When a diameter bisects a chord, the square of the chord is equal tothe rectangle under the doubles of the segments of the diameter. But theproduct 125 (4 × 1280 - 125) is less than 791 × 791. If then the bisectedchord be a side of the decagon, and if the radius be still 1280, the doubleof the lesser segment exceeds 125. 

"3. The rectangle under this doubled segment and the radius is equal to thesquare of the side of an inscribed regular polygon of 20 sides. But theproduct 125 × 1280 is equal to 400 × 400; therefore, the side of thelast-mentioned polygon is greater than 400, if the radius be still 1280. Inother words, if the radius be represented by the new {115} member 16, andtherefore the diameter by 32, this side is greater than 5, and theperimeter exceeds 100. So that, finally, if the diameter be 8, theperimeter of the inscribed regular polygon of 20 sides, and still more thecircumference of the circle, is greater than 25: that is, the circumferenceis more than 3-1/8 diameters. "

The last work in the list was thus noticed in the _Athenæum_, May 27, 1865. 

"Mr. James Smith appears to be tired of waiting for his place in the Budgetof Paradoxes, and accordingly publishes a long letter to Professor DeMorgan, with various prefaces and postscripts. The letter opens by a hintthat the Budget appears at very long intervals, and 'apparently without anysufficient reason for it. ' As Mr. Smith hints that he should like to seeMr. De Morgan, whom he calls an 'elephant of mathematics, ' 'pumping hisbrains' 'behind the scenes'--an odd thing for an elephant to do, and an oddplace to do it in--to get an answer, we think he may mean to hint that theBudget is delayed until the pump has worked successfully. Mr. Smith isinformed that we have had the whole manuscript of the Budget, exceptingonly a final summing-up, in our hands since October, 1863. [This does notrefer to the Supplement. ] There has been no delay: we knew from thebeginning that a series of historical articles would be frequentlyinterrupted by the things of the day. Mr. James Smith lets out that he hasnever been able to get a private line from Mr. De Morgan in answer to hiscommunications: we should have guessed it. He says, 'The Professor is anold bird and not to be easily caught, and by no efforts of mine have I beenable, up to the present moment, either to induce or twit him into adiscussion.... ' Mr. Smith curtails the proverb: old birds are not to becaught with _chaff_, nor with _twit_, which seems to be Mr. Smith's wordfor his own chaff, and, so long as the first letter is sounded, a veryproper word. Why does he not try a little grain of sense? Mr. Smithevidently {116} thinks that, in his character as an elephant, the Professorhas not pumped up brain enough to furnish forth a bird. In serious earnest, Mr. Smith needs no answer. In one thing he excites our curiosity: what ismeant by demonstrating 'geometrically _and_ mathematically?'"

I now proceed to my original treatment of the case. 

Mr. James Smith will, I have no doubt, be the most uneclipsedcircle-squarer of our day. He will not owe this distinction to his being aninfluential and respected member of the commercial world of Liverpool, eventhough the power of publishing which his means give him should induce himto issue a whole library upon one paradox. Neither will he owe it to thepains taken with him by a mathematician who corresponded with him until thejoint letters filled an octavo volume. Neither will he owe it to the noticetaken of him by Sir William Hamilton, of Dublin, who refuted him in amanner intelligible to an ordinary student of Euclid, which refutation hecalls a remarkable paradox easily explainable, but without explaining it. What he will owe it to I proceed to show. 

Until the publication of the _Nut to Crack_ Mr. James Smith stood amongcircle-squarers in general. I might have treated him with ridicule, as Ihave done others: and he says that he does not doubt he shall come in forhis share at the tail end of my Budget. But I can make a better job of himthan so, as Locke would have phrased it: he is such a very striking exampleof something I have said on the use of logic that I prefer to make anexample of his writings. On one point indeed he well deserves the_scutica_, [210] if not the _horribile flagellum_. [211] He tells me that hewill bring his solution to me in such a form as shall compel me to admit itas _un fait accompli_ [_une faute accomplie?_][212] {117} or leave myselfopen to the humiliating charge of mathematical ignorance and folly. He hasalso honored me with some private letters. In the first of these he givesme a "piece of information, " after which he cannot imagine that I, "as anhonest mathematician, " can possibly have the slightest hesitation inadmitting his solution. There is a tolerable reservoir of modest assurancein a man who writes to a perfect stranger with what he takes for anargument, and gives an oblique threat of imputation of dishonesty in casethe argument be not admitted without hesitation; not to speak of the minorcharges of ignorance and folly. All this is blind self-confidence, withoutmixture of malicious meaning; and I rather like it: it makes me understandhow Sam Johnson came to say of his old friend Mrs. Cobb, [213]--"I love MollCobb for her impudence. " I have now done with my friend's _suaviter inmodo_, [214] and proceed to his _fortiter in re_[215]: I shall show that he_has_ convicted himself of ignorance and folly, with an honesty and candorworthy of a better value of [pi]. 

Mr. Smith's method of proving that every circle is 3-1/8 diameters is toassume that it is so, --"if you dislike the term datum, then, by hypothesis, let 8 circumferences be exactly equal to 25 diameters, "--and then to showthat every other supposition is thereby made absurd. The right to thisassumption is enforced in the "Nut" by the following analogy:

"I think you (!) will not dare (!) to dispute my right to this hypothesis, when I can prove by means of it that every other value of [pi] will lead tothe grossest absurdities; unless indeed, you are prepared to dispute theright of Euclid to adopt a false line hypothetically for the purpose {118}of a '_reductio ad absurdum_'[216] demonstration, in pure geometry. "

Euclid assumes what he wants to _disprove_, and shows that his _assumption_leads to absurdity, and so _upsets itself_. Mr. Smith assumes what he wantsto _prove_, and shows that _his_ assumption makes _other propositions_ leadto absurdity. This is enough for all who can reason. Mr. James Smith cannotbe argued with; he has the whip-hand of all the thinkers in the world. Montucla would have said of Mr. Smith what he said of the gentleman whosquared his circle by giving 50 and 49 the same square root, _Il a perdu ledroit d'être frappé de l'évidence_. [217]

It is Mr. Smith's habit, when he finds a conclusion agreeing with its ownassumption, to regard that agreement as proof of the assumption. Thefollowing is the "piece of information" which will settle me, if I behonest. Assuming [pi] to be 3-1/8, he finds out by working instance afterinstance that the mean proportional between one-fifth of the area andone-fifth of eight is the radius. That is, if [pi] = 25/8, sqrt(([pi]r^2)/5 · 8/5) = r. 

This "remarkable general principle" may fail to establish Mr. Smith'squadrature, even in an honest mind, if that mind should happen to knowthat, a and b being any two numbers whatever, we need only assume--

 [pi] = a^2/b, to get at sqrt(([pi]r^2)/a · b/a) = r. 

We naturally ask what sort of glimmer can Mr. Smith have of the subjectwhich he professes to treat? On this point he has given satisfactoryinformation. I had mentioned the old problem of finding two meanproportionals, {119} as a preliminary to the duplication of the cube. Onthis mention Mr. Smith writes as follows. I put a few words in capitals;and I write rq[218] for the sign of the square root, which embarrassessmall type:

"This establishes the following _infallible_ rule, for finding two meanproportionals OF EQUAL VALUE, and is more than a preliminary, to the famousold problem of 'Squaring the circle. ' Let any finite number, say 20, andits fourth part = ¼(20) = 5, be given numbers. Then rq(20 × 5) = rq 100 =10, is their mean proportional. Let this be a given mean proportional TOFIND ANOTHER MEAN PROPORTIONAL OF EQUAL VALUE. Then

 20 × [pi]/4 = 20 × 3. 125/4 = 20 × . 78125 = 15. 625

will be the first number; as

 25 : 16 :: rq 20 : rq 8. 192: and (rq 8. 192)^2 × [pi]/4 = 8. 192 × . 78125 = 6. 4

will be the second number; therefore rq(15. 625 × 6. 4) = rq 100 = 10, is therequired mean proportional.... Now, my good Sir, however competent you maybe to prove every man a fool [not _every_ man, Mr. Smith! only _some_; praylearn logical quantification] who now thinks, or in times gone by hasthought, the 'Squaring of the Circle' _a possibility_; I doubt, and, on theevidence afforded by your Budget, I cannot help doubting, whether you wereever before competent to find two mean proportionals _by my uniquemethod_. "--(_Nut_, pp. 47, 48. ) [That I never was, I solemnly declare!]

All readers can be made to see the following exposure. When 5 and 20 aregiven, x is a mean proportional when in 5, x, 20, 5 is to x as x to 20. Andx must be 10. But x and y are two mean proportionals when in 5, x, y, 20, x{120} is a mean proportional between 5 and y, and y is a mean proportionalbetween x and 20. And these means are x = 5 [cuberoot]4, y = 5[cuberoot]16. But Mr. Smith finds _one_ mean, finds it _again_ in aroundabout way, and produces 10 and 10 as the two (equal!) means, insolution of the "famous old problem. " This is enough: if more were wanted, there is more where this came from. Let it not be forgotten that Mr. Smithhas found a translator abroad, two, perhaps three, followers at home, and--most surprising of all--a real mathematician to try to set him right. And this mathematician did not discover the character of the subsoil of theland he was trying to cultivate until a goodly octavo volume of letters hadpassed and repassed. I have noticed, in more quarters than one, an apparentwant of perception of the _full_ amount of Mr. Smith's ignorance: personswho have not been in contact with the non-geometrical circle-squarers havea kind of doubt as to whether anybody can carry things so far. But I am an"old bird" as Mr. Smith himself calls me; a Simorg, an "all-knowing Bird ofAges" in matters of cyclometry. 

The curious phenomena of thought here exhibited illustrate, as above said, a remark I have long ago made on the effect of proper study of logic. Mostpersons reason well enough on matter to which they are accustomed, and interms with which they are familiar. But in unaccustomed matter, and withuse of strange terms, few except those who are practised in theabstractions of pure logic can be tolerably sure to keep their feet. Andone of the reasons is easily stated: terms which are not quite familiarpartake of the vagueness of the X and Y on which the student of logiclearns to see the formal force of a proposition independently of itsmaterial elements. 

I make the following quotation from my fourth paper on logic in the_Cambridge Transactions_:

"The uncultivated reason proceeds by a process almost entirely material. Though the necessary law of thought {121} must determine the conclusion ofthe ploughboy as much as that of Aristotle himself, the ploughboy'sconclusion will only be tolerably sure when the matter of it is such ascomes within his usual cognizance. He knows that geese being all birds doesnot make all birds geese, but mainly because there are ducks, chickens, partridges, etc. A beginner in geometry, when asked what follows from'Every A is B, ' answers 'Every B is A. ' That is, the necessary laws ofthought, except in minds which have examined their tools, are not very sureto work correct conclusions except upon familiar matter.... As thecultivation of the individual increases, the laws of thought which are ofmost usual application are applied to familiar matter with tolerablesafety. But difficulty and risk of error make a new appearance with a newsubject; and this, in most cases, until new subjects are familiar things, unusual matter common, untried nomenclature habitual; that is, until it isa habit to be occupied upon a novelty. It is observed that many personsreason well in some things and badly in others; and this is attributed tothe consequence of employing the mind too much upon one or another subject. But those who know the truth of the preceding remarks will not have far toseek for what is often, perhaps most often, the true reason.... I maintainthat logic tends to make the power of reason over the unusual andunfamiliar more nearly equal to the power over the usual and familiar thanit would otherwise be. The second is increased; but the first is almostcreated. "

Mr. James Smith, by bringing ignorance, folly, dishonesty into contact withmy name, in the way of conditional insinuation, has done me a good turn: hehas given me right to a freedom of personal remark which I might havedeclined to take in the case of a person who is useful and respected inmatters which he understands. 

Tit for tat is logic all the world over. By the way, what has become of therest of the maxim: we never hear it {122} now. When I was a boy, in someparts of the country at least, it ran thus:

 "Tit for tat; Butter for fat: If you kill my dog, I'll kill your cat. "

He is a glaring instance of the truth of the observations quoted above. Iwill answer for it that, at the Mersey Dock Board, he never dreams ofproving that the balance at the banker's is larger than that in the book byassuming that the larger sum is there, and then proving that the othersupposition--the smaller balance--is upon that assumption, an absurdity. Henever says to another director, How can you dare to refuse me a right toassume the larger balance, when you yourself, the other day, said, --Suppose, for argument's sake, we had 80, 000l. At the banker's, though you knew the book only showed 30, 000l. ? This is the way in which hehas supported his geometrical paradox by Euclid's example: and this is notthe way he reasons at the board; I know it by the character of him as a manof business which has reached my ears from several quarters. But ingeometry and rational arithmetic he is a smatterer, though expert atcomputation; at the board he is a trained man of business. The language ofgeometry is so new to him that he does not know what is meant by "two meanproportionals:" but all the phrases of commerce are rooted in his mind. Heis most unerasably booked in the history of the squaring of the circle, asthe speculator who took a right to assume a proposition for the destructionof other propositions, on the express ground that Euclid assumes aproposition to show that it destroys itself: which is as if the curateshould demand permission to throttle the squire because St. Patrick drovethe vermin to suicide to save themselves from slaughter. He is conspicuousas a speculator who, more visibly than almost any other known to history, reasoned in a circle by way of reasoning on a circle. But {123} what I havechiefly to do with is the force of instance which he has lent to myassertion that men who have not had real training in pure logic are unsafereasoners in matter which is not familiar. It is hard to get first-rateexamples of this, because there are few who find the way to the printeruntil practice and reflection have given security against the grossestslips. I cannot but think that his case will lead many to take what I havesaid into consideration, among those who are competent to think of thegreat mental disciplines. To this end I should desire him to continue hisefforts, to amplify and develop his great principle, that of proving aproposition by assuming it and taking as confirmation every consequencethat does not contradict the assumption. 

Since my Budget commenced, Mr. Smith has written me notes: the portionwhich I have preserved--I suppose several have been mislaid--makes ahundred and seven pages of note-paper, closely written. To all this I havenot answered one word: but I think I cannot have read fewer than fortypages. In the last letter the writer informs me that he will not write atgreater length until I have given him an answer, according to the "rules ofgood society. " Did I not know that for every inch I wrote back he wouldreturn an ell? Surely in vain the net is spread in the eyes of anythingthat hath a wing. There were several good excuses for not writing to Mr. J. Smith: I will mention five. First, I distinctly announced at the beginningof this Budget that I would not communicate with squarers of the circle. Secondly, any answer I might choose to give might with perfect propriety bereserved for this article; had the imputation of incivility been made afterthe first note, I should immediately have replied to this effect: but Ipresumed it was quite understood. Thirdly, Mr. Smith, by his publication ofE. M. 's letters against the wish of the writer, had put himself out of thepale of correspondence. Fourthly, he had also gone beyond the rules of goodsociety in sending {124} letter after letter to a person who had shown byhis silence an intention to avoid correspondence. Fifthly, these same rulesof good society are contrived to be flexible or frangible in extreme cases:otherwise there would be no living under them; and good society would bebad. Father Aldrovand has laid down the necessary distinction--"I tellthee, thou foolish Fleming, the text speaketh but of promises made untoChristians, and there is in the rubric a special exemption of such as aremade to Welchmen. " There is also a rubric to the rules of good society; andsquarers of the circle are among those whom there is special permission notto answer: they are the wild Welchmen of geometry, who are alwaysassailing, but never taking, the Garde Douloureuse[219] of the circle. "Atthis commentary, " proceeds the story, "the Fleming grinned so broadly as toshow his whole case of broad strong white teeth. " I know not whether theWelchman would have done the like, but I hope Mr. James Smith will: and Ihope he has as good a case to show as Wilkin Flammock. For I wish him longlife and long health, and should be very glad to see so much energyemployed in a productive way. I hope he wishes me the same: if not, I willgive him what all his judicious friends will think a good reason for doingso. His pamphlets and letters are all tied up together, and will form acurious lot when death or cessation of power to forage among book-shelvesshall bring my little library to the hammer. And this time may not be faroff: for I was X years old in A. D. X^2; not 4 in A. D. 16, nor 5 in A. D. 25, but still in one case under that law. And now I have made my own age aproblem of quadrature, and Mr. J. Smith may solve it. But I protest againsthis method of assuming a result, and making itself prove itself: he mightin this way, as sure as eggs is eggs (a corruption of X is X), make me1, 864 years old, which is a great deal too much. 

{125}

_April 5, 1864. _--Mr. Smith continues to write me long letters, to which hehints that I am to answer. In his last, of 31 closely written sides ofnote-paper, he informs me, with reference to my obstinate silence, thatthough I think myself and am thought by others to be a mathematicalGoliath, I have resolved to play the mathematical snail, and keep within myshell. A mathematical _snail_! This cannot be the thing so called whichregulates the striking of a clock; for it would mean that I am to make Mr. Smith sound the true time of day, which I would by no means undertake upona clock that gains 19 seconds odd in every hour by false quadrature. But heventures to tell me that pebbles from the sling of simple truth and commonsense will ultimately crack my shell, and put me _hors de combat_. [220] Theconfusion of images is amusing: Goliath turning himself into a snail toavoid [pi] = 3-1/8, and James Smith, Esq. , of the Mersey Dock Board: andput _hors de combat_--which should have been _caché_[221]--by pebbles froma sling. If Goliath had crept into a snail-shell, David would have crackedthe Philistine with his foot. There is something like modesty in theimplication that the crack-shell pebble has not yet taken effect; it mighthave been thought that the slinger would by this time have been singing--

 "And thrice [and one-eighth] I routed all my foes, And thrice [and one-eighth] I slew the slain. "

But he promises to give the public his nut-cracker if I do not, before theBudget is concluded, "unravel" the paradox, which is themathematico-geometrical nut he has given me to crack. Mr. Smith is a crackman: he will crack his own nut; he will crack my shell; in the mean time hecracks himself up. Heaven send he do not crack himself into lateralcontiguity with himself. 

On June 27 I received a letter, in the handwriting of Mr. James Smith, signed Nauticus. I have ascertained {126} that one of the letters to the_Athenæum_ signed Nauticus is in the same handwriting. I make a fewextracts:

"... The important question at issue has been treated by a brace ofmathematical birds with too much levity. It may be said, however, thatsarcasm and ridicule sometimes succeed, where reason fails.... Such acourse is not well suited to a discussion.... For this reason I shall forthe future [this implies there has been a past, so that Nauticus is notbefore me for the first time] endeavor to confine myself to dry reasoningfrom incontrovertible premises. 

... It appears to me that so far as his theory is concerned he comes offunscathed. You might have found "a hole in Smith's circle" (have you seen apamphlet bearing this title? [I never heard of it until now]), but afterall it is quite possible the hole may have been left by design, for thepurpose of entrapping the unwary. "

[On the publication of the above, the author of the pamphlet obliginglyforwarded a copy to me of _A Hole in Smith's Circle_--by a Cantab: Longmanand Co. , 1859, (pp. 15). "It is pity to lose any fun we can get out of theaffair, " says my almamaternal brother: to which I add that in such a casewarning without joke is worse than none at all, as giving a false idea ofthe nature of the danger. The Cantab takes some absurdities on which I havenot dwelt: but there are enough to afford a Cantab from every college hisown separate hunting ground. ]

Does this hint that his mode of proof, namely, assuming the thing to beproved, was a design to entrap the unwary? if so, it bangs Banagher. Washis confounding two mean proportionals with one mean proportional foundtwice over a trick of the same intent? if so, it beats cockfighting. ThatNauticus is Mr. Smith appears from other internal evidence. In 1819, Mr. J. C. Hobhouse[222] was sent to Newgate for a {127} libel on the House ofCommons which was only intended for a libel on Lord Erskine. [223] Theex-Chancellor had taken Mr. Hobhouse to be thinking of him in a certainsentence; this Mr. Hobhouse denied, adding, "There is but one man in thecountry who is always thinking of Lord Erskine. " I say that there is butone man of our day who would couple me and Mr. James Smith as a "brace ofmathematical birds. "

Mr. Smith's "theory" is unscathed by me. Not a doubt about it: but how doeshe himself come off? I should never think of refuting a theory proved byassumption of itself. I left Mr. Smith's [pi] untouched: or, if I put in mythumb and pulled out a plum, it was to give a notion of the cook, not ofthe dish. The "important question at issue" was not the circle: it was, wholly and solely, whether the abbreviation of _James_ might be spelled_Jimm_. [224] This is personal to the verge of scurrility: but in literarycontroversy the challenger names the weapons, and Mr. Smith begins withcharge of ignorance, folly, and dishonesty, by conditional implication. Sothat the question is, not the personality of a word, but its applicabilityto the person designated: it is enough if, as the Latin grammar has it, _Verbum personale concordat cum nominativo_. [225]

I may plead precedent for taking a liberty with the orthography of _Jem_. An instructor of youth was scandalized at the abrupt and irregular--butvery effective--opening of Wordsworth's little piece:

{128}

 "A simple child That lightly draws its breath, And feels its life in every limb, What should it know of death?"

So he mended the matter by instructing his pupils to read the first linethus:

 "A simple child, dear brother ----. "

The brother, we infer from sound, was to be James, and the blank musttherefore be filled up with _Jimb_. 

I will notice one point of the letter, to make a little more distinctionbetween the two birds. Nauticus lays down--quite correctly--that the sineof an angle is less than its circular measure. He then takes 3. 1416 for180°, and finds that 36' is . 010472. But this is exactly what he finds forthe sine of 36' in tables: he concludes that either 3. 1416 or the tablesmust be wrong. He does not know that sines, as well as [pi], areinterminable decimals, of which the tables, to save printing, only take ina finite number. He is a six-figure man: let us go thrice again to make upnine, and we have as follows:

 Circular measure of 36' . 010471975... Sine of 36' . 010471784... Excess of measure over sine . 000000191... 

Mr. Smith invites me to say which is wrong, the quadrature, or the tables:I leave him to guess. He says his assertions "arise naturally andnecessarily out of the arguments of a circle-squarer:" he might just aswell lay down that all the pigs went to market because it is recorded that"_This_ pig went to market. " I must say for circle-squarers that very fewbring their pigs to so poor a market. I answer the above argument becauseit is, of all which Mr. James Smith has produced, the only one which risesto the level of a schoolboy: to meet him halfway I descend to that level. 

Mr. Smith asks me to solve a problem in the _Athenæum_: {129} and I will doit, because the question will illustrate what is _below_ schoolboy level. 

"Let x represent the circular measure of an angle of 15°, and y half thesine of an angle of 30° = area of the square on the radius of a circle ofdiameter unity = . 25. If x - y = xy, firstly, what is the arithmeticalvalue of xy? secondly, what is the angle of which xy represents thecircular measure?"

If x represent 15° and y be ¼, xy represents 3° 45', whether x - y be xy orno. But, y being ¼, x - y is _not_ xy unless x be 1/3, that is, unless 12xor [pi] be 4, which Mr. Smith would not admit. How could a person who hadjust received such a lesson as I had given immediately pray for furtherexposure, furnishing the stuff so liberally as this? Is it possible thatMr. Smith, because he signs himself Nauticus, means to deny his own veryregular, legible, and peculiar hand? It is enough to make the other membersof the Liverpool Dock Board cry, Mersey on the man!

Mr. Smith says that for the future he will give up what he calls sarcasm, and confine himself, "as far as possible, " to what he calls dry reasoningfrom incontrovertible premises. If I have fairly taught him that _his_sarcasm will not succeed, I hope he will find that his wit's end is hislogic's beginning. 

I now reply to a question I have been asked again and again since my lastBudget appeared: Why do you take so much trouble to expose such a reasoneras Mr. Smith? I answer as a deceased friend of mine used to answer on likeoccasions--A man's capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty ofconviction, which does more; and a long purse, which does most of all. Hehas made at least ten publications, full of figures which few readers cancriticize. A great many people are staggered to this extent, that theyimagine there must be {130} the indefinite _something_ in the mysterious_all this_. They are brought to the point of suspicion that themathematicians ought not to treat "all this" with such undisguisedcontempt, at least. Now I have no fear for [pi]: but I do think it possiblethat general opinion might in time demand that the crowd ofcircle-squarers, etc. Should be admitted to the honors of opposition; andthis would be a time-tax of five per cent. , one man with another, uponthose who are better employed. Mr. James Smith may be made useful, in handswhich understand how to do it, towards preventing such opinion fromgrowing. A speculator who expressly assumes what he wants to prove, andargues that all which contradicts it is absurd, _because_ it cannot standside by side with his assumption, is a case which can be exposed to all. And the best person to expose it is one who has lived in the past as wellas the present, who takes misthinking from points of view which none but astudent of history can occupy, and who has something of a turn for thebusiness. 

Whether I have any motive but public good must be referred to those who candecide whether a missionary chooses his pursuit solely to convert theheathen. I shall certainly be thought to have a little of the spirit ofCol. Quagg, who delighted in strapping the Grace-walking Brethren. I mustquote this myself: if I do not, some one else will, and then where am I?The Colonel's principle is described as follows:

"I licks ye because I kin, and because I like, and because ye'se crittersthat licks is good for. Skins ye have on, and skins I'll have off; hard orsoft, wet or dry, spring or fall. Walk in grace if ye like till pumpkins ispeaches; but licked ye must be till your toe-nails drop off and your nosesbleed blue ink. And--licked--they--were--accordingly. "

I am reminded of this by the excessive confidence with which Mr. JamesSmith predicted that he would treat me as Zephaniah Stockdolloger (SamSlick calls it _slockdollager_) treated Goliah Quagg. He has announced his{131} intention of bringing me, with a contrite heart, and cleanshaved, --4159265... Razored down to 25, --to a camp-meeting ofcircle-squarers. But there is this difference: Zephaniah only wanted topass the Colonel's smithy in peace; Mr. James Smith sought a fight with me. As soon as this Budget began to appear, he oiled his own strap, andattempted to treat me as the terrible Colonel would have treated theinoffensive brother. 

He is at liberty to try again. 

THE MOON HOAX. 

 The Moon-hoax; or the discovery that the moon has a vast population of human beings. By Richard Adams Locke. [226] New York, 1859, 8vo. 

This is a reprint of the hoax already mentioned. I suppose R. A. Locke isthe name assumed by M. Nicollet. [227] The publisher informs us that whenthe hoax first appeared day by day in a morning paper, the circulationincreased fivefold, and the paper obtained a permanent footing. Besidesthis, an edition of 60, 000 was sold off in less than one month. 

The discovery was also published under the name of A. R. Grant. [228]Sohncke's[229] _Bibliotheca Mathematica_ confounds this Grant with Prof. R. Grant[230] of Glasgow, the author of the _History of Physical Astronomy_, who is accordingly made to guarantee the discoveries in the moon. I hopeAdams Locke will not merge in J. C. Adams, [231] the co-discoverer ofNeptune. Sohncke gives the titles of {132} three French translations of theMoon hoax at Paris, of one at Bordeaux, and of Italian translations atParma, Palermo, and Milan. 

A Correspondent, who is evidently fully master of details, which he hasgiven at length, informs me that the Moon hoax appeared first in the _NewYork Sun_, of which R. A. Locke was editor. It so much resembled a storythen recently published by Edgar A. Poe, in a Southern paper, "Adventuresof Hans Pfaal, " that some New York journals published the two side by side. Mr. Locke, when he left the _New York Sun_, started another paper, anddiscovered the manuscript of Mungo Park;[232] but this did not deceive. The_Sun_, however, continued its career, and had a great success in an accountof a balloon voyage from England to America, in seventy-five hours, by Mr. Monck Mason, [233] Mr. Harrison Ainsworth, [234] and others. I have no doubtthat M. Nicollet was the author of the Moon hoax, [235] written in a waywhich marks the practised observatory astronomer beyond all doubt, and byevidence seen in the most minute details. Nicollet had an eye to Europe. Isuspect that he took Poe's story, and made it a basis for his own. Mr. Locke, it would seem, when he attempted a fabrication for himself, did notsucceed. 

 The Earth we inhabit, its past, present, and future. By Capt. Drayson. [236] London, 1859, 8vo. 

The earth is growing; absolutely growing larger: its diameter increasesthree-quarters of an inch per mile every year. The foundations of ourbuildings will give way in {133} time: the telegraph cables break, and nocause ever assigned except ships' anchors, and such things. The book is forthose whose common sense is unwarped, who can judge evidence as well as theablest philosopher. The prospect is not a bad one, for population increasesso fast that a larger earth will be wanted in time, unless emigration tothe Moon can be managed, a proposal of which it much surprises me thatBishop Wilkins has a monopoly. 

IMPALEMENT BY REQUEST. 

_Athenæum_, August, 19, 1865. _Notice to Correspondents. _

"R. W. --If you will consult the opening chapter of the Budget of Paradoxes, you will see that the author presents only works in his own library at agiven date; and this for a purpose explained. For ourselves we havecarefully avoided allowing any writers to present themselves in our columnson the ground that the Budget has passed them over. We gather that Mr. DeMorgan contemplates additions at a future time, perhaps in a separate andaugmented work; if so, those who complain that others of no greater claimsthan themselves have been ridiculed may find themselves where they wish tobe. We have done what we can for you by forwarding your letter to Mr. DeMorgan. "

The author of "An Essay on the Constitution of the Earth, " published in1844, demanded of the _Athenæum_, as an _act of fairness_, that a letterfrom him should be published, proving that he had as much right to be"impaled" as Capt. Drayson. He holds, on speculative grounds, what theother claims to have proved by measurement, namely, that the earth isgrowing; and he believes that in time--a good long time, not _our_time--the earth and other planets may grow into suns, with systems of theirown. 

This gentleman sent me a copy of his work, after the commencement of myBudget; but I have no recollection of having received it, and I cannot findit on the (nursery? {134} quarantine?) shelves on which I keep myunestablished discoveries. Had I known of this work in time, (see theIntroduction) I should of course, have impaled it (heraldically) with theother work; but the two are very different. Capt. Drayson professes toprove his point by results of observation; and I think he does not succeed. The author before me only speculates; and a speculator can get anyconclusion into his premises, if he will only build or hire them of shapeand size to suit. It reminds me of a statement I heard years ago, that ascore of persons, or near it, were to dine inside the skull of one of theaboriginal animals, dear little creatures! Whereat I wondered vastly, nothing doubting; facts being stubborn and not easy drove, as Mrs. Gampsaid. But I soon learned that the skull was not a real one, butartificially constructed by the methods--methods which have had strikingverifications, too--which enable zoologists to go the whole hog by help ofa toe or a bit of tail. This took off the edge of the wonder: a hundredpeople can dine inside an inference, if you draw it large enough. Themethod might happen to fail for once: for instance, the toe-bone might havebeen abnormalized by therian or saurian malady; and the possibility of suchfailure, even when of small probability, is of great alleviation. Theauthor before me is, apparently, the sole fabricator of his own premises. With vital force in the earth and continual creation on the part of theoriginal Creator, he expands our bit of a residence as desired. But, as theNewtoness of Cookery observed, First catch your hare. When this is done, when you _have_ a growing earth, you shall dress it with all manner ofproximate causes, and serve it up with a growing Moon for sauce, a growingSun, if it please you, at the other end, and growing planets forside-dishes. Hoping this amount of impalement will be satisfactory, I go onto something else. {135}

THE HAILESEAN SYSTEM OF ASTRONOMY. 

 _The Hailesean System of Astronomy. _ By John Davey Hailes[237] (two pages duodecimo, 1860). 

He offers to _take_ 100, 000l. To 1, 000l. That he shows the sun to be lessthan seven millions of miles from the earth. The earth in the center, revolving eastward, the sun revolving westward, so that they "meet at halfthe circle distance in the 24 hours. " The diameter of the circle being9839458303, the circumference is 30911569920. 

The following written challenge was forwarded to the Council of theAstronomical Society: it will show the "general reader"--and help himtowards earning his name--what sort of things come every now and then toour scientific bodies. I have added punctuation:

 _Challenge. _ 1, 000 to 30, 000. "Leverrier's[238] name stand placed first. Do the worthy Frenchman justice. By awarding him the medal in a trice. Give Adams[239] an extra--of which neck and neck the race. Now I challenge to meet them and the F. R. S. 's all, For good will and _one_ thousand pounds to their _thirty_ thousand withall, That I produce a system, which shall measure the time, When the Sun was vertical to Gibeon, afterward to Syene. To meet any time in London--name your own period, To be decided by a majority of twelve persons--a President, _odd_. That mean, if the twelve equally divide, the President decide, I should prefer the Bishop of London, over the meeting to preside. JOHN DAVY HAILES. " Feb. 17, 1847. "

Mr. Hailes still issues his flying sheets. The last I have met with(October 7, 1863) informs us that the latitude of {136} England is slowlyincreasing, which is the true cause of the alteration in the variation ofthe magnet. 

[Mr. Hailes continues his researches. Witness his new Hailesean system ofAstronomy, displaying Joshua's miracle-time, origin of time from science, with Bible and Egyptian history. Rewards offered for astronomical problems. With magnetism, etc. Etc. Astronomical challenge to all the world. Published at Cambridge, in 1865. The author agrees with Newton in onemarked point. _Errores quam minimi non sunt contemnendi_, [240] says Isaac:meaning in figures, not in orthography. Mr. Hailes enters into the spirit, both positive and negative, of this dictum, by giving the distance of_Sidius_ from the center of the earth at 163, 162, 008 miles 10 feet 8 inches17-28ths of an inch. Of course, he is aware that the center of _figure_ ofthe earth is 17. 1998 inches from the center of _gravity_. Which of the twois he speaking of?]

 The Divine Mystery of Life. London [1861], 18mo. (pp. 32). 

The author has added one class to zoology, which is printed in capitals, asderived from _zoé_, life, not from _zôon_, animal. That class is of_Incorporealia_, order I. , _Infinitum_, of one genus without plurality, _Deus_: order II. , _Finita_, angels good and evil. The rest is all about atriune system, with a diagram. The author is not aware that [Greek: zôon]is not _animal_, but _living being_. Aristotle had classed gods under[Greek: zôa], and has been called to account for it by moderns who havetaken the word to mean _animal_. 

A CHANCE FOR INVENTORS. 

 Explication du Zodiaque de Denderah, des Pyramides, et de Genèse. Par le Capitaine au longcours Justin Roblin. [241] Caen, 1861. 8vo. 

{137}

Capt. Roblin, having discovered the sites of gold and diamond mines by helpof the zodiac of Denderah, offered half to the shareholders of a companywhich he proposed to form. One of our journals, by help of the zodiac ofEsné, offered, at five francs a head, to tell the shareholders the exactamount of gold and diamonds which each would get, and to make up the amountpredicted to those who got less. There are moods of the market in Englandin which this company could have been formed: so we must not laugh at ourneighbors. 

JOHANNES VON GUMPACH. 

 A million's worth of property, and five hundred lives annually lost at sea by the Theory of Gravitation. A letter on the true figure of the earth, addressed to the Astronomer Royal, by Johannes von Gumpach. [242] London, 1861, 8vo. (pp. 54). 

 The true figure and dimensions of the earth, in a letter addressed to the Astronomer Royal. By Joh. Von Gumpach. 2nd ed. Entirely recast. London, 1862, 8vo. (pp. 266). 

 Two issues of a letter published with two different title-pages, one addressed to the Secretary of the Royal Society, the other to the Secretary of the Royal Astronomical Society. It would seem that the same letter is also issued with two other titles, addressed to the British Association and the Royal Geographical Society. By Joh. Von Gumpach. London, 1862, 8vo. 

 Baby-Worlds. An essay on the nascent members of our solar household. By Joh. Von Gumpach. London, 1863, 8vo. 

The earth, it appears, instead of being flattened, is elongated at thepoles: by ignorance of which the loss above mentioned occurs yearly. Thereis, or is to be, a substitute for attraction and an "application hithertoneglected, of a {138} recognized law of optics to the astronomical theory, showing the true orbits of the heavenly bodies to be perfectly circular, and their orbital motions to be perfectly uniform. " all irregularitiesbeing, I suppose, optical delusions. Mr. Von Gumpach is a learned man; whatelse, time must show. 

SLANDER PARADOXES. 

 Perpetuum Mobile: or Search for self-motive Power. By Henry Dircks. [243] London, 1861, 8vo. 

A useful collection on the history of the attempts at perpetual motion, that is, at obtaining the consequences of power without any power toproduce them. September 7, 1863, a correspondent of the _Times_ gave ananecdote of George Stephenson, [244] which he obtained from RobertStephenson. [245] A perpetual motionist wanted to explain his method; towhich George replied--"Sir! I shall believe it when I see you take yourselfup by the waistband, and carry yourself about the room. " Never was theproblem better stated. 

There is a paradox of which I ought to give a specimen, I mean the_slander-paradox_; the case of a person who takes it into his head, uponevidence furnished entirely by the workings of his own thoughts, that someother person has committed a foul act of which the world at large would nomore suppose him guilty than they would suppose that the earth is a flatbordered by ice. If I were to determine on giving cases in which theself-deluded person imagines {139} a conspiracy against _himself_, therewould be no end of choices. Many of the grosser cases are found at last tobe accompanied by mental disorder, and it is difficult to avoid referringthe whole class to something different from simple misuse of the reasoningpower. The first instance is one which puts in a strong light the state ofthings in which we live, brought about by our glorious freedom of thought, speech, and writing. The Government treated it with neglect, the press withsilent contempt, and I will answer for it many of my readers now hear of itfor the first time, when it comes to be enrolled among circle-squarers andearth-stoppers, where, as the old philosopher said, it will not gravitate, being _in proprio loco_. [246]

1862. On new year's day, 1862, when the nation was in the full tide ofsympathy with the Queen, and regret for its own loss, a paper called the_Free Press_ published a number devoted to the consideration of the causesof the death of the Prince Consort. It is so rambling and inconsecutivethat it takes more than one reading to understand it. It is against the_Times_ newspaper. First, the following insinuation:

"To the legal mind, the part of [the part taken by] the _Times_ willpresent a _prima facie_ case of the gravest nature, in the evidentfore-knowledge of the event, and the preparation to turn it to account whenit should have occurred. The article printed on Saturday must have beenwritten on Friday. That article could not have appeared had the Prince beenintended to live. "

Next, it is affirmed that the _Times_ intended to convey the idea that thePrince had been poisoned. 

"Up to this point we are merely dealing with words which the _Times_publishes, and these can leave not a shadow of doubt that there is anintention to promulgate the idea that Prince Albert had been poisoned. "

The article then goes on with a strange olio of {140} insinuations to theeffect that the Prince was the obstacle to Russian intrigue, and that if heshould have been poisoned, --which the writer strongly hints may have beenthe case, --some Minister under the influence of Russia must have done it. Enough for this record. _Un sot trouve toujours un plus sot quil'admire_:[247] who can he be in this case?

THE NEPTUNE CONTROVERSY. 

1846. At the end of this year arose the celebrated controversy relative tothe discovery of Neptune. Those who know it are well aware that Mr. Adams's[248] now undoubted right to rank with Le Verrier[249] was made sureat the very outset by the manner in which Mr. Airy, [250] the AstronomerRoyal, came forward to state what had taken place between himself and Mr. Adams. Those who know all the story about Mr. Airy being arrested in hisprogress by the neglect of Mr. Adams to answer a letter, with all theimputations which might have been thrown upon himself for laxity in thematter, know also that Mr. Airy's conduct exhibited moral courage, honestfeeling, and willingness to sacrifice himself, if need were, to theattainment of the ends of private justice, and the establishment of anational claim. A writer in a magazine, in a long and elaborate article, argued the supposition--put in every way except downright assertion, afterthe fashion of such things--that Mr. Airy had communicated Mr. Adams'sresults to M. Le Verrier, with intention that they should be used. Hispresumption as to motive is that, had Mr. Adams been recognized, "then thediscovery must have been indisputably an _Englishman's_, and thatEnglishman not the Astronomer Royal. " Mr. Adams's conclusions were"retouched in France, and sent {141} over the year after. " The proof givenis that it cannot be "imagined" otherwise. 

"Can it then be imagined that the Astronomer Royal received such resultsfrom Mr. Adams, supported as they were by Professor Challis's[251] valuabletestimony as to their probable accuracy, and did not bring the Frenchastronomer acquainted with them, especially as he was aware that his friendwas engaged in matters bearing directly upon these results?"

The whole argument the author styles "evidence which I consider itdifficult to refute. " He ends by calling upon certain persons, of whom I amone, to "see ample justice done. " This is the duty of every one, accordingto his opportunities. So when the reputed author--the article beinganonymous--was, in 1849, proposed as a Fellow of the Astronomical Society, I joined--if I remember right, I originated--an opposition to his election, until either the authorship should be denied, or a proper retraction made. The friends of the author neither denied the first, nor produced thesecond: and they judged it prudent to withdraw the proposal. Had I heard ofany subsequent repentance, I would have taken some other instance, insteadof this: should I yet hear of such a thing, I will take care to notice itin the continuation of this list, which I confidently expect, life andhealth permitting, to be able to make in a few years. This much may besaid, that the author, in a lecture on the subject, given in 1849, andpublished with his name, did _not_ repeat the charge. 

[The libel was published in the _Mechanics' Magazine_, [252] (vol. For 1846, pp. 604-615): and the editor supported it as follows, (vol. For 1847, p. 476). In answer to Mr. Sheepshanks's charitable hope that he had beenhoaxed, {142} he says: "Mr. Sheepshanks cannot certainly have read thearticle referred to.... Severe and inculpatory it is--unjust some may deemit (though we ourselves are out of the number. )... A 'hoax' forsooth! Maywe be often the dupes of such hoaxes!" He then goes on to describe thearticle as directed against the Astronomer Royal's alleged neglect to giveMr. Adams that "encouragement and protection" which was his due, and _doesnot hint one word_ about the article containing the charge of havingsecretly and fraudulently transmitted news of Mr. Adams's researches toFrance, that an Englishman might not have the honor of the discovery. Mr. Sheepshanks having called this a "deliberate calumny, " without a particleof proof or probability to support it, the editor says "what the reverendgentleman means by this, we are at a loss to understand. " He then proceeds_not_ to remember. I repeat here, what I have said elsewhere, that themanagement of the journal has changed hands; but from 1846 to 1856, it hadthe collar of S. S. (scientific slander). The prayer for more such thingswas answered (See p. 349). ]

JAMES IVORY. [253]

I have said that those who are possessed with the idea of conspiracyagainst themselves are apt to imagine both conspirators and their badmotives and actions. A person who should take up the idea of combinationagainst himself without feeling ill-will and originating accusations wouldbe indeed a paradox. But such a paradox has existed. It is very well known, both in and beyond the scientific world, that the late James Ivory wassubject to the {143} impression of which I am speaking; and the diaries andother sources of anecdote of our day will certainly, sooner or later, makeit a part of his biography. The consequence will be that to his memory willbe attached the unfavorable impression which the usual conduct of suchpersons creates; unless it should happen that some one who knows the realstate of the case puts the two sides of it properly together. Ivory was ofthat note in the scientific world which may be guessed from Laplace'sdescription of him as the first geometer in Britain and one of the first inEurope. Being in possession of accurate knowledge of his peculiarity inmore cases than one; and in one case under his own hand: and having beenable to make full inquiry about him, especially from my friend the lateThomas Galloway[254]--who came after him at Sandhurst--one of the fewpersons with whom he was intimate:--I have decided, after fulldeliberation, to forestall the future biographies. 

That Ivory was haunted by the fear of which I have spoken, to the fullestextent, came to my own public and official knowledge, as Secretary of theAstronomical Society. It was the duty of Mr. Epps, [255] the AssistantSecretary, at the time when Francis Baily[256] first announced hisdiscovery of the Flamsteed Papers, to report to me that Mr. Ivory hadcalled at the Society's apartments to inquire into the contents of thosepapers, and to express his hope that Mr. Baily was not attacking livingpersons under the names of Newton and Flamsteed. [257] Mr. Galloway, to whomI communicated this, immediately went to Mr. Ivory, and succeeded, aftersome explanation, in setting him right. This is but one of many instancesin which a man of thoroughly sound judgment in every other respect seemedto be under a complete chain of delusions about the conduct of {144} othersto himself. But the paradox is this:--I never could learn that Ivory, passing his life under the impression that secret and unprovoked enemieswere at work upon his character, ever originated a charge, imputed a badmotive, or allowed himself an uncourteous expression. Some letters of his, now in my possession, referring to a private matter, are, except in themain impression on which they proceed, unobjectionable in every point: theymight have been written by a cautious friend, whose object was, ifpossible, to prevent a difference from becoming a duel without compromisinghis principal's rights or character. Knowing that in some quarters theknowledge of Ivory's peculiarity is more or less connected with a notionthat the usual consequences followed, I think the preceding statement dueto his memory. 

THREE CLASSES OF JOURNALS. 

In such a record as the present, which mixes up the grossest speculativeabsurdities with every degree of what is better, an instance of anotherkind may find an appropriate place. The faults of journalism, when merelyexposed by other journalism pass by and are no more regarded. A distinctaccount of an undeniable meanness, recorded in a work of amusement andreference both, may have its use: such a thing may act as a warning. Aneditor who is going to indulge his private grudge may be prevented fromcounting upon oblivion as a matter of certainty. 

There are three kinds of journals, with reference to the mode of entranceof contributors. First, as a thing which has been, but which now hardlyexists, there is the journal in which the editor receives a fixed sum to_find the matter_. In such a journal, every article which the editor canget a friend to give him is so much in his own pocket, which has a greattendency to lower the character of the articles; but I am not concernedwith this point. Secondly, there is the journal which is supported byvoluntary contributions of {145} matter, the editor selecting. Thirdly, there is the journal in which the contributor is paid by the proprietors ina manner with which the literary editor has nothing to do. 

The third class is the safe class, as its editors know: and, as a usualrule, they refuse unpaid contributions of the editorial cast. It is saidthat when Canning[258] declined a cheque forwarded for an article in the_Quarterly_, John Murray[259] sent it back with a blunt threat that if hedid not take his money he could never be admitted again. The greatpublisher told him that if men like himself in position worked for nothing, all the men like himself in talent who could not afford it would not workfor the _Quarterly_. If the above did not happen between Canning andMurray, it _must have happened_ between some other two. Now journals of thesecond class--and of the first, if such there be--have a fault to whichthey alone are very liable, to say nothing of the editorial function (seethe paper at the beginning, p. 11 et seq. ), being very much cramped, a sortof gratitude towards effective contributors leads the journal to help theirpersonal likes and dislikes, and to sympathize with them. Moreover, thissort of journal is more accessible than others to articles conveyingpersonal imputation: and when these provoke discussion, the journal is aptto take the part of the assailant to whom it lent itself in the firstinstance. 

THE MECHANICS' MAGAZINE. 

Among the journals which went all lengths with contributors whom theyvalued, was the _Mechanics' Magazine_[260] in the period 1846-56. I cannotsay that matters have not mended in the last ten years: and I draw some{146} presumption that they have mended from my not having heard, since1856, of anything resembling former proceedings. And on actual inquiry, made since the last sentence was written, I find that the property haschanged hands, the editor is no longer the same, and the management is of adifferent stamp. This journal is chiefly supported by voluntary articles:and it is the journal in which, as above noted, the ridiculous chargeagainst the Astronomer Royal was made in 1849. The following instance ofattempt at revenge is so amusing that I select it as the instance of thedefect which I intend to illustrate; for its puerility brings out in betterrelief the points which are not so easily seen in more adult attempts. 

The _Mechanics' Magazine_, which by its connection with engineering, etc. , had always taken somewhat of a mathematical character, began, a littlebefore 1846, to have more to do with abstract science. Observing this, Ibegan to send short communications, which were always thankfully received, inserted, and well spoken of. Any one who looks for my name in that journalin 1846-49, will see nothing but the most respectful and even laudatorymention. In May 1849 occurred the affair at the Astronomical Society, andmy share in forcing the withdrawal of the name of the alleged contributorto the journal. In February 1850 occurred the opportunity of payment. The_Companion to the Almanac_[261] had to be noticed, in which, as then usual, was an article signed with my name. I shall give the review of this articleentire, as a sample of a certain style, as well as an illustration of mypoint. The reader will observe that my name is not mentioned. This wouldnot have done; the readers of the Magazine would have stared to see a nameof not infrequent occurrence in previous years all of a sudden fallen fromthe heaven of respect into the pit of contempt, like Lucifer, son of themorning. But before {147} giving the review, I shall observe that Mr. Adams, in whose _favor_ the attack on the Astronomer Royal was made, didnot appreciate the favor; and of course did not come forward to shield hischampion. This gave deadly offence, as appear from the following passage, (February 16, 1850):

"It was our intention to enter into a comparison of the contents of ourNautical Almanack with those of its rival, the _Connaissance des Temps_;but we shall defer it for the present. The Nautical Almanack for 1851 willcontain Mr. Adams's paper 'On the Perturbation of Uranus'; and when itcomes, in due course, before the public, we are quite sure that thatgentleman will expect that we shall again enter upon the subject withpeculiar delight. Whilst we have a thorough loathing for mean, cowardly, crawlers--we have an especial pleasure in maintaining the claims of men whoare truly grateful as well as highly talented: Mr. Adams, therefore, willfind that he cannot be disappointed--and the occasion will afford us anopportunity for making the comparison to which we have adverted. "

This passage illustrates what I have said on the editorial function (Vol. I, p. 15). What precedes and follows has some criticism on the Government, the Astronomer Royal, etc. , but reserved in allusion, oblique in sarcasm, and not fiercely uncourteous. The coarseness of the passage I have quotedshows editorial insertion, which is also shown by its blunder. The inserteris waiting for the Almanac of 1851 that he may review Mr. Adams's paper, which is to be contained in it. His own contributor, only two sentencesbefore the insertion, had said, "The Nautical Almanac, we believe, ispublished three or four years in advance. " In fact, the Almanac for1851--with Mr. Adams's paper at the end--was published at the end of 1847or very beginning of 1848; it had therefore been more than two years beforethe public when the passage quoted was written. And probably every personin the country who was fit to review Mr. Adams's {148} paper--and most ofthose who were fit to read it--knew that it had been widely circulated, inrevise, at the end of 1846: my copy has written on it, "2d revise, December27, 1846, at noon, " in the handwriting of the Superintendent of theAlmanac; and I know that there was an extensive issue of these revises, brought out by the Le-Verrier-and-Adams discussion. I now give the reviewof myself, (February 23, 1850):

"_The British Almanack and Companion. _

"The Companion to this Almanack, for some years after its firstpublication, annually contained scientific articles by Sir J. Lubbock[262]and others of a high order and great interest; we have now, however, closedthe publication as a scientific one in remembrance of what it was, and notin consequence of what it is. Its list of contributors on science, hasgrown 'small by degrees and beautifully less, ' until it has dwindled downto one--'a last rose of summer left withering alone. ' The one contributorhas contributed one paper 'On Ancient and Modern Usage in Reckoning. '

"The learned critic's _chef d'oeuvre_, is considered, by competent judges, to be an Essay on _Old Almanacks_ printed a few years ago in this annual, and supposed to be written with the view of surpassing a profound memoir onthe same subject by James O. Halliwell, [263] Esq. , F. R. And A. S. S. , but thetremendous effort which the learned writer then made to excel many titledcompetitors for honors in the antique line appears to have had a sad effectupon his mental powers--at any rate, his efforts have since yearly becomeduller and duller; happily, at last, we should suppose, 'the ancient {149}and modern usage in reckoning' indicates the lowest point to which the _visinertia_ of the learned writer's peculiar genius can force him. 

"We will give a few extracts from the article. 

"The learned author says, 'Those who are accustomed to settle the meaningof ancient phrases by self-examination will find some _strange_ conclusionsarrived at by us. ' The writer never wrote a more correct sentence--itadmits of no kind of dispute. 

"'Language and counting, ' says the learned author, 'both came before thelogical discussion of either. It is not allowable to argue that somethingis or was, because it ought to be or ought to have been. That two negativesmake an affirmative, ought to be; if _no_ man have done _nothing_, the manwho has done nothing does not exist, and _every_ man has done _something_. But in Greek, and in uneducated English, it is unquestionable that 'no manhas done nothing' is only an emphatic way of saying that no man has done_anything_; and it would be absurd to reason that it could not have beenso, because it should not. '--p. 5. 

"'But there _is_ another difference between old and new times, yet moreremarkable, for we have _nothing_ of it now: whereas in things indivisiblewe count with our fathers, and should say in buying an acre of land, thatthe result has no parts, and that the purchaser, till he owns all theground, owns none, the change of possession being instantaneous. Thissecond difference lies in the habit of considering nothing, nought, zero, cipher, or whatever it may be called, to be at the beginning of the scaleof numbers. Count four days from Monday: we should now say Tuesday, Wednesday, Thursday, Friday; formerly, it would have been Monday, Tuesday, Wednesday, Thursday. Had we asked, what at that rate is the first day fromMonday, all would have stared at a phrase they had never heard. Those whowere capable of extending language would have said, Why it must be Mondayitself: the rest would have said, there can {150} be no first day fromMonday, for the day after is Tuesday, which must be the second day: Monday, one; Tuesday, two, '--p. 10. 

"We assure our readers that the whole article is equally lucid, and itslogic alike formal. 

"There are some exceedingly valuable footnotes; we give one of the mostinteresting, taken from the learned Mr. Halliwell's profound book onNursery Rhymes[264]--a celebrated production, for which it is supposed theauthor was made F. R. S. 

 "'_One's nine_, Two's some, Three's a many, Four's a penny, Five's a little hundred. '

'The last line refers to five score, the so-called hundred being moreusually six score. The first line, looked at etymologically, is _one is notone_, and the change of thought by which _nine_, the decimal of _one_, aimsto be associated with the decimal of _plurality_ is curious:'--Very. 

"This valuable and profound essay will very probably be transferred to thenext edition of the learned Mr. Halliwell's rare work, of kindred worth, entitled 'RARA MATHEMATICA, ' it will then be deservedly handed down toposterity as a covering for cheap trunks--a most appropriate archive forsuch a treasure. "

In December, 1846, the _Mechanics' Magazine_ published a libel on Airy inthe matter of the discovery of Neptune. In May, 1849, one * * * was to havebeen brought forward for election at the Astronomical Society, and wasopposed by me and others, on the ground that he was the probable author ofthis libel, and that he would not, perhaps could {151} not, deny it. [N. B. I no more doubt that he was the author then I doubt that I am the author ofthis sentence. ][265]

Accordingly, * * * was withdrawn, and a discussion took place, for whichsee the _Athenæum_, No. 1126, May 26, 1849, p. 544. The _Mechanics'Magazine_ was very sore, but up to this day has never ventured beyond anattack on Airy, private whisperings against Adams--(see _ante_, p. 147), --and the above against myself. In due time, I doubt not my name willappear as one of the _âmes damnées_[266] of the _Mechanics' Magazine_. [267]

T. S. DAVIES ON EUCLID. 

First, as to Mr. Halliwell. The late Thomas Stephens Davies, [268] excellentin geometry, and most learned in its history, was also a good hand atenmity, though not implacable. He and Mr. Halliwell, who had long beforebeen very much one, were, at this date, very much two. I do not think T. S. Davies wrote this article; and I think that by giving my reasons I shall doservice to his memory. It must have been written at the beginning ofFebruary; and within three days of that time T. S. Davies was making overto me, by his own free act, to be kept until claimed by the relatives, whatall who knew even his writings knew that he considered as the most preciousdeposit he had ever had in his keeping--Horner's[269] papers. His letterannouncing the transmission is dated February 2, 1850. This is a strongpoint; but there is another quite as strong. Euclid and {152} his writingswere matters on which T. S. Davies knew neither fear nor favor: he couldnot have written lightly about a man who stood high with him as a judge ofEuclid. Now in this very letter of Feb. 2, there is a sentence which Ihighly value, because, as aforesaid, it is on a point on which he wouldnever have yielded anything, to which he had paid life-long attention, andon which he had the bias of having long stood alone. In fact, knowing--andwhat I shall quote confirms me, --that in the matter of Euclid his hand wasagainst every man, I expected, when I sent him a copy of my 22-columnarticle, "Eucleides" in Smith's _Dictionary_, [270] to have received back acriticism, that would have blown me out of the water: and I thought it notunlikely that a man so well up in the subject might have made me feeldemolished on some points. Instead of this, I got the following: "Althoughon one or two minor points I do not quite accord with your views, yet as awhole and without regard to any minor points, I think you are the first whohas succeeded in a delineation of Euclid as a geometer. " All this dulyconsidered, it is utterly incredible that T. S. Davies should have writtenthe review in question. And yet Mr. Halliwell is treated just as T. S. Davies would have treated him, as to tone and spirit. The inference in mymind is that we have here a marked instance of the joining of hatreds whichtakes place in journals supported by voluntary contributions of matter. Should anything ever have revived this article--and no one ever knows whatmight have been fished up from the forgotten mass of journals--thetreatment of Mr. Halliwell would certainly have thrown a suspicion on T. S. Davies, a large and regular contributor to the Magazine. It is good serviceto his memory to point out what makes it incredible that he should havewritten so unworthy an article. 

The fault is this. There are four extracts: the first {153} three areperfectly well printed. The printing of the _Mechanics' Magazine_ was verygood. I was always exceedingly satisfied with the manner in which myarticles appeared, without my seeing proof. Most likely these extracts wereprinted from my printed paper; if not the extractor was a good copier. Iknow this by a test which has often served me. I use the subjunctive--"ifno man _have_ done nothing, " an ordinary transcriber, narrating a quotationalmost always lets his own habit write _has_. The fourth extract has threealterations, all tending to make me ridiculous. _None_ is altered, in twoplaces, into _nine_, _denial_ into _decimal_, and _comes_ into _aims_; sothat "none, the denial of one, comes to be associated with the denial ofplurality, " reads as "nine, the decimal of one, aims to be associated withthe decimal of plurality. " This is intentional; had it been a compositor'sreading of bad handwriting, these would not have been the only mistakes; tosay nothing of the corrector of the press. And both the compositor andreader would have guessed, from the first line being translated into "oneis not one, " that it must have been "one's none, " not "one's nine. " But itwas not intended that the gem should be recovered from the unfathomed cave, and set in a Budget of Paradoxes. 

We have had plenty of slander-paradox. I now give a halfpennyworth of breadto all this sack, an instance of the paradox of benevolence, in which anindividual runs counter to all the ideas of his time, and sees his way intothe next century. At Amiens, at the end of the last century, an institutionwas endowed by a M. De Morgan, to whom I hope I am of kin, but I cannottrace it; the name is common at Amiens. It was the first of the kind I everheard of. It is a Salle d'Asyle for children, who are taught and washed andtaken care of during the hours in which their parents must be at work. Thefounder was a large wholesale grocer and colonial importer, who was made aBaron by Napoleon I for his commercial success and his charities. {154}

JAS. SMITH AGAIN. 

1862. Mr. Smith replies to me, still signing himself Nauticus: I give anextract:

"By hypothesis [what, again!] let 14° 24' be the chord of an arc of 15°[but I wont, says 14° 24'], and consequently equal to a side of a regularpolygon of 24 sides inscribed in the circle. Then 4 times 14° 24' = 57° 36'= the radius of the circle ... "

That is, four times the chord of an arc is the chord of four times the arc:and the sum of four sides of a certain pentagon is equal to the fifth. Thisis the capital of the column, the crown of the arch, the apex of thepyramid, the watershed of the elevation. Oh! J. S. ! J. S. ! groansGeometry--_Summum J. S. Summa injuria_![271] The other J. S. , JosephScaliger, [272] as already mentioned, had his own way of denying that astraight line is always the shortest distance between two points. Aparallel might be instituted, but not in half a column. And J. S. The_second_ has been so tightly handled that he may now be dismissed, with aninscription for his circular shield, obtained by changing _Lexica contexat_into _Circus quadrandus_ in an epigram of J. S. The _first_:

 "Si quem dura manet sententia judicis, olim Damnatum ærumnis suppliciisque caput, Hunc neque fabrili lassent ergastula massa, Nec rigidas vexent fossa metalla manus. Circus quadrandus: nam--cætera quid moror?--omnes Poenarum facies hic labor unus habet. "[273]

{155}

I had written as far as _damnatum_ when in came the letter of Nauticus as aprinted slip, with a request that I would consider the slip as a 'revisedcopy. ' Not a word of alteration in the part I have quoted! And in theevening came a letter desiring that I would alter a gross error; but notthe one above: this is revising without revision! If there were cyclometersenough of this stamp, they would, as cultivation progresses--and really, with John Stuart Mill in for Westminster, it seems on the move, eventhough, as I learn while correcting the proof, Gladstone be out fromOxford; for Oxford is no worse than in 1829, while Westminster is far abovewhat she ever has been: election time excuses even such a parenthesis asthis--be engaged to amuse those who can afford it with paralogism at theirmeals, after the manner of the other jokers who wore the caps and bells. The rich would then order their dinners with _panem et Circenses_, --up withthe victuals and the circle-games--as the poor did in the days of old. 

Mr. Smith is determined that half a column shall not do. Not a day withoutsomething from him: letter, printed proof, pamphlet. In what is the last atthis moment of writing he tells me that part of the title of a work of hiswill be "Professor De Morgan in the pillory without hope of escape. " Andwhere will he be himself? This I detected by an effort of reasoning which Inever could have made except by following in his steps. In all mattersconnected with [pi] the letters l and g are closely related: this appearsin the well-known formula for the time of oscillation [pi] [sqrt](l : g). Hence g may be written for l, but only once: do it twice, and you requirethe time to be [pi] [sqrt](l^2 : g^2). This may be reinforced by observingthat if as a datum, or if you dislike that word, by hypothesis, the first lbe a g, it is absurd that it should be an l. Write g for the first l, andwe have _un fait accompli_. I shall be in pillory; and overhead, in acloud, will sit Mr. James Smith on one stick laid across two others, undera nimbus of 3-1/8 diameters to {156} the circumference--in [pi]-glory. Ohfor a drawing of this scene! Mr. De Morgan presents his compliments to Mr. James Smith, and requests the honor of an exchange of photographs. 

_July 26. _--Another printed letter. --Mr. James Smith begs for a distinctanswer to the following plain question: "Have I not in this communicationbrought under your notice _truths_ that were never before dreamed of inyour geometrical and mathematical philosophy?" To which, he having takenthe precaution to print the word _truths_ in italics, I can conscientiouslyanswer, Yes, you have. And now I shall take no more notice of these_truths_, until I receive something which surpasses all that has yet beendone. 

A FEW SMALL PARADOXERS. 

 The Circle secerned from the Square; and its area gauged in terms of a triangle common to both. By Wm. Houlston, [274] Esq. London and Jersey, 1862, 4to. 

Mr. Houlston squares at about four poetical quotations in a page, andbrings out [pi] = 3. 14213.... His frontispiece is a variegated diagram, having parts designated Inigo and Outigo. All which relieves the subject, but does not remove the error. 

 Considerations respecting the figure of the Earth.... By C. F. Bakewell. [275] London, 1862, 8vo. 

Newton and others think that in a revolving sphere the {157} loose surfacematter will tend to the equator: Mr. Bakewell thinks it will tend to thepoles. 

 On eccentric and centric force: a new theory of projection. By H. F. A. Pratt, M. D. [276] London, 1862, 8vo. 

Dr. Pratt not only upsets Newton, but cuts away the very ground he standson: for he destroys the first law of motion, and will not have the naturaltendency of matter in motion to be rectilinear. This, as we have seen, wasJohn Walsh's[277] notion. In a more recent work "On Orbital Motion, "London, 1863, 8vo. , Dr. Pratt insists on another of Walsh's notions, namely, that the precession of the equinoxes is caused by the motion of thesolar system round a distant central sun. In this last work the authorrefers to a few notes, which completely destroy the theory of gravitationin terms "perfectly intelligible as well to the unlearned as to thelearned": to me they are quite unintelligible, which rather tends toconfirm a notion I have long had, that I am neither one thing nor theother. There is an ambiguity of phrase which delights a writer on logic, always on the look-out for specimens of _homonymia_ or _æquivocatio_. Theauthor, as a physician, is accustomed to "appeal from mere formulæ":accordingly, he sets at nought the whole of the mathematics, which he doesnot understand. This equivocation between the formula of the physician andthat of the mathematician is as good, though not so perceptible to theworld at large, as that made by Mr. Briggs's friend in _Punch's_ picture, which I cut out to paste into my Logic. Mr. Briggs wrote for a couple of_bruisers_, meaning to prepare oats for his horses: his friend sent him theWhitechapel Chicken and the Bayswater Slasher, with the gloves, all ready. 

{158}

 On matter and ether, and the secret laws of physical change. By T. R. Birks, M. A. [278] Cambridge, 1862, 8vo. 

Bold efforts are made at molecular theories, and the one before me is ablyaimed. When the Newton of this subject shall be seated in his place, bookslike the present will be sharply looked into, to see what amount ofanticipation they have made. 

DR. THORN AND MR. BIDEN. 

 The history of the 'thorn tree and bush' from the earliest to the present time: in which is clearly and plainly shown the descent of her most gracious Majesty and her Anglo-Saxon people from the half tribe of Ephraim, and possibly from the half tribe of Manasseh; and consequently her right and title to possess, at the present moment, for herself and for them, a share or shares of the desolate cities and places in the land of their forefathers! By Theta, M. D. [279] (Private circulation. ) London, 1862, 8vo. 

This is much about _Thorn_, and its connected words, Thor, Thoth, Theta, etc. It is a very mysterious vagary. The author of it is the person whom Ihave described elsewhere as having for his device the round man in thethree-cornered hole, the writer of the little heap of satirical anonymousletters about the Beast and 666. By accident I discovered the writer: sothat if there be any more thorns to crackle under the pot, they need not beanonymous. 

Nor will they be anonymous. Since I wrote the above, I have received_onymous_ letters, as _ominous_ as the rest. The writer, William Thorn, M. D. , is obliged to reveal {159} himself, since it is his object to provethat he himself is one 666. By using W for double Vau (or 12) he cooks thenumber out of his own name. But he says it is the number not of a beast butof a man, and adds, "Thereby hangs a tale!" which sounds likecontradiction. He informs me that he will talk the matter over with me: butI shall certainly have nothing to say to a gentleman of his number; it isbest to keep on the safe side. 

In one letter I am informed that not a line should I have had, but for my"sneer at 666, " which, therefore, I am well pleased to have given. I amalso told that my name means the "'garden of death, ' that place in whichthe tree of knowledge was plucked, and so you are like your name 'dead' tothe fact that you are an Israelite, like those in Ezekiel 37 ch. " Somehints are given that I shall not fare well in the next world, which any onewho reads the chapter in Ezekiel will see is quite against his comparison. The reader must not imagine that my prognosticator means _Morgan_ to be acorruption of _Mortjardin_; he proves his point by Hebrew: but anyphilologist would tell him the true derivation of the name, and how_Glamorgan_ came to get it. It will be of much comfort to those young menwho have not got through to know that the tree of knowledge itself was oncein the same case. And so good bye to 666 for the present, and theassumption that the enigma is to be solved by the united numeral forces ofthe letters of a word. 

It is worthy of note that, as soon as my Budget commenced, two guardianspirits started up, fellow men as to the flesh, both totally unknown to me:they have stuck to me from first to last. James Smith, Esq. , finallyNauticus, watches over my character in this world, and would fain preserveme from ignorance, folly, and dishonesty, by inclosing me in a magic circleof 3-1/8 diameters in circumference. The round man in the three-corneredhole, finally William Thorn, M. D. , takes charge of my future destiny, {160}and tries to bring me to the truth by unfolding a score of meanings--allright--of 666. He hints that I, and my wife, are servants of Satan: atleast he desires us both to remember that we cannot serve God and Satan;and he can hardly mean that we are serving the first, and that he wouldhave us serve the second. As becomes an interpreter of the Apocalypse, heuses seven different seals; but not more than one to one letter. If hisseals be all signet-rings, he must be what Aristophanes calls asphragidonychargocometical fellow. But--and many thanks to him for thesame--though an M. D. , he has not sent me a single vial. And so much for mytree of secular knowledge and my tree of spiritual life: I dismiss themwith thanks from myself and thanks from my reader. The dual of thePythagorean system was Isis and Diana; of the Jewish law, Moses and Aaron;and of the City of London, Gog and Magog; of the Paradoxiad, James Smith, Esq. , and William Thorn, M. D. 

_September, 1866. _ Mr. James Biden[280] has favored me with some of hispublications. He is a rival of Dr. Thorn; a prophet by name-right andcrest-right. He is of royal descent through the De Biduns. He is the_watchman_ of Ezekiel: God has told him so. He is the author of _The TrueChurch_, a phrase which seems to have a book-meaning and a mission-meaning. He shall speak for himself:

"A crest of the Bidens has significance. It is a lion rampant betweenwings--wings in Scripture denote the flight of time. Thus the beasts orliving creatures of the Revelations have each six wings, intimating acondition of mankind up to and towards the close of six thousand years ofBible teaching. The two wings of the crest would thus intimate powertowards the expiration of 2000 years, as time is marked in the history ofGreat Britain. 

{161}

"In a recent publication, _The Pestilence, Why Inflicted_, are given manyreasons why the writer thinks himself to be the appointed watchman foretoldby Ezekiel, chapters iii. And xxxiii. Among the reasons are many propheciesfulfilled in him. Of these it is now needful to note two as bearingespecially on the subject of the reign of Darius. 

"1. --In Daniel it is said, 'Darius the Median took the kingdom, being aboutthreescore and two years old. '--Daniel v. 31. 

"When 'Belshazzar' the king of the Chaldeans is found wanting, Darius takesthe kingdom. It is not given him by the popular voice; he asserts hisright, and this is not denied. He takes it when about sixty-two years ofage. The language of Daniel is prophetic, and Darius has in another anantitype. The writer was born July 18th, 1803; and the claim was assertedat the close of 1865, when he was about sixty-two years of age. 

"The claims which have been asserted demand a settled faith, and whichcould only be reached through a long course of divine teaching. "

When I was a little boy at school, one of my school-fellows took it intohis head to set up a lottery of marbles: the thing took, and he made astony profit. Soon, one after another, every boy had his lottery, and itwas, "I won't put into yours unless you put into mine. " This knocked up thescheme. It will be the same with the prophets. Dr. Thorn, Mr. Biden, Mrs. Cottle, [281] etc. Will grow imitators, until we are all pointed out in theBible: but A will not admit B's claim unless B admits his. For myself, aselsewhere shown, I am the first Beast in the Revelations. 

Every contraband prophet gets a few followers: it is a great point to makethese sequacious people into Buridan's asses, which they will become whenprophets are so numerous that there is no choosing. 

{162}

SIR G. C. LEWIS. 

 An historical survey of the Astronomy of the Ancients. By the Rt. Hon. Sir G. C. Lewis. [282] 8vo. 1862. 

There are few men of our day whom I admire more than the late Sir G. Lewis:he was honest, earnest, sagacious, learned, and industrious. He probablysacrificed his life to his conjunction of literature and politics: and hestood high as a minister of state in addition to his character as a man ofletters. The work above named is of great value, and will be read for itsintrinsic merit, consulted for its crowd of valuable references, quoted forits aid to one side of many a discussion, and opposed for its force againstthe other. Its author was also a wit and a satirist. I know of threeclassical satires of our day which are inimitable imitations: Mr. Malden's[283] _Pragmatized Legends_, Mr. Mansel's[284] _Phrontisterion_, and Sir G. Cornewall Lewis's _Inscriptio Antiqua_. In this last, HEYDIDDLEDIDDLETHECATANDTHEFIDDLE etc. Is treated as an Oscan inscription, and rendered into Latin by approved methods. As few readers have seen it, Igive the result:

 "Hejus dedit libenter, dedit libenter. Deus propitius [est], deus [donatori] libenter favet. Deus in viarum {163} juncturâ ovorum dape [colitur], deus mundi. Deus in litatione voluit, benigno animo, hædum, taurum intra fines [loci sacri] portandos. Deus, bis lustratus, beat fossam sacræ libationis. "[285]

How then comes the history of astronomy among the paradoxes? Simply becausethe author, so admirably when writing about what he knew, did not know whathe did not know, and blundered like a circle-squarer. And why should thefaults of so good a writer be recorded in such a list as the present? Forthree reasons: First, and foremost, because if the exposure be not made bysome one, the errors will gradually ooze out, and the work will get thecharacter of inaccurate. Nothing hurts a book of which few can fathom thedepths so much as a plain blunder or two on the surface. Secondly, becausethe reviews either passed over these errors or treated them too gently, rather implying their existence than exposing them. Thirdly, because theystrongly illustrate the melancholy truth, that no one knows enough to writeabout what he does not know. The distinctness of the errors is a merit; itproceeds from the clear-headedness of the author. The suppression in thejournals may be due partly to admiration of the talent and energy whichlived two difficult lives at once, partly to respect for high position inpublic affairs, partly to some of the critics being themselves men oflearning only, unable to detect the errors. But we know that action andreaction are equal and contrary. If our generation take no notice ofdefects, and allow them to go down undetected among merits, the nextgeneration will discover them, will perhaps believe us incapable ofdetecting them, at least will pronounce our judgment good for nothing, andwill form an {164} opinion in which the merits will be underrated: so ithas been, is, and will be. The best thing that can be done for the memoryof the author is to remove the unsound part that the remainder may thrive. The errors do not affect the work; they occur in passages which might verywell have been omitted: and I consider that, in making them conspicuous, Iam but cutting away a deleterious fungus from a noble tree. 

(P. 154). The periodic times of the five planets were stated byEudoxus, [286] as we learn from Simplicius;[287] the following is hisstatement, to which the true times are subjoined, for the sake ofcomparison:

 STATEMENT OF EUDOXUS TRUE TIME Mercury 1 year -- 87d. 23h. Venus 1 " -- 224d. 16h. Mars 2 " 1y. 321d. 23h. Jupiter 12 " 11y. 315d. 14h. Saturn 30 " 29y. 174d. 1h. 

Upon this determination two remarks may be made. First, the error withrespect to Mercury and Venus is considerable; with respect to Mercury, itis, in round numbers, 365 instead of 88 days, more than four times toomuch. Aristotle remarks that Eudoxus distinguishes Mercury and Venus fromthe other three planets by giving them one sphere each, with the poles incommon. The proximity of Mercury to the sun would render its coursedifficult to observe and to measure, but the cause of the large error withrespect to Venus (130 days) is not apparent. 

{165}

Sir G. Lewis takes Eudoxus as making the planets move round the sun; he hasaccordingly compared the _geocentric_ periods of Eudoxus with our_heliocentric_ periods. What greater blunder can be made by a writer onancient astronomy than giving Eudoxus the Copernican system? If Mercurywere a black spot in the middle of the sun it would of course move roundthe earth in a year, or appear to do so: let it swing a little on one sideand the other of the sun, and the average period is still a year, withslight departures both ways. The same for Venus, with larger departures. Say that a person not much accustomed to the distinction might for oncewrite down the mistake; how are we to explain its remaining in the mind ina permanent form, and being made a ground for such speculation as that ofthe difficulty of observing Mercury leading to a period four times what itought to be, corrected in proof and published by an industrious andthoughtful person? Only in one way: the writer was quite out of his depth. This one case is conclusive; be it said with all respect for the realstaple of the work and of the author. He knew well the difference of thesystems, but not the effect of the difference: he is another instance ofwhat I have had to illustrate by help of a very different person, that itis difficult to reason well upon matter which is not familiar. 

(P. 254). Copernicus, in fact, supposed the axis of the earth to be alwaysturned towards the Sun.^{(169)} [(169). See Delambre, _Hist. Astr. Mod. _, Vol. I, p. 96]. It was reserved to Kepler to propound the hypothesis of theconstant parallelism of the earth's axis to itself. 

If there be one thing more prominent than another in the work of Copernicushimself, in the popular explanations of it, and in the page ofDelambre[288] cited, it is that the _parallelism of the earth's axis_ is aglaring part of the {166} theory of Copernicus. What Kepler[289] did was tothrow away, as unnecessary, the method by which Copernicus, _per fas etnefas_, [290] secured it. Copernicus, thinking of the earth's orbitalrevolution as those would think who were accustomed to the _solidorbs_--and much as the stoppers of the moon's rotation do now: why do theynot strengthen themselves with Copernicus?--thought that the earth's axiswould always incline the same end towards the sun, unless measures weretaken to prevent it. He _did_ take measures: he invented a _compensating_conical motion of the axis to preserve the parallelism; and, which is oneof the most remarkable points of his system, he obtained the precession ofthe equinoxes by giving the necessary trifle more than compensation. Whatstares us in the face at the beginning of the paragraph to which the authorrefers?

"C'est donc pour arriver à ce parallelisme, ou pour le conserver, queCopernic a cru devoir recourir à ce mouvement égal et opposé qui détruitl'effet qu'il attribue si gratuitement au premier, de déranger leparallelisme. "[291]

Parallelism at any price, is the motto of Copernicus: you need not pay sodear, is the remark of Kepler. 

The opinions given by Sir G. Lewis about the effects of modern astronomy, which he does not understand and singularly undervalues, will now be seento be of no authority. He fancies that--to give an instance--for thedetermination of a ship's place, the invention of chronometers has been farmore important than any improvement in astronomical theory (p. 254). Not tospeak of latitude, --though the omission is not without importance, --heought to have known that longitude is found by the difference between whato'clock it is at Greenwich and at the ship's place, at {167} one absolutemoment of time. Now if a chronometer were quite perfect--which nochronometer is, be it said--and would truly tell Greenwich mean time allover the world, it ought to have been clear that just as good a watch iswanted for the time at _the place of observation_, before the longitude ofthat place with respect to Greenwich can be found. There is no such watch, except the starry heaven itself: and that watch can only be read byastronomical observation, aided by the best knowledge of the heavenlymotions. 

I think I have done Sir G. Lewis's very excellent book more good than allthe reviewers put together. 

I will give an old instance in which literature got into confusion aboutastronomy. Theophrastus, [292] who is either the culprit or his historian, attributes to Meton, [293] the contriver of the lunar calendar of nineteenyears, which lasts to this day, that his solstices were determined for himby a certain Phaeinus of Elis on Mount Lycabettus. Nobody else mentionsthis astronomer: though it is pretty certain that Meton himself made morethan one appointment with him for the purpose of observing solstices; andwe may be sure that if either were behind his time, it was Meton. For_Phaeinus Helius_ is the shining sun himself; and in the astronomical poetAratus[294] we read about the nineteen years of the shining sun:

 [Greek: Enneakaideka kukla phaeinou êelioio]. [295]

Some man of letters must have turned Apollo into Phaeinus of Elis; andthere he is in the histories of astronomy to {168} this day. Salmasius[296]will have Aratus to have meant him, and proposes to read [Greek: êleioio]:he did not observe that Phaeinus is a very common adjective of Aratus, andthat, if his conjecture were right, this Phaeinus would be the onlynon-mythical man in the poems of Aratus. 

[When I read Sir George Lewis's book, the points which I have criticizedstruck me as not to be wondered at, but I did not remember why at the time. A Chancellor of the Exchequer and a writer on ancient astronomy are birdsof such different trees that the second did not recall the first. In 1855 Iwas one of a deputation of about twenty persons who waited on Sir G. Lewis, as Chancellor of the Exchequer, on the subject of a decimal coinage. Thedeputation was one of much force: Mr. Airy, with myself and others, represented mathematics; William Brown, [297] whose dealings with the UnitedStates were reckoned by yearly millions, counted duodecimally in Englandand decimally in America, was the best, but not the only, representative ofcommerce. There were bullionists, accountants, retailers, etc. Sir G. L. Walked into the room, took his seat, and without waiting one moment, beganto read the deputation a smart lecture on the evils of a decimal coinage;it would require alteration of all the tables, it would impede calculation, etc. Etc. Of those arguments against it which weighed with many of betterknowledge than his, he obviously knew nothing. The members of thedeputation began to make their statements, and met with curious denials. Heinterrupted me with "Surely there is no doubt that the calculations of ourbooks of arithmetic are easier {169} than those in the French books. " Hewas not aware that the _universally admitted_ superiority of decimal_calculation_ made many of those who prefer our system for the market andthe counter cast a longing and lingering look towards decimals. My answerand the smiles which he saw around, made him give a queer puzzled look, which seemed to say, "I may be out of my depth here!" His manner changed, and he listened. I saw both the slap-dash mode in which he dealt withsubjects on which he had not thought, and the temperament which admittedsuspicion when the means of knowledge came in his way. Having seen his twophases, I wonder neither at his more than usual exhibition of shallownesswhen shallow, nor at the intensity of the contrast when he had greaterdepth. ]

DECIMAL COINAGE. 

Among the paradoxers are the political paradoxers who care not how far theygo in debate, their only object being to carry the House with them for thecurrent evening. What I have said of editors I repeat of them. Thepreservation of a very marked instance, the association of politicalrecklessness with cyclometrical and Apocalyptic absurdity, may have atendency to warn, not indeed any hardened public-man and sinner, but someyoung minds which have yearnings towards politics, and are in formation ofhabits. 

In the debate on decimal coinage of July 12, 1855, Mr. Lowe, [298] thenmember for Kidderminster, an effective speaker and a smart man, exhibitedhimself in a speech on which I wrote a comment for the Decimal Association. I have seldom seen a more wretched attempt to distort the points of apublic question than the whole of this speech. Looking at the intelligenceshown by the speaker on other occasions, {170} it is clear that if charity, instead of believing all things, believed only all things but one, he mighttremble for his political character; for the honesty of his intention onthis occasion might be the incredible exception. I give a few paragraphswith comments:

"In commenting on the humorous, but still argumentative speech of Mr. Lowe, the member for Kidderminster, we may observe, in general, that it consistsof points which have been several times set forth, and several timesanswered. Mr. Lowe has seen these answers, but does not allude to them, farless attempt to meet them. There are, no doubt, individuals, who show intheir public speaking the outward and visible signs of a greater degree ofacuteness than they can summon to guide their private thinking. If Mr. Lowebe not one of these, if the power of his mind in the closet be at allcomparable to the power of his tongue in the House, it may be suspectedthat his reserve with respect to what has been put forward by the veryparties against whom he was contending, arises from one or both of twothings--a high opinion of the arguments which he ignored--a low opinion ofthe generality of the persons whom he addressed. [Both, I doubt not]. 

"Did they calculate in florins In the name of common sense, ?" how can it be objected to a system that people do not use it before it is introduced? Let the decimal system be completed, and calculation shall be made in florins; that is, florins shall take their proper place. If florins were introduced _now_, there must be a column for the odd shilling. "He was glad that some hon. If the hon. Gentleman makegentleman had derived benefit this assertion of himself, itfrom the issue of florins. His is not for us to gainsay it. Only experience of their It only proves that he is oneconvenience was, that when he of that class of {171} men whoought to have received are described in the old song, half-a-crown, he had generally of which one couplet runsreceived a florin, and when he thus:ought to have paid a florin, he had generally paid I sold my cow to buy me ahalf-a-crown. " (Hear, hear, calf;and laughter. ) I never make a bargain but I lose half, With a etc. Etc. Etc. 

But he cannot mean that Englishmen in general are so easily managed. And asto Jonathan, who is but John lengthened out a little, he would see creationwhittled into chips before he would even split what may henceforth becalled the Kidderminster difference. The House, not unmoved--for itlaughed--with sly humor decided that the introduction of the florin hadbeen "eminently successful and satisfactory. "

The truth is that Mr. Lowe here attacks nothing except the coexistence ofthe florin and half-crown. We are endeavoring to abolish the half-crown. Let Mr. Lowe join us; and he will, if we succeed, be relieved from thepressure on his pocket which must arise from having the turn of the marketalways against him. 

"From a florin they get to 2 Note the sophism of expressing2-5ths of a penny, but who our coin in terms of theever bought anything, who ever penny, which we abandon, reckoned or wished to reckon instead of the florin, whichin such a coin as that?" we retain. Remember that this(Hear, hear. ) 2 2-5ths is the hundredth part of the pound, which is called, as yet, a _cent_. Nobody buys anything at a cent, because the cent is not yet introduced. Nobody reckons in cents for the same reason. Everybody wishes to reckon in cents, who wishes to combine the advantage of decimal reckoning with the preservation of the pound as {172} the highest unit of account; amongst others, a majority of the House of Commons, the Bank of England, the majority of London bankers, the Chambers of Commerce in various places, etc. Etc. Etc. "Such a coin could never come Does 2½d. Never pass from handinto general circulation to hand? And is 2½d. Sobecause it represents nothing precisely the modulus ofwhich corresponds with any of popular wants, that anthe wants of the people. " alteration of 4 per cent. Would make it useless? Of all the values which 2½d. Measures, from three pounds of potatoes down to certain arguments used in the House of Commons, there is not one for which a cent would not do just as well. Mr. Lowe has fallen into the misconception of the person who admired the dispensation of Providence by which large rivers are made to run through cities so great and towns so many. If the cent were to be introduced to-morrow, straightway the buns and cakes, the soda-water bottles, the short omnibus fares, the bunches of radishes, etc. Etc. Etc. , would adapt themselves to the coin. "If the proposed system were The confusion of ideas hereadopted, they would all be exhibited is most instructive. Compelled to live in decimals The speaker is under thefor ever; if a man dined at a impression that _we_ arepublic house he would have to introducing fractions: thepay for his dinner in decimal truth is, that we only want tofractions. (Hear, hear. ) He abandon the _more difficult_objected to that, for he fractions which we _have got_, thought that a man ought to be and to introduce _easierable to pay for his dinner in fractions_. Does he deny this?integers. " (Hear, hear, and a Let us trace his denial to itslaugh. ) legitimate consequences. A man ought to pay for his dinner in integers. 

{173}

Now, if Mr. Lowe insists on it that our integer is the pound, he is boundto admit that the present integer is the pound, of which a shilling, etc. , are fractions. The next time he has a chop and a pint of stout in the city, the waiter should say--"A pound, sir, to you, " and should add, "Please toremember the waiter in integers. " Mr. Lowe fancies that when he pays oneand sixpence, he pays in integers, and so he does, if his integer be apenny or a sixpence. Let him bring his mind to contemplate a mil as theinteger, the lowest integer, and the seven cents five mils which he wouldpay under the new system would be payment in integers also. But, as ithappens with some others, he looks _up_ the present system, withCocker, [299] and Walkingame, [300] and always looks _down_ the proposedsystem. The word _decimal_ is obstinately associated with _fractions_, forwhich there is no need. Hence it becomes so much of a bugbear, that, toparody the lines of Pope, which probably suggested one of Mr. Lowe'sphrases--

 "Dinner he finds too painful an endeavor, Condemned to pay in decimals for ever. "

"The present system, however, A pleasant sum even for anhad not yet been changed into accomplished mathematician. Decimal system. That change What does divided by themight appear very easy to decimal of a pound mean?accomplished mathematicians Perhaps it means _reduced_ toand men of science, but it was the decimal of a pound! Mr. One which it would be very Lowe supposes, as many othersdifficult to carry out. (Hear, do, that, after the change, hear). What would have to be all calculations will bedone? Every sum would have to _proposed in old money_, andbe reduced into a vulgar then _converted into new_. Hefraction of a pound, and then cannot hit the {174} idea thatdivided by the decimal of a the new coins will take thepound--a pleasant sum for an place of the old. This lack ofold applewoman to work out!" apprehension will presently(Hear, hear, and laughter. ) appear further. "It would not be an agreeable Let the members be assuredtask, even for some members of that nine half-pence will be, that House, to reduce 4½d. , or for every practical purpose, nine half-pence, to mils. " 18 mils. But now to the fact(Hear, hear. ) asserted. Davies Gilbert[301] used to maintain that during the long period he sat in the House, he never knew more than three men in it, at one time, who had a tolerable notion of fractions. [I heard him give the names of three at the time when he spoke: they were Warburton, [302] Pollock, [303] and Hume. [304] He himself was then out of Parliament. ] Joseph Hume affirmed that he had never met with more than ten members who were arithmeticians. But both these gentlemen had a high standard. Mr. Lowe has given a much more damaging opinion. He evidently means that the general run of members could not do his question. It is done as follows: Since farthings gain on mils, at the rate of a whole mil in 24 farthings (24 farthings being 25 mils), it is clear that 18 farthings being three-quarters of 24 farthings, will gain three-quarters of a mil; that is, 18 farthings are eighteen {175} mils and three-quarters of a mil. Any number of farthings is as many mils and as many twenty-fourths of a mil. To a certain extent, we feel able to protest against the manner in which Kidderminster has treated the other constituencies. We do not hold it impossible to give the Members of the House in general a sufficient knowledge of the meaning and consequences of the _decimal_ succession of units, tens, hundreds, thousands, etc. ; and we believe that there are in the House itself competent men, in number enough to teach all the rest. All that is wanted is the power of starting from the known to arrive at the unknown. Now there is one kind of decimals with which every member is acquainted--the _Chiltern Hundreds_. If public opinion would enable the competent minority to start from this in their teaching, not as a basis, but as an alternative, in three weeks the fundamentals would be acquired, and members in general would be as fit to turn 4½d. Into mils, as any boys on the lower forms of a commercial school. 

 For a long period of years, allusion to the general ignorance of arithmetic, has been a standing mode of argument, and has always been well received: whenever one member describes others as _knownothings_, those others cry _Hear_ to the country in a transport of delight. In the meanwhile the country is gradually arriving at the conclusion that a true joke is no joke. "The main objection was, if Fine words, wrongly used. Thethey went below 6d. , that the new coins are commensurablenew scale of coins would not with, and in a finite ratiobe commensurate in any finite to, the old ones. The farthingratio with anything in this is to the mil as 25 to 24. Thenew currency of mils. " speaker has something here in the bud, which we shall presently meet with in the flower; and fallacies are more easily nipped in flower than in bud. {176}"No less than five of our This dreadful change of valuepresent coins must be called consists in sixpence farthingin, or else--which would be going to the half-shillingworse--new values must be instead of sixpence. Whethergiven to them. " the new farthings be called mils or not is of no consequence. "If a poor man put a penny in Mr. Lowe, who cannot pass ahis pocket, it would come out half-crown for more than aa coin of different value, florin, or get in a florin atwhich he would not understand. Less than half-a-crown, hasSuppose he owed another man a such a high faith in thepenny, how was he to pay him ? sterner stuff of his fellowWas he to pay him in mils? countrymen, that he believesFour mils would be too little, any two of them would go toand five mils would be too fisty cuffs for the 25th partmuch. The hon. Gentlemen said of a farthing. He reasonsthere would be only a mil thus: He has often heard inbetween them. That was exactly the streets, "I'd fight youit. He believed there would be for the fiftieth part of aa 'mill' between them. " (Much farden:" and having (that is, laughter. ) for a Member) a notion both of fractions and logic, he infers that those who would fight for the 50th of a farthing would, _a fortiori_, fight for a 25th. His mistake arises from his not knowing that when a person offers to fight another for 1/200d. , he really means to fight for love; and that the stake is merely a matter of form, a feigned issue, a _pro forma_ report of progress. Do the Members of the House think they have all the forms to themselves?"What would be the present We should hardly believe allexpression for four-pence? this to be uttered in earnest, Why, 0. 166 (a laugh); for if we had not known {177} thatthreepence? . 0125; for a several persons who have notpenny? . 004166, and so on _ad Mr. Lowe's humor, neverthelessinfinitum_ (a laugh); for a have his impressions on thishalf-penny? . 002083 _ad point. It must therefore beinfinitum_. (A laugh). What answered; but how is this towould be the present be done seriously?expression for a farthing?Why, . 0010416 _ad infinitum_. _Dialogue between a member of(A laugh). And this was the Parliament and an orange-boy, system which was to cause such three days after thea saving in figures, and these introduction of the completewere the quantities into which decimal system. The member, the poor would have to reduce going down to the House, wantsthe current coin of the realm. Oranges to sustain his voice(Cheers). With every respect in a two hours' speech onfor decimal fractions, of moving that 100000l. Be placedwhich he boasted no profound at the disposal of Herknowledge, he doubted whether Majesty, to supply the poorthe poor were equal to mental with ready-reckoners. _arithmetic of this kind, (hear, hear) and he hoped the _Boy. _ Fine oranges! two aadoption of the system would penny! two a penny! {178}be deferred until there weresome proof that they would be _Member. _ Here boy, two! Now, able to understand it; for, how am I to pay you?after all, this was thequestion of the poor, and the _Boy. _ Give you change, yourwhole weight of the change honor. Would fall upon them. Let therich by all means have _Member. _ Ah! but how? Where'spermission to perplex your ready-reckoner?themselves by any division ofa pound they pleased; but do _Boy. _ I sells a better sortnot let them, by any nor them. Mine's real Cheyny. Experiment like this, imposedifficulties upon the poor and _Member. _ But you see acompel men to carry farthing is now . 0014166666ready-reckoners in their _ad infinitum_, and if wepocket to give them all these multiply this by 4----fractional quantities. " (Hear, hear. ) _Boy. _ Hold hard, Guv'ner; I sees what you're arter. Now what'll you stand if I puts you up to it? which Bill Smith he put me up in two minutes, cause he goes to the Ragged School. 

 _Member. _ You don't mean that you do without a book!

 _Boy. _ Book be blowed. Come now, old un, here's summut for both on us. I got a florin, you gives me a half-a-crown for it, and I larns you the new money, gives you your oranges, and calls you a brick into the bargain. 

 _Member_ (_to himself_). Never had such a chance of getting off half-a-crown for value since that ---- fellow Bowring carried his crochet. (_Aloud. _) Well, boy, it's a bargain. Now!

 _Boy. _ Why, look 'e here, my trump, its a farden more to the tizzy--that's what it is. 

 _Member_. What's that?

 _Boy. _ Why, you knows a sixpence when you sees it. (_Aside_). Blest if I think he does! Well, its six browns and a farden now. A lady buys two oranges, and forks {179} out a sixpence; well in coorse, I hands over fippence farden astead of fippence. I always gives a farden more change, and takes according. 

 _Member_ (_in utter surprise, lets his oranges tumble into the gutter_). Never mind! They won't be wanted now. (_Walks off one way. Boy makes a pass of naso-digital mesmerism, and walks off the other way_). 

To the poor, who keep no books, the whole secret is "Sixpence farthing tothe half shilling, twelve pence halfpenny to the shilling. " The _newtwopence halfpenny_, or cent, will be at once five to the shilling. 

In conclusion, we remark that three very common misconceptions run throughthe hon. Member's argument; and, combined in different proportions, givevariety to his patterns. 

First, he will have it that we design to bring the uneducated into contactwith _decimal fractions_. If it be so, it will only be as M. Jourdain wasbrought into contact with prose. In fact, _Quoi! quand je dis, Nicole, apportez-moi mes pantoufles, c'est de la prose?_[305] may be rendered:"What! do you mean that _ten to the florin is a cent a piece_ must becalled decimal reckoning?" If we had to comfort a poor man, horror-struckby the threat of _decimals_, we should tell him what manner of fractionshad been inflicted upon him hitherto; nothing less awful than_quarto-duodecimo-vicesimals_, we should assure him. 

Secondly, he assumes that the penny, such as it now is, will remain, as acoin of estimation, after it has ceased to be a coin of exchange; and thatthe mass of the people will continue to think of prices in old pence, andto calculate them in new ones, or else in new mils. No answer is requiredto this, beyond the mere statement of the nature of the assumption anddenial. 

{180}

Thirdly, he attributes to the uneducated community a want of perception andof operative power which really does not belong to them. The evidenceoffered to the Committee of the House shows that no fear is entertained onthis point by those who come most in contact with farthing purchasers. Andthis would seem to be a rule, --that is, fear of the intelligence of thelower orders in the minds of those who are not in daily communication withthem, no fear at all in the minds of those who are. 

A remarkable instance of this distinction happened five-and-twenty yearsago. The Admiralty requested the Astronomical Society to report on thealterations which should be made in the _Nautical Almanac_, the seaman'sguide-book over the ocean. The greatest alteration proposed was thedescription of celestial phenomena in _mean_ (or clock time), instead of_apparent_ (or sundial) time, till then always employed. This change wouldrequire that in a great many operations the seaman should let alone what heformerly altered by addition or subtraction, and alter by addition orsubtraction what he formerly let alone; provided always that what heformerly altered by addition he should, when he altered at all, alter bysubtraction, and _vice versa_. This was a tolerably difficult change foruneducated skippers, working by rules they had only learned by rote. TheAstronomical Society appointed a Committee of forty, of whom nine werenaval officers or merchant seamen [I was on this Committee]. Some men ofscience were much afraid of the change. They could not trust an ignorantskipper or mate to make those alterations in their routine, on thecorrectness of which the ship might depend. Had the Committee consisted ofmen of science only, the change might never have been ventured on. But thenaval men laughed, and said there was nothing to fear; and on theirauthority the alteration was made. The upshot was, that, after the newalmanacs appeared, not a word of complaint was ever heard on the matter. Had the House of Commons had to {181} decide this question, with Mr. Loweto quote the description given by Basil Hall[306] (who, by the way, was oneof the Committee) of an observation on which the safety of the shipdepended, worked out by the light of a lantern in a gale of wind off a leeshore, this simple and useful change might at this moment have been in thehands of its tenth Government Commission. 

[_Aug. 14, 1866. _ The Committee was appointed in the spring of 1830: itconsisted of forty members. Death, of course, has been busy; there are nowleft Lord Shaftesbury, [307] Mr. Babbage, [308] Sir John Herschel, [309] SirThomas Maclear[310] (Astronomer Royal at the Cape of Good Hope), Dr. Robinson[311] (of Armagh), Sir James South, [312] Lord Wrottesley, [313] andmyself]. 

{182}

THE TONAL SYSTEM. 

 Project of a new system of arithmetic, weight, measure, and coins, proposed to be called the tonal system, with sixteen to the base. By J. W. Mystrom. [314] Philadelphia, 1862, 8vo. 

That is to say, sixteen is to take the place of ten, and to be written 10. The whole language is to be changed; every man of us is to besixteen-stringed Jack and every woman sixteen-stringed Jill. Our old _one_, _two_, _three_, up to sixteen, are to be (_Noll_ going for nothing, whichwill please those who dislike the memory of _Old Noll_) replaced by An, De, Ti, Go, Su, By, Ra, Me, Ni, Ko, Hu, Vy, La, Po, Fy, Ton; and then Ton-an, Ton-de, etc. For 17, 18, etc. The number which in the system has the symbol

 28(13)5(11)7(14)0(15)

(using our present compounds instead of new types) is to be pronounced

 Detam-memill-lasan-suton-hubong-ramill-posanfy. 

The year is to have sixteen months, and here they are:

 Anuary, Debrian, Timander, Gostus, Suvenary, Bylian, Ratamber, Mesudius, Nictoary, Kolumbian, Husamber, Vyctorius, Lamboary, Polian, Fylander, Tonborius. 

Surely An-month, De-month, etc. Would do as well. Probably the wants ofpoetry were considered. But what are we to do with our old poets? Forexample--

 "It was a night of lovely June, High rose in cloudless blue the moon. "

Let us translate--

 "It was a night of lovely Nictoary, High rose in cloudless blue the (what, in the name of all that is absurd?). "

And again, _Fylander_ thrown into our December! What is {183} to become ofthose lines of Praed, which I remember coming out when I was atCambridge, --

 "Oh! now's the time of all the year for flowers and fun, the Maydays; To trim your whiskers, curl your hair, and sinivate the ladies. "

If I were asked which I preferred, this system or that of BaronFerrari[315] already mentioned, proceeding by _twelves_, I should reply, with Candide, when he had the option given of running the gauntlet or beingshot: Les volontés sont libres, et je ne veux ni l'un ni l'autre. [316] Wecan imagine a speculator providing such a system for Utopia as it would bein the mind of a Laputan: but to explain how an engineer who has surveyedmankind from Philadelphia to Rostof on the Don should for a momententertain the idea of such a system being actually adopted, would beat ajury of solar-system-makers, though they were shut up from the beginning ofAnuary to the end of Tonborius. When I see such a scheme as this imaginedto be practicable, I admire the wisdom of Providence in providing thequadrature of the circle, etc. , to open a harmless sphere of action to thepossessors of the kind of ingenuity which it displays. Those who cultivatemathematics have a right to speak strongly on such efforts of arithmetic asthis: for, to my knowledge, persons who have no knowledge are frequentlydisposed to imagine that their makers are true brothers of the craft, alittle more intelligible than the rest. 

SOME SMALL PARADOXERS. 

 Vis inertiae victa, [317] or Fallacies affecting science. By James Reddie. [318] London, 1862, 8vo. 

{184}

An attack on the Newtonian mechanics; revolution by gravitationdemonstrably impossible; much to be said for the earth being the immovablecenter. A good analysis of contents at the beginning, a thing seldom found. The author has followed up his attack in a paper submitted to the BritishAssociation, but which it appears the Association declined to consider. Itis entitled--

 _Victoria Toto Coelo_; or, Modern Astronomy recast. London, 1863, 8vo. 

At the end is a criticism of Sir G. Lewis's _History of Ancient Astronomy_. 

 On the definition and nature of the Science of Political Economy. By H. Dunning Macleod, [319] Esq. Cambridge, 1862, 8vo. 

A paper read--but, according to the report, not understood--at the BritishAssociation. There is a notion that political economy is entirelymathematical; and its negative quantity is strongly recommended for study:it contains "the whole of the Funds, Credit, 32 parts out of 33 of thevalue of Land.... " The mathematics are described as consisting of--first, number, or Arithmetic; secondly, the theory of dependent quantities, subdivided into dependence by cause and effect, and dependence bysimultaneous variations; thirdly, "independent quantities or unconnectedevents, which is the theory of probabilities. " I am not ashamed, having theBritish Association as a co-non-intelligent, to say I do not understandthis: there is a paradox in it, and the author should give furtherexplanation, especially of his negative quantity. Mr. Macleod has gained{185} praise from great names for his political economy; but this, Isuspect, must have been for other parts of his system. 

 On the principles and practice of just Intonation, with a view to the abolition of temperament.... By General Perronet Thompson. [320] Sixth Edition. London, 1862, 8vo. 

Here is General Thompson again, with another paradox: but always master ofthe subject, always well up in what his predecessors have done, and alwaysaiming at a useful end. He desires to abolish temperament by additionalkeys, and has constructed an enharmonic organ with forty sounds in theoctave. If this can be introduced, I, for one, shall delight to hear it:but there are very great difficulties in the way, greater than stood evenin the way of the repeal of the bread-tax. 

In a paper on the beats of organ-pipes and on temperament published someyears ago, I said that equal temperament appeared to me insipid, and not soagreeable as the effect of the instrument when in progress towards beingwhat is called out of tune, before it becomes offensively wrong. There isthroughout that period unequal temperament, determined by accident. GeneralThompson, taking me one way, says I have launched a declaration which islikely to make an epoch in musical practice; a public musical critic, taking me another way, quizzes me for preferring music _out of tune_. I donot think I deserve either one remark or the other. My opponent critic, Isuspect, takes _equally tempered_ and _in tune_ to be phrases of onemeaning. But by equal temperament is meant equal distribution among all thekeys of the error which an instrument _must_ have, which, with twelvesounds only in the octave, professes to be fit for all the keys. I amreminded of the equal temperament which was once applied to the postmen'sjackets. The coats were all made for the average man: the {186} consequencewas that all the tall men had their tails too short; all the short men hadthem too long. Some one innocently asked why the tall men did not changecoats with the short ones. 

 A diagram illustrating a discovery in the relation of circles to right-lined geometrical figures. London, 1863, 12mo. 

The circle is divided into equal sectors, which are joined head and tail:but a property is supposed which is not true. 

 An attempt to assign the square roots of negative powers; or what is [sqrt] -1? By F. H. Laing. [321] London, 1863, 8vo. 

If I understand the author, -a and +a are the square roots of -a^2, asproved by multiplying them together. The author seems quite unaware of whathas been done in the last fifty years. 

BYRNE'S DUAL ARITHMETIC. 

 Dual Arithmetic. A new art. By Oliver Byrne. [322] London, 1863, 8vo. 

The plan is to throw numbers into the form a(1. 1)^{b} (1. 01)^{c}(1. 001)^{d}... And to operate with this form. This is an ingenious andelaborate speculation; and I have no doubt the author has practised hismethod until he could surprise any one else by his use of it. But I doubtif he will persuade others to use it. As asked of Wilkins's universallanguage, Where is the second man to come from?

An effective predecessor in the same line of invention {187} was the lateMr. Thomas Weddle, [323] in his "New, simple, and general method of solvingnumeric equations of all orders, " 4to, 1842. The Royal Society, to whichthis paper was offered, declined to print it: they ought to have printed anorganized method, which, without subsidiary tables, showed them, in sixquarto pages, the solution (x=8. 367975431) of the equation

 1379. 664 x^{622} + 2686034 × 10^{432} x^{152} - 17290224 × 10^{518} x^{60} + 2524156 × 10^{574} = 0. 

The method proceeds by successive factors of the form, a being the firstapproximation, a × 1. B × 1. 0c × 1. 00d.... In my copy I find a fewcorrections made by me at the time in Mr. Weddle's announcement. "It wasread before that learned body [the R. S. ] and they were pleased [but] totransmit their thanks to the author. The en[dis]couragement which hereceived induces [obliges] him to lay the result of his enquiries in thisimportant branch of mathematics before the public [, at his own expense; hebeing an usher in a school at Newcastle]. " Which is most satirical, Mr. Weddle or myself? The Society, in the account which it gave of this paper, described it as a "new and remarkably simple method" possessing "severalimportant advantages. " Mr. Rutherford's[324] extended value of [pi] wasread at the very next meeting, and was printed in the _Transactions_; andvery properly: Mr. Weddle's paper was excluded, and very very improperly. 

HORNER'S METHOD. 

I think it may be admited that the indisposition to look at and encourageimprovements of calculation which once {188} marked the Royal Society is nolonger in existence. But not without severe lessons. They had the luck toaccept Horner's[325] now celebrated paper, containing the method which isfar on the way to become universal: but they refused the paper in whichHorner developed his views of this and other subjects: it was printed byT. S. Davies[326] after Horner's death. I make myself responsible for thestatement that the Society could not reject this paper, yet felt unwillingto print it, and suggested that it should be withdrawn; which was done. 

But the severest lesson was the loss of _Barrett's Method_, [327] now theuniversal instrument of the actuary in his highest calculations. It waspresented to the Royal Society, and refused admission into the_Transactions_: Francis Baily[328] printed it. The Society is now betterinformed: "_live and learn_, " meaning "_must live, so better learn_, " oughtto be the especial motto of a corporation, and is generally acted on, moreor less. 

Horner's method begins to be introduced at Cambridge: it was published in1820. I remember that when I first went to Cambridge (in 1823) I heard mytutor say, in conversation, there is no doubt that the true method ofsolving equations is the one which was published a few years ago in the_Philosophical Transactions_. I wondered it was not taught, but presumedthat it belonged to the higher mathematics. This Horner himself had in hishead: and in a sense it is true; for all lower branches belong to thehigher: but he would have stared to have been told that he, Horner, {189}was without a European predecessor, and in the distinctive part of hisdiscovery was heir-at-law to the namelessBrahmin--Tartar--Antenoachian--what you please--who concocted theextraction of the square root. 

It was somewhat more than twenty years after I had thus heard a Cambridgetutor show sense of the true place of Horner's method, that a pupil of minewho had passed on to Cambridge was desired by his college tutor to solve acertain cubic equation--one of an integer root of two figures. In a minutethe work and answer were presented, by Horner's method. "How!" said thetutor, "this can't be, you know. " "There is the answer, Sir!" said mypupil, greatly amused, for my pupils learnt, not only Horner's method, butthe estimation it held at Cambridge. "Yes!" said the tutor, "there is theanswer certainly; but it _stands to reason_ that a cubic equation cannot besolved in this space. " He then sat down, went through a process about tentimes as long, and then said with triumph: "There! that is the way to solvea cubic equation!"

I think the tutor in this case was never matched, except by the countryorganist. A master of the instrument went into the organ-loft duringservice, and asked the organist to let him _play the congregation out_;consent was given. The stranger, when the time came, began a voluntarywhich made the people open their ears, and wonder who had got into theloft: they kept their places to enjoy the treat. When the organist sawthis, he pushed the interloper off the stool, with "You'll never play 'emout this side Christmas. " He then began his own drone, and the congregationbegan to move quietly away. "There, " said he, "that's the way to play 'emout!"

I have not scrupled to bear hard on my own university, on the RoyalSociety, and on other respectable existences: being very much the friend ofall. I will now clear the Royal Society from a very small and obscureslander, simply because I know how. This dissertation began with {190} thework of Mr. Oliver Byrne, the dual arithmetician, etc. This writerpublished, in 1849, a method of calculating logarithms. [329] First, a longlist of instances in which, as he alleges, foreign discoverers have beenpillaged by Englishmen, or turned into Englishmen: for example, O'Neill, [330] so called by Mr. Byrne, the rectifier of the semi-cubicalparabola claimed by the Saxons under the name of _Neal_: the grandfather ofthis mathematician was conspicuous enough as _Neal_; he was archbishop ofYork. This list, says the writer, might be continued without end; but hehas mercy, and finishes with his own case, as follows:--"About twenty yearsago, I discovered this method of directly calculating logarithms. I couldgenerally find the logarithm of any number in a minute or two without theuse of books or tables. The importance of the discovery subjected me to allsorts of prying. Some asserted that I committed a table of logarithms tomemory; others attributed it to a peculiar mental property; and whenSocieties and individuals failed to extract my secret, they never failed totraduce the inventor and the invention. Among the learned Societies, theRoyal Society of London played a very base part. When I have more space andtime at my disposal, I will revert to this subject again. "

Such a trumpery story as this remains unnoticed at the time; but when allare gone, a stray copy from a stall falls into hands which, not knowingwhat to make of it, make history of it. It is a very curious distortion. The reader may take it on my authority, that the Royal Society played nopart, good or bad, nor had the option of playing a part. {191} But I myself_pars magna fui_:[331] and when the author has "space and time" at hisdisposal, he must not take all of them; I shall want a little of both. 

ARE ATOMS WORLDS?

 The mystery of being; or are ultimate atoms inhabited worlds? By Nicholas Odgers. [332] Redruth and London, 1863, 8vo. 

This book, as a paradox, beats quadrature, duplication, trisection, philosopher's stone, perpetual motion, magic, astrology, mesmerism, clairvoyance, spiritualism, homoeopathy, hydropathy, kinesipathy, Essaysand Reviews, and Bishop Colenso, [333] all put together. Of all thesuppositions I have given as actually argued, this is the one which ishardest to deny, and hardest to admit. Reserving the question--as beyondhuman discussion--whether our particles of carbon, etc. Are _clusters_ ofworlds, the author produces his reasons for thinking that they are at leastsingle worlds. Of course--though not mentioned--the possibility is to beadded of the same thing being true of the particles which make up ourparticles, and so down, for ever: and, on the other hand, of our planetsand stars as being particles in some larger universe, and so up, for ever. 

 "Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so _ad infinitum. _ And the great fleas themselves, in turn, have greater fleas to go on; While these again have greater still, and greater still, and so on. "[334]

I have often had the notion that all the nebulæ we see, including our own, which we call the Milky Way, may be particles of snuff in the box of agiant of a proportionately {192} larger universe. Of course the minim oftime--a million of years or whatever the geologists make it[335]--which ourlittle affair has lasted, is but a very small fraction of a second to thegreat creature in whose nose we shall all be in a few tens of thousands ofmillions of millions of millions of years. 

All this is quite possible, and the probabilities for and against are quiteout of reach. Perhaps also all the worlds, both above and below us, arefac-similes of our own. If so, away goes free will for good and all;unless, indeed, we underpin our system with the hypothesis that all thefac-simile bodies of different sizes are actuated by a common soul. Theseacute supplementary notions of mine go far to get rid of the difficultywhich some have found in the common theory that the soul inhabits the body:it has been stated that there is, somewhere or another, a world of soulswhich communicate with their bodies by wondrous filaments of a natureneither mental nor material, but of a _tertium quid_ fit to be ago-between; as it were a corporispiritual copper encased in aspiritucorporeal gutta-percha. My theory is that every soul is everywhere_in posse_, as the schoolmen said, but not anywhere _in actu_, except whereit finds one of its bodies. These _a priori_ difficulties being thusremoved, the system of particle-worlds is reduced to a dry question offact, and remitted to the decision of the microscope. And a grand field maythus be opened, as optical science progresses! For the worlds are notfac-similes of ours in time: there is not a moment of _our_ past, and not amoment of _our_ future, but is the _present_ of one or more of theparticles. A will write the death of Cæsar, and B the building of thePyramids, by actual observation of the processes with a power of a thousandmillions; C will discover the commencement of the Millennium, and D the{193} termination of Ersch and Gruber's Lexicon, [336] as mere physicalphenomena. Against this glorious future there is a sad omen: the initialsof the forerunner of this discovery are--NO!

THE SUPERNATURAL. 

 The History of the Supernatural in all ages and nations, and in all Churches, Christian and Pagan: demonstrating a universal faith. By Wm. Howitt. [337] London, 2 vols. 8vo. 1863. 

Mr. Howitt is a preacher of spiritualism. He cements an enormous collectionof alleged facts with a vivid outpouring of exhortation, and an unsparingflow of sarcasm against the scorners of all classes. He and the Rev. J. Smith[338] (_ante_, 1854) are the most thoroughgoing universalists of allthe writers I know on spiritualism. If either can insert the small end ofthe wedge, he will not let you off one fraction of the conclusion that allcountries, in all ages, have been the theaters of one vast spiritualdisplay. And I suspect that this consequence cannot be avoided, if any partof the system be of truly spiritual origin. Mr. Howitt treats thephilosophers either as ignorant babies, or as conscious spirit-fearers: andseems much inclined to accuse the world at large of dreading, lest by theactual presence of the other world their Christianity should imbibe aspiritual element which would unfit it for the purposes of their lives. 

{194}

FROM MATTER TO SPIRIT. 

 From Matter to Spirit. By C. D. With a preface by A. B. [339] London, 1863, 8vo. 

This is a work on Spiritual Manifestations. The author upholds the factsfor spiritual phenomena: the prefator suspends his opinion as to the cause, though he upholds the facts. The work begins systematically with the lowerclass of phenomena, proceeds to the higher class, and offers a theory, suggested by the facts, of the connection of the present and future life. Iagree in the main with A. B. ; but can, of course, make none but horrescentreference to his treatment of the smaller philosophers. This is always theway with your paradoxers: they behave towards orthodoxy as the thresherfish behaves towards the whale. But if true, as is said, that the drubbingclears the great fish of parasites which he could not otherwise get rid of, he ought to bear no malice. This preface retorts a little of that contemptwhich the "philosophical world" has bestowed with heaped measure upon thosewho have believed their senses, and have drawn natural, even if hasty, inferences. There is philosophercraft as well as priestcraft, both from onesource, both of one spirit. In English cities and towns, the minister ofreligion has been tamed: so many weapons are bared against him when heobtrudes his office in a dictatory manner, that, as a rule, there is nomore quiet and modest member of society than the urban clergyman. Domination over religious belief is reserved for the exclusive use of thosewho admit the right: the rare exception to this mode of behavior is laughedat as a bigot, or shunned as a nuisance. But the overbearing minister ofnature, who snaps you with _unphilosophical_ as the clergyman oncefrightened you with _infidel_, is still a recognized member of society, wants taming, and will get it. He wears the priest's cast-off {195}clothes, dyed to escape detection: the better sort of philosophers wouldgladly set him to square the circle. 

The book just named appeared about the same time as this Budget began inthe _Athenæum_. It was commonly attributed, the book to my wife, thepreface to myself. Some time after, our names were actually announced bythe publisher, who ought to know. It will be held to confirm this statementthat I announce our having in our possession some twenty reviews ofdifferent lengths, and of all characters: who ever collects a number ofreviews of a book, except the author?

A great many of these reviews settle the matter _a priori_. If there hadbeen spirits in the matter, they would have done this, and they would nothave done that. Jean Meslier[340] said there could be no God over all, for, _if_ there had been one, He would have established a universal religion. IfJ. M. _knew_ that, J. M. Was right: but if J. M. Did not know that, thenJ. M. Was on the "high priori road, " and may be left to his course. Thesame to all who know what spirits would do and would not do. 

A. B. Very distinctly said that he knew some of the asserted facts, believed others on testimony, but did not pretend to know whether they werecaused by spirits, or had some unknown and unimagined origin. This he saidas clearly as I could have said it myself. But a great many persons cannotunderstand such a frame of mind: their own apparatus is a kind ofspirit-level, and their conclusion on any subject is the little bubble, which is always at one end or the other. Many of the reviewers declare thatA. B. Is a secret believer in the spirit-hypothesis: and one of them wishesthat he had "endorsed his opinion more boldly. " According to this reviewer, any one who writes "I boldly {196} say I am unable to choose, " contradictshimself. In truth, a person who does say it has a good deal of courage, foreach side believes that he secretly favors the other; and both look uponhim as a coward. In spite of all this, A. B. Boldly repeats that he feelsassured of many of the facts of _spiritualism_, and that he cannot pretendto affirm or deny anything about their cause. 

The great bulk of the illogical part of the educated community--whethermajority or minority I know not; perhaps six of one and half-a-dozen of theother--have not power to make a distinction, cannot be made to take adistinction, and of course, never attempt to shake a distinction. With themall such things are evasions, subterfuges, come-offs, loopholes, etc. Theywould hang a man for horse-stealing under a statute against sheep-stealing;and would laugh at you if you quibbled about the distinction between ahorse and a sheep. I divide the illogical--I mean people who have not thatamount of natural use of sound inference which is really not uncommon--intothree classes:--First class, three varieties: the Niddy, the Noddy, and theNoodle. Second class, three varieties: the Niddy-Noddy, the Niddy-Noodle, and the Noddy-Noodle. Third class, undivided: the Niddy-Noddy-Noodle. Noperson has a right to be angry with me for more than one of thesesubdivisions. 

The want of distinction was illustrated to me, when a boy, about 1820, bythe report of a trial which I shall never forget: boys read newspapers morekeenly than men. Every now and then a bench of country magistrates ratherastonishes the town populations, accustomed to rub their brains[341]against one another. Such a story as the following would, {197} in our day, bring down grave remarks from above: but I write of the olden (orEldon[342]) time, when nothing but conviction in a court of record woulddisplace a magistrate. In that day the third-class amalgamator of distinctthings was often on the bench of quarter-sessions. 

An attorney was charged with having been out at night poaching. A clear_alibi_ was established; and perjury had certainly been committed. Thewhole gave reason to suspect that some ill-willers thought the benchdisliked the attorney so much that any conviction was certain on anyevidence. The bench did dislike the attorney: but not to the extent ofthinking he could snare any partridges in the fields while he was asleep inbed, except the dream-partridges which are not always protected by thedream-laws. So the chairman said, "Mr. ----, you are discharged; but youshould consider this one of the most fortunate days of your life. " Theattorney indignantly remonstrated, but the magistrate was right; for hesaid, "Mr. ----, you have frequently been employed to defend poachers: haveyou been careful to impress upon them the enormity of their practices?" Itappeared in a wrangling conversation that the magistrates saw little moraldifference between poaching and being a poacher's professional defenderwithout lecturing him on his wickedness: but they admitted with reluctance, that there was a legal distinction; and the brain of N^3 could no furthergo. This is nearly fifty years ago; and Westernism was not quite extinct. If the present lords of the hills and the valleys want to shine, let thempublish a true history of their own order. I am just old enough to remembersome of the last of the squires and parsons who protested against teachingthe poor to read and write. They now write books for the working classes, give them lectures, and the like. There is now no class, as a class, morehighly educated, broadly educated, and deeply educated, {198} than thosewho were, in old times, best described as partridge-popping squireens. Ihave myself, when a boy, heard Old Booby speaking with pride of Young Boobyas having too high a spirit to be confined to books: and I suspected thathis dislike to teaching the poor arose in fact from a feeling that theywould, if taught a little, pass his heir. 

A. B. Recommended the spirit-theory as an hypothesis on which to groundinquiry; that is, as the means of suggestion for the direction of inquiry. Every person who knows anything of the progress of physics understands whatis meant; but not the reviewers I speak of. Many of them consider A. B. As_adopting_ the spirit-hypothesis. The whole book was written, as both theauthors point out, to suggest inquiry to those who are curious; C. D. Firmly believing, A. B. As above. Neither C. D. Nor A. B. Make any otherpretence. Both dwell upon the absence of authentications and thesuppression of names as utterly preventive of anything like proof. AndA. B. Says that his reader "will give him credit, if not himself a goose, for seeing that the tender of an anonymous cheque would be of equal effect, whether drawn on the Bank of England or on Aldgate Pump. " By this test anumber of the reviewers are found to be geese: for they take the authors asoffering proof, and insist, against the authors, on the very point on whichthe authors had themselves insisted beforehand. 

Leaving aside imperceptions of this kind, I proceed to notice a clericaland medical review. I have lived much in the middle ages, especially sincethe invention of printing; and from thence I have brought away a highrespect for and grateful recollection of--the priest in everything buttheology, and the physician in everything but medicine. The professionalharness was unfavorable to all progress, except on a beaten road; theprofessional blinkers prevented all but the beaten road from being seen:the professional reins were pulled at the slightest attempt to quickenpace, even on the permitted path; and the {199} professional whip washeavily laid on at the slightest attempt to diverge. But when theintelligent man of either class turned his attention out of his ordinarywork, he had, in most cases, the freshness and vigor of a boy at play, andlike the boy, he felt his freedom all the more from the contrast ofschool-restraint. 

In the case of medicine, and physics generally, the learned were, in someessential points, more rational than many of their present impugners. Theypass for having put _a priori_ obstacles in the way of progress: they mightrather be reproved for too much belief in progress obtained by _a priori_means. They would have shouted with laughter at a dunce who--in a review Iread, but without making a note--declared that he would not believe hissenses except when what they showed him was capable of explanation uponsome known principle. I have seen such stuff as this attributed to theschoolmen; but only by those who knew nothing about them. The following, which I wrote some years ago, will give a notion of a distinction worthremembering. It is addressed to the authorities of the College ofPhysicians. 

"The ignominy of the word _empiric_ dates from the ages in which scholasticphilosophy deduced physical consequences _a priori_;--the ages in which, because a lion is strong, rubbing with lion's fat would have been held aninfallible tonic. In those happy days, if a physician had given decoctionof a certain bark, only because in numberless instances that decoction hadbeen found to strengthen the patient, he would have been a miserableempiric. Not that the colleges would have passed over his returns becausethey were empirical: they knew better. They were as skilful in findingcauses for facts, as facts for causes. The president and the elects of thatday would have walked out into the forest with a rope, and would havepulled heartily at the tree which yielded the bark: nor would they everhave left it until they had pulled out a legitimate {200} reason. If thetree had resisted all their efforts, they would have said, 'Ah! no wondernow; the bark of a strong tree makes a strong man. ' But if they had managedto serve the tree as you would like to serve homoeopathy, then it wouldhave been 'We might have guessed it; all the _virtus roborativa_ hassettled in the bark. ' They admitted, as we know from Molière, the _virtusdormitiva_[343] of opium, for no other reason than that opium _facitdormire_. [344] Had the medicine not been previously _known_, they would, strange as it may seem to modern pharmacopoeists, have accorded a _virtusdormitiva_ to the new _facit dormire_. On this point they have beenmisapprehended. They were prone to infer _facit_ from a _virtus_ imagined_a priori_; and they were ready in supplying _facit_ in favor of anorthodox _virtus_. They might have gone so far, for example, underpre-notional impressions, as the alliterative allopath, who, whenmaintenance of truth was busy opposing the progress of science called_vaccination_, declared that some of its patients coughed like cows, andbellowed like bulls; but they never refused to find _virtus_ when _facit_came upon them, no matter whence. They would rather have accepted Tenterdensteeple than have rejected the Goodwin Sands. They would have laughed theirmodern imitators to scorn: but as they are not here, we do it for them. 

"The man of our day--the _a priori_ philosopher--tries the question whetheropium can cause sleep by finding out in the recesses of his own noddlewhether the drug can have a dormitive power: Well! but did not theschoolman do the same? He did; but mark the distinction. The schoolman hadrecourse to first principles, when there was no opium to try it by: our mansettles the point in the same way _with a lump of opium before him_. Theschoolman shifted his principles with his facts: the man of ourdrawing-rooms will fight facts with his principles, just as an old {201}physician would have done in actual practice, with the rod of his _Church_at his back. 

"The story about Galileo--which seems to have been either a joke madeagainst him, or by him--illustrates this. _Nature abhors a vacuum_ was theexplanation of the water rising in a pump: but they found that the waterwould not rise more than 32 feet. They asked for explanation: what does thesatirist make the schoolmen say? That the stoppage is _not_ a fact, becausenature abhors a vacuum? No! but that the principle should be that natureabhors a vacuum as far as 32 feet. And this is what would have been done. 

"There are still among us both priests and physicians who would havebelonged, had they lived three or four centuries ago, to the glorious bandof whom I have spoken, the majority of the intelligent, working well formankind out of the professional pursuit. But we have a great many who havehelped to abase their classes. Go where we may, we find specimens of thelower orders of the ministry of religion and the ministry of health showingthemselves smaller than the small of other pursuits. And how is this?First, because each profession is entered upon a mere working smack of itsknowledge, without any depth of education, general or professional. Notthat this is the whole explanation, nor in itself objectionable: the greatmass of the world must be tended, soul and body, by those who are neitherHookers[345] nor Harveys[346]: let such persons not venture _ultracrepidam_, and they are useful and respectable. But, secondly, there is avast upheaving of thought from the depths of commonplace learning. I am aclergyman! Sir! I am a medical man! Sir! and forthwith the nature of thingsis picked to pieces, and there is a race, with the last the winner, betweenPhilosophy mounted on Folly's donkey, and Folly mounted on Philosophy'sdonkey. How fortunate {202} it is for Law that her battles are fought bypoliticians in the Houses of Parliament. Not that it is better done: butthen _politics_ bears the blame. "

I now come to the medical review. After a quantity of remark which has beenalready disposed of, the writer shows Greek learning, a field in which theold physician would have had a little knowledge. A. B. , for the joke'ssake, had left untranslated, as being too deep, a remarkably easy sentenceof Aristotle, to the effect that what has happened was possible, for ifimpossible it would not have happened. The reviewer, in "simpleastonishment, "--it was simple--at the pretended incapacity--I was told byA. B. That the joke was intended to draw out a reviewer--translates:--Hesays that this sentence is A. B. 's summing up of the evidence ofSpiritualism. Now, being a sort of _alter ego_[347] of A. B. , I do declarethat he is not such a fool as to rest the evidence of Spiritualism--the_spirit explanation_--upon the occurrence of certain facts proving thepossibility of those very facts. In truth, A. B. Refuses to receivespiritualism, while he receives the facts: this is the gist of his wholepreface, which simply admits spiritualism among the qualified candidates, and does not know what others there may be. 

The reviewer speaks of Aristotle as "that clear thinker and concisewriter. " I strongly suspect that his knowledge of Aristotle was limited tothe single sentence which he had translated or got translated. Aristotle isconcise in _phrase_, not in book, and is powerful and profound in thought:but no one who knows that his writing, all we have of him, is the veryopposite of clear, will pretend to decide that he thought clearly. As hiswriting, so probably was his thought; and his books are, if not anythingbut clear, at least anything good but clear. Nobody thinks them clearexcept a person who always clears difficulties: which I have no doubt wasthe reviewer's habit; that is, if he ever took the field {203} at all. Thegentleman who read Euclid, all except the As and Bs and the pictures ofscratches and scrawls, is the type of a numerous class. 

The reviewer finds that the word _amosgepotically_, used by A. B. , isutterly mysterious and incomprehensible. He hopes his translation of thebit of Greek will shield him from imputation of ignorance: and thinks theword may be referred to the "obscure dialect" out of which sprung_aneroid_, _kalos geusis sauce_, and _Anaxyridian trousers_. To lump thefirst two phrases with the third smacks of ignorance in a Greek critic; for[Greek: anaxuridia], _breeches_, would have turned up in the lexicon; and_kalos geusis_, though absurd, is not obscure. And [Greek: amôsgepôs], _somehow or other_, is as easily found as [Greek: anaxuridia]. The word_aneroid_, I admit, has puzzled better scholars than the critic: but neverone who knows the unscholarlike way in which words ending in [Greek: eidês]have been rendered. The _aneroid barometer_ does _not_ use a column of airin the same way as the old instrument. Now [Greek: aeroeidês]--properly_like_ the atmosphere--is by scientific non-scholarship rendered having todo with the atmosphere; and [Greek: anaeroeidês]--say _anaëroid_--denieshaving to do with the atmosphere; a nice thing to say of an instrumentwhich is to measure the weight of the atmosphere. One more absurdity, andwe have _aneroid_, and there you are. The critic ends with a declarationthat nothing in the book shakes his faith in a _Quarterly_ reviewer whosaid that suspension of opinion, until further evidence arrives, isjustifiable: a strange summing up for an article which insists upon utterrejection being unavoidable. [348] The expressed aim of both A. B. And C. D. Was to excite inquiry, and get further evidence: until this is done, neither asks for a verdict. 

Oh where! and oh where! is old Medicine's learning gone! There _was_ somein the days of yore, when Popery {204} was on! And it's oh! for some Greek, just to find a word upon! The reviewer who, lexicon in hand, can neithermake out _anaxyridical_, _amosgepotical_, _kalos geusis_, nor distinguishthem from _aneroid_, cannot be trusted when he says he has translated asentence of Aristotle. He may have done it; but, as he says ofspiritualism, we must suspend our opinion until further evidence shallarrive. 

We now come to the theological review. I have before alluded to the faultsof logic which are Protestant necessities: but I never said that Protestantargument had _nothing but_ paralogism. The writer before me attains thiscompleteness: from beginning to end he is of that confusion and perversionwhich, as applied to interpretation of the New Testament, is so common asto pass unnoticed by sermon-hearers; but which, when applied out of church, is exposed with laughter in all subjects except theology. I shall take oneinstance, putting some words in italics. 

_A. B. _ _Theological Critic. _

My state of mind, which refers ... He proceeds to argue thatthe whole _either_ to unseen he himself is outside itsintelligence, _or something sacred pale because he referswhich man has never had any all these strange phenomena toconception of_, proves me to _unseen spiritualbe out of the pale of the intelligence_. Royal Society. 

The possibility of a _yet unimagined_ cause is insisted on in severalplaces. On this ground it is argued by A. B. That spiritualists are"incautious" for giving in at once to the spirit doctrine. But, it is said, they may be justified by the philosophers, who make the flint _axes_, asthey call them, to be the works of men, because no one can see _what elsethey can be_. This kind of adoption, _condemned_ as a conclusion, is_approved_ as a provisional theory, suggestive of direction of inquiry:experience having shown that {205} inquiry directed by a _wrong_ theory hasled to more good than inquiry without any theory at all. All this A. B. Hasfully set forth, in several pages. On it the reviewer remarks that "withinfinite satisfaction he tries to justify his view of the case by urgingthat there is no other way of accounting for it; after the fashion of thephilosophers of our own day, who conclude that certain flints found in thedrift are the work of men, because the geologist does not see what elsethey can be. " After this twist of meaning, the reviewer proceeds to say, and A. B. Would certainly join him, "There is no need to combat any suchmode of reasoning as this, because it would apply with equal force andjustice to any theory whatever, however fantastic, profane, or silly. " Andso, having shown how the reviewer has hung himself, I leave himfunipendulous. 

One instance more, and I have done. A reviewer, not theological, speakingof the common argument that things which are derided are not _therefore_ tobe rejected, writes as follows:--"It might as well be said that they wholaughed at Jenner[349] and vaccination were, in a certain but veryunsatisfactory way, witnesses to the possible excellence of the system ofSt. John Long. "[350] Of course it _might_: and of course it _is_ said byall people of common sense. In introducing the word "possible, " thereviewer has hit the point: I suspect that this word was introduced duringrevision, to put the sentence into fighting order; hurry preventing itbeing seen that the sentence was thus made to fight on the wrong side. Jenner, who was laughed at, was right; therefore, it is notimpossible--that is, it is _possible_--that a derided system may be right. Mark the three gradations: _in medio tutissimus ibis_. [351]

{206}

_Reviewer. _--If a system be derided, it is no ground of suspense thatderided systems have turned out true: if it were, you would suspend youropinion about St. John Long on account of Jenner. --_Ans. _ You ought to doso, as to _possibility_; and _before examination_; not with the notion thatJ. Proves St. J. _probable_; only _possible_. 

_Common Sense. _--The past emergence of truths out of derided systems provesthat there is a practical certainty of like occurrence to come. But, inasmuch as a hundred speculative fooleries are started for one truth, themind is permitted to approach the examination of any one given novelty witha bias against it of a hundred to one: and this permission is given becauseso it will be, leave or no leave. Every one has licence not to jump overthe moon. 

_Paradoxer. _--Great men have been derided, and I am derided: which provesthat my system ought to be adopted. This is a summary of all the degrees inwhich paradoxers contend for the former derision of truths now established, giving their systems _probability_. I annex a paragraph which D [e &c. ]inserted in the _Athenæum_ of October 23, 1847. 

"_Discoverers and Discoveries. _

"Aristotle once sent his servant to the cellar to fetch wine:--and thefellow brought him back small beer. The Stagirite (who knew the difference)called him a blockhead. 'Sir, ' said the man, 'all I can say is, that Ifound it in the cellar. ' The philosopher muttered to himself that anaffirmative conclusion could not be proved in the second figure, --and Mrs. Aristotle, who was by, was not less effective in her remark, that smallbeer was not wine because it was in the same cellar. Both were rightenough: and our philosophers might take a lesson from either--for theyinsinuate an affirmative conclusion in the second figure. Great discoverershave been little valued by established {207} schools, --and they are littlevalued. The results of true science are strange at first, --and so aretheir's. Many great men have opposed existing notions, --and so do they. Allgreat men were obscure at first, --and they are obscure. Thinking mendoubt, --and they doubt. Their small beer, I grant, has come out of the samecellar as the wine; but this is not enough. If they had let it stand awhilein the old wine-casks, it might have imbibed a little of the flavor. "

There are better reviews than I have noticed; which, though entirelydissenting, are unassailable on their own principles. What I have givenrepresents five-sixths of the whole. But it must be confessed that thefraction of fairness and moderation and suspended opinion which thedoctrine of _Spirit Manifestations_ has met with--even in the lowerreviews--is strikingly large compared to what would have been the casefifty years ago. It is to be hoped that our popular and periodicalliteratures are giving us one thinker created for twenty geesedouble-feathered: if this hope be realized, we shall do! Seeing all that Isee, I am not prepared to go the length of a friend of mine who, afterreading a good specimen of the lower reviewing, exclaimed--Oh! if all thefools in the world could be rolled up into one fool, what a reviewer hewould make!

 Calendrier Universel et Perpétuel; par le Commandeur P. J. Arson. [352] Publié par ses Enfans (Oeuvre posthume). Nice, 1863, 4to. 

I shall not give any account of this curious calendar, with all its changesand symbols. But there is one proposal, which, could we alter the generalnotions of time--a thing of very dubious possibility--would be convenient. The week is made to wax and wane, culminating on the Sunday, {208} whichcomes in the middle. Thursday, Friday, Saturday, are ascending or waxingdays; Monday, Tuesday, Wednesday, are descending or waning days. Our sixdays, lumped together after the great distinguishing day, Sunday, are toomany to be distinctly thought of together: a division of three precedingand three following the day of most note would be much more easily used. But all this comes too late. It may be, nevertheless, that some individualsmay be able to adjust their affairs with advantage by referring Thursday, Friday, Saturday, to the following Sunday, and Monday, Tuesday, Wednesday, to the preceding Sunday. But M. Arson's proposal to alter the names of thedays is no more necessary than it is practicable. 

CYCLOMETRY. 

I am not to enter anything I do not possess. The reader therefore will notlearn from me the feats of many a man-at-arms in these subjects. He must becontent, unless he will bestir himself for himself, not to know how Mr. Patrick Cody trisects the angle at Mullinavat, or Professor Recalcatisquares the circle at Milan. But this last is to be done by subscription, at five francs a head: a banker is named who guarantees restitution if thesolution be not perfectly rigorous; the banker himself, I suppose, is thejudge. I have heard of a man of business who settled the circle in thisway: if it can be reduced to a debtor and creditor account, it cancertainly be done; if not, it is not worth doing. Montucla will give theaccounts of the lawsuits which wagers on the problem have produced inFrance. 

Neither will I enter at length upon the success of the new squarer whoadvertises (Nov. 1863) in a country paper that, having read that thecircular ratio was undetermined, "I thought it very strange that so manygreat scholars in all ages should have failed in finding the true ratio, and have been determined to try myself.... I am about to secure the {209}benefit of the discovery, so until then the public cannot know my new andtrue ratio. " I have been informed that this trial makes the diameter to thecircumference as 64 to 201, giving [pi] = 3. 140625 exactly. The result wasobtained by the discoverer in three weeks after he first heard of theexistence of the difficulty. This quadrator has since published a littleslip, and entered it at Stationers' Hall. He says he has done it by actualmeasurement; and I hear from a private source that he uses a disk of 12inches diameter, which he rolls upon a straight rail. Mr. James Smith didthe same at one time; as did also his partisan at Bordeaux. We have, then, both 3. 125 and 3. 140625, by actual measurement. The second result is morethan the first by about one part in 200. The second rolling is a verycreditable one; it is about as much below the mark as Archimedes was aboveit. Its performer is a joiner, who evidently knows well what he is aboutwhen he measures; he is not wrong by 1 in 3, 000. 

The reader will smile at the quiet self-sufficiency with which "I have beendetermined to try myself" follows the information that "so many greatscholars in all ages" have failed. It is an admirable spirit, whenaccompanied by common sense and uncommon self-knowledge. When I was anundergraduate there was a little attendant in the library who gave me thefollowing, --"As to cleaning this library, Sir, if I have spoken to theMaster once about it, I have spoken fifty times: but it is of no use; hewill not employ _littery_ men; and so I am obliged to look after itmyself. "

I do not think I have mentioned the bright form of quadrature in which asquare is made equal to a circle by making each side equal to a quarter ofthe circumference. The last squarer of this kind whom I have seen figuresin the last number of the _Athenæum_ for 1855: he says the thing is nolonger a _problem_, but an _axiom_. He does not know that the area of thecircle is greater than that of any other figure of the same circuit. Thisany one might see without {210} mathematics. How is it possible that thefigure of greatest area should have any one length in its circuit unlike inform to any other part of the same length?

The feeling which tempts persons to this problem is that which, in romance, made it impossible for a knight to pass a castle which belonged to a giantor an enchanter. I once gave a lecture on the subject: a gentleman who wasintroduced to it by what I said remarked, loud enough to be heard by allaround, "Only prove to me that it is impossible, and I will set about itthis very evening. "

This rinderpest of geometry cannot be cured, when once it has seated itselfin the system: all that can be done is to apply what the learned callprophylactics to those who are yet sound. When once the virus gets into thebrain, the victim goes round the flame like a moth; first one way and thenthe other, beginning where he ended, and ending where he begun: thusverifying the old line

 "In girum imus nocte, ecce! et consumimur igni. "[353]

Every mathematician knows that scores of methods, differing altogether fromeach other in process, all end in this mysterious 3. 14159... , which insistson calling itself the circumference to a unit of diameter. A reader who iscompetent to follow processes of arithmetic may be easily satisfied thatsuch methods do actually exist. I will give a sketch, carried out to a fewfigures, of three: the first two I never met with in my reading; the thirdis the old method of Vieta. [354] [I find that both the first and secondmethods are contained in a theorem of Euler. ]

What Mr. James Smith says of these methods is worth noting. He says I havegiven three "_fancy_ proofs" of the value of [pi]: he evidently takes me tobe offering demonstration. He proceeds thus:--

"His first proof is traceable to the diameter of a circle {211} of radius1. His second, to the side of any inscribed equilateral triangle to acircle of radius 1. His third, to a radius of a circle of diameter 1. Now, it may be frankly admitted that we can arrive at the same result by manyother modes of arithmetical calculation, all of which may be shown to havesome sort of relation to a circle; but, after all, these results are mereexhibitions of the properties of numbers, and have no more to do with theratio of diameter to circumference in a circle than the price of sugar withthe mean height of spring tides. (_Corr. _ Oct. 21, 1865). "

I quote this because it is one of the few cases--other than absoluteassumption of the conclusion--in which Mr. Smith's conclusions would betrue if his premise were true. Had I given what follows as _proof_, itwould have been properly remarked, that I had only exhibited properties ofnumbers. But I took care to tell my reader that I was only going to showhim _methods_ which end in 3. 14159.... The proofs that these methodsestablish the value of [pi] are for those who will read and can understand. 

 200000000 31415 3799 66666667 2817 26666667 1363 11428571 661 5079365 321 2308802 156 1065601 76 497281 37 234014 18 110849 9 52785 5 25245 2 12118 1 5834 -------- -------- ------- 314153799 31415 9265

{212}

1. Take any diameter, double it, take 1-3d of that double, 2-5ths of thelast, 3-7ths of the last, 4-9ths of the last, 5-11ths of the last, and soon. The sum of all is the circumference of that diameter. The preceding isthe process when the diameter is a hundred millions: the errors arisingfrom rejection of fractions being lessened by proceeding on a thousandmillions, and striking off one figure. Here 200 etc. Is double of thediameter; 666 etc. Is 1-3rd of 200 etc. ; 266 etc. Is 2-5ths of 666 etc. ;114 etc. Is 3-7ths of 266 etc. ; 507 etc. Is 4-9ths of 114 etc. ; and so on. 

2. To the square root of 3 add its half. Take _half_ the third part ofthis; half 2-5ths of the last; half 3-7ths of the last; and so on. The sumis the circumference to a unit of diameter. 

 Square root of 3.... 1. 73205081 . 86602540 ------------ 2. 59807621 . 43301270 . 08660254 1855768 412393 93726 21629 5047 1188 281 67 16 4 1 ------------ 3. 14159265

3. Take the square root of ½; the square root of half of one more thanthis; the square root of half of one more {213} than the last; and so on, until we come as near to unity as the number of figures chosen will permit. Multiply all the results together, and divide 2 by the product: thequotient is an approximation to the circumference when the diameter isunity. Taking aim at four figures, that is, working to five figures tosecure accuracy in the fourth, we have . 70712 for the square root of ½;. 92390 for the square root of half one more than . 70712; and so on, through. 98080, . 99520, . 99880, . 99970, . 99992, . 99998. The product of the eightresults is . 63667; divide 2 by this, and the quotient is 3. 1413... , ofwhich four figures are correct. Had the product been . 636363... Instead of. 63667... , the famous result of Archimedes, 22-7ths, would have beenaccurately true. It is singular that no cyclometer maintains thatArchimedes hit it exactly. 

A literary journal could hardly admit as much as the preceding, if it stoodalone. But in my present undertaking it passes as the halfpennyworth ofbread to many gallons of sack. Many more methods might be given, all endingin the same result, let that result mean what it may. 

Now since dozens of methods, to which dozens more might be added atpleasure, concur in giving one and the same result; and since these methodsare declared by all who have shown knowledge of mathematics to be_demonstrated_: it is not asking too much of a person who has just a littleknowledge of the first elements that he should learn more, and put his handupon the error, before he intrudes his assertion of the existence of errorupon those who have given more time and attention to it than himself, andwho are in possession, over and above many demonstrations, of manyconsequences verifying each other, of which he can know nothing. This isall that is required. Let any one square the circle, and persuade hisfriends, if he and they please: let him print, and let all read who choose. But let him abstain from intruding himself upon those who have beensatisfied by existing demonstration, until he is prepared {214} to lay hisfinger on the point in which existing demonstration is wrong. Let him alsosay what this mysterious 3. 14159... Really is, which comes in at every doorand window, and down every chimney, calling itself the circumference to aunit of diameter. This most impudent and successful impostor holds falsetitle-deeds in his hands, and invites examination: surely those who canfind out the rightful owner are equally able to detect the forgery. All thequadrators are agreed that, be the right what it may, 3. 14159... Is wrong. It would be well if they would put their heads together, and say what thiswrong result really means. The mathematicians of all ages have tried allmanner of processes, with one object in view, and by methods which areadmitted to yield demonstration in countless cases. They have all arrivedat one result. A large number of opponents unite in declaring this resultwrong, and all agree in two points: first, in differing among themselves;secondly, in declining to point out what that curious result really iswhich the mathematical methods all agree in giving. 

Most of the quadrators are not aware that it has been fully demonstratedthat no two numbers whatsoever can represent the ratio of the diameter tothe circumference with perfect accuracy. When therefore we are told thateither 8 to 25 or 64 to 201 is the true ratio, we know that it is no suchthing, without the necessity of examination. The point that is left open, as not fully demonstrated to be impossible, is the _geometrical_quadrature, the determination of the circumference by the straight line andcircle, used as in Euclid. The general run of circle-squarers, hearing thatthe quadrature is not pronounced to be _demonstratively_ impossible, imagine that the _arithmetical_ quadrature is open to their ingenuity. Before attempting the arithmetical problem, they ought to acquire knowledgeenough to read Lambert's[355] demonstration (last given in Brewster's[356]translation {215} of Legendre's[357] Geometry) and, if they can, to refuteit. [It will be given in an Appendix. ] Probably some have begun this way, and have caught a Tartar who has refused to let them go: I have never heardof any one who, in producing his own demonstration, has laid his finger onthe faulty part of Lambert's investigation. This is the answer to those whothink that the mathematicians treat the arithmetical squarers too lightly, and that as some person may succeed at last, all attempts should beexamined. Those who have so thought, not knowing that there isdemonstration on the point, will probably admit that a person whocontradicts a theorem of which the demonstration has been acknowledged fora century by all who have alluded to it as read by themselves, mayreasonably be required to point out the error before he demands attentionto his own result. 

_Apopempsis of the Tutelaries. _--Again and again I am told that I spend toomuch time and trouble upon my two tutelaries: but when I come to mysumming-up I shall make it appear that I have a purpose. Some say I am toohard upon them: but this is quite a mistake. Both of them beat littleOliver himself in the art and science of asking for more; but withoutOliver's excuse, for I had given good allowance. Both began with me, not Iwith them: and both knew what they had to expect when they applied for asecond helping. 

On July 31, the Monday after the publication of my remarks on my 666correspondent, I found _three_ notes in separate envelopes, addressed to meat "7A, University College. " When I saw the three new digits I was takenrhythmopoetic, as follows--

 Here's the Doctor again with his figs, and by Heavens! He was always at sixes, and now he's at sevens. 

To understand this fully the reader must know that the greater part ofApocalyptic interpretation has long been condensed, in my mind, into theTurkish street-cry--In the {216} name of the Prophet! figs! I make a fewextracts. The reader will observe that Dr. Thorn grumbles at his _private_letters being _publicly_ ridiculed. A man was summoned for a glutolacticassault; he complained of the publication of his proceeding: I kicked etc. _in confidence_, he said. 

"After reading your last, which tries in every way to hold me up to publicridicule for daring to write you privately ['that you would be d----d, 'omitted by accident] one would say, Why have anything to do with such atesty person? [Wrong word; no testy person can manage cool and consecutiveridicule. Quære, what is this word? Is it anything but a corruption of theobsolete word _tetchy_ of the same meaning? Some think _touchy_ is ourmodern form of _tetchy_, which I greatly doubt]. My answer is, the poor manis lamentably ignorant; he is not only so, but 'out of the way' [quitetrue; my readers know me by this time for an out-of-the-way person. Whatother could tackle my squad of paradoxers? What other would undertake thejob?] Can he be brought back and form one of those who in Ezekiel 37 ch. Have the Spirit breathed into them and live.... Have I any other feelingtowards you except that of peace and goodwill? [Not to your distinctknowledge; but in all those who send people to 'the other place' forcontempt of their interpretations, there is a lurking wish which is fatherto the thought; 'you _will_ be d----d' and 'you _be_ d--d' are Siamesetwins]. Of course your sneer at 666 brought plain words; but when menmeddle with what they do not understand (not having the double _Vahu_) theymust be dealt with faithfully by those who do.... [They must; whichjustifies the Budget of Paradoxes: but no occasion to send them anywhere;no preachee and floggee too, as the negro said]. Many will find the textProv. I. 26 fully realized. [All this contains distinct assumption of aright 'of course' to declare accursed those who do not respect the writer'svagary].... If I could but get the [Hebrew: A], the Ox-head, which in OldHebrew was just the Latin Digamma, F, out {217} of your name, and couldthen Thau you with the Thau of Ezekiel ix, 4, the [chi], then you wouldbear the number of a man! But this is too hard for me, although not so forthe Lord! Jer. Xxxii. 17.... And now a word: is ridicule the right thing inso solemn a matter as the discussion of Holy Writ? [Is food for ridiculethe right thing? Did I discuss Holy Writ? I did not: I concussed profanescribble. Even the Doctor did not _discuss_; he only enunciated anddenunciated out of the mass of inferences which a mystical head has foundpremises for in the Bible]. "

 M 40 O 70 R 100 G 6 N 50 ---- 266 [Hebrew: t]=[chi] 400

[That ill opinions are near relations of ill wishes, will be detected bythose who are on the look out. The following was taken down in a ScotchChurch by Mr. Cobden, [358] who handed it to a Roman friend of mine, for hisdelectation (in 1855): "Lord, we thank thee that thou hast brought the Popeinto trouble; and we pray that thou wouldst be mercifully pleased toincrease the same. "]

Here is a martyr who quarrels with his crown; a missionary who reviles hispersecutor: send him to New Zealand, and he would disagree with the Maoriswho ate him. Man of unilateral reciprocity! have you, who write to astranger with hints that that stranger and his wife are children ofperdition, the bad taste to complain of a facer in return? As JamesSmith[359]--the Attorney-wit, not the Dock-cyclometer--said, or nearlysaid, "A pretty thing, forsooth! Is he to burn, all scalding hot, Me and my wife, and am I not To job him out a tooth?"

{218}

Those who think parody vulgar will be pleased to substitute for the above aquotation from Butler[360]:--

 "There's nothing so absurd or vain Or barbarous or inhumane, But if it lay the least pretence To piety and godliness, Or tender-hearted conscience, And zeal for gospel truths profess, -- Does sacred instantly commence, And all that dare but question it are straight Pronounced th' uncircumcised and reprobate, As malefactors that escape and fly Into a sanctuary for defence, Must not be brought to justice thence, Although their crimes be ne'er so great and high. And he that dares presume to do't Is sentenced and delivered up To Satan that engaged him to't. "

THE NUMBER OF THE BEAST. 

Of all the drolleries of controversy none is more amusing than the mannerin which those who provoke a combat expect to lay down the laws ofretaliation. You must not strike this way! you must not parry that way! Ifyou don't take care, we shall never meddle with you again! We were not_prepared_ for such as this! Why did we have anything to do with such atesty person? M. Jourdain must needs show Nicole, his servant-maid, howgood a thing it was to be sure of fighting without being killed, by careand tierce. [361] "Et cela n'est il pas beau d'être assuré de son fait quandon se bat contre quelqu'un? Là, pousse moi un peu, pour voir. NICOLE. Ehbien! quoi? M. JOURDAIN. Tout beau. Hola! {219} Ho! doucement. Diantre soitla coquine! NICOLE. Vous me dites de pousser. M. JOURDAIN. Oui; mais tu mepousses en tierce, avant que de pousser en quarte, et tu n'as pas lapatience que je pare. "

His colleague, my secular tutelary, who also made an anachronistic onset, with his repartees and his retorts, before there was anything to fire at, takes what I give by way of subsequent provocation with a good humor whichwould make a convert of me if he could afford . 01659265 ... Of a grain oflogic. He instantly sent me his photograph for the asking, and anotherletter in proof. The Thor-hammerer does nothing but grumble, except when hetells a good story, which he says he had from Dr. Abernethy. [362] A Mr. James Dunlop was popping at the Papists with a 666-rifled gun, when Dr. Chalmers[363] quietly said, "Why, Dunlop, you bear it yourself, " and handedhim a paper on which the numerals in

 I A C O B V S D V N L O P V S 1 100 5 500 5 50 5

were added up. This is almost as good as the _Filii Dei Vicarius_, thenumerical letters of which also make 666. No more of these crazy--I firstwrote _puerile_, but why should young cricketers be libelled?--attempts toextract religious use from numerical vagaries, and to make God over all aproposer of _salvation conundrums_: and no more of the trumpery hints aboutfuture destiny which is too great a compliment to call blasphemous. If theDoctor will cipher upon the letter in [Greek: en hôi metrôi metreitemetrêthêsetai humin][364] with _double Vahu_ cubic measure, he will perhapslearn to leave off trying to frighten me into gathering grapes from thorns. 

Mystical hermeneutics may be put to good use by out-of-the-way people. Theymay be made to call the attention {220} of the many to a distinction wellknown among the learned. The books of the New Testament have been for 1, 500years divided into two classes: the _acknowledged_ ([Greek:homologoumena]), which it has always been paradox not to receive; and the_controverted_ ([Greek: antilegomena]), about which there has always beenthat difference of opinion which no scholar overlooks, however he maydecide for himself after balance of evidence. Eusebius, [365] who first (l. 3, c. 25) recorded the distinction--which was much insisted on by the earlyProtestants--states the books which are questioned as doubtful, but whichyet are approved and acknowledged by _many_--or _the many_, it is not easyto say which he means--to be the Epistles of James and Jude, the second ofPeter and the second and third of John. In other places he speaksdoubtingly of the Epistle to the Hebrews. The Apocalypse he does not evenadmit into this class, for he proceeds as follows--I use the second editionof the English folio translation (1709), to avert suspicion of bias frommyself:--

"Among the _spurious_ [[Greek: nothoi]] let there be ranked both the workentitled the _Acts of Paul_, and the book called _Pastor_, and the_Revelation of Peter_: and moreover, that which is called the _Epistle ofBarnabas_, and that named the _Doctrines of the Apostles_: and moreover, asI said, the _Revelation of John_ (if you think good), which some, as I havesaid, do reject, but others allow of, and admit among those books which arereceived as unquestionable and undoubted. "

Eusebius, though he will not admit the Apocalypse even into the_controverted_ list, but gives permission to call it _spurious_, yetqualifies his permission in a manner which almost annihilates thedistinctive force of [Greek: nothos], and gives the book a claim to rank(if you think good, again) in the controverted list. And this is theimpression received by {221} the mind of Lardner, who gives Eusebius fullyand fairly, but when he sums up, considers his author as admitting theApocalypse into the second list. A stick may easily be found to beat thefather of ecclesiastical history. There are whole faggots in writers asopposite as Baronius and Gibbon, who are perhaps his two most celebratedsons. But we can hardly imagine him totally misrepresenting the state ofopinion of those for whom and among whom he wrote. The usual plan, that ofmaking an author take the views of his readers, is more easy in his casethan in that of any other writer: for, as the riddle says, he isYou-see-by-us; and to this reading of his name he has often been subjected. Dr. Nathaniel Lardner, [366] who, though heterodox in doctrine, tries hardto be orthodox as to the Canon, is "sometimes apt to think" that the listshould be collected and divided as in Eusebius. He would have no one of thecontroverted books to be allowed, by itself, to establish any doctrine. Even without going so far, a due use of early opinion and long continueddiscussion would perhaps prevent rational people from being induced bythose who have the _double Vahu_ to place the Apocalypse _above_ theGospels, which all the Bivahuites do in effect, and some are said to havedone in express words. But my especial purpose is to point out that an easyway of getting rid of 665 out of 666 of the mystics is to require them toestablish the Apocalypse before they begin. See if they even know so muchas that there is a crowd of testimonies for and against, running throughthe first four centuries, which makes this book the most difficult of thewhole Canon. Try this method, and you will escape beautiful, as the Frenchsay. Dean Alford, [367] in Vol. IV, p. 8, of his New Testament, gives anelaborate handling of this question. He concludes by saying that he cannot{222} venture to refuse his consent to the tradition that the Apostle isthe author. This modified adherence, or non-nonadherence, pretty wellrepresents the feeling of orthodox Protestants, when learning and commonsense come together. 

I have often, in former days, had the attempt made to place the Apocalypseon my neck as containing prophecies yet unfulfilled. The preceding methodprevents success; and so does the following. It may almost be taken forgranted that theological system-fighters do not read the New Testament:they hunt it for detached texts; they listen to it in church in that stateof quiescent nonentity which is called reverent attention: but they neverread it. When it is brought forward, you must pretend to find it necessaryto turn to the book itself: you must read "The revelation ... To show untohis servants _things which must shortly come to pass_.... Blessed is hethat readeth ... _for the time is at hand_. " You must then ask your mysticwhether things deferred for 1800 years were shortly to come to pass, etc. ?You must tell him that the Greek [Greek: en tachei], rendered "shortly, " isas strong a phrase as the language has to signify _soon_. The interpreterwill probably look as if he had never read this opening: the chances arethat he takes up the book to see whether you have been committing a fraud. He will then give you some exquisite evasion: I have heard it pleaded thatthe above was a _mere preamble_. This word _mere_ is all-sufficient: itturns anything into nothing. Perhaps he will say that the argument is thatof the Papists: if so, tell him that there is no Christian sect but bearstrue witness against some one or more absurdities in other sects. 

An anonyme suggests that [Greek: en tachei] may not be "soon, " it may be"quickly, without reference to time when:" he continues thus, "May not timebe 'at hand' when it is ready to come, no matter how long delayed?" I nowunderstand what *** and *** meant when they borrowed my books and promisedto return them quickly, it was "without {223} reference to time when. " Asto time at _hand_--provided you make a long _arm_--I admire the quirk, butcannot receive it: the word is [Greek: engus], which is a word of_closeness_ in time, in place, in reckoning, in kindred, etc. 

Another gentleman is not surprised that Apocalyptic reading leads to adoubt of the "canonicity" of the book: it ought not to rest on churchtestimony, but on visible miracle. He offers me, or any reader of the_Athenæum_, the "sight of a miracle to that effect, and within forty-eighthours' journey (fare paid). " I seldom travel, and my first thought waswhether my carpet-bag would be found without a regular hunt: but, onreading further, I found that it was only a concordance that would bewanted. Forty hours' collection and numerical calculation of Greek nounswould make it--should I happen to agree with the writer--many hundredmillions to one that Revelation xiii is superhuman. There is but one verse(the fifth) which the writer does not see verified. I looked at this verse, and was much startled. The Budget began in October 1863: should it lastuntil March 1867--it is now August 1866--it is clear that I am the firstBeast, and my paradoxers are the saints whom I persecute. 

[The Budget _did_ terminate in March 1867: I hope the gentleman will besatisfied with the resulting interpretation. ]

The same opponent is surprised that I should suppose a thing which "comesto pass" must be completed, and cannot contain what is to happen 1800 yearsafter. All who have any knowledge of English idiom know that a thing_comes_ to pass when it happens, and _came_ to pass afterwards. But as theoriginal is Greek, we must look at the Greek: it is [Greek: dei genesthai]for "must come to pass, " and we know that [Greek: egeneto] is what isusually translated "came to pass. " No word of more finished completionexists in Greek. 

And now for a last round of biter-bit with the Thor-hammerer, of whom, asin the other case, I shall take no {224} more notice until he can contriveto surpass himself, which I doubt his being able to do. He informs me thatby changing A into [Hebrew: t] in my name he can make a 666 of _me_;adding, "This is too hard for me, although not so for the Lord!" Sheernonsense! He could just as easily have directed to "Prof. De Morg[Hebrew:t]n" as have assigned me apartment 7A in University College. It would havebeen seen for whom it was intended: and if not, it would still have reachedme, for my colleagues have for many a year handed all out-of-the-way thingsover to me. There is no 7A: but 7 is the Museum of Materia Medica. I tookthe only hint which the address gave: I inquired for hellebore, but theytold me it was not now recognized, that the old notion of its value wasquite obsolete, and that they had nothing which was considered a specificin senary or septenary cases. The great platitude is the reference of sucha difficulty as writing [Hebrew: t] for A to the Almighty! Not childish, but fatuous: real childishness is delightful. I knew an infant to whom, before he could speak plain, his parents had attempted to give notions ofthe Divine attributes: a wise plan, many think. His father had dandled himup-side-down, ending with, There now! Papa could not dance on his head! Themannikin made a solemn face, and said, _But Dod tood_! I think the Doctorhas rather mistaken the way of becoming as a little child, intended inMatt. Xviii. 3: let us hope the will may be taken for the deed. 

Two poets have given images of transition from infancy to manhood:Dryden, --for the Hind is Dryden himself on all fours! and Wordsworth, inhis own character of broad-nailed, featherless biped:

 "The priest continues what the nurse began, And thus the child imposes on the man. " "The child's the father of the man, And I could wish my days to be Bound each to each by natural piety. "

{225}

In Wordsworth's aspiration it is meant that sense and piety should growtogether: in Dryden's description a combination of Mysticism And Bigotry(can this be the _double Vahu_?), personified as "the priest, "--who alwayscatches it on this score, though the same spirit is found in allassociations, --succeeds the boguey-teaching of the nurse. Never was thecontrast of smile and scowl, of light and darkness, better seen than in thetwo pictures. But an acrostic distinction may be drawn. When mysticismpredominates over bigotry, we have the grotesque picturesque, and thenatural order of words gives us _Mab_, an appropriate suggestion. But whenbigotry has the upper hand, we see _Bam_, which is just as appropriate; forbigotry nearly always deals with facts and logic so as to require theapplication of at least one of the minor words by which dishonesty issignified. I think that M is the Doctor's initial, and that Queen Mabtickles him in his sleep with the sharp end of a 6. 

(_Monday, August 21. _) Three weeks having elapsed without notice from me ofthe Doctor, I receive a reminder of his existence, in which I find that asI am the Daniel who judges the Magi of Babylon, it is to be pointed outthat Daniel "bore a certain number, that of a man (beloved), Daniel, ch. 10. V. 11, and which you certainly do not. " Then, "by Greek power, "Belteshazzar is made = 666. Here is another awkward imitation of the way ofa baby child. When you have sported with the tiny creature until it runsaway offended, by the time you have got into conversation again you willfind the game is to be renewed: a little head peeps out from a hiding-placewith "I don't love you. " The proper rejoinder is, "Very well! then I'llhave pussy. " But in the case before me there is a rule of three sums to do;as baby : Pussy Dr. :: 666 : the answer required. I will work it out, if Ican. 

The squaring of the circle and the discovery of the Beast are the twogoals--and goals also--of many unbalanced intellects, and of a fewinstances of the better kind. {226} I might have said more of 666, but I amnot deep in its bibliography. A work has come into my hands which containsa large number of noted cases: to some of my readers it will be a treat tosee the collection; and the sight will perhaps be of some use to those whohave read controversy on the few celebrated cases which are of generalnotoriety. It is written by a learned decipherer, a man who really knew thehistory of the subject, the Rev. David Thom, [368] of Bold Street Chapel, Liverpool, who died, I am told, a few years ago. 

Anybody who reads his book will be inclined to parody a criticism which wasonce made on Paley's[369] Evidences--"Well! if there be anything inChristianity, this man is no fool. " And, if he should chance to rememberit, he will be strongly reminded of a sentence in my opening chapter, --"Themanner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has notmade a sufficient knowledge of what has been done by others, _especially asto the mode of doing it_, a preliminary to inventing knowledge forhimself. " And this is reinforced by the fact that Mr. Thom, though ascholar, was not conspicuous for learning, except in this his greatpursuit. He was a paradoxer on other points. He reconciled Calvinism andeternal reprobation with Universalism and final salvation; showing thesetwo doctrines to be all one. 

This gentleman must not be confounded with the Rev. John Hamilton Thom[370](no relation), at or near the same {227} time and until recently, ofRenshaw Street Chapel, Liverpool who was one of the minority in theLiverpool controversy when, nearly thirty years ago, _three_ hereticalUnitarian schooners exchanged shotted sermons with _thirteen_ Orthodoxships of the line, and put up their challengers' dander--an Americancorruption of _d--d anger_--to such an extent, by quiet and respectfulargument, that those opponents actually addressed a printed intercession tothe Almighty for the Unitarian triad, as for "Jews, Turks, Infidels, andHeretics. " So much for the distinction, which both gentlemen would thank mefor making very clear: I take it quite for granted that a guesser at 666would feel horrified at being taken for a Unitarian, and that a Unitarianwould feel queerified at being taken for a guesser at 666. Mr. David Thom'sbook is _The Number and Names of the Apocalyptic Beasts_, Part I, 1848, 8vo. : I think the second part was never published. I give the Greek andLatin solutions, omitting the Hebrew: as usual, all the Greek letters arenumeral, but only M D C L X V I of the Latin. I do not give either thedecipherers or their reasons: I have not room for this; nor would I, if Icould, bias my reader for one rather than another. 

D. F. Julianus Cæsar Atheus (or Aug. [371]); Diocles Augustus; Ludovicus;Silvester Secundus; Linus Secundus; {228} Vicarius Filii Dei; Doctor et RexLatinus; Paulo V. Vice-Deo; Vicarius Generalis Dei in Terris; IpseCatholicæ Ecclesiæ Visibile Caput; Dux Cleri; Una, Vera, Catholica, Infallibilis Ecclesia; Auctoritas politica ecclesiasticaque Papalis (Latinawill also do); Lutherus Ductor Gregis; Calvinus tristis fidei interpres;Dic Lux ; Ludvvic; Will. Laud; [Greek: Lateinos];[372] [Greek: hê latinêbasileia]; [Greek: ekklêsia italika]; [Greek: euanthas]; [Greek: teitan];[Greek: arnoume]; [Greek: lampetis]; [Greek: ho nikêtês]; [Greek: kakoshodêgos]; [Greek: alêthês blaberos]; [Greek: palai baskanos]; [Greek: amnosadikos]; [Greek: antemos]; [Greek: gensêrikos]; [Greek: euinas]; [Greek:Benediktos]; [Greek: Bonibazios g. Papa x. ê. E. E. A. ], meaning BonifaceIII. Pope 68th, bishop of bishops the first! [Greek: oulpios]; [Greek: dioseimi hê hêras]; [Greek: hê missa hê papikê]; [Greek: loutherana]; [Greek:saxoneios]; [Greek: Bezza antitheos] (Beza); [Greek: hê alazoneia biou];[Greek: Maometis]; [Greek: Maometês b. ]; [Greek: theos eimi epi gaiês];[Greek: iapetos]; [Greek: papeiskos]; [Greek: dioklasianos]; [Greek:cheina]; [Greek: braski]; [Greek: Ion Paune]; [Greek: koupoks]; (cowpox, [Greek: s] being the _vau_; certainly the {229} vaccinated have the mark ofthe Beast); [Greek: Bonnepartê]; [Greek: N. Bonêparte]; [Greek: euporia];[Greek: paradosis]; [Greek: to megathêrion]. 

All sects fasten this number on their opponents. It is found in _MartinLauter_, affirmed to be the true way of writing the name, by carryingnumbers through the Roman Alphabet. Some Jews, according to Mr. Thorn, found it in [Hebrew: JSHW NTSRJ] _Jesus of Nazareth_. I find on inquirythat this satire was actually put forth by some medieval rabbis, but thatit is not idiomatic: it represents quite fairly "Jesus Nazarene, " but theHebrew wants an article quite as much as the English wants "the. "

Mr. David Thom's own solution hits hard at all sides: he finds a 666 forboth beasts; [Greek: hê phrên] (the mind) for the first, and [Greek:ekklêsiai sarkikai] (fleshly churches) for the second. A solution whichembodies all mental philosophy in one beast and all dogmatic theology inthe other, is very tempting: for in these are the two great supports ofAntichrist. It will not, however, mislead me, who have known the trueexplanation a long time. The three sixes indicate that any two of the threesubdivisions, Roman, Greek, and Protestant, are, in corruption ofChristianity, six of one and half a dozen of the other: the distinctions ofunits, tens, hundreds, are nothing but the old way (1 Samuel xviii. 7, andConcordance at _ten_, _hundred_, _thousand_) of symbolizing differences ofnumber in the subdivisions. 

It may be good to know that, even in speculations on 666, there aredifferent degrees of unreason. All the diviners, when they get a colleagueor an opponent, at once proceed to reckon him up: but some do it in playand some in earnest. Mr. David Thom found a young gentleman of the name St. Claire busy at the Beast number: he forthwith added the letters in [Greek:st klaire] and found 666: this was good fun. But my spiritual tutelary, when he found that he could not make a beast of me, except by changing[Hebrew: A] into [Hebrew: T], solemnly referred the difficulty to theAlmighty: this was poor earnest. {230}

I am glad I did not notice, in time to insert it in the _Athenæum_, a veryremarkable paradoxer brought forward by Mr. Thom, his friend Mr. Wapshare[373]: it is a little too strong for the general public. In the_Athenæum_ they would have seen and read it: but this book will be avoidedby the weaker brethren. It is as follows:

"God, the Elohim, was six days in creating all things, and having made MANhe entered into his rest. He is no more seen as a Creator, as Elohim, butas Jehovah, the _Lord_ of the Sabbath, and the Spirit of life in MAN, whichSpirit worketh _sin in the flesh_; for the Spirit of Love, in all flesh, isLust, or the spirit of a beast, So Rom. Vii. And which Spirit is_crucified_ in the flesh. He then, as Jehovah--as the power of the Law, _in_ and _over_ all flesh, John viii. 44--increases that which he has madeas the Elohim, and his power shall last for 6 days, or 6 periods of time, computed at a millennium of years; and at the end of which six days, he whois the Spirit of all flesh shall manifest himself as the Holy Spirit ofAlmighty Love, and of all truth; and so shall the Church have her Sabbathof Rest--all contention being at an end. This is, as well as I may nowexpress it, my solution of the mystery in Hebrew, and in Greek, and also inLatin, IHS. For he that was lifted up _is_ King of the Jews, and is theLord of all Life, working in us, both to will and to do; as is manifest inthe Jews--they slaying him that his blood might be _good_ for the healingof the nations, of all people and tongues. As the Father of all _natural_flesh, he is the Spirit of Lust, as in all _beasts_; as the Father, or Kingof the Jews, he is the Devil, as he himself witnesseth in John viii. , already referred to. As lifted up, he is transformed into the Spirit ofLove, a light to the Gentiles, and the glory of his people Israel.... Forthere is but ONE God, ONE Lord, ONE Spirit, ONE body, etc. And he who wasSatan, the Spirit of life in that body, is, in {231} Christ crucified, seenin the Spirit that is in all, and through all and over all, God blessed forever. "

All this seems well meant, and Mr. Thom prints it as convinced of itspiety, and "pronounces no opinion. " Mystics of all sorts! see what you maycome to, or what may come to you! I have inserted the above for your good. 

There is nothing in this world so steady as some of the paradoxers. Theyare like the spiders who go on spinning after they have web enough to catchall the flies in the neighborhood, if the flies would but come. They arelike the wild bees who go on making honey which they never can eat, proving_sic vos non vobis_ to be a physical necessity of their own contriving. Butnobody robs their hives: no, unlike the bees, they go about offering theirware to any who will take it as a gift. I had just written the lastsentence (Oct. 30, 1866, 8. 45 A. M. ) when in comes the second note receivedthis morning from Dr. Thorn: at 1. 30 P. M. Came in a third. These arise outof the above account of the Rev. D. Thom, published Oct. 27: three noteshad arrived before. 

For curiosity I give one day's allowance, supposing these to be all: moremay arrive before night. 

29th Oct. 1866. 

"Dear Sir, --

In re [swastika]. [374]

"So that 'Zaphnath Paaneah' may be after all the revealer of the 'NorthernTau' [Greek: Phaneroô]--To make manifest, shew, or explain; and this maysatisfy the House of Joseph in Amos 5^c. While Belteshazzar = 666 may bealso satisfactory to the House of David, and so we may have Zech. 10^c. 6^v. In operation when Ezekiel 37^c. 16^v. Has been realised;--but there, what is the use of writing, it is all Coptic {232} to a man who has not[swastika], The Thau of the North, the double Vahu [Hebrew: W\qamats\W]. Look at Jeremiah 3^c. 8^v. And then to Psalm 83 for 'hidden ones' [Hebrew:TS\sheva\PW\dagesh\N\segol\Y Y\sheva\HW\qamats\H]--The Zephoni Jehovah, andsay whether they have any connection with the Zephon _Thau_. The Hammer ofThor of Jeremiah 23^c. 29^v. As I gave you in No. 3 of my present edition. 

Yours truly

LE CHEVALIER AU CIN. "

_By Greek Power. _

 C = 20 H = 8 E = 5 V = 6 A = 1 L = 30 I = 10 E = 5 R = 100

 A = 1 U = 400

 C = 20 I = 10 N = 50 ---- 666

There will be thousands of Morgans who will be among the wise and prudentof Hosea 14^c. 9^v. When the Seventh Angel sounds, let me number _that One_by Greek, Rev. 17^c. 1^v: {233}

 S = 200 E = 5 × V = 6 E = 5 N = 50 T = 300 H = 8

 A = 1 N = 50 × G = 6 E = 5 L = 30 ---- 666

 V and G = 12 ought to be equal to one Gammadion or ^3[swastika]3 × 4 = 12, what say you?

London, October 29, 1866. 

"Dear Sir, --

In re [swastika] versus [maltese cross]. 

However pretentious the X or [maltese cross] may be, and it is peculiarlyso just now in this land; after all it is only made of two Roman V's--andso is only = [ one inverted](10)--and therefore is not the perfect number12 of Revel^n, but is the mark of the goddess _Decima_!

Yours truly

WM. THORN. "

Had the _one_ who sent forth a pastoral (Romish) the other day, remainedamongst the faithful expectants, see how he would have numbered, whereas hesold himself for the privilege of signing

[maltese cross] HENRY E. MANNING. [375]

{234}

_By English Key. _

 H = 8 E = 5 N = 40 R = 80 Y = 140

 E = 5 D = 4 W = 120 A = 1 R = 80 D = 4

 M = 30 A = 1 N = 40 N = 40 I = 9 N = 40 G = 7 [swastika] = 12 ---- 666

 Can you now understand the difference between [swastika] and [maltese cross] or X? Look to my challenge. 

Cutting from newspaper:--

ITALY. 

Rome (_via_ Marseilles), October

Mr. Gladstone has paid a visit to the Pope. 

_By Greek Power. _

 G = 6 L = 30 A = 1 D = 4 S = 200 T = 300 O = 70 N = 50 E = 5 ---- 666

And what then [swastika]?

{235}

In other letters _John Stuart Mill_ is 666 if the _a_ be left out;_Chasuble_ is perfect. _John Brighte_[376] is a _fait accompli_; and I amasked whether intellect can account for the final e. Very easily: thisBeast is not the M. P. , but another person who spells his name differently. But if John Sturt Mill and John Brighte choose so to write themselves, theymay. 

A curious collection; a mystical phantasmagoria! There are those who willtry to find meaning: there are those who will try to find purpose. 

 "And some they said--What are you at? And some--What are you arter?"

My account of Mr. Thom and his 666 appeared on October 27: and on the 29thI received from the editor a copy of Mr. Thom's sermons published in 1863(he died Feb. 27, 1862) with best wishes for my health and happiness. Theeditor does not name himself in the book; but he signed his name in mycopy: and may my circumference never be more than 3-1/8 of my diameter ifthe signature, name and writing both, were not that of my [circle square]ing friend Mr. James Smith! And so I have come in contact with him on 666as well as on [pi]! I should have nothing left to live for, had I nothappened to hear that he has a perpetual motion on hand. I returned thanksand kind regards: and Miss Miggs's words--"Here's forgivenesses ofinjuries! here's amicablenesses!"--rang in my ears. But I was made slightlyuncomfortable: how could the war go on after this armistice? Could I evermake it understood that the truce only extended to the double Vahu andthings thereunto relating? It was once held by seafaring men that there wasno peace with Spaniards beyond the line: I was determined that there mustbe no concord with J. S. Inside the circle; that this must be a specialexception, like Father Huddleston {236} and old Grouse in the gun-room. Iwas not long in anxiety; twenty-four hours after the book of sermons therecame a copy of the threatened exposure--_The British Association inJeopardy, and Professor De Morgan in the Pillory without hope of escape_. By James Smith, Esq. London and Liverpool, 8vo. , 1866 (pp. 94). Thisexposure consists of reprints from the _Athenæum_ and _Correspondent_: ofthings new there is but one. In a short preface Mr. J. S. Particularlyrecommends to "_read to the end_. " At the end is an appendix of two pages, in type as large as the work; a very prominent peroration. It is an articlefrom the _Athenæum_, left out of its place. In the last sentence Mr. J. Smith, who had asked whether his character as an honest Geometer andMathematician was not at stake, is warned against the _fallacia pluriuminterrogationum_. [377] He is told that there is not a more honestwhat's-his-name in the world: but that as to the counter which he calls hischaracter as a mathematician, he is assured that it has been staked yearsago, and lost. And thus truth has the last word. There is no occasion tosay much about reprints. One of them is a letter [that given above] ofAugust 25, 1865, written by Mr. J. S. To the _Correspondent_. It is one ofhis quadratures; and the joke is that I am made to be the writer: itappears as what Mr. J. S. Hopes I shall have the sense to write in the_Athenæum_ and forestall him. When I saw myself thus quoted--yes! quoted!double commas, first person--I felt as I suppose did Wm. Wilberforce[378]when he set eyes on the affectionate benediction of the potato whichwaggish comrades had imposed on a raw Irish reporter as part of his speech. I felt as Martin[379] of {237} Galway--kind friend of the poor dumbcreatures!--when he was told that the newspapers had put him in Italics. "Iappeal to you, Mr. Speaker! I appeal to the House! Did I speak in Italics?Do I ever speak in Italics?" I appeal to editor and readers, whether I eversquared the circle until a week or two ago, when I gave my charitable modeof reconciling the discrepant cyclometers. 

The absurdity of the imitation of symbolic reasoning is so lusciously rich, that I shall insert it when I make up my final book. Somebody masteredSpanish merely to read Don Quixote: it would be worth while to learn alittle algebra merely to enjoy this a b-istical attack on the windmills. The principle is, Prove something in as roundabout a way as possible, mention the circle once or twice irrelevantly in the course of your proof, and then make an act of Q. E. D. In words at length. The following ishardly caricature:--

To prove that 2 and 2 make 5. Let a = 2, b = 5: let c = 658, the number ofthe House: let d = 666, the number of the Beast. Then of necessity d = a +b + c + 1; so that 1 is a harmonious and logical quantification of thenumber of which we are to take care. Now, b, the middle of our digitalsystem, is, by mathematical and geometrical combination, a mean between 5 +1 and 2 + 2. Let 1 be removed to be taken care of, a thing no realmathematician can refuse without serious injury to his mathematical andgeometrical reputation. It follows of necessity that 2 + 2 = 5, _quod eratdemonstrumhorrendum_. If Simpkin & Marshall have not, after my notice, toaccount for a gross of copies more than would have gone off without me, theworld is not worthy of its James Smith!

The only fault of the above is, that there is more {238} connection than inthe process of Faber Cyclometricus: so much, in fact, that the blunders arevisible. The utter irrelevance of premises to conclusion cannot beexhibited with the requisite obscurity by any one who is able to followreasoning: it is high art displayed in a certain toning down of the _ægrisomnia_, which brings them to a certain look of reproach to reasoning whichI can only burlesque. Mr. J. S. Produces something which resembles argumentmuch as a chimpanzee in dolor, because balked of his dinner, resembles athinking man at his studies. My humble attempt at imitation of him is morelike a monkey hanging by his tail from a tree and trying to crack acocoa-nut by his chatter. 

I could forgive Mr. J. S. Anything, properly headed. I would allow him toprove--_for himself_--that the Quadrature of the Circle is the child of aprivate marriage between the Bull Unigenitus and the Pragmatic Sanction, claiming tithe of onions for repeal of the Mortmain Act, before the Bishopsin Committee under the kitchen table: his mode of imitating reason would dothis with ease. But when he puts his imitation into my mouth, to make mewhat _he_ calls a "real mathematician, " my soul rises in epigram againsthim. I say with the doll's dressmaker--such a job makes me feel like apuppet's tailor myself--"He ought to have a little pepper? just a fewgrains? I think the young man's tricks and manners make a claim upon hisfriends for a little pepper?" De Fauré[380] and Joseph Scaliger[381] comeinto my head: my reader may look back for them. 

 "Three circlesquarers to the manner born, Switzerland, France, and England did adorn, De Fauré in equations did surpass, Joseph at contradictions was an ass. Groaned Folly, I'm used up! What shall I do To make James Smith? Grinned Momus, _Join the two_!"

{239}

As to my _locus poenitentiæ_, [382] the reader who is fit to enjoy theletter I have already alluded to will see that I have a soft and easyposition; that the thing is really a _pillowry_; and that I am, likePerrette's pot of milk, "Bien posé sur un coussinet. "[383]

Joanna Southcott[384] never had a follower who believed in her with morehumble piety than Mr. James Smith believes in himself. After all that hashappened to him, he asks me with high confidence to "favor the writer witha proof" that I still continue of opinion that "the best of the argument isin my jokes, and the best of the joke is in his arguments. " I will not sofavor him. At the very outset I told him in plain English that he has thewhiphand of all the reasoners in the world, and in plain French that _il aperdu le droit d'être frappé de l'évidence_[385]; I might have said_pendu_. [386] To which I now add, in plain Latin, _Sapienti pauca, indoctonihil_. [387] The law of Chancery says that he who will have equity must doequity: the law of reasoning says that he who will have proof must seeproof. 

The introduction of things quite irrelevant, by way of reproach, is anargument in universal request: and it often happens that the argument soproduced really tells against the producer. So common is it that we forgethow boyish it is; but we are strikingly reminded when it actually comesfrom a boy. In a certain police court, certain small boys were arraignedfor conspiring to hoot an obnoxious individual on his way from one of theirschool exhibitions. This proceeding was necessary, because there seemed tobe a permanent conspiracy to annoy the gentleman; and the {240} masters didnot feel able to interfere in what took place outside the school. So theboys were arraigned; and their friends, as silly in their way asthemselves, allowed one of them to make the defence, instead of employingcounsel; and did not even give them any useful hints. The defence was asfollows; and any one who does not see how richly it sets off the defencesof bigger boys in bigger matters has much to learn. The innocent convictionthat there was answer in the latter part is delightful. Of course fine andrecognizance followed. 

A---- said the boys had received great provocation from B----. He wasconstantly threatening them with a horsewhip which he carried in his hand[the boy did not say what had passed to induce him to take such a weapon], and he had repeatedly insulted the master, which the boys could not stand. B---- had in his own drawing-room told him (A----) that he had drawn hissword against the master and thrown away the scabbard. B---- knew well thatif he came to the college he would catch it, and then he went off through aside door--which was no sign of pluck; and then he brought Mrs. B---- withhim, thinking that her presence would protect him. 

My readers may expect a word on Mr. Thom's sermons, after my account of hisqueer doings about 666. He is evidently an honest and devout man, muchwanting in discrimination. He has a sermon about private _judgment_, inwhich he halts between the logical and legal meanings of the word. Heloathes those who apply their private judgment to the word of God: here hemeans those who decide what it _ought to be_. He seems in other placesaware that the theological phrase means taking right to determine what it_is_. He uses his own private judgment very freely, and is strong in theconclusion that others ought not to use theirs except as he tells them how;he leaves all the rest of mankind free to think with him. In this he is notoriginal: his fame must rest on his senary tripod. {241}

JAMES SMITH ONCE MORE. 

Mr. James Smith's procedures are not caricature of reasoning; they arecaricature of blundering. The old way of proving that 2 = 1 is solemnearnest compared with his demonstrations. As follows:[388]

 Let x = 1 Then x^2 = x And x^2 - 1 = x - 1 Divide both sides by x - 1; then x + 1 = 1; but x = 1, whence 2 = 1. 

When a man is regularly snubbed, bullied, blown up, walked into, and putdown, there is usually some reaction in his favor, a kind of deostracism, which cannot bear to hear him always called the blunderer. I hope it willbe so in this case. There is nothing I more desire than to see _sects_ ofparadoxers. There are fully five thousand adults in England who ought to bethe followers of some one false quadrature. And I have most hope of 3-1/8, because I think Mr. James Smith better fitted to be the leader of anorganized infatuation than any one I know of. He wants no pity, and willget none. He has energy, means, good humor, strong conviction, character, and popularity in his own circle. And, most indispensable point of all, hesticks at nothing;

 "In coelum jusseris, ibit. "[389]

When my instructor found I did not print an acceptance of what I havequoted, he addressed me as follows (_Corr. _, Sept 23):--

"In this life, however, we must do our duty, and, when {242} necessary, usethe rod, not in a spirit of revenge, but for the benefit of the culprit andthe good of society. Now, Sir, the opportunity has been thrown in your wayof slipping out of the pillory without risk of serious injury; but, like anobstinate urchin, you have chosen to quarrel with your opportunity andremain there, and thus you compel me to deal with you as schoolmasters usedto do with stupid boys in bygone days--that is to say, you force me to theuse of the critic's rod, compel me to put you where little Jack Horner sat, and, as a warning to other naughty boys, to ornament you with a dunce'scap. The task I set you was a very simple one, as I shall make manifest atthe proper time. "

In one or more places, as well as this, Mr. Smith shows that he does notknow the legend of little Jack Horner, whom he imagines to be put in thecorner as a bad boy. This is curious; for there had been many allusions tothe story in the journal he was writing in, and the Christmas pie hadbecome altered into the Seaforth [pi]. 

Mr. Smith is satisfied at last that--what between argument and punishmenthe has convinced me. He says (_Corr. _, Jan. 27, 1866): "I tell him withouthesitation that he knows the true ratio of diameter to circumference aswell as I do, and if he be wise he will admit it. " I should hope I do, andbetter; but there is no occasion to admit what everybody knows. 

I have often wished that we could have a slight glimpse of the receptionwhich was given to some of the old cyclometers: but we have nothing, exceptthe grave disapprobation of historians. I am resolved to give the NewZealander a chance of knowing a little more than this about one of them atleast; and, by the fortunate entrance into life of the _Correspondent_, Iam able to do it. I omit sober mathematical answers, of which there wereseveral. The following letter is grave earnest:

"Sir, --I have watched Mr. James Smith's writings on this subject from thefirst, and I did hope that, as the more {243} he departs from truth themore easy it must be to refute him, [this by no means always true] some ofyour correspondents would by this time have done so. I own that I am unableto detect the fallacy of his argument; and I am quite certain that '[Pi]'is wrong, in No. 23, where he declares that Mr. Smith is 'ignorant of thevery elements of mathematical truth. ' I have observed an immense amount ofgeometrical reasoning on his part, and I cannot see that it is either fairor honest to deny this, which may be regarded as the 'elements' ofmathematical truth. Would it not be better for '[Pi]' to answer Mr. Smith, to refute his arguments, to point out their fallacies, and to save learnersfrom error, than to plunge into gross insult and unmanly abuse? Would itnot be well, also, that Professor De Morgan should favour us with a littlereasoning?

"I have hitherto seen no attempt to overthrow Mr. Smith's arguments; Itrust that this will not continue, since the subject is one of immenseimportance to science in general, especially to nautical science, and allthat thereto belongs. 

Yours, etc. , A CAPTAIN, R. N. "

On looking at this homoeopathic treatment of the 3-1/8quadrature--remember, homoeopathic, _similia similibus_, [390] notinfinitesimal--and at the imputation thrown upon it, I asked myself, what_is_ vulgarity? No two agree, except in this, that every one sees vulgarityin what is directed against himself. Mark the world, and see if anything beso common as the description of the other side's remarks as "vulgar attemptat wit. " "I suppose you think that very witty:" the answer is "No myfriend! your remark shows that you feel it as wit, so that the purpose isanswered; I keep my razor for something else than cutting blocks;" I aminclined to think that "out of place" is a necessary attribute of truevulgarity. And further, it is to be noticed that nothing is {244}unproducible--_salvo pudore_[391]--which has classical authority, modern orancient, in its favor. "He is a vulgar fellow; I asked him what he wasupon, and what do you think he answered, My legs!"--"Well, and has he notjustification? what do you find in Terence? _Quid agitur? Statur. _"[392] Ido not even blench from my principle where I find that it brings what iscalled "taking a sight" within permissible forms of expression: Rabelaisnot only establishes its antiquity, but makes it English. Our oldtranslation[393] has it thus (book 2. Ch. 19):

"Then made the Englishman this sign. His left hand, all open, he lifted upinto the air, then instantly shut into his fist the four fingers thereof;and his thumb extended at length he placed upon the tip of his nose. Presently after he lifted up his right hand all open and abased and bent itdownwards, putting the thumb thereof in the very place where the littlefinger of the left hand did close in the fist, and the four right handfingers he softly moved in the air. Then contrarily he did with the righthand what he had done with the left, and with the left what he had donewith the right. "

An impressive sight! The making of a fist of the left hand is a greataddition of power, and should be followed in modern practice. The gentlesullation of the front fingers, with the clenched fist behind them, says asplainly as possible, Put _suaviter in modo_ in the van, but don't forget tohave _fortiter in re_[394] in the rear. 

{245}

My Budget was announced (March 23, 1867) for completion on the 30th. Mr. James Smith wrote five letters, one before the completion, four after it;the five contained 68 pages of quarto letter paper. Mr. J. S. Had picked upa clerical correspondent, with whom he was in the heat of battle. 

"_March 27. _--Dear Sir. Very truly yours. Duty; for my own sake; just timeleft to retrieve my errors; sends copy of letter to clergyman; new proofnever before thought of; merest tyro would laugh if I were to stifle it, whether by rhodomontade or silent contempt; keep your temper. I shall beconvinced; and if world be right in supposing me incapable of a foul act, Ishall proclaim glorious discovery in the _Athenæum_. 

"_April 15. _--Sir, ... My dear Sir, Your sincere tutelary. Copy of anotherletter to clergyman; discovery tested by logarithms; reasons such as nonebut a knave or a sinner can resist. Let me advise you to take counselbefore it is too late! Keep your temper. Let not your _pride_ get thebetter of your discretion! Screw up your courage, my good friend, and_resolve_ to show the world that you are an _honest_ man.... 

"_April 20. _--Sir ... Your very sincere and favorite tutelary. I have longplayed the _cur_, snapping and snarling... ; suddenly lost my power, andbecame _half-starved_ dog without _spirit_ to bark; try if air cannotrestore me; calls himself the _thistle_ in allusion to my other tutelary, the _thorn_; Would I prefer his next work to be, 'A whip for theMathematical Cur, Prof. De M. ' In some previous letter which I havemislaid, he told me his next would be 'a muzzle for the Mathematical Bulldog, Prof. De M. '

"_April 23. _--Sir. Very sincerely yours. More letters to clergyman; you mayas well knock your head against a stone wall to improve your intellect asattempt to controvert my proofs. [I thought so too; and tried neither]. {246}

"_May 6. _--My dear Sir. Very sincerely yours. All to myself, and nothing tonote. 

"_July 2. _--No more in this interval. All that precedes is a desperateattempt to induce me to continue my descriptions: notoriety at any price. "

I dare say the matter is finished: the record of so marked an instance ofself-delusion will be useful. 

I append to the foregoing a letter from Dr. Whewell[395] to Mr. JamesSmith. The Master of Trinity was conspicuous as a rough customer, anintellectual bully, an overbearing disputant: the character was as wellestablished as that of Sam Johnson. But there was a marked difference. Itwas said of Johnson that if his pistol missed fire, he would knock you downwith the butt end of it: but Whewell, in like case, always acknowledged themiss, and loaded again or not, as the case might be. He reminded me ofDennis Brulgruddery, who says to Dan, Pacify me with a good reason, andyou'll find me a dutiful master. I knew him from the time when he was myteacher at Cambridge, more than forty years. As a teacher, he was anythingbut dictatorial, and he was perfectly accessible to proposal of objections. He came in contact with me in his slashing way twice in our after jointlives, and on both occasions he acknowledged himself overcome, by thatchange of manner, and apologetic mode of continuance, which I had seen himemploy towards others under like circumstances. 

I had expressed my wish to have a _thermometer of probability_, withimpossibility at one end, as 2 and 2 make 5, and necessity at the other, as2 and 2 make 4, and a graduated rise of examples between them. Down came ablow: "What! put necessary and contingent propositions together! It'sabsurd!" I pointed out that the two kinds of necessity are but suchextremes of probability as 0 and [infinity] are of number, and illustratedby an urn with 1 white and _n_ black {247} balls, _n_ increasing withoutlimit. It was frankly seen, and the point yielded; a large company waspresent. 

Again, in a large party, after dinner, and politics being the subject, Iwas proceeding, in discussion with Mr. Whewell, with "I think"... --"Ugh!_you_ think!" was the answer. I repeated my phrase, and gave as a reasonthe words which Lord Grey[396] had used in the House of Lords the nightbefore (the celebrated advice to the Bishops to set their houses in order). He had not heard of this, and his manner changed in an instant: he was therational discutient all the rest of the evening, having previously beennothing but a disputant with all the distinctions strongly marked. 

I have said that Whewell was gentle with his pupils; it was the same withall who wanted teaching: it was only on an armed enemy that he drew hisweapon. The letter which he wrote to Mr. J. Smith is an instance: and as itapplies with perfect fidelity to the efforts of unreasoning abovedescribed, I give it here. Mr. James Smith is skilfully exposed, and feltit; as is proved by "putting the writer in the stocks. "

"The Lodge, Cambridge, September 14th, 1862. 

"Sir, --I have received your explanation of your proposition that thecircumference of the circle is to its diameter as 25 to 8. I am afraid Ishall disappoint you by saying that I see no force in your proof: and Ishould hope that you will see that there is no force in it if you considerthis: In the whole course of the proof, though the word cycle occurs, thereis no property of the circle employed. You may do this: you may put theword _hexagon_ or _dodecagon_, or any other word describing a polygon inthe place of _Circle_ in your proof, and the proof would be just as good asbefore. Does not this satisfy you that you cannot have proved a property ofthat special figure--a circle? {248}

"Or you may do this: calculate the side of a polygon of 24 sides inscribedin a circle. I think you are a Mathematician enough to do this. You willfind that if the radius of the circle be one, the side of this polygon is. 264 etc. Now, the arc which this side subtends is according to yourproposition 3. 125/12 = . 2604, and therefore the chord is greater than itsarc, which you will allow is impossible. 

"I shall be glad if these arguments satisfy you, and

"I am, Sir, your obedient Servant, "W. WHEWELL. "

AN M. P. 'S ARITHMETIC. 

In the debate of May, 1866, on Electoral Qualifications, a question aroseabout arithmetical capability. Mr. Gladstone asked how many members of theHouse could divide 1330l. 7s. 6d. By 2l. 13s. 8d. Six hundred andfifty-eight, answered one member; the thing cannot be done, answeredanother. There is an old paradox to which this relates: it arises out ofthe ignorance of the distinction between abstract and concrete arithmetic. _Magnitude_ may be divided by _magnitude_; and the answer is number: howoften does 12d. Contain 4d. ; answer three times. _Magnitude_ may be dividedby _number_, and the answer is _magnitude_: 12d. Is divided in four equalparts, what is each part? Answer three _pence_. The honorable objector, whose name I suppress, trusting that he has mended his ways, gave thefollowing utterance:

"With regard to the division sum, it was quite possible to divide by a sum, but not by money. How could any one divide money by 2l. 16s. 8d. ?(Laughter. ) The question might be asked, 'How many times 2s. Will go into1l. ?' but that was not dividing by money; it was simply dividing 20 by 2. He might be asked, 'How many times will 6s. 8d. Go into a pound?' but itwas only required to divide 240 by 80. If the right hon. Gentleman were toask the hon. {249} member for Brighton (Professor Fawcett), [397] or anyother authority, he would receive the same answer--viz. , that it waspossible to divide by a sum, but not by money. (Hear. )"

I shall leave all comment for the second edition, if I publish one. [398] Ishall be sure to have something to laugh at. Anything said from arespectable quarter, or supposed to be said, is sure to find defenders. SamJohnson, a sound arithmetician, comparing himself, and what he alone haddone in three years, with forty French Academicians and their forty years, said it proved that an Englishman is to a Frenchman as 40 × 40 to 3, or as1600 to 3. Boswell, who was no great hand at arithmetic, made him say thatan Englishman is to a Frenchman as 3 to 1600. When I pointed this out, thesupposed Johnson was defended through thick and thin in _Notes andQueries_. 

I am now curious to see whether the following will find a palliator. It isfrom "Tristram Shandy, " book V. Chapter 3. There are two curious idioms, "for for" and "half in half"; but these have nothing to do with my point:

"A blessing which tied up my father's tongue, and a misfortune which set itloose with a good grace, were pretty equal: sometimes, indeed, themisfortune was the better of the two; for, for instance, where the pleasureof harangue was as _ten_, and the pain of the misfortune but as _five_, myfather gained half in half; and consequently was as well again off as if ithad never befallen him. "

This is a jolly confusion of ideas; and wants nothing but a defender tomake it perfect. A person who invests five {250} with a return of ten, andone who loses five with one hand and gains ten with the other, both leaveoff five richer than they began, no doubt. The first gains "half in half, "more properly "half _on_ half, " that is, of the return, 10, the second 5 isgain upon the first 5 invested. "Half _in_ half" is a queer way of sayingcent. Per cent. If the 5l. Invested be all the man had in the world, hecomes out, after the gain, twice as well off as he began, with reference tohis whole fortune. But it is very odd to say that balance of 5l. Gain is_twice_ as good as if nothing had befallen, either loss or gain. Amathematician thinks 5 an infinite number of times as great as 0. The wholeconfusion is not so apparent when money is in question: for money is moneywhether gained or lost. But though pleasure and pain stand to one anotherin the same algebraical relation as money gained and lost, yet there ismore than algebra can take account of in the difference. 

Next, Ri. Milward[399] (Richard, no doubt, but it cannot be proved) whopublished Selden's[400] Table Talk, which he had collected while serving asamanuensis, makes Selden say, "A subsidy was counted the fifth part of aman's estate; and so fifty subsidies is five and forty times more than aman is worth. " For _times_ read _subsidies_, which seems part of theconfusion, and there remains the making all the subsidies equal to thefirst, though the whole of which they are to be the fifths is perpetuallydiminished. 

Thirdly, there is the confusion of the great misomath {251} of our own day, who discovered two quantities which he avers to be identically the same, but the greater the one the less the other. He had a truth in his mind, which his notions of quantity were inadequate to clothe in language. Thiserroneous phraseology has not found a defender; and I am almost inclined tosay, with Falstaff, The poor abuses of the time want countenance. 

ERRONEOUS ARITHMETICAL NOTIONS. 

"Shallow numerists, " as Cocker[401] is made to call them, have long been atwork upon the question how to _multiply_ money by money. It is, I haveobserved, a very common way of amusing the tedium of a sea voyage: I havehad more than one bet referred to me. Because an oblong of five inches byfour inches contains 5 × 4 or 20 _square_ inches, people say that fiveinches multiplied by four inches _is_ twenty _square_ inches: and, thinkingthat they have multiplied length by length, they stare when they are toldthat money cannot be multiplied by money. One of my betters made it anargument for the thing being impossible, that there is no _square money_:what could I do but suggest that postage-stamps should be made legaltender. Multiplication must be _repetition_: the repeating process must beindicated by _number_ of times. I once had difficulty in persuading anotherof my betters that if you repeat five shillings as often as there are hairsin a horse's tail, you do not _multiply five shillings by ahorsetail_. [402]

I am very sorry to say that these wrong notions have found support--I thinkthey do so no longer--in the University of Cambridge. In 1856 or 1857, anexaminer was displaced by a vote of the Senate. The pretext was that he wastoo severe an examiner: but it was well known that {252} greatdissatisfaction had been expressed, far and wide through the Colleges, atan absurd question which he had given. He actually proposed such a fractionas

 6s. 3d. --------. 17s. 4d. 

As common sense gained a hearing very soon, there is no occasion to saymore. In 1858, it was proposed at a college examination, to divide 22557days, 20 hours, 20 minutes, 48 seconds, by 57 minutes, 12 seconds, and alsoto explain the fraction

 32l. 18s. 8d. -------------. 62l. 12s. 9d. 

All paradoxy, in matters of demonstration, arises out of muddle about firstprinciples. Who can say how much of it is to be laid at the door of theUniversity of Cambridge, for not taking care of the elements ofarithmetical thought?

ON LITERARY BARGAINS. 

The phenomena of the two ends of society, when brought together, giveinteresting comparisons: I mean the early beginnings of thought andliterature, and our own high and finished state, as we think it. There isone very remarkable point. In the early day, the letter was matter of theclosest adherence, and implied meanings were not admitted. 

The blessing of Isaac meant for Esau, went to false Jacob, in spite of theimposition; and the writer of Genesis seems to intend to give the notionthat Isaac had no power to pronounce it null and void. And "Jacob's policy, whereby he became rich"--as the chapter-heading puts it--in speckled andspotted stock, is not considered as a violation of the agreement, whichcontemplated natural proportions. In {253} the story of Lycurgus thelawgiver is held to have behaved fairly when he bound the Spartans to obeyhis laws until he returned--intimating a short absence--he intending neverto return. And Vishnoo, when he asked the usurper for three steps ofterritory as a dwarf, and then enlarged himself until he could bring heavenand earth under the bargain, was thought clever, certainly, but quite fair. 

There is nothing of this kind recognized in our day: so far good. But thereis a bad contrary: the age is apt, in interpretation, to upset the letterin favor of the view--very often the after thought--of one side only. Thecase of John Palmer, [403] the improver of the mail coach system, issmothered. He was to have an office and a salary, and 2½ per cent for lifeon the increased _revenue_ of the Post-Office. His rights turned out solarge, that Government would not pay them. For misconduct, real orpretended, they turned him out of his _office_: but his bargain as to thepercentage had nothing to do with his future conduct; it was payment forhis _plan_. I know nothing, except from the debates of 1808 in the twoHouses: if any one can redeem the credit of the nation, the field is open. When I was young, the old stagers spoke of this transaction sparingly, anddismissed it speedily. 

The government did not choose to remember what private persons mustremember, and are made to remember, if needful. When Dr. Lardner[404] madehis bargain with the {254} publishers for the _Cabinet Cyclopædia_ heproposed that he, as editor, should have a certain sum for every hundredsold above a certain number: the publishers, who did not think there wasany chance of reaching the turning sale of this stipulation, readilyconsented. But it turned out that Dr. Lardner saw further than they: thereturns under this stipulation gave him a very handsome addition to hisother receipts. The publishers stared; but they paid. They had no idea ofstanding out that the amount was too much for an editor; they knew that, though the editor had a percentage, they had all the rest; and they wouldnot have felt aggrieved if he had received ten times as much. Butgovernments, which cannot be brought to book before a sworn jury, are ruledonly by public opinion. John Palmer's day was also the day of Thomas FyshePalmer, [405] and the governments, in their prosecutions for sedition, knewthat these would have a reflex action upon the minds of all who wrote aboutpublic affairs. 

DECLARATION OF BELIEF

1864-65. --It often happens that persons combine to maintain and enforce anopinion; but it is, in our state of society, a paradox to unite for thesole purpose of blaming the opposite side. To invite educated men to dothis, and above all, men of learning or science, is the next paradoxicalthing of all. But this was done by a small combination in 1864. They gottogether and drew up a _declaration_, to be signed by "students of thenatural sciences, " who were to express their "sincere regret thatresearches into {255} scientific truth are perverted by some in our owntimes into occasion for casting doubt upon the truth and authenticity ofthe Holy Scriptures. " In words of ambiguous sophistry, they proceeded torequest, in effect, that people would be pleased to adopt the views ofchurches as to the _complete_ inspiration of all the canonical books. Thegreat question whether the Word of God is _in_ the Bible, or whether theWord of God is _all_ the Bible, was quietly taken for granted in favor ofthe second view; to the end that men of science might be induced to blamethose who took the first view. The first public attention was drawn to thesubject by Sir John Herschel, [406] who in refusing to sign the writ sent tohim, administered a rebuke in the _Athenæum_, which would have opened mosteyes to see that the case was hopeless. The words of a man whose _suaviterin modo_ makes his _fortiter in re_[407] cut blocks with a razor are worthpreserving:

"I consider the act of calling upon me publicly to avow or disavow, toapprove or disapprove, in writing, any religious doctrine or statement, however carefully or cautiously drawn up (in other words, to append my nameto a religious manifesto) to be an infringement of that social forbearancewhich guards the freedom of religious opinion in this country with especialsanctity.... I consider this movement simply mischievous, having a directtendency (by putting forward a new Shibboleth, a new verbal test ofreligious partisanship) to add a fresh element of discord to the alreadytoo discordant relations of the Christian world.... But no nicety ofwording, no artifice of human language, will suffice to discriminate thehundredth part of the shades of meaning in which the most world-widedifferences of thought on such subjects may be involved; or prevent themost gentle worded and apparently justifiable expression of regret, soembodied, from grating on the {256} feelings of thousands of estimable andwell-intentioned men with all the harshness of controversial hostility. "

Other doses were administered by Sir J. Bowring, [408] Sir W. RowanHamilton, [409] and myself. The signed declaration was promised forChristmas, 1864: but nothing presentable was then ready; and it was nearMidsummer, 1865, before it was published. Persons often incautiously puttheir names without seeing the _character_ of a document, because theycoincide in its _opinions_. In this way, probably, fifteen respectablenames were procured before printing; and these, when committed, were hawkedas part of an application to "solicit the favor" of other signatures. It islikely enough no one of the fifteen saw that the declaration was, not_maintenance_ of their own opinion, but _regret_ (a civil word for _blame_)that others should _think differently_. 

When the list appeared, there were no fewer than 716 names! But analysisshowed that this roll was not a specimen of the mature science of thecountry. The collection was very miscellaneous: 38 were designated as"students of the College of Chemistry, " meaning young men who attendedlectures in that college. But as all the Royal Society had been applied to, a test results as follows. Of Fellows of the Royal Society, 600 in number, 62 gave their signatures; of writers in the _Philosophical Transactions_, 166 in number, 19 gave their signatures. Roughly speaking, then, only oneout of ten could be got to express disapprobation of the free comparison ofthe results of science with the statements of the canonical books. And I amsatisfied that many of these thought they were signing only a declarationof difference of opinion, not of blame for that difference. The number ofpersons is not small who, when it comes to signing printed documents, wouldput their names to a declaration that the coffee-pot ought to be takendown-stairs, meaning that the teapot ought to be brought {257} up-stairs. And many of them would defend it. Some would say that the two things arenot contradictory; which, with a snort or two of contempt, would be veryeffective. Others would, in the candid and quiet tone, point out that it isall one, because coffee is usually taken before tea, and it keeps the tableclear to send away the coffee-pot before the teapot is brought up. 

The original signatures were decently interred in the Bodleian Library: andthe advocates of scattering indefinite blame for indefinite sins of opinionamong indefinite persons are, I understand, divided in opinion about thetime at which the next attempt shall be made upon men of scientificstudies: some are for the Greek Calends, and others for the RomanOlympiads. But, with their usual love of indefiniteness, they havedetermined that the choice shall be argued upon the basis that which comesfirst cannot be settled, and is of no consequence. 

I give the declaration entire, as a curiosity: and parallel with it I givea substitute which was proposed in the _Athenæum_, as worthy to be signedboth by students of theology, and by students of science, especially inpast time. When a new attempt is made, it will be worth while to look atboth:

_Declaration. _ _Proposed Substitute. _

We, the undersigned Students We, the undersigned Studentsof the Natural Sciences, of Theology and of Nature, desire to express our sincere desire to express our sincereregret, that researches into regret, that common notions ofscientific truth are perverted religious truth are pervertedby some in our own times into by some in our own times intooccasion for casting doubt occasion for casting reproachupon the Truth and upon the advocates ofAuthenticity of the Holy demonstrated or highlyScriptures. Probable scientific theories. 

 {258}We conceive that it is We conceive that it isimpossible for the Word of impossible for the Word ofGod, as written in the book of God, as correctly read in thenature, and God's Word written Book of Nature, and the Wordin Holy Scripture, to of God, as truly interpretedcontradict one another, out of the Holy Scripture, tohowever much they may appear contradict one another, to differ. However much they may appear to differ. We are not forgetful that We are not forgetful thatPhysical Science is not neither theologicalcomplete, but is only in a interpretation nor physicalcondition of progress, and knowledge is yet complete, butthat at present our finite that both are in a conditionreason enables us only to see of progress; and that atas through a glass darkly, present our finite reason enables us only to see both one and the other as through a glass darkly [the writers of the original declaration have distinctively applied to physical science the phrase by which St. Paul denotes the imperfections of theological vision, which they tacitly assume to be quite perfect], and we confidently believe, and we confidently believe, that a time will come when the that a time will come when thetwo records will be seen to two records will be seen toagree in every particular. We agree in every particular. Wecannot but deplore that cannot but deplore thatNatural Science should be Religion should be looked uponlooked upon with suspicion by with suspicion by some andmany who do not make a study Science by others, of theof it, merely on account of students of either who do notthe unadvised manner in which make a study of the {259}some are placing it in other, merely on account ofopposition to Holy Writ. The unadvised manner in which some are placing Religion in opposition to Science, and some are placing Science in opposition to Religion. We believe that it is the duty We believe that it is the dutyof every Scientific Student to of every theological studentinvestigate nature simply for to investigate the Scripture, the purpose of elucidating and of every scientifictruth, student to investigate Nature, simply for the purpose of elucidating truth. And that if he finds that some And if either should find thatof his results appear to be in some of his results appear tocontradiction to the Written be in contradiction, whetherWord, or rather to his own to Scripture or to Nature, or_interpretations_ of it, which rather to his ownmay be erroneous, he should _interpretation_ of one or thenot presumptuously affirm that other, which may be erroneous, his own conclusions must be he should not affirm as withright, and the statements of certainty that his ownScripture wrong; conclusion must be right, and the other interpretation wrong:rather, leave the two side by but should leave the two sideside till it shall please God by side for further inquiryto allow us to see the manner into both, until it shallin which they may be please God to allow us toreconciled; arrive at the manner in which they may be reconciled. And, instead of insisting upon In the mean while, instead ofthe seeming differences insisting, and least of allbetween Science and the with acrimony or injuriousScriptures, it would be as {260} statements about others, well to rest in faith upon the upon the seeming differencespoints in which they agree. Between Science and the Scriptures, it would be a thousand times better to rest in faith as to our future state, in hope as to our coming knowledge, and in charity as to our present differences. 

The distinctness of the fallacies is creditable to the composers, and showsthat scientific habits tend to clearness, even to sophistry. Nowhere doesit so plainly stand out that the _Written Word_ means the sense in whichthe accuser takes it, while the sense of the other side is _theirinterpretation_. The infallible church on one side, arrayed againstheretical pravity on the other, is seen in all subjects in which mendiffer. At school there were various games in which one or anotheradvantage was the right of those who first called for it. In adult argumentthe same thing is often attempted: we often hear--I cried _Church_ first!

I end with the answer which I myself gave to the application: its revivalmay possibly save me from a repetition of the like. If there be anything Ihate more than another it is the proposal to place any persons, especiallythose who allow freedom to me, under any abridgment of their liberty tothink, to infer, and to publish. If they break the law, take the law; butdo not make the law: [Greek: agoraioi agontai enkaleitôsan allêlois. ][410]I would rather be asked to take shares in an argyrosteretic company (withlimited liability) for breaking into houses by night on fork and spoonerrands. I should put aside this proposal with _nothing but laughter_. Itwas a joke against Sam Rogers[411] that his appearance was very like thatof a corpse. The _John Bull_ {261} newspaper--suppose we now say TheodoreHook[412]--averred that when he hailed a coach one night in St. Paul'sChurchyard, the jarvey said, "Ho! ho! my man; I'm not going to be taken inthat way: go back to your grave!" This is the answer I shall make for thefuture to any relics of a former time who shall want to call me off thestand for their own purposes. What obligation have I to admit that theybelong to our world?

"SCRIPTURE AND SCIENCE. 

"_The Writ De Hæretico Commiserando. _[413]

Nov. 14, 1864. 

"This document was sent to me four days ago. It 'solicits the favor'--Ithought at first it was a grocer's supplication for tea and sugarpatronage--of my signature to expression of 'sincere regret' that somepersons unnamed--general warrants are illegal--differ from what I amsupposed--by persons whom it does not concern--to hold about Scripture andScience in their real or alleged discrepancies. 

"No such favor from me: for three reasons. First, I agree with Sir. J. Herschel that the solicitation is an intrusion to be publicly repelled. Secondly, I do _not_ regret that others should differ from me, think what Imay: those others are as good as I, and as well able to think, and as muchentitled to their conclusions. Thirdly, even if I did regret, I should beashamed to put my name to bad chemistry made to do duty for good reasoning. The declaration is an awkward attempt to saturate sophism with truism; butthe sophism is left largely in excess. 

{262}

"I owe the inquisitors a grudge for taking down my conceit of myself. Fortwo months I have crowed in my own mind over my friend Sir J. Herschel, fancying that the promoters instinctively knew better than to bring theirfallacies before a writer on logic. Ah! my dear Sir John! thought I, if youhad shown yourself to be well up in _Barbara Celarent_, [414] and had everand anon astonished the natives with the distinction between _simpliciter_and _secundum quid_, no autograph-hunters would have baited a trap with_non sequitur_[415] to catch your signature. What can I say now? I hide mydiminished head, diminished by the horns which I have been compelled todraw in. 

"Those who make personal solicitation for support to an opinion aboutreligion are bound to know their men. The king had a right to BrotherNeale's money, because Brother Neale offered it. Had he put his hand intopurse after purse by way of finding out all who were of Brother Neale'smind, he would have been justly met by a rap on the knuckles whenever hemissed his mark. 

"The kind of test before me is the utmost our time will allow of thatinquisition into opinion which has been the curse of Christianity eversince the State took Providence under its protection. The writ _de hæreticocommiserando_ is little more than the smell of the empty cask: and thosewho issue it may represent the old woman with her

 "O suavis anima, quale in te dicam bonum Antehac fuisse; tales cum sint reliquiæ. "[416]

It is no excuse that the illegitimate bantling is a very little one. Itsparents may think themselves hardly treated when they are called linealsuccessors of Tony Fire-the-faggot: {263} but, degenerate though they be, such is their ancestry. Let every allowance be made for them: but theirunholy fire must be trodden out; so long as a spark is left, nothing butfuel is wanted to make a blaze. If this cannot be done, let the flame beconfined to theology, though even there it burns with diminished vigor: andlet charity, candor, sense, and ridicule, be ready to play upon it wheneverthere is any chance of its extending to literature and science. 

"What would be the consequence if this test-signing absurdity were to grow?Deep would call unto deep; counter-declaration would answer declaration, each stronger than the one before. The moves would go on like the disputeof two German students, of whom each is bound to a sharper retort on agraduated scale, until at last comes _dummer Junge_![417]--and then theymust fight. There is a gentleman in the upper fifteen of the signers of thewrit--the hawking of whose names appears to me very bad taste--whom I metin cordial cooperation for many a year at a scientific board. All I knewabout his religion was that he, as a clergyman, must in some sense or otherreceive the 39 Articles:--all that he could know about mine was that I wassome kind of heretic, or so reputed. If we had come to signing oppositemanifestoes, turn-about, we might have found ourselves in the lowest depthsof party discussion at our very council-table. I trust the list ofsubscribers to the declaration, when it comes to be published, will showthat the bulk of those who have really added to our knowledge have seen thething in its true light. 

"The promoters--I say nothing about the subscribers--of the movement will, I trust, not feel aggrieved at the course I have taken or the remarks Ihave made. Walter Scott says that before we judge Napoleon by thetemptation to which he yielded, we ought to remember how much he may haveresisted: I invite them to apply this rule to myself; they can have no ideaof the feeling with which I {264} contemplate all attempts to repressfreedom of inquiry, nor of the loathing with which I recoil from theproposal to be art and part. They have asked me to give a public opinionupon a certain point. It is true that they have had the kindness to tenderboth the opinion they wish me to form, and the shape in which they wouldhave it appear: I will let them draw me out, but I will not let them takeme in. If they will put an asterisk to my name, and this letter to theasterisk, they are welcome to my signature. As I do not expect them torelish this proposal, I will not solicit the favor of its adoption. Butthey have given a right to think, for they have asked me to think; topublish, for they have asked me to allow them to publish; to blame them, for they have asked me to blame their betters. Should they venture to findfault because my direction of disapproval, publicly given, is half arevolution different from theirs, they will be known as having presented aloaded document at the head of a traveler in the highway of discussion, with--Your signature or your silence!"

THE FLY-LEAF PARADOX. 

The paradox being the proposition of something which runs counter to whatwould generally be thought likely, may present itself in many ways. Thereis a _fly-leaf paradox_, which puzzled me for many years, until I found aprobable solution. I frequently saw, in the blank leaves of old books, learned books, Bibles of a time when a Bible was very costly, etc. , thename of an owner who, by the handwriting and spelling, must have been anilliterate person or a child, followed by the date of the book itself. Accordingly, this uneducated person or young child seemed to be the firstowner, which in many cases was not credible. Looking one day at aBarker's[418] Bible of 1599, I saw an {265} inscription in a child'swriting, which certainly belonged to a much later date. It was "MarthaTaylor, her book, giuen me by Granny Scott to keep for her sake. " With thisthe usual verses, followed by 1599, the date of the book. But it so chancedthat the blank page opposite the title, on which the above was written, wasa verso of the last leaf of a prayer book, which had been bound before theBible; and on the recto of this leaf was a colophon, with the date 1632. Itstruck me immediately that uneducated persons and children, having seendates written under names, and not being quite up in chronology, didfrequently finish off with the date of the book, which stared them in theface. 

Always write in your books. You may be a silly person--for though yourreading my book is rather a contrary presumption, yet it is notconclusive--and your observations may be silly or irrelevant, but youcannot tell what use they may be of long after you are gone whereBudgeteers cease from troubling. 

I picked up the following book, printed by J. Franklin[419] at Boston, during the period in which his younger brother Benjamin was his apprentice. And as Benjamin was apprenticed very early, and is recorded as havinglearned the mechanical art very rapidly, there is some presumption thatpart of it may be his work, though he was but thirteen at the time. As thisset of editions of Hodder[420] (by {266} Mose[421]) is not mentioned, to myknowledge, I give the title in full:

 "Hodder's Arithmetick: or that necessary art made most easy: Being explained in a way familiar to the capacity of any that desire to learn it in a little time. By James Hodder, Writing-master. The Five and twentieth edition, revised, augmented, and above a thousand faults amended, by Henry Mose, late servant and successor to the author. Boston: printed by J. Franklin, for S. Phillips, N. Buttolph, B. Elliot, D. Henchman, G. Phillips, J. Elliot, and E. Negus, booksellers in Boston, and sold at their shops. 1719. "

The book is a very small octavo, the type and execution are creditable, thewoodcut at the beginning is clumsy. It is a perfect copy, page for page, ofthe English editions of Mose's Hodder, of which the one called seventeenthis of London, 1690. There is not a syllable to show that the edition abovedescribed might not be of Boston in England. Presumptions, but not verystrong ones, might be derived from the name of _Franklin_, and from thelarge number of booksellers who combined in the undertaking. It chanced, however, that a former owner had made the following note in my copy:

 "Wednessday, July y^e 14, 1796, att ten in y^e forenoon we sail^d from Boston, came too twice, once in King Rode, and once in y^e Narrows. Sail^d by y^e lighthouse in y^e even^g. "

{267}

No ordinary map would decide these points: so I had to apply to my friendSir Francis Beaufort, [422] and the charts at the Admiralty decidedimmediately for Massachusetts. 

PARADOXES OF ORTHOGRAPHY AND COMPUTATION. 

The French are able paradoxers in their spelling of foreign names. The AbbéSabatier de Castres, [423] in 1772, gives an account of an imaginarydialogue between Swif, Adisson, Otwai, and Bolingbrocke. I had hoped thatthis was a thing of former days, like the literal roasting of heretics; butthe charity which hopeth all things must hope for disappointments. Lookingat a recent work on the history of the popes, I found referred to, in thematter of Urban VIII[424] and Galileo, references to the works of twoEnglishmen, the Rev. Win Worewel and the Rev. Raden Powen. [Wm. Whewell andBaden Powell]. [425]

I must not forget the "moderate computation" paradox. This is the way bywhich large figures are usually obtained. Anything surprisingly great isgot by the "lowest computation, " anything as surprisingly small by the"utmost computation"; and these are the two great subdivisions of "moderatecomputation. " In this way we learn that 70, 000 persons were executed in onereign, and 150, 000 persons {268} burned for witchcraft in one century. Sometimes this computation is very close. By a card before me it appearsthat all the Christians, including those dispersed in heathen countries, those of Great Britain and Ireland excepted, are 198, 728, 000 people, andpay their clergy 8, 852, 000l. But 6, 400, 000 people pay the clergy of theAnglo-Irish Establishment 8, 896, 000l. ; and 14, 600, 000 of otherdenominations pay 1, 024, 000l. When I read moderate computations, I alwaysthink of Voltaire and the "mémoires du fameux évêque de Chiapa, parlesquels il paraît qu'il avait égorgé, ou brulé, ou noyé dix millionsd'infidèles en Amérique pour les convertir. Je crus que cet évêqueexaggérait; mais quand on réduisait ces sacrifices à cinq millions devictimes, cela serait encore admirable. "[426]

CENTRIFUGAL FORCE. 

My Budget has been arranged by authors. This is the only plan, for much ofthe remark is personal: the peculiarities of the paradoxer are a large partof the interest of the paradox. As to subject-matter, there are pointswhich stand strongly out; the quadrature of the circle, for instance. Butthere are others which cannot be drawn out so as to be conspicuous in areview of writers: as one instance, I may take the _centrifugal force_. 

When I was about nine years old I was taken to hear a course of lectures, given by an itinerant lecturer in a country town, to get as much as I couldof the second half of a good, sound, philosophical omniscience. The firsthalf (and sometimes more) comes by nature. To this end I smelt chemicals, learned that they were different kinds of _gin_, saw young wags try to kissthe girls under the excuse of what was called _laughing gas_--which I wassure {269} was not to blame for more than five per cent of the requisiteassurance--and so forth. This was all well so far as it went; but there wasalso the excessive notion of creative power exhibited in the millions ofmiles of the solar system, of which power I wondered they did not give astill grander idea by expressing the distances in inches. But even this wasnothing to the ingenious contrivance of the centrifugal force. "You haveheard what I have said of the wonderful centripetal force, by which DivineWisdom has retained the planets in their orbits round the Sun. But, ladiesand gentlemen, it must be clear to you that if there were no other force inaction, this centripetal force would draw our earth and the other planetsinto the Sun, and universal ruin would ensue. To prevent such acatastrophe, the same wisdom has implanted a centrifugal force of the sameamount, and directly opposite, " etc. I had never heard of Alfonso X ofCastile, [427] but I ventured to think that if Divine Wisdom had just letthe planets alone it would come to the same thing, with equal and oppositetroubles saved. The paradoxers deal largely in speculation conducted uponthe above explanation. They provide external agents for what they call thecentrifugal force. Some make the sun's rays keep the planets off, without athought about what would become of our poor eyes if the _push_ of the lightwhich falls on the earth were a counterpoise to all its gravitation. Thetrue explanation cannot be given here, for want of room. 

CAMBRIDGE POETS. 

Sometimes a person who has a point to carry will assert a singular fact orprediction for the sake of his point; and {270} this paradox has almostobtained the sole use of the name. Persons who have reputation to care forshould beware how they adopt this plan, which now and then eventuates aspanker, as the American editor said. Lord Byron, in "English Bards, etc. "(1809), ridiculing Cambridge poetry, wrote as follows:

 "But where fair Isis rolls her purer wave, The partial muse delighted loves to lave; On her green banks a greener wreath she wove, To crown the bards that haunt her classic grove; Where Richards[428] wakes a genuine poet's fires, And modern Britons glory in their sires. "[429]

There is some account of the Rev. Geo. Richards, Fellow of Oriel and Vicarof Bampton, (M. A. In 1791) in the _Living Authors_ by Watkins[430] andShoberl[431] (1816). In Rivers's _Living Authors_, of 1798, which is bestfitted for citation, as being published before Lord Byron wrote, he isspoken of in high terms. The _Aboriginal Britons_ was an Oxford (special)prize poem, of 1791. Charles Lamb mentions Richards as his school-fellow atChrist's Hospital, "author of the _Aboriginal Britons_, the most spiritedof the Oxford Prize Poems: a pale, studious Grecian. "

As I never heard of Richards as a poet, [432] I conclude that his fame isdefunct, except in what may prove to be a very ambiguous kind ofimmortality, conferred by Lord Byron. The awkwardness of a case which timehas broken down {271} is increased by the eulogist himself adding sopowerful a name to the list of Cambridge poets, that his college has placedhis statue in the library, more conspicuously than that of Newton in thechapel; and this although the greatness of poetic fame had some seriousdrawbacks in the moral character of some of his writings. And it will befound on inquiry that Byron, to get his instance against Cambridge, had togo back eighteen years, passing over seven intermediate productions, ofwhich he had either never heard, or which he would not cite as waking agenuine poet's fires. 

The conclusion seems to be that the _Aboriginal Britons_ is a remarkableyouthful production, not equalled by subsequent efforts. 

To enhance the position in which the satirist placed himself, two thingsshould be remembered. First, the glowing and justifiable terms in whichByron had spoken, --a hundred and odd lines before he found it convenient tosay no Cambridge poet could compare with Richards, --of a Cambridge poet whodied only three years before Byron wrote, and produced greatly admiredworks while actually studying in the University. The fame of KirkeWhite[433] still lives; and future literary critics may perhaps compare hiswritings and those of Richards, simply by reason of the curious relation inwhich they are here placed alongside of each other. And it is much toByron's credit that, in speaking of the deceased Cambridge poet, he forgothis own argument and its exigencies, and proved himself only a paradoxer_pro re nata_. 

Secondly, Byron was very unfortunate in another passage of the same poem:

{272}

 "What varied wonders tempt us as they pass! The cow-pox, tractors, galvanism, and gas. In turns appear, to make the vulgar stare, Till the swoln bubble bursts--and all is air!"

Three of the bubbles have burst to mighty ends. The metallic tractors aredisused; but the force which, if anything, they put in action, is at thisday, under the name of mesmerism, used, prohibited, respected, scorned, assailed, defended, asserted, denied, declared utterly obscure, anduniversally known. It was hard lines to select for candidates for oblivionnot one of whom got in. I shall myself, I am assured, be some day cited forlaughing at the great discovery of ----: the blank is left for my reader tofill up in his own way; but I think I shall not be so unlucky in fourdifferent ways. 

FALSIFIED PREDICTION. 

The narration before the fact, as prophecy has been called, sometimes quiteas true as the narration after the fact, is very ridiculous when it iswrong. Why, the pre-narrator could not know; the post-narrator might haveknown. A good collection of unlucky predictions might be made: I hardlyknow one so fit to go with Byron's as that of the Rev. Daniel Rivers, already quoted, about Johnson's biographers. Peter Pindar[434] may beexcused, as personal satire was his object, for addressing Boswell and Mrs. Piozzi[435] as follows:

 "Instead of adding splendor to his name, Your books are downright gibbets to his fame; You never with posterity can thrive, 'Tis by the Rambler's death alone you live. "

But Rivers, in prose narrative, was not so excusable. He says:

{273}

"As admirers of the learning and moral excellence of their hero, we glow atalmost every page with indignation that his weaknesses and his failingsshould be disclosed to public view.... Johnson, after the luster he hadreflected on the name of Thrale ... Was to have his memory tortured andabused by her detested itch for scribbling. More injury, we will venture toaffirm, has been done to the fame of Johnson by this Lady and her latebiographical helpmate, than his most avowed enemies have been able toeffect: and if his character becomes unpopular with some of his successors, it is to those gossiping friends he is indebted for the favor. "

Poor dear old Sam! the best known dead man alive! clever, good-hearted, logical, ugly bear! Where would he have been if it had not been for Boswelland Thrale, and their imitators? What would biography have been if Boswellhad not shown how to write a life?

Rivers is to be commended for not throwing a single Stone at Mrs. Thrale'ssecond marriage. This poor lady begins to receive a little justice. Theliterary world seems to have found out that a blue-stocking dame who keepsopen house for a set among them has a right, if it so please her, to marryagain without taking measures to carry on the cake-shop. I was before myage in this respect: as a boy-reader of Boswell, and a few other thingsthat fell in my way, I came to a clearness that the conduct of societytowards Mrs. Piozzi was _blackguard_. She wanted nothing but what was inthat day a woman's only efficient protection, a male relation with a braceof pistols, and a competent notion of using them. 

BYRON AND WORDSWORTH. 

Byron's mistake about Hallam in the Pindar story may be worth placing amongabsurdities. For elucidation, suppose that some poet were now to speak--{274}

 "Of man's first disobedience, and the fruit Eve gave to Adam in his birthday suit--"

and some critic were to call it nonsense, would that critic be laughing atMilton? Payne Knight, [436] in his _Taste_, translated part of Gray's _Bard_into Greek. Some of his lines are

 [Greek: therma d' ho tengôn dakrua stonachais] [Greek: oulon melos phoberai] [Greek: êeide phônai. ]

Literally thus:

 "Wetting warm tears with groans, Continuous chant with fearful Voice he sang. "

On which Hallam remarks: "The twelfth line [our first] is nonsense. " And soit is, a poet can no more wet his tears with his groans than wet his alewith his whistle. Now this first line is from Pindar, but is only part ofthe sense; in full it is:

 [Greek: therma de tengôn dakrua stonachais] [Greek: horthion phônase. ]

Pindar's [Greek: tengôn] must be Englished by _shedding_, and he standsalone in this use. He says, "shedding warm tears, he cried out loud, withgroans. " Byron speaks of

 "Classic Hallam, much renowned for Greek:"

and represents him as criticising _the Greek_ of all Payne's lines, and notdiscovering that "the lines" were Pindar's {275} until after publication. Byron was too much of a scholar to make this blunder himself: he eitheraccepted the facts from report, or else took satirical licence. And whynot? If you want to laugh at a person, and he will not give occasion, whosefault is it that you are obliged to make it? Hallam did criticise some ofPayne Knight's Greek; but with the caution of his character, he remarkedthat possibly some of these queer phrases might be "critic-traps" justifiedby some one use of some one author. I remember well having a Latin essay towrite at Cambridge, in which I took care to insert a few monstrous andunusual idioms from Cicero: a person with a Nizolius, [437] and withoutscruples may get scores of them. So when my tutor raised his voice againstthese oddities, I was up to him, for I came down upon him with Cicero, chapter and verse, and got round him. And so my own solecisms, many ofthem, passed unchallenged. 

Byron had more good in his nature than he was fond of letting out: whetherhe was a soured misanthrope, or whether his _vein_ lay that way in poetry, and he felt it necessary to fit his demeanor to it, are matters far beyondme. Mr. Crabb Robinson[438] told me the following story more than once. Hewas at Charles Lamb's chambers in the Temple when Wordsworth came in, withthe new _Edinburgh Review_ in his hand, and fume on his countenance. "Thesereviewers, " said he, "put me out of patience! Here is a young man--they sayhe is a lord--who has written a volume of poetry; and these fellows, justbecause he is a lord, set upon him, laugh at him, and sneer at his writing. The young man will do something, if he goes on as he has begun. But thesereviewers seem to think {276} that nobody may write poetry, unless he livesin a garret. " Crabb Robinson told this long after to Lady Byron, who said, "Ah! if Byron had known that, he would never have attacked Wordsworth. Hewent one day to meet Wordsworth at dinner; when he came home I said, 'Well, how did the young poet get on with the old one?' 'Why, to tell you thetruth, ' said he, 'I had but one feeling from the beginning of the visit tothe end, and that was--_reverence_!'" Lady Byron told my wife that herhusband had a very great respect for Wordsworth. I suppose he would havesaid--as the Archangel said to his Satan--"Our difference is po[li =e]tical. "

I suspect that Fielding would, if all were known, be ranked among theunlucky railers at supposed paradox. In his _Miscellanies_ (1742, 8vo) hewrote a satire on the Chrysippus or Guinea, an animal which multipliesitself by division, like the polypus. This he supposes to have been drawnup by Petrus Gualterus, meaning the famous usurer, Peter Walter. He callsit a paper "proper to be read before the R----l Society": and next year, 1743, a quarto reprint was made to resemble a paper in the _PhilosophicalTransactions_. So far as I can make out, one object is ridicule of what thezoologists said about the polypus: a reprint in the form of the_Transactions_ was certainly satire on the Society, not on Peter Walter andhis knack of multiplying guineas. 

Old poets have recognized the quadrature of the circle as a well-knowndifficulty. Dante compares himself, when bewildered, to a geometer whocannot find the principle on which the circle is to be measured:

 "Quale è 'l geometra che tutto s' affige Per misurar lo cerchio, e non ritruova, Pensando qual principio ond' egli indige. "[439]

{277} And Quarles[440] speaks as follows of the _summum bonum_:

 "Or is't a tart idea, to procure An edge, and keep the practic soul in ure, Like that dear chymic dust, or puzzling quadrature?"

The poetic notion of the quadrature must not be forgotten. Aristophanes, inthe _Birds_, introduces a geometer who announces his intention to _make asquare circle_. Pope, in the _Dunciad_, delivers himself as follows, with aGreek pronunciation rather strange in a translator of Homer. Probably Poperecognized, as a general rule, the very common practice of throwing backthe accent in defiance of quantity, seen in o´rator, au´ditor, se´nator, ca´tenary, etc. 

 "Mad _Mathesis_ alone was unconfined, Too mad for mere material chains to bind, -- Now to pure space lifts her ecstatic stare, Now, running round the circle, finds it square. "

The author's note explains that this "regards the wild and fruitlessattempts of squaring the circle. " The poetic idea seems to be that thegeometers try to make a square circle. Disraeli quotes it as "finds _its_square, " but the originals do not support this reading. 

DE BECOURT. 

I have come in the way of a work, entitled _The Grave of HumanPhilosophies_ (1827), translated from the French of R. De Bécourt[441] byA. Dalmas. It supports, but I suspect not very accurately, the views of theold Hindu books. {278} That the sun is only 450 miles from us, and only 40miles in diameter, may be passed over; my affair is with the state of mindinto which persons of M. Bécourt's temperament are brought by a fancy. Hefully grants, as certain, four millions of years as the duration of theHindu race, and 1956 as that of the universe. It must be admitted he is notwholly wrong in saying that our errors about the universe proceed from ourignorance of its origin, antiquity, organization, laws, and finaldestination. Living in an age of light, he "avails himself of thatopportunity" to remove this veil of darkness, etc. The system of theBrahmins is the only true one: he adds that it has never before beenattempted, as it could not be obtained except by him. The author requestsus first, to lay aside prejudice; next, to read all he says in the order inwhich he says it: we may then pronounce judgment upon a work which beginsby taking the Brahmins for granted. All the paradoxers make the samerequests. They do not see that compliance would bring thousands of systemsbefore the world every year: we have scores as it is. How is a poor candidinquirer to choose. Fortunately, the mind has its grand jury as well as itslittle one: and it will not put a book upon its trial without a _primafacie_ case in its favor. And with most of those who really search forthemselves, that case is never made out without evidence of knowledge, standing out clear and strong, in the book to be examined. 

BEQUEST OF A QUADRATURE. 

There is much private history which will never come to light, _caret quiavate sacro_, [442] because no Budgeteer comes across it. Many years ago aman of business, whose life was passed in banking, amused his leisure withquadrature, was successful of course, and bequeathed the result in a sealedbook, which the legatee was enjoined not to sell {279} under a thousandpounds. The true ratio was 3. 1416: I have the anecdote from the legatee'sexecutor, who opened the book. That a banker should square the circle isvery credible: but how could a City man come by the notion that a thousandpounds could be got for it? A friend of mine, one of the twins of myzodiac, will spend a thousand pounds, if he have not done it already, inblack and white cyclometry: but I will answer for it that he, a man ofsound business notions, never entertained the idea of [pi] recouping him, as they now say. I speak of individual success: of course if a company wereformed, especially if it were of unlimited lie-ability, the shares would betaken. No offence; there is nothing but what a pun will either sanctify, justify, or nullify:

 "It comes o'er the soul like the sweet South That breathes upon a bank of _vile hits_. "

The shares would be at a premium of 3-1/8 on the day after issue. If theypresented me with the number of shares I deserve, for suggestion andadvertisement, I should stand up for the Archpriest of St. Vitus[443] and3-1/5, with a view to a little more gold on the bridge. 

I now insert a couple of reviews, one about Cyclopædias, one aboutepistolary collections. Should any reader wish for explanation of thisinsertion, I ask him to reflect a moment, and imagine me set to justify allthe additions now before him! In truth these reviews are the repositoriesof many odds and ends: they were not made to the books; the materials werein my notes, and the books came as to a ready-made clothes shop, and foundwhat would fit them. Many remember Curll's[444] bequest of some very goodtitles {280} which only wanted treatises written to them. Well! here weresome tolerable reviews--as times go--which only wanted books fitted tothem. Accordingly, some tags were made to join on the books; and then asthe reader sees. 

I should find it hard to explain why the insertion is made in this placerather than another. But again, suppose I were put to make such anexplanation throughout the volume. The improver who laid out grounds andalways studied what he called _unexpectedness_, was asked what name he gaveit for those who walked over his grounds a second time. He was silenced;but I have an answer: It is that which is given by the very procedure oftaking up my book a second time. 

REVIEW OF CYCLOPÆDIAS. 

 October 19, 1861. _The English Cyclopædia. _ Conducted by Charles Knight. [445] 22 vols. : viz. , _Geography_, 4 vols. ; _Biography_, 6 vols. ; _Natural History_, 4 vols. ; _Arts and Sciences_, 8 vols. (Bradbury & Evans. )

 _The Encyclopædia Britannica: a Dictionary of Arts, Sciences, and General Literature. _ Eighth Edition. 21 vols. And Index. (Black. )

The two editions above described are completed at the same time: and theystand at the head of the two great branches into which pantologicalundertakings are divided, as at once the largest and the best of theirclasses. 

When the works are brought together, the first thing that strikes the eyeis the syllable of difference in the names. The word _Cyclopædia_ is a bitof modern purism. Though [Greek: enkuklopaideia][446] is not absolutelyGreek of Greece, we learn from both Pliny[447] and Quintilian[448] that thecircle {281} of the sciences was so called by the Greeks, andVitruvius[449] has thence naturalized _encyclium_ in Latin. Nevertheless weadmit that the initial _en_ would have euphonized but badly with the word_Penny_: and the _English Cyclopædia_ is the augmented, revised, anddistributed edition of the _Penny Cyclopædia_. It has indeed been said thatCyclopædia should mean the education _of_ a circle, just as Cyropædia isthe education _of_ Cyrus. But this is easily upset by Aristotle's word[Greek: kuklophoria], [450] motion _in_ a circle, and by many other cases, for which see the lexicon. 

The earliest printed Encyclopædia of this kind was perhaps the famous"myrrour of the worlde, " which Caxton[451] translated from the French andprinted in 1480. The original Latin is of the thirteenth century, orearlier. This is a collection of very short treatises. In or shortly after1496 appeared the _Margarita Philosophica_ of Gregory Reisch, [452] the samewe must suppose, who was confessor to the Emperor Maximilian. [453] This isagain a collection of treatises, of much more pretension: and theestimation formed of it is proved by the number of editions it wentthrough. In 1531 appeared the little collection of _works_ ofRingelberg, [454] which is truly called an Encyclopædia by {282} Morhof, though the thumbs and fingers of the two hands will meet over the length ofits one volume. There are more small collections; but we pass on to thefirst work to which the name of _Encyclopædia_ is given. This is aponderous _Scientiarum Omnium Encyclopædia_ of Alsted, [455] in four foliovolumes, commonly bound in two: published in 1629 and again in 1649; thetrue parent of all the Encyclopædias, or collections of treatises, or worksin which that character predominates. The first great _dictionary_ mayperhaps be taken to be Hofman's _Lexicon Universale_[456] (1677); butChambers's[457] (so called) _Dictionary_ (1728) has a better claim. And wesupport our proposed nomenclature by observing that Alsted accidentallycalled his work _En_cyclopædia, and Chambers simply Cyclopædia. 

We shall make one little extract from the _myrrour_, and one fromRingelberg. Caxton's author makes a singular remark for his time; and onewell worthy of attention. The grammar rules of a language, he says, musthave been invented by foreigners: "And whan any suche tonge was perfytelyhad and usyd amonge any people, than other people not used to the sametonge caused rulys to be made wherby they myght lerne the same tonge ... And suche rulys be called the gramer of that tonge. " Ringelberg says thatif the right nostril bleed, the little finger of the right hand should becrooked, and squeezed with great force; and the same for the left. 

{283}

We pass on to _the_ Encyclopédie, [458] commenced in 1751; the work whichhas, in many minds, connected the word _encyclopædist_ with that of_infidel_. Readers of our day are surprised when they look into this work, and wonder what has become of all the irreligion. The truth is, that thework--though denounced _ab ovo_[459] on account of the character of itssupporters--was neither adapted, nor intended, to excite any particularremark on the subject: no work of which D'Alembert[460] was co-editor wouldhave been started on any such plan. For, first, he was a real _sceptic_:that is, doubtful, with a mind not made up. Next, he valued his quiet morethan anything; and would as soon have gone to sleep over an hornet's nestas have contemplated a systematic attack upon either religion orgovernment. As to Diderot[461]--of whose varied career of thought it isdifficult to fix the character of any one moment, but who is veryfrequently taken among us for a pure atheist--we will quote one sentencefrom the article "_Encyclopédie_, " which he wrote himself:--"Dans le moral, il n'y a que Dieu qui doit servir de modèle a 1'homme; dans les art, que lanature. "[462]

A great many readers in our country have but a very hazy idea of thedifference between the political Encyclopædia, as we may call it, and the_Encyclopédie Méthodique_, [463] which we always take to be meant--whetherrightly or not we cannot tell--when we hear of the "great FrenchEncyclopædia. " This work, which takes much from its {284} predecessor, professing to correct it, was begun in 1792, and finished in 1832. Thereare 166 volumes of text, and 6439 plates, which are sometimes incorporatedwith the text, sometimes make about 40 more volumes. This is still themonster production of the kind; though probably the German Cyclopædia ofErsch and Gruber, [464] which was begun in 1818, and is still in progress, will beat it in size. The great French work is a collection ofdictionaries; it consists of Cyclopædias of all the separate branches ofknowledge. It is not a work, but a collection of works, one or anotherdepartment is to be bought from time to time; but we never heard of acomplete set for sale in one lot. As ships grow longer and longer, thequestion arises what limit there is to the length. One answer is, that itwill never do to try such a length that the stern will be rotten before theprow is finished. This wholesome rule has not been attended to in thematter before us; the earlier parts of the great French work wereantiquated before the whole were completed: something of the kind willhappen to that of Ersch and Gruber. 

The production of a great dictionary of either of the kinds is far from aneasy task. There is one way of managing the _En_cyclopædia which has beenlargely resorted to; indeed, we may say that no such work has been freefrom it. This plan is to throw all the attention upon the great treatises, and to resort to paste and scissors, or some process of equally easycharacter, for the smaller articles. However it may be done, it has beenthe rule that the Encyclopædia of treatises should have its supplementalDictionary of a very incomplete character. It is true that the treatisesare intended to do a good deal; and that the Index, if it be good, knitsthe treatises and the dictionary into one whole of reference. Still thereare two stools, and between them a great deal will fall to the ground. Thedictionary portion of the _Britannica_ is not to be compared with its {285}treatises; the part called Miscellaneous and Lexicographical in the_Metropolitana_[465] is a great failure. The defect is incompleteness. Thebiographical portion, for example, of the Britannica is very defective: ofmany names of note in literature and science, which become known to thereader from the treatises, there is no account whatever in the dictionary. So that the reader who has learnt the results of a life in astronomy, forexample, must go to some other work to know when that life began and ended. This defect has run through all the editions; it is in the casting of thework. The reader must learn to take the results at their true value, whichis not small. He must accustom himself to regard the Britannica as asplendid body of treatises on all that can be called heads of knowledge, both greater and smaller; with help from the accompanying dictionary, butnot of the most complete character. Practically, we believe, this defectcannot be avoided: two plans of essentially different structure cannot beassociated on the condition of each or either being allowed to abbreviatethe other. 

The defect of all others which it is most difficult to avoid is inequalityof performance. Take any dictionary you please, of any kind which requiresthe association of a number of contributors, and this defect must result. We do not merely mean that some will do their work better than others; thisof course: we mean that there will be structural differences of execution, affecting the relative extent of the different parts of the whole, as wellas every other point by which a work can be judged. A wise editor will notattempt any strong measures of correction: he will remember that if someportions be below the rest, which is a disadvantage, it follows that someportions must be above the rest, which is an advantage. The only practicallevel, if {286} level there must be, is that of mediocrity, if not ofabsolute worthlessness: any attempt to secure equality of strength willresult in equality of weakness. Efficient development may be cut down intomeager brevity, and in this way only can apparent equality of plan besecured throughout. It is far preferable to count upon differences ofexecution, and to proceed upon the acknowledged expectation that theprominent merits of the work will be settled by the accidental character ofthe contributors; it being held impossible that any editorial efforts cansecure a uniform standard of goodness. Wherever the greatest power isfound, it should be suffered to produce its natural effect. There are, indeed, critics who think that the merit of a book, like the strength of achain, is that of its weakest part: but there are others who know that theparallel does not hold, and who will remember that the union of manywriters must show exaggeration of the inequalities which almost alwaysexist in the production of one person. The true plan is to foster all thegood that can be got, and to give development in the directions in whichmost resources are found: a Cyclopædia, like a plant, should grow towardsthe light. 

The _Penny Cyclopædia_ had its share of this kind of defect or excellence, according to the way in which the measure is taken. The circumstance is notso much noticed as might be expected, and this because many a person is inthe habit of using such a dictionary chiefly with relation to one subject, his own; and more still want it for the pure dictionary purpose, which doesnot go much beyond the meaning of the word. But the person of full andvaried reference feels the differences; and criticism makes capital ofthem. The Useful Knowledge Society was always odious to the organs ofreligious bigotry; and one of them, adverting to the fact that geographywas treated with great ability, and most unusual fullness, in the _PennyCyclopædia_, announced it by making it the sole merit of {287} the workthat, with sufficient addition, it would make a tolerably good gazetteer. 

Some of our readers may still have hanging about them the feelings derivedfrom this old repugnance of a class to all that did not associate directdoctrinal teaching of religion with every attempt to communicate knowledge. I will take one more instance, by way of pointing out the extent to whichstupidity can go. If there be an astronomical fact of the telescopiccharacter which, next after Saturn's ring and Jupiter's satellites, wasknown to all the world, it was the existence of multitudes of double stars, treble stars, etc. A respectable quarterly of the theological cast, whichin mercy we refrain from naming, was ignorant of this commonknowledge, --imagined that the mention of such systems was a blunder of oneof the writers in the _Penny Cyclopædia_, and lashed the presumed ignoranceof the statement in the following words, delivered in April, 1837:

 "We have forgotten the name of that Sidrophel who lately discovered that the fixed stars were not single stars, but appear in the heavens like soles at Billingsgate, in pairs; while a second astronomer, under the influence of that competition in trade which the political economists tell us is so advantageous to the public, professes to show us, through his superior telescope, that the apparently single stars are really three. Before such wondrous mandarins of science, how continually must _homunculi_ like ourselves keep in the background, lest we come between the wind and their nobility. "

Certainly these little men ought to have kept in the background; but theydid not: and the growing reputation of the work which they assailed haschronicled them in literary history; grubs in amber. 

This important matter of inequality, which has led us so far, is one towhich the _Encyclopædia_ is as subject as the _Cyclopædia_; but it is notso easily recognized as a fault. {288} We receive the first book as mainlya collection of treatises: we know their authors, and we treat them asindividuals. We see, for instance, the names of two leading writers onOptics, Brewster[466] and Herschel. [467] It would not at all surprise us ifeither of these writers should be found criticising the other by name, eventhough the very view opposed should be contained in the same _Encyclopædia_with the criticism. And in like manner, we should hold it no wonder if wefound some third writer not comparable to either of those we have named. Itis not so in the _Cyclopædia_: here we do not know the author, except byinference from a list of which we never think while consulting the work. Wedo not dissent from this or that author: we blame the book. 

The _Encyclopædia Britannica_ is an old friend. Though it holds a proudplace in our present literature, yet the time was when it stood by itself, more complete and more clear than anything which was to be found elsewhere. There must be studious men alive in plenty who remember when they werestudious boys, what a literary luxury it was to pass a few days in thehouse of a friend who had a copy of this work. The present edition is aworthy successor of those which went before. The last three editions, terminating in 1824, 1842, and 1861, seem to show that a lunar cycle cannotpass without an amended and augmented edition. Detailed criticism is out ofthe question; but we may notice the effective continuance of the plan ofgiving general historical dissertations on the progress of knowledge. Ofsome of these dissertations we have had to take separate notice; and allwill be referred to in our ordinary treatment of current literature. [468]

The literary excellence of these two extensive undertakings is of the samehigh character. To many this will {289} need justification: they will noteasily concede to the cheap and recent work a right to stand on the sameshelf with the old and tried magazine, newly replenished with the best ofeverything. Those who are cognizant by use of the kind of material whichfills the _Penny Cyclopædia_ will need no further evidence: to others weshall quote a very remarkable and certainly very complete testimony. The_Cyclopædia of the Physical Sciences_, published by Dr. Nichol[469] in 1857(noticed by us, April 4), is one of the most original of our specialdictionaries. The following is an extract from the editor's preface:

 "When I assented to Mr. Griffin's proposal that I should edit such a Cyclopædia, I had it in my mind that I might make the _scissors_ eminently effective. Alas! on narrowly examining our best Cyclopædias, I found that the scissors had become blunted through too frequent and vigorous use. One great exception exists: viz. , the _Penny Cyclopædia_ of Charles Knight. [470] The cheapest and the least pretending, it is really the most philosophical of our _scientific_ dictionaries. It is not made up of a series of treatises, some good and many indifferent, but is a thorough _Dictionary_, well proportioned and generally written by the best men of the time. The more closely it is examined, the more deeply will our obligation be felt to the intelligence and conscientiousness of its projector and editor. "

After Dr. Nichol's candid and amusing announcement of his scissorialpurpose, it is but fair to state that nothing of the kind was ultimatelycarried into effect, even upon the work in which he found so much topraise. I quote this testimony because it is of a peculiar kind. 

{290}

The success of the _Penny Magazine_ led Mr. Charles Knight in 1832 topropose to the Useful Knowledge Society a Cyclopædia in weekly pennynumbers. These two works stamp the name of the projector on the literatureof our day in very legible characters. Eight volumes of 480 pages each werecontemplated; and Mr. Long[471] and Mr. Knight were to take the jointmanagement. The plan embraced a popular account of Art and Science, withvery brief biographical and geographical information. The early numbers ofthe work had some of the _Penny Magazine_ character: no one can look at thepictures of the Abbot and Abbess in their robes without seeing this. By thetime the second volume was completed, it was clearly seen that the plan wasworking out its own extension: a great development of design was submittedto, and Mr. Long became sole editor. Contributors could not be found tomake articles of the requisite power in the assigned space. One of themtold us that when he heard of the eight volumes, happening to want a shelfto be near at hand for containing the work as it went on, he ordered it tobe made to hold twenty-five volumes easily. But the inexorable logic offacts beat him after all: for the complete work contained twenty-sixvolumes and two thick volumes of Supplement. 

The penny issue was brought to an end by the state of the law, whichrequired, in 1833, that the first and last page of everything soldseparately should contain the name and address of the printer. The pennynumbers contained this imprint on the fold of the outer leaf: and _quitam_[472] informations were laid against the agents in various towns. {291}It became necessary to call in the stock; and the penny issue wasabandoned. Monthly parts were substituted, which varied in bulk, as thedemands of the plan became more urgent, and in price from one sixpence tothree. The second volume of Supplement appeared in 1846, and during thefourteen years of issue no one monthly part was ever behind its time. Thisresult is mainly due to the peculiar qualities of Mr. Long, who unites thetalents of the scholar and the editor in a degree which is altogetherunusual. If any one should imagine that a mixed mass of contributors is apunctual piece of machinery, let him take to editing upon that hypothesis, and he shall see what he shall see and learn what he shall learn. 

The _English_ contains about ten per cent more matter than the _PennyCyclopædia_ and its supplements; including the third supplementary volumeof 1848, which we now mention for the first time. The literary work of thetwo editions cost within 500l. And 50, 000l. : that of the two editions ofthe _Britannica_ cost 41, 000l. But then it is to be remembered that the_Britannica_ had matter to begin upon, which had been paid for in theformer editions. Roughly speaking, it is probable that the authorship of apage of the same size would have cost nearly the same in one as in theother. 

The longest articles in the _Penny Cyclopædia_ were "Rome" in 98 columnsand "Yorkshire" in 86 columns. The only article which can be called atreatise is the Astronomer Royal's "Gravitation, " founded on the method ofNewton in the eleventh section, but carried to a much greater extent. Inthe _English Cyclopædia_, the longest article of geography is "Asia, " in 45columns. In natural history the antelopes demand 36 columns. In biography, "Wellington" uses up 42 columns, and his great military opponent 41columns. In the division of Arts and Sciences, which includes much of asocial and commercial character, the length of articles often depends uponthe state of the {292} times with regard to the subject. Our readers wouldnot hit the longest article of this department in twenty guesses: it is"Deaf and Dumb" in 60 columns. As other specimens, we may cite Astronomy, 19; Banking, 36; Blind, 24; British Museum, 35; Cotton, 27; Drama, 26;Gravitation, 50; Libraries, 50; Painting, 34; Railways, 18; Sculpture, 36;Steam, etc. , 37; Table, 40; Telegraph, 30; Welsh language and literature, 39; Wool, 21. These are the long articles of special subdivisions: thewords under which the _En_cyclopædia gives treatises are not so prominent. As in Algebra, 10; Chemistry, 12; Geometry, 8; Logic, 14; Mathematics, 5;Music, 9. But the difference between the collection of treatises and thedictionary may be illustrated thus: though "Mathematics" have only fivecolumns, "Mathematics, recent terminology of, " has eight: and this articlewe believe to be by Mr. Cayley, [473] who certainly ought to know hissubject, being himself a large manufacturer of the new terms which heexplains. Again, though "Music" _in genere_, as the schoolmen said, hasonly nine columns, "Temperament and Tuning, " has eight, and "Chord" alonehas two. And so on. 

In a dictionary of this kind it is difficult to make a total clearance of_personality_: by which we mean that exhibition of peculiar opinion whichis offensive to taste when it is shifted from the individual on thecorporate book. The treatise of the known author may, as we have said, carry that author's controversies on its own shoulders: and even hiscrotchets, if we may use such a word. But {293} the dictionary should notput itself into antagonism with general feeling, nor even with the feelingsof classes. We refer particularly to the ordinary and editorial teaching ofthe article. If, indeed, the writer, being at issue with mankind, shouldconfess the difference, and give abstract of his full grounds, the case isaltered: the editor then, as it were, admits a correspondent to a statementof his own individual views. The dictionary portion of the Britannica isquite clear of any lapses on this point, so far as we know: the treatisesand dissertations rest upon their authors. The Penny Cyclopædia was all butclear: and great need was there that it should have been so. The UsefulKnowledge Society, starting on the principle of perfect neutrality inpolitics and religion, was obliged to keep strict watch against theentrance of all attempt even to look over the hedge. There were two--webelieve only two--instances of what we have called personality. The firstwas in the article "Bunyan. " It is worth while to extract all that issaid--in an article of thirty lines--about a writer who is all butuniversally held to be the greatest master of allegory that ever wrote:

 "His works were collected in two volumes, folio, 1736-7: among them 'The Pilgrim's Progress' has attained the greatest notoriety. If a judgment is to be formed of the merits of a book by the number of times it has been reprinted, and the many languages into which it has been translated, no production in English literature is superior to this coarse allegory. On a composition which has been extolled by Dr. Johnson, and which in our own times has received a very high critical opinion in its favor [probably Southey], it is hazardous to venture a disapproval, and we, perhaps, speak the opinion of a small minority when we confess that to us it appears to be mean, jejune and wearisome. "

--If the unfortunate critic who thus individualized himself had been asedulous reader of Bunyan, his power over {294} English would not have beenso _jejune_ as to have needed that fearful word. This little bit ofcriticism excited much amusement at the time of its publication: but it wasso thoroughly exceptional and individual that it was seldom or nevercharged on the book. The second instance occurred in the article"Socinians. " It had been arranged that the head-words of Christian sectsshould be intrusted to members of the sects themselves, on theunderstanding that the articles should simply set forth the accounts whichthe sects themselves give of their own doctrines. Thus the article on theRoman Church was written by Dr. Wiseman. [474] But the Unitarians were notallowed to come within the rule: as in other quarters, they were treated asthe gypsies of Christianity. Under the head "Socinians"--a name repudiatedby themselves--an opponent was allowed not merely to state their allegeddoctrines in his own way, but to apply strong terms, such as "audaciousunfairness, " to some of their doings. The protests which were made againstthis invasion of the understanding produced, in due time, the article"Unitarians, " written by one of that persuasion. We need not say that theseerrors have been amended in the English Cyclopædia: and our chief purposein mentioning them is to remark, that this is all we can find on the pointsin question against twenty-eight large volumes produced by an editor whosetask was monthly, and whose issue was never delayed a single hour. How muchwas arrested before publication none but himself can say. We have notalluded to one or two remonstrances on questions of absolute fact, whichare beside the present purpose. 

Both kinds of encyclopædic works have been fashioned upon predecessors, from the very earliest which had a predecessor to be founded upon; and theundertakings before us will be themselves the ancestors of a line ofsuccessors. Those who write in such collections should be {295} carefulwhat they say, for no one can tell how long a mis-statement may live. Onthis point we will give the history of a pair of epithets. When thehistorian De Thou[475] died, and left the splendid library which wascatalogued by Bouillaud[476] and the brothers Dupuis[477] (Bullialdus andPuteanus), there was a manuscript of De Thou's friend Vieta, [478] the_Harmonicon Coeleste_, of which it is on record, under Bouillaud's hand, that he himself lent it to Cosmo de' Medici, [479] to which must be addedthat M. Libri[480] found it in the Magliabecchi Library at Florence in ourown day. Bouillaud, it seems, entirely forgot what he had done. Something, probably, that Peter Dupuis said to Bouillaud, while they were at work onthe catalogue, remained on his memory, and was published by him in 1645, long after; to the effect that Dupuis lent the manuscript to Mersenne, [481]from whom it was procured by some intending plagiarist, who would not giveit back. This was repeated by Sherburne, [482] in 1675, who speaks of thework, which "being communicated to Mersennus was, by some perfidiousacquaintance of that honest-minded person, surreptitiously taken from him, and irrecoverably lost or suppressed, to the unspeakable detriment of thelettered world. " Now let the {296} reader look through the dictionaries ofthe last century and the present, scientific or general, at the article, "Vieta, " and he will be amused with the constant recurrence of"honest-minded" Mersenne, and his "surreptitious" acquaintance. We cannothave seen less than thirty copies of these epithets. 

REVIEW OF MACCLESFIELD LETTERS. 

 October 18, 1862. _Correspondence of Scientific Men of the Seventeenth Century, in the Collection of the Earl of Macclesfield. _[483] 2 vols. (Oxford, University Press. )

Though the title-page of this collection bears the date 1841, it is onlyjust completed by the publication of its Table of Contents and Index. Without these, a work of the kind is useless for consultation, and cannotmake its way. The reason of the delay will appear: its effect is well knownto us. We have found inquirers into the history of science singularlyignorant of things which this collection might have taught them. 

In the same year, 1841, the Historical Society of Science, which had but abrief existence, published a collection of letters, eighty-three in number, edited by Mr. Halliwell, [484] of English men of science, which dovetailswith the one before us, and is for the most part of a prior date. The twoshould be bound up together. The smaller collection runs from 1562 to 1682;the larger, from 1606 to past 1700. We shall speak of the two as the Museumcollection and the Macclesfield collection. And near them should be placed, in every scientific library, the valuable collection published, by Mr. Edleston, [485] for Trinity College, in 1850. 

{297}

The history of these letters runs back to famous John Collins, theattorney-general of the mathematics, as he has been called, who wrote toeverybody, heard from everybody, and sent copies of everybody's letter toeverybody else. He was in England what Mersenne[486] was in France: asearly as 1671, E. Bernard[487] addresses him as "the very Mersennus andintelligence of this age. " John Collins[488] was never more than accountantto the Excise Office, to which he was promoted from teaching writing andciphering, at the Restoration: he died in 1682. We have had a man of thesame office in our own day, the late Prof. Schumacher, [489] who made thelittle Danish Observatory of Altona the junction of all the lines by whichastronomical information was conveyed from one country to another. When thecollision took place between Denmark and the Duchies, the EnglishGovernment, moved by the Astronomical Society, instructed its diplomaticagents to represent strongly to the Danish Government, when occasion shouldarise, the great importance of the Observatory of Altona to theastronomical communications of the whole world. But Schumacher had his owncelebrated journal, the _Astronomische Nachrichten_, by which to work outpart of his plan; private correspondence was his supplementary assistant. Collins had only correspondence to rely on. Nothing is better known thanthat it was Collins's collection which furnished the materials put forwardby the Committee of the Royal Society in 1712, as a defence of Newtonagainst the partisans of Leibnitz. The noted _Commercium Epistolicum_ isbut the abbreviation of a title which runs on with "D. Johannis Collins etaliorum ... "

The whole of this collection passed into the hands of {298} WilliamJones, [490] the father of the Indian Judge of the same name, who died in1749. Jones was originally a teacher, but was presented with a valuablesinecure by the interest of George, second Earl of Macclesfield, the moverof the bill for the change of style in Britain, who died President of theRoyal Society. This change of style may perhaps be traced to the union ofenergies which were brought into concert by the accident of a commonteacher: Lord Macclesfield and Lord Chesterfield, [491] the mover and theseconder, and Daval, [492] who drew the bill, were pupils of De Moivre. [493]Jones, who was a respectable mathematician though not an inventor, collected the largest mathematical library of his day, and became possessorof the papers of Collins, which contained those of Oughtred[494] andothers. Some of these papers passed into the custody of the Royal Society:but the bulk was either bequeathed to, or purchased by, Lord Macclesfield;and thus they found their way to Shirburn Castle, where they still remain. 

A little before 1836, this collection attracted the attention of asearching inquirer into points of mathematical history, the late ProfessorRigaud, [495] who died in 1839. He examined the whole collection of letters, obtained Lord Macclesfield's consent to their publication, and induced theOxford Press to bear the expense. It must be particularly remembered thatthere still remains at Shirburn Castle a {299} valuable mass ofnon-epistolary manuscripts. So far as we can see, the best chance of afurther examination and publication lies in public encouragement of thecollection now before us: the Oxford Press might be induced to extend itsoperations if it were found that the results were really of interest to theliterary and scientific world. Rigaud died before the work was completed, and the publication was actually made by one of his sons, S. JordanRigaud, [496] who died Bishop of Antigua. But this publication was littlenoticed, for the reasons given. The completion now published consists of asufficient table of contents, of the briefest kind, by Professor De Morgan, and an excellent index by the Rev. John Rigaud. [497] The work is now fairlystarted on its career. 

If we were charged to write a volume with the title "Small things in theirconnection with great, " we could not do better than choose the small partof this collection of letters as our basis. The names, as well as thecontents, are both great and small: the great names, those which are knownto every mathematician who has any infusion of the history of his pursuit, are Briggs, [498] Oughtred, Charles Cavendish, [499] Gascoigne, [500] SethWard, [501] Wallis, [502] {300} Hu[y]gens, [503] Collins, [504] WilliamPetty, [505] Hooke, [506] Boyle, [507] Pell, [508] Oldenburg, [509]Brancker, [510] Slusius, [511] Bertit, [512] Bernard, [513] Borelli, [514]Mouton, [515] Pardies, [516] Fermat, [517] Towneley, [518] Auzout, [519] {301}D. Gregory, [520] Halley, [521] Machin, [522] Montmort, [523] Cotes, [524]Jones, [525] Saunderson, [526] Reyneau, [527] Brook Taylor, [528]Maupertuis, [529] Bouguer, [530] La Condamine, [531] Folkes, [532]Macclesfield, [533] {302} Baker, [534] Barrow, [535] Flamsteed, [536] LordBrounker, [537] J. Gregory, [538] Newton[539] and Keill. [540] To these theMuseum collection adds the names of Thomas Digges, [541] Dee, [542] TychoBrahe, [543] Harriot, [544] Lydyat, [545] Briggs, [546] Warner, [547] Tarporley, Pell, [548] Lilly, [549] Oldenburg, [550] Collins, [551] Morland. [552]

{303}

The first who appears on the scene is the celebrated Oughtred, who isrelated to have died of joy at the Restoration: but it should be added, byway of excuse, that he was eighty-six years old. He is an animal of extinctrace, an Eton mathematician. Few Eton men, even of the minority which knowswhat a sliding rule is, are aware that the inventor was of their own schooland college: but they may be excused, for Dr. Hutton, [553] so far as hisDictionary bears witness, seems not to have known it any more than they. Aglance at one of his letters reminds us of a letter from the AstronomerRoyal on the discovery of Neptune, which we printed March 20, 1847. Mr. Airy[554] there contends, and proves it both by Leverrier[555] and byAdams, [556] that the limited publication of a private letter is moreefficient than the more general publication of a printed memoir. The samemay be true of a dead letter, as opposed to a dead book. Our eye was caughtby a letter of Oughtred (1629), containing systematic use of contractionsfor the words _sine_, _cosine_, etc. , prefixed to the symbol of the angle. This is so very important a step, simple as it is, that Euler[557] isjustly held to have greatly advanced trigonometry by its introduction. Nobody that we know of has noticed that Oughtred was master of theimprovement, and willing to have taught it, if people would have learnt. After looking at his dead letter, we naturally turned to his dead book ontrigonometry, and there we found the abbreviations _s_, _sco_, _t_, _tco_, _se_, _seco_, regularly established as part of the system of the work. Butnot one of those who have investigated the contending claims of Euler andThomas {304} Simpson[558] has chanced to know of Oughtred's"Trigonometrie": and the present revival is due to his letter, not to hisbook. 

A casual reader, turning over the pages, would imagine that almost all theletters had been printed, either in the General Dictionary, or inBirch, [559] etc. : so often does the supplementary remark begin with "thisletter has been printed in ----. " For ourselves we thought, until wecounted, that a large majority of the letters had been given, either inwhole or in part. But the positive strikes the mind more forcibly than thenegative: we find that all of which any portion has been in type makes upvery little more than a quarter; the cases in which the whole letter isgiven being a minority of this quarter. The person who has been bestransacked is Flamsteed: of 36 letters from him, 34 had been previouslygiven in whole or in part. Of 59 letters to and from Newton, only 17 havebeen culled. 

The letters have been modernized in spelling, and, to some extent, inalgebraical notation; it also seems that conjectural methods of introducinginterpolations into the text have been necessary. For all this we aresorry: the scientific value of the collection is little altered, but itsliterary value is somewhat lowered. But it could not be helped: theprinters could not work from the originals, and Professor Rigaud had tocopy everything himself. A fac-simile must have been the work of more timethan he had to give: had he attempted it, his death would have cut shortthe whole undertaking, instead of allowing him to prepare everything but apreface, and to superintend the printing of one of the volumes. We may alsoadd, that we believe we have notices of _all_ the letters in theMacclesfield collection. We judge this because several which are tootrivial to print are numbered and described; and those would certainly nothave been noticed if _any_ omissions had {305} been made. And we know thatevery letter was removed from Shirburn Castle to Oxford. 

Two persons emerge from oblivion in this series of letters. The first isMichael Dary, [560] an obscure mathematician, who was in correspondence withNewton and other stars. He was a gauger at Bristol, by the interest ofCollins; afterwards a candidate for the mathematical school at Christ'sHospital, with a certificate from Newton: he was then a gunner in theTower, and is lastly described by Wallis as "Mr. Dary, the tobacco-cutter, a knowing man in algebra. " In 1674, Dary writes to Newton at Cambridge, asfollows:--"Although I sent you three papers yesterday, I cannot refrainfrom sending you this. I have had fresh thoughts this morning. " Two monthsafterwards poor Newton writes to Collins, "Mr. Dary is very solicitousabout mathematics": but in spite of the persecution, he subscribes himselfto Dary "your loving friend. " Dary's _problem_ is that of finding the rateof interest of an annuity of which the value and term are given. Dary's_theorem_, which he seems to have invented specially for the solution ofhis problem, though it is of wide range, can be exhibited to mathematicalreaders even in our columns. In modern language, it is that the limit of[phi]^{_n_}_x_, when _n_ increases without limit, is a solution of [phi]_x_= _x_. We have mentioned the I. Newton to whom Dary looked up; we add aword about the one on whom he looked down. Dr. John Newton, [561] a sedulouspublisher of logarithms, tables of interest, etc. , who began his careerbefore Isaac Newton, sometimes puzzles those who do not know him, whendescribed as I. Newton. The scientific world was of opinion that all thatwas valuable in one of his works was taken from Dary's privatecommunications. 

{306}

The second character above alluded to is one who carried mathematicalresearches a far greater length than Newton himself: the assistance whichhe rendered in this respect, even to Newton, has never been acknowledged inmodern times: though the work before us shows that his contemporaries werefully aware of it, and never thought of concealing it. In his theory ofgravitation, in which, so far as he went, we have every reason to believehe was prior to Newton, he did not extend his calculations to the distanceof the moon; his views in this matter were purely terrestrial, and led himto charge according to weight. He was John Stiles, the London and Cambridgecarrier: his name is a household word in the Macclesfield Letters, and iseven enshrined in the depths of Birch's quartos. Dary informs Newton--letus do his memory this justice--that he had paid John Stiles for thecarriage. At the time when the railroad to Cambridge was opened, acorrespondent recommended the directors, in our columns, to call an engineby the name of John Stiles, and never to let that name go off the road. Wedo not know whether the advice was followed: if not, we repeat it. 

Little points of life and manners come out occasionally. Baker, the authorof a work on algebra much esteemed at the time, wrote to Collins that theircircumstances are alike, "having a just and equal number of chargeableolive-branches, and being in the same predicament and blessed condemnationwith you, not more preaching than unpaid, and preaching the art ofcontentment to others, am forced to practise it. " But the last sentence ofhis letter runs as follows: "I have sent by the bearer ... Twentyshillings, as a token to you; desiring you to accept of it, as a smalltaste from Yours, Thos. Baker. " In our day, men of a station to pay parishtaxes do not offer their friends hard money to buy liquor. ButFlamsteed[562] writes to Collins as follows: "Last week he sent us down thecounterpart, which {307} my father has scaled, and I return up to you bythe carrier, with 5l. To be paid to Mr. Leneve for the writing, I haveadded 2s. 6d. Over, which will pay the expenses and serve to drink, withhim. " This would seem as odd to us as it would have seemed thirty years agothat half-a-crown should pay carriage for a deed from Derby to London, andleave margin for a bottle of wine: in our day, the Post-office and theFrench treaty would just manage it between them. But Flamsteed does notlimit his friend to one bottle; he adds, "If you expend more than thehalf-crown, I will make it good after Whitsuntide. " Collins does notremember exactly where he had met James Gregory, and mentions two equallylikely places thus: "Sir, it was once my good hap to meet with you in analehouse or in Sion College. " There is a little proof how universally thedinner-hour was twelve o'clock. Astronomers well know the method of findingtime by equal altitudes of the sun before and after noon: Huyghens calls it"le moyen de deux égales hauteurs du soleil devant et après _dîner_. "[563]

There is one mention of "Mr. Cocker, [564] our famous English graver andwriter, now a schoolmaster at Northampton. " This is the true Cocker: hisgenuine works are specimens of writing, such as engraved copy-books, including some on arithmetic, with copper-plate questions and space for theworking; also a book of forms for law-stationers, with specimens of legalhandwriting. It is recorded somewhere that Cocker and another, whose namewe forget, competed with the Italians in the beauty of their flourishes. This was his real fame: and in these matters he was great. The eighthedition of his book of law forms (1675), published shortly after Cocker'sdeath, has a preface signed "J. H. " This was John Hawkins, who becamepossessed of Cocker's papers--at least he said so--and {308} subsequentlyforged the famous Arithmetic, [565] a second work on Decimal Arithmetic, andan English dictionary, all attributed to Cocker. The proofs of this are setout in De Morgan's _Arithmetical Books_. Among many other corroborativecircumstances, the clumsy forger, after declaring that Cocker to his dyingday resisted strong solicitation to publish his Arithmetic, makes him writein the preface _Ille ego qui quondam_[566] of this kind: "I have beeninstrumental to the benefit of many, by virtue of those useful arts, writing and engraving; and do _now_, with the same _wonted alacrity_, castthis my arithmetical mite into the public treasury. " The book itself is notcomparable in merit to at least half-a-dozen others. How then comes Cockerto be the impersonation of Arithmetic? Unless some one can show proof, which we have never found, that he was so before 1756, the matter is to beaccounted for thus. 

Arthur Murphy, [567] the dramatist, was by taste a man of letters, and endedby being the translator of Tacitus; though many do not know that the twoare one. His friends had tried to make him a man of business; and no doubthe had been well plied with commercial arithmetic. His first dramaticperformance, the farce of "The Apprentice, " produced in 1756, is about anidle young man who must needs turn actor. Two of the best known books ofthe day in arithmetic were those of Cocker and Wingate. [568] Murphy chooses_Wingate_ to be the name of an old merchant who {309} delights in vulgarfractions, and _Cocker_ to be his arithmetical catchword--"You readShakespeare! get Cocker's Arithmetic! you may buy it for a shilling on anystall; best book that ever was wrote!" and so on. The farce became verypopular, and, as we believe, was the means of elevating Cocker to hispresent pedestal, where Wingate would have been, if his name had had thedroller sound of the two to English ears. 

A notoriety of an older day turns up, Major-General Lambert. [569] Thecommon story is that he was banished to Guernsey, where he passed thirtyyears in confinement, rearing and painting flowers. But Baker, in 1678, represents him as a prisoner at Plymouth, sending equations for solution asa challenge: probably his place of confinement was varied, and hisoccupation also. 

[General Lambert was removed to Plymouth, probably about 1668. His daughtercaptured the son of the Governor of Guernsey, who therefore probably wasreckoned an unsafe custodier thenceforward; though he assured the king thathe had turned the young couple out of doors, and had never given them apenny. Great importance was attached to Lambert's safe detention: probablythe remaining republicans looked upon him as to be their next Cromwell, ifsuch a thing were to be. There were standing orders to shoot him at once onthe first appearance of any enemy before the island. See _Notes andQueries_, 3d S. Iv. 89. ]

Collins informs James Gregory that "some of the Royal Academy wrote over toMr. Oldenburg, who was desired to impart the same to the Council of theRoyal Society, that the French King was willing to allow pensions to one ortwo learned Englishmen, but they never made any answer {310} to such aproposal. " This was written in 1671, and the thing probably happenedseveral years before. Mr. De Morgan communicated the account of theproposal to Lord Macaulay, who replied that he did not think that anyEnglishman _received_ a literary pension from Louis; but that there is acurious letter, about 1664, from the French Ambassador, in which he saysthat he has, by his master's orders, been making inquiries as to the stateof learning in England, and that he is sorry to find that the best writeris _the infamous Miltonus_. On two such independent testimonies it may beheld proved that the French King had attempted to buy a little adherencefrom English literature and science; and the silent contempt of the RoyalSociety is an honorable fact in their history. 

Another little bit of politics is as follows. Oughtred is informed that"Mr. Foster, [570] our Lecturer on Astronomy at Gresham College, is put outbecause he will not kneel down at the communion-table. A Scotsman [MungoMurray], one that is _verbi bis minister_, [571] is now lecturer in Mr. Foster's place. " Ward in his work on the Gresham Professors, [572]suppresses the reason, and the suppression lowers the character of hisbook. Foster was expelled in 1636, and re-elected on a vacancy in 1641, when Puritanism had gained strength. 

The correspondence of Newton would require deeper sifting than could begiven in such an article as the present. The first of the letters (1669) iscurious, as presenting the {311} appearance of forms belonging to the greatcalculus which, in this paragraph, we ought to call that of fluxions. Wefind, of the date February 18, 1669-70, what we believe is the earliestmanifestation of that morbid part of Newton's temperament which has been sovariously represented. He had solved a problem--being that which we havecalled Dary's--on which he writes as follows: "The solution of the annuityproblem, if it will be of any use, you have my leave to insert into the_Philosophical Transactions_, so it be without my name to it. For I see notwhat there is desirable in public esteem, were I able to acquire andmaintain it. It would perhaps increase my acquaintance, the thing which Ichiefly study to decline. "

Three letters touch upon "the experiment of glass rubbed to cause variousmotions in bits of paper underneath": they are supplements to the accountgiven by Newton to the Royal Society, and printed by Birch. It was Newton, so far as appears, who added _glass_ to the substances known to beelectric. Soon afterwards we come to a little bit of the history of theappointment to the Mint. It has appeared from the researches of late yearsthat Newton was long an aspirant for public employment: the only coolnesswhich is known to have taken place between him and Charles Montague[573][Halifax] arose out of his imagining that his friend was not in earnestabout getting him into the public service. March 14, 1696, Newton writesthus to Halley: "And if the rumour of preferment for me in the Mint shouldhereafter, upon the death of Mr. Hoar [the comptroller], or any otheroccasion, be revived, I pray that you would {312} endeavour to obviate itby acquainting your friends that I neither _put in_ for _any_ place in theMint, nor would meddle with _Mr. Hoar's place_, were it offered to me. "This means that Mr. Hoar's place had been suggested, which Newton seems tohave declined. Five days afterwards, Montague writes to Newton that he isto have the _Wardenship_. It is fair to Newton to say that in allprobability this was not--or only in a smaller degree--a question ofpersonal dignity, or of salary. It must by this time have been clear to himthat the minister, though long bound to make him an object of patronage, was actually seeking him for the Mint, because he wanted both Newton's nameand his talents for business--which he knew to be great--in the weighty anddangerous operation of restoring the coinage. It may have been, andprobably was, the case that Newton had a tolerably accurate notion of whathe would have to do, and of what degree of power would be necessary toenable him to do it in his own way. 

We have said that the non-epistolary manuscripts are still unexamined. There is a chance that one of them may answer a question of two centuries'standing, which is worth answering, because it has been so often asked. About 1640, Warner, [574] afterwards assisted by Pell, [575] commenced atable of _antilogarithms_, of the kind which Dodson[576] afterwardsconstructed anew and published. In the Museum collection there is inquiryafter inquiry from Charles Cavendish, [577] first, as to when the_Analogics_, as he called them, would be finished; next, when they would beprinted. Pell answers, in 1644, that Warner left his papers to a kinsman, who had become bankrupt, and proceeds thus:

"I am not a little afraid that all Mr. Warner's papers, {313} and no smallshare of my labours therein, are seazed upon, and most unmathematicallydivided between the sequestrators and creditors, who (not being able toballance the account where there appeare so many numbers, and much troubledat the sight of so many crosses and circles in the superstitious Algebraand that black art of Geometry) will, no doubt, determine once in theirlives to become figure-casters, and so vote them all to be throwen into thefire, if some good body doe not reprieve them for pye-bottoms, for whichpurposes you know analogicall numbers are incomparably apt, if they beaccurately calculated. "

Pell afterwards told Wallis[578] that the papers had fallen into the handsof Dr. Busby, [579] and Collins[580] writes that they were left in the handsof Dr. Thorndike, [581] a prebendary of Westminster; whence Rigaud[582]seems to say that Thorndike had left them to Dr. Busby. Birch[583] saysthat he procured for the Royal Society four boxes from Busby's trustees, containing papers of Warner and Pell: but there is no other tradition ofsuch things in the Society. But in the Birch manuscripts at the BritishMuseum, there turns up, as printed in what we call the Museum collection, alist of Warner's papers, with _Collins's_ receipt to Dr. Thorndike at thebottom, and engagement to restore them on demand. The date is December 14, 1667; Wallis's statement being in 1693. It is possible that Busby may be amistake altogether: he was very unlikely to have had charge of anymathematical papers: there may have been a confusion between the Prebendaryof Westminster and the Head Master of Westminster School. If so, in allprobability Thorndike handed {314} the cumbrous lot over to the notoriouscollector of mathematical papers, blessing himself that he got rid of themin a manner which would insure their return if he were called upon by theowners to restore them. It is much against this hypothesis that Dodson, whocertainly recalculated, can say nothing more about Warner than a repetitionof Wallis's story: though, had Collins kept the papers, they would probablyhave been in Jones's possession at the very time when Dodson, who was afriend of Jones and a user of his library, was engaged on his owncomputations. But even books, and still more manuscripts, are oftensingularly overlooked; and it remains not very improbable that Warner'stable is now at Shirburn Castle, among the unexamined manuscripts. 

CYCLOMETRY AND STEEL PENS. 

_Redit labor actus in orbem. _[584] Among the matters which have come to mesince the Budget opened, there is a pamphlet of quadrature of two pages anda half from Professor Recalcati, [585] already mentioned. It ends with"Quelque objection qu'on fasse touchant les raisonnements ci-dessus ontombera toujours dans l'absurde. "[586] A civil engineer--so he says--hasmade the quadrature "no longer a problem, but an axiom. " As follows: "Takethe quadrant of a circle whose circumference is given, square the quadrantwhich gives the true square of the circle. Because 30 ÷ 4 = 7. 5 × 7. 5 =56. 25 = the positive square of a circle whose circumference is 30. "Brevity, the soul of wit, is the "wings of mighty-winds" to quadrature, andsends it "flying all abroad. " A _surbodhicary_--something like M. A. OrLL. D. , I understand--at Calcutta, published in 1863 the division of an{315} angle into any odd number of parts, demonstration and all in--whenthe diagram is omitted--one page, good-sized, well-leaded type, smallduodecimo. But in the Preface he acknowledges "sheer inability" to executehis task. Mr. William Dean, of Todmorden, in 1863, announced 3-9/64 asproved both practically and geometrically: he has been already mentionedanonymously. Next I have the tract of Don Juan Larriva, published at Leiriain 1856, and dedicated to Queen Victoria. Mr. W. Peters, [587] alreadymentioned, who has for some months been circulating diagrams on a card, publishes (August, 1865) _The Circle Squared_. He agrees with theArchpriest of St. Vitus. He hints that a larger publication will dependpartly on the support he receives, and partly on the castigation, for whichlast, of course, he looks to me. Cyclometers have their several styles ofwit; so have anticyclometers too, for that matter. Mr. Peters will notallow me any extra-journal being: I am essentially a quotation from the_Athenæum_; "A. De Morgan" _et præterea nihil_. [588] If he had to pay forkeeping me set up, he would find out his mistake, and would be glad tocompound handsomely for a stereotype. Next comes a magnificent sheet ofpasteboard, printed on both sides. Having glanced at it and detectedquadrature, I began methodically at the beginning--"By Royal Command, " withthe lion and unicorn, and all that comes between. Mercy on us! thought I tomyself: has Her Majesty referred the question to the Judicial Committee ofthe Privy Council, where all the great difficulties go now-a-days, and isthis proclamation the result? On reading further I was relieved by findingthat the first side is entirely an advertisement of Joseph Gillott's[589]steel pens, with engraving of his {316} premises, and notice of novelapplication of his unrivalled machinery. The second side begins with "thecircle rectified" by W. E. Walker, [590] who finds [pi] =3. 141594789624155.... This is an off-shoot from an accurate geometricalrectification, on which is to be presumed Mr. Gillott's new machinery isfounded. I have no doubt that Mr. Walker's error, which is only in thesixth place of decimals, will not hurt the pens, unless it be by theslightest possible increase of the tendency to open at the points. Thisarises from Mr. Walker having rectified above proof by . 000002136034362.... 

Lastly, I, even I myself, who have long felt that I was a quadrature belowpar, have solved the problem by means which, in the present state of thelaw of libel, I dare not divulge. But the result is permitted; and it goesfar to explain all the discordances. The ratio of the circumference to thediameter is not always the same! Not that it varies with the radius; thegeometers are right enough on that point: but it varies with the time, in amanner depending upon the difference of the true longitudes of the Sun andMoon. A friend of mine--at least until he misbehaved--insisted on the meanright ascensions: but I served him as Abraham served his guest inFranklin's parable. The true formula is, A and a being the Sun's and Moon'slongitudes, [pi] = 3-13/80 + 3/80 cos(A - a). 

Mr. James Smith obtained his quadrature at full moon; the Archpriest of St. Vitus and some others at new moon. Until I can venture to publish thedemonstration, I recommend the reader to do as I do, which is to adopt3. 14159... , and to think of the matter only at the two points of the lunarmonth at which it is correct. The _Nautical Almanac_ will no doubt givethese points in a short time: I am in correspondence with the Admiralty, with nothing {317} to get over except what I must call a perverse notion onthe part of the Superintendent of the _Almanac_, who suspects onecorrection depending on the Moon's latitude; and the Astronomer Royal leanstowards another depending on the date of the Queen's accession. I have nopatience with these men: what can the Moon's node of the Queen's reignpossibly have to do with the ratio in question? But this is the way withall the regular men of science; Newton is to them etc. Etc. Etc. Etc. 

The following method of finding the circumference of a circle (taken from apaper by Mr. S. Drach[591] in the _Phil. Mag. _, Jan. 1863, Suppl. ) is asaccurate as the use of 3. 14159265. From three diameters deduct8-thousandths and 7-millionths of a diameter; to the result add five percent. We have then not quite enough; but the shortcoming is at the rate ofabout an inch and a sixtieth of an inch in 14, 000 miles. 

JACOB BEHMEN. 

Though I have met with nothing but a little tract from the school of JacobBehmen[592] (or Böhme; I keep to the old English version of his name), yetthere has been more, and of a more recent date. I am told of an"Introduction to Theosophy [_Theo_ private, I suppose, as in theological];or, the Science of the Mystery of Christ, " published in 1854, mostly fromthe writings of William Law[593]: and also of a volume of 688 pages, of thesame year, printed for private circulation, containing notes for abiography of William Law. The editor of the first work wishes to grow "a{318} generation of perfect Christians" by founding a Theosophic College, for which he requests the public to raise a hundred thousand pounds. Thereis a good account of Jacob Behmen in the _Penny Cyclopædia_. The authormentions inaccurate accounts, one of which he quotes, as follows: "Hederived all his mystical and rapturous doctrine from Wood's[594] _AthenæOxonienses_, Vol. I, p. 610, and _Hist. Et Antiq. Acad. Oxon. _, Vol. II, p. 308. " On which the author remarks that Wood was born after Behmen's death. There must have been a few words which slipped out: what is meant is thatBehmen "derived his doctrine from _Robert Fludd_, [595] _for whom see_Wood's etc. Etc. " Even this is absurd enough: for Behmen began to publishin 1610, and Fludd in 1616. Fludd was a Rosicrucian, and a mystic of adifferent type from Behmen. I have some of his works, and could produce outof them paradoxes enough, according to our ways of thinking, to fit out ahost. But the Rosicrucian system was a recognized school of its day, andFludd, a man of great learning, had abettors enough in all which headvanced, and predecessors in most of it. 

[A Correspondent has recently sent a short summary of the claims of JacobBehmen to rank higher than I have placed him. I shall gladly insert thissummary in the book I contemplate, as a statement of what is said of Behmenfar less liable to suspicion of exaggeration than anything I could write. Ishall add a few extracts from Behmen himself, in support of his right to bein my list. ]

"_Jacob Behmen. _--That Prof. De Morgan classes Jacob Behmen amongparadoxers can only be attributed to the fact of his being avowedlyunacquainted with the writings {319} of that author. Perhaps you may thinka few words from one who knows them well of sufficient interest to thelearned Professor, and your readers in general, to be worthy of space inyour columns. The metaphysical system of Behmen--the most perfect and onlytrue one--still awaits a qualified commentator. Behmen's countryman, Dionysius Andreas Freher, [596] who spent the greater part of his life inthis country, and whose exposition of Behmen exists only in MS. , fillingmany volumes, written in English, with the exception of two, written inGerman, with numerous beautiful, highly ingenious, and elaborateillustrations, --copies of some of which are in the British Museum, but allthe originals of which are in the possession of the gentleman who is theeditor of the two works alluded to by Professor De Morgan, --this Freher wasthe first to philosophically expound Behmen's system, which was afterwards, with the help of these MSS. , as it were, popularized by William Law; butboth Freher and Law confined themselves chiefly to its theological aspect. In Behmen, however, is to be found, not only the true ground of alltheology, but also that of all physical science. He demonstrated with afullness, accuracy, completeness and certainty that leave nothing to bedesired, the innermost ground of Deity and Nature; and, confining myself tothe latter, I can from my own knowledge assert, that in Behmen's writingsis to be found the true and clear demonstration of every physical fact thathas been discovered since his day. Thus, the science of electricity, whichwas not yet in existence when he wrote, is there anticipated; and not onlydoes Behmen describe all the now known phenomena of that force, but he evengives us the origin, generation and birth of electricity itself. Again, positive evidence can be adduced that Newton derived all his knowledge ofgravitation and its {320} laws from Behmen, with whom gravitation orattraction is, and very properly so, as he shows us, the first of the sevenproperties of Nature. The theory defended by Mr. Grove, [597] at theNottingham meeting of last year, that all the apparently distinct causes ofmoral and physical phenomena are but so many manifestations of one centralforce, and that Continuity is the law of nature, is clearly laid down, andits truth demonstrated, by Behmen, as well as the distinction betweenspirit and matter, and that the moral and material world is pervaded by asublime unity. And though all this was not admitted in Behmen's days, because science was not then sufficiently advanced to understand the deepsense of our author, many of his passages, then unintelligible, orapparently absurd, read by the light of the present age, are found tocontain the positive enunciation of principles at whose discovery andestablishment science has only just arrived by wearisome and painfulinvestigations. Every new scientific discovery goes to prove his profoundand intuitive insight into the most secret workings of nature; and ifscientific men, instead of sharing the prejudice arising from ignorance ofBehmen's system, would place themselves on the vantage ground it affords, they would at once find themselves on an eminence whence they could beholdall the arcana of nature. Behmen's system, in fact, shows us the _inside_of things, while modern physical science is content with looking at the_outside_. Behmen traces back every outward manifestation or development toits one central root, --to that one central energy which, as yet, is onlysuspected; every link in the chain of his demonstration is perfect, andthere is not one link wanting. He carries us from the out-births of thecircumference, along the radius to the center, {321} or point, and beyondthat even to the zero, demonstrating the constitution of the zero, ornothing, with mathematical precision. C. W. H. "

And so Behmen is no subject for the Budget! I waited until I should chanceto light on one of his volumes, knowing that any volume would do, andalmost any page. My first hap was on the second volume of the edition of1664 (4to, published by M. Richardson) and opening near the beginning, aturn or two brought me to page 13, where I saw about _sulphur_ and_mercurius_ as follows:

"Thus SUL is the soul, in an herb it is the oil, and in man also, accordingto the spirit of _this_ world in the third principle, which is continuallygenerated out of the anguish of the will in the mind, and theBrimstone-worm is the Spirit, which hath the fire and _burneth_: PHUR isthe sour wheel in itself which causeth that. 

"_Mercurius_ comprehendeth all the four forms, even as the life springethup, and yet hath not its dark beginning in the Center as the PHUR hath, butafter the flash of fire, when the sour dark form is terrified, where thehardness is turned into pliant sharpness, and where the second will (_viz. _the will of nature, which is called the Anguish) ariseth, there Mercuriushath its original. For MER is the shivering wheel, very horrible, sharp, venomous, and hostile; which assimulateth it thus in the sourness in theflash of fire, where the sour wrathful life _ariseth_. The syllable CU isthe pressing out, of the _Anxious_ will of the mind, from Nature: which isclimbing up, and _willeth_ to be out aloft. RI is the comprehension of theflash of fire, which in MER giveth a clear sound and tune. For the flashmaketh the tune, and it is the Salt-Spirit which _soundeth_, and its form(or quality) is gritty like sand, and herein arise noises, sounds andvoices, and thus CU comprehendeth the flash, and so the pressure is as a_wind_ which thrusteth, and giveth a spirit to the flash, so that it livethand burneth. Thus the {322} syllable US is called the burning fire, whichwith the spirit continually driveth itself forth: and the syllable CUpresseth continually upon the flash. "

Shades of Tauler[598] and Paracelsus, [599] how strangely you do mix! Wellmay Hallam call Germany the native soil of Mysticism. Had Behmen been theleast of a scholar, he would not have divided _sulph-ur_ and _merc-ur-i-us_as he has done: and the inflexion _us_, that boy of all work, would havebeen rejected. I think it will be held that a writer from whom hundreds ofpages like the above could be brought together, is fit for the Budget. IfSampson Arnold Mackay[600] had tied his etymologies to a mysticalChristology, instead of a mystical infidelity, he might have had a schoolof followers. The nonsense about Newton borrowing gravitation from Behmenpasses only with those who know neither what Newton did, nor what was donebefore him. 

The above reminds me of a class of paradoxers whom I wonder that I forgot;they are without exception the greatest bores of all, because they can putthe small end of their paradox into any literary conversation whatever. Imean the people who have heard the local pronunciation of celebrated names, and attempt not only to imitate it, but to impose on others their brokenGerman or Arabic, or what not. They also learn the vernacular names ofthose who are generally spoken of in their Latin forms; at least, theylearn a few cases, and hawk them as evidences of erudition. They aremiserably mistaken: scholarship, as a rule, {323} always accepts thevernacular form of a name which has vernacular celebrity. Hallam writesBehmen: his index-maker, rather superfluously, gives "_Behmen_ or Boehm. "And he retains Melanchthon, [601] the name given by Reuchlin[602] to hislittle kinsman Schwartzerd, because the world has adopted it: but he willnone of Capnio, the name which Reuchlin fitted on to himself, because theworld has not adopted it. He calls the old forms pedantry: but he sees thatthe rejection of well-established results of pedantry would be greaterpedantry still. The paradoxers assume the question that it is more_correct_ to sound a man by lame imitation of his own countrymen than asusual in the country in which the sound is to be made. Against them are, first, the world at large; next, an overpowering majority of those who knowsomething about surnames and their history. Some thirty years ago--afact--there appeared at the police-office a complainant who found his ownlaw. In the course of his argument, he asked, "What does Kittysay?"--"Who's Kitty?" said the magistrate, "your wife, or yournurse?"--"Sir! I mean Kitty, the celebrated lawyer. "--"Oh!" said themagistrate, "I suspect you mean Mr. Chitty, [603] the author of the greatwork on pleading. "--"I do sir! But Chitty is an Italian name, and ought tobe pronounced _Kitty_. " This man was a full-blown flower: but there is manya modest bud; and all ought either to blush when seen or to waste theirpronunciation on the desert air. 

{324}

A PLEA FOR KING CUSTOM. 

I stand up for King Custom, or _Usus_, as Horace called him, with whom is_arbitrium_ the decision, and _jus_ the right, and _norma_ the way ofdeciding, simply because he has _potestas_ the power. He may admit one andanother principle to advise: but Custom is not a constitutional king; hemay listen to his cabinet, but he decides for himself: and if the ministryshould resign, he blesses his stars and does without them. We have aglorious liberty in England of owning neither dictionary, grammar, norspelling-book: as many as choose write by either of the three, and decideall disputed points their own way, those following them who please. 

Throughout this book I have called people by the names which denote them intheir books, or by our vernacular names. This is the intelligible way ofproceeding. I might, for instance (Vol. I, p. 44), have spoken of Charlesde Bovelles, [604] of Lefèvre d'Étaples, [605] of Pèlerin, [606] and ofEtienne. [607] But I prefer the old plan. Those who like another planbetter, are welcome to substitute with a pen, when they know what to write;when they do not, it is clear that they would not have understood me if Ihad given modern names. 

The principal advisers of King Custom are as follows. First, there isEtymology, the _chiffonnier_, or general rag-merchant, who has made such afortune of late years in his own business that he begins to be consideredhighly respectable. He gives advice which is more thought of than followed, partly on account of the fearful extremes into which he runs. He latelyasked some boys of sixteen, at a matriculation examination in _English_, towhat branch of {325} the Indo-Germanic family they felt inclined to referthe Pushto language, and what changes in the force of the letters tookplace in passing from Greek into Moeso-Gothic. Because all syllables wereonce words, he is a little inclined to insist that they shall be so still. He would gladly rule English with a Saxon rod, which might be permittedwith a certain discretion which he has never attained: and when opposed, hedefends himself with analogies of the Aryan family until those who hear himlong for the discovery of an Athanasyus. He will transport a word beyondseas--he is recorder of Rhematopolis--on circumstantial evidence whichlooks like mystery gone mad; but, strange to say, something very oftencomes to light after sentence is passed which proves the soundness of theconviction. 

The next adviser is Logic, a swearing old justice of peace, quorum, androtulorum, whose excesses brought on such a fit of the gout that for manyyears he was unable to move. He is now mending, and his friends say he hassown his wild oats. He has some influence with the educated subjects ofCustom, and will have more, if he can learn the line at which interferenceought to stop: with them he has succeeded in making an affirmative of twonegatives; but the vulgar won't never have nothing to say to him. He hasalways railed at Milton for writing that Eve was the fairest of herdaughters; but has never satisfactorily shown what Milton ought to havesaid instead. 

The third adviser has more influence with the mass of the subjects of KingCustom than the other two put together; his name is Fiddlefaddle, thetoy-shop keeper; and the other two put him forward to do their worst work. In return, he often uses their names without authority. He took Etymologyto witness that _means_ to an end must be plural: and he would have any onemethod to be a _mean_. But Etymology proved him wrong, King Custom referredhim to his Catechism, in which is "a means whereby we receive the same, "and Analogy--a subordinate of {326} Etymology--asked whether he thought ita great _new_ to hear that he was wrong. It was either this Fiddlefaddle, or Lindley Murray[608] his traveler, who persuaded the Miss Slipslops, ofthe Ladies Seminary, to put "The Misses Slipslop" over the gate. Sixtyyears ago, this bagman called at all the girls' schools, and got many ofthe teachers to insist on the pupils saying "Is it not" and "Can I not" for"Isn't it" and "Can't I": of which it came that the poor girls weredreadfully laughed at by their irreverent brothers when they went home forthe holidays. Had this bad adviser not been severely checked, he might bythis time have proposed our saying "The Queen's of England son, " declaring, in the name of Logic, that the prince was the Queen's son, not England's. 

Lastly, there is Typography the metallurgist, an executive officer who isalways at work in secret, and whose lawless mode of advising is often doneby carrying his notions into effect without leave given. He it is who neverceases suggesting that the same word is not to occur in a second placewithin sight of the first. When the Authorized Version was first printed, he began this trick at the passage, "Let there be light, and there waslight;" he drew a line on the proof under the second _light_, and wrote"_luminosity?_" opposite. He is strongest in the punctuations and othersigns; he has a pepper-box full of commas always by his side. He putseverything under marks of quotation which he has ever heard before. Anearnest preacher, in a very moving sermon, used the phrase Alas! and alacka day! Typography stuck up the inverted commas because he had read the oldAnglo-Indian toast, "A lass and a lac a day!" If any one should have thesense to leave out of his Greek {327} the unmeaning scratches which theycall accents, he goes to a lexicon and puts them in. He is powerful inroutine; but when two routines interlace or overlap, he frequently takesthe wrong one. 

Subject to bad advice, and sometimes misled for a season, King Custom goeson his quiet way and is sure to be right at last. 

 "Treason does never prosper: what's the reason? Why, when it prospers, none dare call it treason. "

Language is in constant fermentation, and all that is thrown in, so far asit is not fit to assimilate, is thrown off; and this without any obviousstruggle. In the meanwhile every one who has read good authors, fromShakspeare downward, knows what is and what is not English; and knows, also, that our language is not one and indivisible. Two very differentturns of phrase may both be equally good, and as good as can be: we may berelieved of the consequences of contempt of one court by _habeas corpus_issuing out of another. 

TEST OF LANGUAGE. 

Hallam remarks that the Authorized Version of the Bible is not in thelanguage of the time of James the First: that it is not the English ofRaleigh or of Bacon. Here arises the question whether Raleigh and Bacon arethe true expositors of the language of their time; and whether they werenot rather the incipient promoters of a change which was successfullyresisted by--among other things--the Authorized Version of the Testaments. I am not prepared to concede that I should have given to the English whichwould have been fashioned upon that of Bacon by imitators, such as theyusually are, the admiration which is forced from me by Bacon's English fromBacon's pen. On this point we have a notable parallel. Samuel Johnson {328}commands our admiration, at least in his matured style: but we nauseate hisfollowers. It is an opinion of mine that the works of the leading writersof an age are seldom the proper specimens of the language of their day, when that language is in its state of progression. I judge of a language bythe colloquial idiom of educated men: that is, I take this to be the bestmedium between the extreme cases of one who is ignorant of grammar and onewho is perched upon a style. Dialogue is what I want to judge by, and plaindialogue: so I choose Robert Recorde[609] and his pupil in the _Castle ofKnowledge_, written before 1556. When Dr. Robert gets into his altitudes ofinstruction, he differs from his own common phraseology as much as probablydid Bacon when he wrote morals and philosophy. But every now and then Icome to a little plain talk about a common thing, of which I propose toshow a specimen. Anything can be made to look old by such changes as_makes_ into _maketh_, with a little old spelling. I shall invert thesechanges, using the newer form of inflexion, and the modern spelling: withno other variation whatever. 

"_Scholar. _ Yet the reason of that is easy enough to be conceived, for whenthe day is at the longest the Sun must needs shine the more time, and somust it needs shine the less time when the day is at the shortest: thisreason I have heard many men declare. 

_Master. _ That may be called a crabbed reason, for it {329} goes backwardlike a crab. The day makes not the Sun to shine, but the Sun shining makesthe day. And so the length of the day makes not the Sun to shine long, neither the shortness of the day causes not [_sic_] the Sun to shine thelesser time, but contrariwise the long shining of the Sun makes the longday, and the short shining of the Sun makes the lesser day: else answer mewhat makes the days long or short?

_Scholar. _ I have heard wise men say that Summer makes the long days, andWinter makes the long nights. 

_Master. _ They might have said more wisely, that long days make summer andshort days make winter. 

_Scholar. _ Why, all that seems one thing to me. 

_Master. _ Is it all one to say, God made the earth, and the earth made God?Covetousness overcomes all men, and all men overcome covetousness?

_Scholar. _ No, not so; for here the effect is turned to be the cause, andthe agent is made the patient. 

_Master. _ So is it to say Summer makes long days, when you should say: Longdays make summer. 

_Scholar. _ I perceive it now: but I was so blinded with the vulgar error, that if you had demanded of me further what did make the summer, I had beenlike to have answered that green leaves do make summer; and the sooner byremembrance of an old saying that a year should come in which the summershould not be known but by the green leaves. 

_Master. _ Yet this saying does not import that green leaves do make summer, but that they betoken summer; so are they the sign and not the cause ofsummer. "

I have taken a whole page of our author, without omission, that the readermay see that I do not pick out sentences convenient for my purpose. I havedone nothing but alter the third person of the verb and the spelling: butgreat is the effect thereof. We say "the Sun shining makes the day";Recorde, "the Sonne shynynge maketh the daye. " {330} These points apart, wesee a resemblance between our English and that of three hundred years ago, in the common talk of educated persons, which will allow us to affirm thatthe language of the authorized Bible must have been very close to that ofits time. For I cannot admit that much change can have taken place in fiftyyears: and the language of the version represents both our common Englishand that of Recorde with very close approximation. Take sentences fromBacon and Raleigh, and it will be apparent that these writers will be heldto differ from all three, Recorde, the version, and ourselves, bydifferences of the same character. But we speak of Recorde's conversation, and of our own. We conclude that it is the plain and almost colloquialcharacter of the Authorized Version which distinguishes it from the Englishof Bacon and Raleigh, by approximating it to the common idiom of the time. If any one will cast an eye upon the letters of instruction written byCecil[610] and the Bishop of London to the translators themselves, or tothe general directions sent to them in the King's name, he will find thatthese plain business compositions differ from the English of Bacon andRaleigh by the same sort of differences which distinguish the versionitself. 

PRONUNCIATION. 

The foreign word, or the word of a district, or class of people, passesinto the general vernacular; but it is long before the specially learnedwill acknowledge the right of those with whom they come in contact tofollow general usage. The rule is simple: so long as a word is technical orlocal, those who know its technical or local pronunciation may reasonablyemploy it. But when the word has become general, the specialist is not verywise if he refuse to follow {331} the mass, and perfectly foolish if heinsist on others following him. There have been a few who demanded thatEuler should be pronounced in the German fashion:[611] Euler has long beenthe property of the world at large; what does it matter how his owncountrymen pronounce the letters? Shall we insist on the French pronouncing_Newton_ without that final _tong_ which they never fail to give him? Theywould be wise enough to laugh at us if we did. We remember that a pedantwho was insisting on all the pronunciations being retained, was met by amaxim in contradiction, invented at the moment, and fathered uponKaen-foo-tzee, [612] an authority which he was challenged to dispute. Whomdid you speak of? said the bewildered man of accuracy. Learn your ownsystem, was the answer, before you impose it on others; Confucius says thattoo. [613]

The old English has _fote_, _fode_, _loke_, _coke_, _roke_, etc. , for_foot_, etc. And _above_ rhymes in Chaucer to _remove_. Suspecting that thebroader sounds are the older, we may surmise that _remove_ and _food_ haveretained their old sounds, and that _cook_, once _coke_, would have rhymedto our _Luke_, the vowel being brought a little nearer, perhaps, to the _o_in our present _coke_, the fuel, probably so called as used by cooks. Ifthis be so, the Chief Justice _Cook_[614] of our lawyers, and the _Coke_(pronounced like the fuel) of the greater part of the world, are equallywrong. The lawyer has no right whatever to fasten his pronunciation uponus: even leaving aside the general custom, he cannot prove himself right, and is probably wrong. Those who {332} know the village of Rokeby(pronounced Rookby) despise the world for not knowing how to name WalterScott's poem: that same world never asked a question about the matter, andthe reception of the parody of _Jokeby_, which soon appeared, was asufficient indication of their notion. Those who would fasten the hodiernalsound upon us may be reminded that the question is, not what they call itnow, but what it was called in Cromwell's time. Throw away general usage asa lawgiver, and this is the point which emerges. Probably _R[=u]ke-by_would be right, with a little turning of the Italian [=u] towards [=o] ofmodern English. 

[Some of the above is from an old review. I do not always notice suchinsertions: I take nothing but my own writings. A friend once said to me, "Ah! you got that out of the _Athenæum_!" "Excuse me, " said I. "the_Athenæum_ got that out of me!"]

APOLOGIES TO CLUVIER. 

It is part of my function to do justice to any cyclometers whose methodshave been wrongly described by any orthodox sneerers (myself included). Inthis character I must notice _Dethlevus Cluverius_, [615] as the LeipzigActs call him (probably Dethleu Cluvier), grandson of the celebratedgeographer, Philip Cluvier. The grandson was a Fellow of the Royal Society, elected on the same day as Halley, [616] November 30, 1678: I suppose helived in England. This {333} man is quizzed in the Leipzig Acts for 1686;and, if Montucla insinuate rightly, by Leibnitz, who is further suspectedof wanting to embroil Cluvier with his own opponent Nieuwentiit, [617] onthe matter of infinitesimals. So far good: I have nothing against Leibnitz, who though he was ironical, told us what he laughed at. But Montucla hasbehaved very unfairly: he represents Cluvier as placing the essence of hismethod in the solution of the problem _construere mundum divinæ mentianalogum_, to construct a world corresponding to the divine mind. Nothingto begin with: no way of proceeding. Now, it ought to have been _ex datalinea construere_, [618] etc. : there is a given line, which is something togo on. Further, there is a way of proceeding: it is to find the product of1, 2, 3, 4, etc. For ever. Moreover, Montucla charges Cluvier with_unsquaring_ the parabola, which Archimedes had squared as tight as aglove. But he never mentions how very nearly Cluvier agrees with the Greek:they only differ by 1 divided by 3n^2, where n is the infinite number ofparts of which a parabola is composed. This must have been the conceit thattickled Leibnitz, and made him wish that Cluvier and Nieuwentiit shouldfight it out. Cluvier, was admitted, on terms of irony, into the LeipzigActs: he appeared on a more serious footing in London. It is very rare forone cyclometer to refute another: _les corsaires ne se battent pas_. [619]The only instance I recall is that of M. Cluvier, who (_Phil. Trans. _, 1686, No. 185) refuted M. Mallemont de Messange, [620] who {334} publishedat Paris in 1686. He does it in a very serious style, and shows himself amathematician. And yet in the year in which, in the _Phil. Trans. _, he wasa geometer, and one who rebukes his squarer for quoting Matthew xi. 25, inthat very year he was the visionary who, in the Leipzig Acts, professed tobuild a world resembling the divine mind by multiplying together 1, 2, 3, 4, etc. Up to infinity. 

THE RAINBOW PARADOX. 

There is a very pretty opening for a paradox which has never found itsparadoxer in print. The philosophers teach that the rainbow is notmaterial: it comes from rain-drops, but those rain-drops do not _take_color. They only _give_ it, as lenses and mirrors; and each one drop gives_all_ the colors, but throws them in different directions. Accordingly, thesame drop which furnishes red light to one spectator will furnish violet toanother, properly placed. Enter the paradoxer whom I have to invent. Thephilosopher has gulled you nicely. Look into the water, and you will seethe reflected rainbow: take a looking-glass held sideways, and you seeanother reflection. How could this be, if there were nothing colored toreflect? The paradoxer's facts are true: and what are called the reflectedrainbows are _other_ rainbows, caused by those _other_ drops which areplaced so as to give the colors to the eye after reflection, at the wateror the looking-glass. A few years ago an artist exhibited a picture with arainbow and its apparent reflection: he simply copied what he had seen. When his picture was examined, some started the idea that there could be noreflection of a rainbow; they were right: they inferred that the artist hadmade a mistake; they were wrong. When it was explained, some agreed andsome dissented. Wanted, {335} immediately, an able paradoxer: testimonialsto be forwarded to either end of the rainbow, No. 1. No circle-squarer needapply, His Variegatedness having been pleased to adopt 3. 14159... From Noahdownwards. 

TYCHO BRAHE REVIVED. 

The system of Tycho Brahé, [621] with some alteration and addition, has beenrevived and contended for in our own day by a Dane, W. Zytphen, [622] whohas published _The Motion of the Sun in the Universe_, (second edition)Copenhagen, 1865, 8vo, and _Le Mouvement Sidéral_, 1865, 8vo. I make anextract. 

"How can one explain Copernically that the velocity of the Moon must beadded to the velocity of the Earth on the one place in the Earth's orbit, to learn how far the Moon has advanced from one fixed star to another; butin another place in the orbit these velocities must be subtracted (themovements taking place in opposite directions) to attain the same result?In the Copernican and other systems, it is well known that the Moon, abstracting from the insignificant excentricity of the orbit, always intwenty-four hours performs an equally long distance. Why has Copernicusnever been denominated Fundamentus or Fundator? Because he has neverconvinced anybody so thoroughly that this otherwise so natural epithet hasoccurred to the mind. "

Really the second question is more effective against Newton than againstCopernicus; for it upsets gravity: the first is of great depth. 

{336}

JAMES SMITH WILL NOT DOWN. 

The _Correspondent_ journal makes a little episode in the history of myBudget (born May, 1865, died April, 1866). It consisted entirely of letterswritten by correspondents. In August, a correspondent who signed "FairPlay"--and who I was afterwards told was a lady--thought it would be a goodjoke to bring in the Cyclometers. Accordingly a letter was written, complaining that though Mr. Sylvester's[623] demonstration of Newton'stheorem--then attracting public attention--was duly lauded, the possiblygreater discovery of the quadrature seemed to be blushing unseen, andwasting etc. It went on as follows:

"Prof. De Morgan, who, from his position in the scientific world, mightfairly afford to look favourably on less practised efforts than his own, seems to delight in ridiculing the discoverer. Science is, of course, avery respectable person when he comes out and makes himself useful in theworld [it must have been a lady; each sex gives science to the other]: butwhen, like a monk of the Middle Ages, he shuts himself up [it must havebeen a lady; they always snub the bachelors] in his cloistered cell, repeating his mumpsimus from day to day, and despising the labourers on theoutside, we begin to think of Galileo, [624] Jenner, [625] Harvey, [626] andother glorious trios, who have been contemned ... "

The writer then called upon Mr. James Smith[627] to come {337} forward. Theirony was not seen; and that day fortnight appeared the first of more thanthirty letters from his pen. Mr. Smith was followed by Mr. Reddie, [628]Zadkiel, [629] and others, on their several subjects. To some of the lettersI have referred; to others I shall come. The _Correspondent_ was to becomea first-class scientific journal; the time had arrived at which truth hadan organ: and I received formal notice that I could not stifle it bysilence, nor convert it into falsehood by ridicule. When my reader sees myextracts, he will readily believe my declaration that I should have beenthe last to stifle a publication which was every week what James Mill[630]would call a dose of capital for my Budget. A few anti-paradoxers broughtin common sense: but to the mass of the readers of the journal it allseemed to be the difference between Tweedledum and Tweedledee. Some saidthat the influx of scientific paradoxes killed the journal: but my beliefis that they made it last longer than it otherwise would have done. Twentyyears ago I recommended the paradoxers to combine and publish their viewsin a common journal: with a catholic editor, who had no pet theory, but astern determination not to exclude anything merely for absurdity. I suspectit would answer very well. A strong title, or motto, would be wanted: notso coarse as was roared out in a Cambridge mob when I was anundergraduate--"No King! No Church! No House of Lords! No nothing, blastme!"--but something on that _principle_. 

At the end of 1867 I addressed the following letter to the _Athenæum_:

PSEUDOMATH, PHILOMATH, AND GRAPHOMATH. 

_December 31, 1867_

Many thanks for the present of Mr. James Smith's letters {338} of Sept. 28and of Oct. 10 and 12. He asks where you will be if you read and digest hisletters: you probably will be somewhere first. He afterwards asks what theWE of the _Athenæum_ will be if, finding it impossible to controvert, itshould refuse to print. I answer for you, that We-We of the _Athenæum_, notbeing Wa-Wa the wild goose, so conspicuous in "Hiawatha, " will leave whatcontroverts itself to print itself, if it please. 

_Philomath_ is a good old word, easier to write and speak than_mathematician_. It wants the words between which I have placed it. Theyare not well formed: _pseudomathete_ and _graphomathete_ would be better:but they will do. I give an instance of each. 

The _pseudomath_ is a person who handles mathematics as the monkey handledthe razor. The creature tried to shave himself as he had seen his masterdo; but, not having any notion of the angle at which the razor was to beheld, he cut his own throat. He never tried a second time, poor animal! butthe pseudomath keeps on at his work, proclaims himself clean-shaved, andall the rest of the world hairy. So great is the difference between moraland physical phenomena! Mr. James Smith is, beyond doubt, the greatpseudomath of our time. His 3-1/8 is the least of a wonderful chain ofdiscoveries. His books, like Whitbread's barrels, will one day reach fromSimpkin & Marshall's to Kew, placed upright, or to Windsor laidlength-ways. The Queen will run away on their near approach, as BishopHatto did from the rats: but Mr. James Smith will follow her were it toJohn o' Groats. 

The _philomath_, for my present purpose, must be exhibited as giving alesson to presumption. The following anecdote is found in Thiébault's[631]_Souvenirs de vingt ans de séjours à Berlin_, published in 1804. The bookitself got a high character for truth. In 1807 Marshal Mollendorff[632]{339} answered an inquiry of the Duc de Bassano, [633] by saying that it wasthe most veracious of books, written by the most honest of men. Thiébaultdoes not claim personal knowledge of the anecdote, but he vouches for itsbeing received as true all over the north of Europe. [634]

Diderot[635] paid a visit to Russia at the invitation of Catherine theSecond. At that time he was an atheist, or at least talked atheism: itwould be easy to prove him either one thing or the other from his writings. His lively sallies on this subject much amused the Empress, and all theyounger part of her Court. But some of the older courtiers suggested thatit was hardly prudent to allow such unreserved exhibitions. The Empressthought so too, but did not like to muzzle her guest by an expressprohibition: so a plot was contrived. The scorner was informed that aneminent mathematician had an algebraical proof of the existence of God, which he would communicate before the whole Court, if agreeable. Diderotgladly consented. The mathematician, who is not named, was Euler. [636] Hecame to Diderot with the gravest air, and in a tone of perfect convictionsaid, "_Monsieur!_

 (a + b^n)/n = x

_donc Dieu existe; répondez!_"[637] Diderot, to whom algebra was Hebrew, though this is expressed in a very roundabout way by Thiébault--and whom wemay suppose to have expected some verbal argument of alleged algebraicalcloseness, was disconcerted; while peals of laughter sounded on all sides. Next day he asked permission to return to France, which was granted. Analgebraist would have {340} turned the tables completely, by saying, "Monsieur! vous savez bien que votre raisonnement demande le développementde x suivant les puissances entières de n". [638] Goldsmith could not haveseen the anecdote, or he might have been supposed to have drawn from it ahint as to the way in which the Squire demolished poor Moses. 

The _graphomath_ is a person who, having no mathematics, attempts todescribe a mathematician. Novelists perform in this way: even Walter Scottnow and then burns his fingers. His dreaming calculator, Davy Ramsay, swears "by the bones of the immortal Napier. " Scott thought that the thephilomaths worshiped relics: so they do, in one sense. Look intoHutton's[639] Dictionary for _Napier's Bones_, and you shall learn allabout the little knick-knacks by which he did multiplication and division. But never a bone of his own did he contribute; he preferred elephants'tusks. The author of _Headlong Hall_[640] makes a grand error, which isquite high science: he says that Laplace proved the precession of theequinoxes to be a periodical inequality. He should have said the variationof the obliquity. But the finest instance is the following: Mr. Warren, [641] in his well-wrought tale of the martyr-philosopher, wasincautious enough to invent the symbols by which his _savant_ satisfiedhimself Laplace[642] was right on a doubtful point. And this is what he puttogether--

 [sqrt]-3a^2, [rectangle]y^2 / z^2 + 9 - n = 9, n × log e. 

Now, to Diderot and the mass of mankind this might be Laplace all over:and, in a forged note of Pascal, would {341} prove him quite up togravitation. But I know of nothing like it, except in the lately receivedstory of the American orator, who was called on for some Latin, andperorated thus: "Committing the destiny of the country to your hands, Gentlemen, I may without fear declare, in the language of the noble Romanpoet, E pluribus unum, Multum in parvo, Ultima Thule, Sine qua non. "[643]

But the American got nearer to Horace than the martyr-philosopher toLaplace. For all the words are in Horace, except _Thule_, which might havebeen there. But [rectangle] is not a symbol wanted by Laplace; nor can wesee how it could have been; in fact, it is not recognized in algebra. As tothe junctions, etc. , Laplace and Horace are about equally well imitated. 

Further thanks for Mr. Smith's letters to you of Oct. 15, 18, 19, 28, andNov. 4, 15. The last of these letters has two curious discoveries. First, Mr. Smith declares that he has _seen_ the editor of the _Athenæum_: inseveral previous letters he mentions a name. If he knew a little ofjournalism he would be aware that editors are a peculiar race, obtained bynatural selection. They are never seen, even by their officials; only hearddown a pipe. Secondly, "an ellipse or oval" is composed of four arcs ofcircles. Mr. Smith has got hold of the construction I was taught, when aboy, for a pretty four-arc oval. But my teachers knew better than to callit an ellipse: Mr. Smith does not; but he produces from it suchconfirmation of 3-1/8 as would convince any _honest_ editor. 

Surely the cyclometer is a Darwinite development of a spider, who is alwaysat circles, and always begins again when his web is brushed away. Heinforms you that he {342} has been privileged to discover truths unknown tothe scientific world. This we know; but he proceeds to show that he isequally fortunate in art. He goes on to say that he will make use of you tobring those truths to light, "just as an artist makes use of a dummy forthe purpose of arranging his drapery. " The painter's lay-figure is forflowing robes; the hairdresser's dummy is for curly locks. Mr. James Smithshould read Sam Weller's pathetic story of the "four wax dummies. " As to_his_ use of a dummy, it is quite correct. When I was at UniversityCollege, I walked one day into a room in which my Latin colleague wasexamining. One of the questions was, "Give the lives and fates of Sp. Mælius, [644] and Sp. Cassius. "[645] Umph! said I, surely all know thatSpurius Mælius was whipped for adulterating flour, and that Spurius Cassiuswas hanged for passing bad money. Now, a robe arranged on a dummy wouldlook just like the toga of Cassius on the gallows. Accordingly, Mr. Smithis right in the drapery-hanger which he has chosen: he has been detected inthe attempt to pass bad circles. He complains bitterly that his geometry, instead of being read and understood by you, is handed over to me to betreated after my scurrilous fashion. It is clear enough that he wouldrather be handled in this way than not handled at all, or why does he go onwriting? He must know by this time that it is a part of the institutionthat his "untruthful and absurd trash" shall be distilled into mine at therate of about 3-1/8 pages of the first to one column of the second. Yourreaders will never know how much they gain by the process, until Mr. JamesSmith publishes it all in a big book, or until they get hold of what he hasalready published. I have six pounds avoirdupois of pamphlets and letters;and there is more than half a pound of letters {343} written to you in thelast two months. Your compositor must feel aggrieved by the rejection ofthese clearly written documents, without erasures, and on one side only. Your correspondent has all the makings of a good contributor, except theknowledge of his subject and the sense to get it. He is, in fact, only amask: of whom the fox

 "O quanta species, inquit, cerebrum non habet. "[646]

I do not despair of Mr. Smith on any question which does not involve thatunfortunate two-stick wicket at which he persists in bowling. He haspublished many papers; he has forwarded them to mathematicians: and hecannot get answers; perhaps not even readers. Does he think that he wouldget more notice if you were to print him in your journal? Who would studyhis columns? Not the mathematician, we know; and he knows. Would others?His balls are aimed too wide to be blocked by any one who is near thewicket. He has long ceased to be worth the answer which a new invader mayget. Rowan Hamilton, [647] years ago, completely knocked him over; and hehas never attempted to point out any error in the short and easy method bywhich that powerful investigator condescended to show that, be right whomay, he must be wrong. There are some persons who feel inclined to thinkthat Mr. Smith should be argued with: let those persons understand that hehas been argued with, refuted, and has never attempted to stick a pen intothe refutation. He stated that it was a remarkable paradox, easilyexplicable; and that is all. After this evasion, Mr. James Smith is belowthe necessity of being told that he is unworthy of answer. His friendscomplain that I do nothing but _chaff_ him. Absurd! I winnow him; and ifnothing but chaff results, whose fault is that? I am usefully employed: forhe is the type of a class which ought to be known, and which I have donemuch to make known. 

{344}

Nothing came of this until July 1869, when I received a reprint of theabove letter, with a comment, described as Appendix D of a work in courseof publication on the geometry of the circle. The _Athenæum_ journalreceived the same: but the Editor, in his private capacity, received thewhole work, being _The Geometry of the Circle and Mathematics as applied toGeometry by Mathematicians, shown to be a mockery, delusion, and a snare_, Liverpool, 8vo, 1869. Mr. J. S. Here appears in deep fight with ProfessorWhitworth, [648] and Mr. Wilson, [649] the author of the alleged amendment ofEuclid. How these accomplished mathematicians could be inveigled intocontinued discussion is inexplicable. Mr. Whitworth began by complaining ofMr. Smith's attacks upon mathematicians, continued to correspond after hewas convinced that J. S. Proved an arc and its chord to be equal, and onlyretreated when J. S. Charged him with believing in 3-1/8, and refusingacknowledgment. Mr. Wilson was introduced to J. S. By a volunteer defenseof his geometry from the assaults of the _Athenæum_. This the editor wouldnot publish; so J. S. Sent a copy to Mr. Wilson himself. Somecorrespondence ensued, but Mr. Wilson soon found out his man, and withdrew. 

There is a little derision of the _Athenæum_ and a merited punishment for"that unscrupulous critic and contemptible mathematical twaddler, DeMorgan. "

MR. REDDIE'S ASTRONOMY. 

At p. 183 I mentioned Mr. Reddie, [650] the author of _Vis Inertiæ Victa_and of _Victoria toto coelo_, [651] which last is not {345} an address tothe whole heaven, either from a Roman Goddess or a British Queen, whatevera scholar may suppose. Between these Mr. Reddie has published _TheMechanics of the Heavens_, 8vo, 1862: this I never saw until he sent it tome, with an invitation to notice it, he very well knowing that it wouldcatch. His speculations do battle with common notions of mathematics and ofmechanics, which, to use a feminine idiom, he blasphemes so you can'tthink! and I suspect that if you do not blaspheme them too, _you_ can'tthink. He appeals to the "truly scientific, " and would be glad to havereaders who have read what he controverts, i. E. , Newton's _Principia_: Iwish he may get them; I mean I hope he may obtain them. To none but thesewould an account of his speculations be intelligible: I accordinglydisposed of him in a very short paragraph of description. Now manyparadoxers desire notice, even though it be disparaging. I have lettersfrom more than one--besides what have been sent to the Editor of the_Athenæum_--complaining that they are not laughed at; although they deserveit, they tell me, as much as some whom I have inserted. Mr. Reddie informsme that I have not said a single word against his books, though I havegiven nearly a column to sixteen-string arithmetic, and as much toanimalcule universes. What need to say anything to readers of Newtonagainst a book from which I quoted that revolution by gravitation is_demonstrably_ impossible? It would be as useless as evidence against a manwho has pleaded guilty. Mr. Reddie derisively thanks me for "smallmercies"; he wrote me private letters; he published them, and more, in the_Correspondent_. He gave me, _pro viribus suis_, [652] such a dressing youcan't think, both for my Budget non-notice, and for reviews which heassumed me to have written. He outlawed himself by declaring(_Correspondent_, Nov. 11, 1856) that I--in a review--had made a quotationwhich was "garbled, evidently on purpose {346} to make it appear that" he"was advocating solely a geocentric hypothesis, which is not true. " Infact, he did his best to get larger "mercy. " And he shall have it; and at alength which shall content him, unless his mecometer be an insatiableapparatus. But I fear that in other respects I shall no more satisfy himthan the Irish drummer satisfied the poor culprit when, after several timeschanging the direction of the stroke at earnest entreaty, he was at lastprovoked to call out, "Bad cess to ye, ye spalpeen! strike where one will, there's no _plasing_ ye!"

Mr. Reddie attaches much force to Berkeley's[653] old arguments against thedoctrine of fluxions, and advances objections to Newton's second section, which he takes to be new. To me they appear "such as have been often made, "to copy a description given in a review: though I have no doubt Mr. Reddiegot them out of himself. But the whole matter comes to this: Mr. Reddiechallenged answer, especially from the British Association, and got none. He presumes that this is because he is right, and cannot be answered: theAssociation is willing to risk itself upon the counter-notion that he iswrong, and need not be answered; because so wrong that none who couldunderstand an answer would be likely to want one. 

Mr. Reddie demands my attention to a point which had already particularlystruck me, as giving the means of showing to _all_ readers the kind ofconfusion into which paradoxers are apt to fall, in spite of the clearestinstruction. It is a very honest blunder, and requires notice: it mayotherwise mislead some, who may suppose that no one able to read could bemistaken about so simple a matter, {347} let him be ever so wrong aboutNewton. According to his own mis-statement, in less than five months hemade the Astronomer Royal abandon the theory of the solar motion in space. The announcement is made in August, 1865, as follows: the italics are notmine:

 "The third (_Victoria ... _), although only published in September, 1863, has already had its triumph. _It is the book that forced the Astronomer Royal of England, after publicly teaching the contrary for years, to come to the conclusion, "strange as it may appear, " that "the whole question of solar motion in space is at the present time in doubt and abeyance. "_ This admission is made in the Annual Report of the Council of the Royal Astronomical Society, published in the Society's _Monthly Notices_ for February, 1864. "

It is added that solar motion is "full of self-contradiction, which "theastronomers" simply overlooked, but which they dare not now deny afterbeing once pointed out. "

The following is another of his accounts of the matter, given in the_Correspondent_, No. 18, 1865:

 "... You ought, when you came to put me in the 'Budget, ' to have been aware of the Report of the Council of the Royal Astronomical Society, where it appears that Professor Airy, [654] with a better appreciation of my demonstrations, had admitted--'strange, ' say the Council, 'as it may appear, '--that 'the whole question of solar motion in space [and here Mr. Reddie omits some words] is now in _doubt and abeyance_. ' You were culpable as a public teacher of no little pretensions, if you were 'unaware' of this. If aware of it, you ought not to have suppressed such an important testimony to my really having been 'very successful' in drawing the teeth of the pegtops, though you thought them so firmly fixed. And if you still suppress {348} it, in your Appendix, or when you reprint your 'Budget, ' you will then be guilty of a _suppressio veri_, also of further injury to me, who have never injured you.... "

Mr. Reddie must have been very well satisfied in his own mind before heventured such a challenge, with an answer from me looming in the distance. The following is the passage of the Report of the Council, etc. , from whichhe quotes:

 "And yet, strange to say, notwithstanding the near coincidence of all the results of the before-mentioned independent methods of investigation, the inevitable logical inference deduced by Mr. Airy is, that the whole question of solar motion in space, _so far at least as accounting for the proper motion of the stars is concerned_, [I have put in italics the words omitted by Mr. Reddie] appears to remain at this moment in doubt and abeyance. "

Mr. Reddie has forked me, as he thinks, on a dilemma: if unaware, culpableignorance; if aware, suppressive intention. But the thing is a _trilemma_, and the third horn, on which I elect to be placed, is surmounted by adoubly-stuffed seat. First, Mr. Airy has not changed his opinion about the_fact_ of solar motion in space, but only suspends it as to the sufficiencyof present means to give the amount and direction of the motion. Secondly, all that is alluded to in the Astronomical Report was said and printedbefore the Victoria proclamation appeared. So that the author, instead ofdrawing the tooth of the Astronomer Royal's pegtop, has burnt his owndoll's nose. 

William Herschel, [655] and after him about six other astronomers, had aimedat determining, by the proper motions of the stars, the point of theheavens towards which the solar system is moving: their results weretolerably accordant. Mr. Airy, in 1859, proposed an improved method, and, applying it to stars of large proper motion, produced {349} much the sameresult as Herschel. Mr. E. Dunkin, [656] one of Mr. Airy's staff atGreenwich, applied Mr. Airy's method to a very large number of stars, andproduced, again, nearly the same result as before. This paper was read tothe Astronomical Society in _March_, 1863, was printed in abstract in the_Notice_ of that month, was printed in full in the volume then current, andwas referred to in the Annual Report of the Council in _February_, 1864, under the name of "the Astronomer Royal's elaborate investigation, asexhibited by Mr. Dunkin. " Both Mr. Airy and Mr. Dunkin express grave doubtsas to the sufficiency of the data: and, regarding the coincidence of allthe results as highly curious, feel it necessary to wait for calculationsmade on better data. The report of the Council states these doubts. Mr. Reddie, who only published in _September_, 1863, happened to see the Reportof February, 1864, assumes that the doubts were then first expressed, anddeclares that his book of September had the triumph of forcing theAstronomer Royal to abandon the _fact_ of motion of the solar system by theFebruary following. Had Mr. Reddie, when he saw that the Council wereavowedly describing a memoir presented some time before, taken theprecaution to find out _when_ that memoir was presented, he would perhapshave seen that doubts of the results obtained, expressed by one astronomerin March, 1863, and by another in 1859, could not have been due to hispublication of September, 1863. And any one else would have learnt thatneither astronomer doubts the _solar motion_, though both doubt thesufficiency of present means to determine its _amount_ and _direction_. This is implied in the omitted words, which Mr. Reddie--whose omissionwould have been dishonest if he had seen their meaning--no doubt took forpleonasm, superfluity, overmuchness. The rashness which pushed him headlong{350} into the quillet that _his_ thunderbolt had stopped the chariot ofthe Sun and knocked the Greenwich Phaeton off the box, is the same whichbetrayed him into yet grander error--which deserves the full word, _quidlibet_--about the _Principia_ of Newton. There has been no change ofopinion at all. When a person undertakes a long investigation, his opinionis that, at a certain date, there is _prima facie_ ground for thinking asound result may be obtained. Should it happen that the investigation endsin doubt upon the sufficiency of the grounds, the investigator is not putin the wrong. He knew beforehand that there was an alternative: and hetakes the horn of the alternative indicated by his calculations. The twosides of this case present an instructive contrast. Eight astronomersproduce nearly the same result, and yet the last two doubt the sufficiencyof their means: compare them with the what's-his-name who rushes in wherething-em-bobs fear to tread. 

I was not aware, until I had written what precedes, that Mr. Airy had givena sufficient answer on the point. Mr. Reddie says (_Correspondent_, Jan. 20, 1866):

 "I claim to have forced Professor Airy to give up the notion of 'solar motion in space' altogether, for he admits it to be 'at present in doubt and abeyance. ' I first made that claim in a letter addressed to the Astronomer Royal himself in June, 1864, and in replying, very courteously, to other portions of my letter, he did not gainsay that part of it. "

Mr. Reddie is not ready at reading satire, or he never would have so missedthe meaning of the courteous reply on one point, and the total silence uponanother. Mr. Airy must be one of those peculiar persons who, when they donot think an assertion worth notice, let it alone, without noticing it by anotification of non-notice. He would never commit the bull of "Sir! I willnot say a word on that subject. " He would put it thus, "Sir! I will onlysay ten words on that subject, "--and, having thus said them, would {351}proceed to something else. He assumed, as a matter of form, that Mr. Reddiewould draw the proper inference from his silence: and this because he didnot care whether or no the assumption was correct. 

The _Mechanics of the Heavens_, which Mr. Reddie sends to be noticed, shallbe noticed, so far as an extract goes:

"My connection with this subject is, indeed, very simply explained. Inendeavoring to understand the laws of physical astronomy as generallytaught, I happened to entertain some doubt whether gravitating bodies couldrevolve, and having afterwards imbibed some vague idea that the laws of theuniverse were chemical and physical rather than mechanical, and somehowconnected with electricity and magnetism as opposing correlativeforces--most probably suggested to my mind, as to many others, by thetranscendent discoveries made in electro-magnetism by ProfessorFaraday[657]--my former doubts about gravitation were revived, and I wasled very naturally to try and discover whether a gravitating body reallycould revolve; and I became convinced it could not, before I had everpresumed to look into the demonstrations of the _Principia_. "

This is enough against the book, without a word from me: I insert it onlyto show those who know the subject what manner of writer Mr. Reddie is. Itis clear that "presumed" is a slip of the pen; it should have been_condescended_. 

Mr. Reddie represents me as dreaming over paltry paradoxes. He is right;many of my paradoxes are paltry: he is wrong; I am wide awake to them. Asingle moth, beetle, or butterfly, may be a paltry thing; but when acabinet is arranged by genus and species, we then begin to admire the {352}infinite variety of a system constructed on a wonderful sameness of leadingcharacteristics. And why should paradoxes be denied that collectiveimportance, paltry as many of them may individually be, which is accordedto moths, beetles, or butterflies? Mr. Reddie himself sees that "there is amethod in" my "mode of dealing with paradoxes. " I hope I have atoned forthe scantiness of my former article, and put the demonstrated impossibilityof gravitation on that level with Hubongramillposanfy arithmetic andinhabited atoms which the demonstrator--not quite without reason--claimsfor it. 

In the Introduction to a collected edition of the three works, Mr. Reddiedescribes his _Mechanism of the Heavens_, from which I have just quoted, as--

"a public challenge offered to the British Association and themathematicians at Cambridge, in August, 1862, calling upon them to point toa single demonstration in the _Principia_ or elsewhere, which even attemptsto prove that Universal Gravitation is possible, or to show that agravitating body could possibly revolve about a center of attraction. Thechallenge was not accepted, and never will be. No such demonstrationexists. And the public must judge for themselves as to the character of aso-called "certain science, " which thus shrinks from rigid examination, anddares not defend itself when publicly attacked: also of the character ofits teachers, who can be content to remain dumb under such circumstances. "

ON PARADOXERS IN GENERAL. 

The above is the commonplace talk of the class, of which I proceed to speakwithout more application to this paradoxer than to that. It reminds one ofthe funny young rascals who used, in times not yet quite forgotten, toabuse the passengers, as long as they could keep up with the {353} stagecoach; dropping off at last with "Why don't you get down and thrash us?You're afraid, you're afraid!" They will allow the public to judge forthemselves, but with somewhat of the feeling of the worthy uncle in _TomJones_, who, though he would let young people choose for themselves, would_have them_ choose wisely. They try to be so awfully moral and so ghastlysatirical that they must be answered: and they are best answered in theirown division. We have all heard of the way in which sailors cat's-pawed themonkeys: they taunted the dwellers in the trees with stones, and themonkeys taunted them with cocoa-nuts in return. But these were sillydendrobats: had they belonged to the British Association they would havesaid--No! No! dear friends; it is not in the itinerary: if you want nuts, you must climb, as we do. The public has referred the question to Time: theprocedure of this great king I venture to describe, from precedents, by anadaptation of some smart anapæstic tetrameters--your anapæst is the footfor satire to halt on, both in Greek and English--which I read about twentyyears ago, and with the point of which I was much tickled. Poetasters werelaughed at; but Mr. Slum, whom I employed--Mr. Charles Dickens obliged mewith his address--converted the idea into that of a hit atmathematicasters, as easily as he turned the Warren acrostic into Jarley. As he observed, when I settled his little account, it is cheaper than anyprose, though the broom was not stolen quite ready made:

 _Forty stripes save one for the smaller Paradoxers. _

 Hark to the wisdom the sages preach Who never have learnt what they try to teach. We are the lights of the age, they say! We are the men, and the thinkers we! So we build up guess-work the livelong day, In a topsy-turvy sort of way, Some with and some wanting _a_ plus b. Let the British Association fuss; What are theirs to the feats to be wrought by us? {354} Shall the earth stand still? Will the round come square? Must Isaac's book be the nest of a mare? Ought the moon to be taught by the laws of space To turn half round without right-about-face? Our whimsey crotchets will manage it all; Deep! Deep! posterity will them call! Though the world, for the present, lets them fall Down! Down! to the twopenny box of the stall!

 Thus they--But the marplot Time stands by, With a knowing wink in his funny old eye. He grasps by the top an immense fool's cap, Which he calls a philosophaster-trap: And rightly enough, for while these little men Croak loud as a concert of frogs in a fen, He first singles out one, and then another, Down goes the cap--lo! a moment's pother, A spirit like that which a rushlight utters As just at the last it kicks and gutters: When the cruel smotherer is raised again Only snuff, and but little of that, will remain. 

 But though _uno avulso_ thus comes every day _Non deficit alter_ is also in play: For the vacant parts are, one and all, Soon taken by puppets just as small; Who chirp, chirp, chirp, with a grasshopper's glee, We're the lamps of the Universe, We! We! We! But Time, whose speech is never long, -- He hasn't time for it--stops the song And says--Lilliput lamps! leave the twopenny boxes, And shine in the Budget of Paradoxes!

When a paradoxer parades capital letters and diagrams which are as good asNewton's to all who know nothing about it, some persons wonder why sciencedoes not rise and triturate the whole thing. This is why: all who are fitto read the refutation are satisfied already, and can, if they please, detect the paradoxer for themselves. Those who are not fit to do this wouldnot know the difference between the true answer and the new capitals anddiagrams on which the delighted paradoxer would declare {355} that he hadcrumbled the philosophers, and not they him. Trust him for having the lastword: and what matters it whether he crow the unanswerable sooner or later?There are but two courses to take. One is to wait until he has committedhimself in something which all can understand, as Mr. Reddie has done inhis fancy about the Astronomer Royal's change of opinion: he can then beput in his true place. The other is to construct a Budget of Paradoxes, that the world may see how the thing is always going on, and that thepicture I have concocted by cribbing and spoiling a bit of poetry is drawnfrom life. He who wonders at there being no answer has seen one or two: hedoes not know that there are always fifty with equal claims, each of whomregards his being ranked with the rest as forty-nine distinct and severalslanders upon himself, the great Mully Ully Gue. And the fifty would soonbe five hundred if any notice were taken of them. They call mankind towitness that science _will not_ defend itself, though publicly attacked interms which might sting a pickpocket into standing up for his character:science, in return, allows mankind to witness or not, at pleasure, that it_does not_ defend itself, and yet receives no injury from centuries ofassault. Demonstrative reason never raises the cry of _Church in Danger_!and it cannot have any Dictionary of Heresies except a Budget of Paradoxes. Mistaken claimants are left to Time and his extinguisher, with theapprobation of all thinking non-claimants: there is no need of a successionof exposures. Time gets through the job in his own workmanlike manner asalready described. 

On looking back more than twenty years, I find among my cuttings thefollowing passage, relating to a person who had signalized himself by aneffort to teach comets to the conductor of the _Nautical Almanac_:

"Our brethren of the literary class have not the least idea of the smallamount of appearance of knowledge {356} which sets up the scientificcharlatan. Their world is large, and there are many who have that moderateknowledge, and perception of what is knowledge, before which extremeignorance is detected in its first prank. There is a public of moderatecultivation, for the most part sound in its judgment, always ready in itsdecisions. Accordingly, all their successful pretenders have _somepretension_. It is not so in science. Those who have a right to judge arefewer and farther between. The consequence is, that many scientificpretenders have _nothing but pretension_. "

This is nearly as applicable now as then. It is impossible to make thosewho have not studied for themselves fully aware of the truth of what I havequoted. The best chance is collection of cases; in fact, a Budget ofParadoxes. Those who have no knowledge of the subject can thus argue fromthe seen to the unseen. All can feel the impracticability of theHubongramillposanfy numeration, and the absurdity of the equality ofcontour of a regular pentagon and hexagon in one and the same circle. Manymay accordingly be satisfied, on the assurance of those who have studied, that there is as much of impracticability, or as much of absurdity, inthings which are hidden under

 "Sines, tangents, secants, radius, cosines Subtangents, segments and all those signs; Enough to prove that he who read 'em Was just as mad as he who made 'em. "

Not that I mean to be disrespectful to mathematical terms: they are shortand easily explained, and compete favorably with those of most othersubjects: for instance, with

 "Horse-pleas, traverses, demurrers, Jeofails, imparlances, and errors, Averments, bars, and protestandos, And puis d'arreign continuandos. "

{357}

From which it appears that, taking the selections made by satirists for oursamples, there are, one with another, four letters more in a law term thanin one of mathematics. But pleading has been simplified of late years. 

All paradoxers can publish; and any one who likes may read. But this is notenough; they find that they cannot publish, or those who can find they are_not_ read, and they lay their plans athwart the noses of those who, theythink, ought to read. To recommend them to be content with publication, like other authors, is an affront: of this I will give the reader anamusing instance. My good nature, of which I keep a stock, though I do notuse it all up in this Budget, prompts me to conceal the name. 

I received the following letter, accompanied by a prospectus of a work onmetaphysics, physics, astronomy, etc. The author is evidently one whom Ishould delight to honor:

"Sir, --A friend of mine has mentioned your name in terms of panigeric[_sic_], as being of high standing in mathematics, and of greatly originalthought. I send you the enclosed without comment; and, assuming that thebent of your mind is in free inquiry, shall feel a pleasure in showing youmy portfolio, which, as a mathematician, you will acknowledge to be deeplyinteresting, even in an educational point of view. The work is complete, and the system so far perfected as to place it above criticism; and, so faras regards astronomy, as will Ptolemy beyond rivalry [_sic_: no doubt somewords omitted]. Believe me to be, Sir, with the profoundest respect, etc. The work is the result of thirty-five years' travel and observation, labor, expense, and self-abnegation. "

I replied to the effect that my time was fully occupied, and that I wasobliged to decline discussion with many persons who have views of theirown; that the proper way is to publish, so that those who choose may readwhen they can find leisure. I added that I should advise a precursor in theshape of a small pamphlet, as two octavo volumes {358} would be too muchfor most persons. This was sound advice; but it is not the first, second, or third time that it has proved very unpalatable. I received the followinganswer, to which I take the liberty of prefixing a bit of leonine wisdom:

 "Si doceas stultum, lætum non dat tibi vultum; Odit te multum; vellet te scire sepultum. [658]"

"Sir, --I pray you pardon the error I unintentionally have fallen into;deceived by the F. R. S. [I am not F. R. S. ] I took you to be a man of science[_omnis homo est animal, Sortes est homo, ergo Sortes est animal_][659]instead of the mere mathematician, or human calculating-machine. Believeme, Sir, you also have mistaken your mission, as I have mine. I wrote toyou as I would to any other man well up in mathematics, with the intent tocall your attention to a singular fact of omission by Euclid, and othergreat mathematicians: and, in selecting you, I did you an honor which, fromwhat I have just now heard, was entirely out of place. I think, consideringthe nature of the work set forth in the prospectus, you are guilty of bothfolly and presumption, in assuming the character of a patron; for your ownsense ought to have assured you that was such my object I should not havesought him in a De Morgan, who exists only by patronage of others. On theother hand, I deem it to be an unpardonable piece of presumption inoffering your advice upon a subject the magnitude, importance, and realutility of which you know nothing about: by doing so you have offered me adirect insult. The system is a manual of Philosophy, a one inseparablewhole of metaphysics and physic; embracing points the most interesting, laws the most important, {359} doctrines the most essential to advance manin accordance with the spirit of the times. I may not live to see it inprint; for, at ----, life at best is uncertain: but, live or die, beassured Sir, it is not my intention to debase the work by seekingpatronage, or pandering to the public taste. Your advice was the lessneeded, seeing I am an old-established ----. I remain, etc. --P. S. You willoblige me by returning the prospectus of my work. "

My reader will, I am sure, not take this transition from the "profoundestrespect" to the loftiest insolence for an _apocraphical_ correspondence, touse a word I find in the Prospectus: on my honor it is genuine. He will bebetter employed in discovering whether I exist by patronizing others, or bybeing patronized by them. I make any one who can find it out a fair offer:I will give him my patronage if I turn out to be Bufo, on condition hegives me his, if I turn out to be Bavius. [660] I need hardly say that Iconsidered the last letter to be one of those to which no answer is so goodas no answer. 

These letters remind me in one respect of the correspondents of thenewspapers. My other party wrote because a friend had pointed me out: buthe would not have written if he had known what another friend told him justin time for the second letter. The man who sends his complaint to thenewspaper very often says, in effect, "Don't imagine, Sir, that I read yourcolumns; but a friend who sometimes does has told me ... " It is wordedthus: "My attention {360} has been directed to an article in your paper of... " Many thanks to my friend's friends for not mentioning the Budget: hadmy friend's attention been directed to it I might have lost a strikingexample of the paradoxer in search of a patron. That my Friend was on thisscent in the first letter is revealed in the second. Language was given toman to conceal his thoughts; but it is not every one who can do it. 

Among the most valuable information which my readers will get from me iscomparison of the reactions of paradoxers, when not admitted to argument, or when laughed at. Of course, they are misrepresented; and at this theyare angry, or which is the same thing, take great pains to assure thereader that they are not. So far natural, and so far good; anything shortof concession of a case which must be seriously met by counter-reasons issure to be misrepresentation. My friend Mr. James Smith and my friend Mr. Reddie are both terribly misrepresented: they resent it by someinsinuations in which it is not easy to detect whether I am a conscioussmotherer of truth, or only muddle-headed and ignorant. [This was writtenbefore I received my last communication from Mr. James Smith. He tells methat I am wrong in saying that his work in which I stand in the pillory isall reprint: I have no doubt I confounded some of it with some of themanuscript or slips which I had received from my much not-agreed-withcorrespondent. He adds that my mistake was intentional, and that my reasonis obvious to the reader. This _is_ information, as the sea-serpent saidwhen he read in the newspaper that he had a mane and tusks. ]

THE DOUBLE VAHU PROCESS. 

My friend Dr. Thorn[661] sees deeper into my mystery. By the way, he stillsends an occasional touch at the old {361} subject; and he wants meparticularly to tell my readers that the Latin numeral letters, if M beleft out, give 666. And so they do: witness DCLXVI. A person who thinks ofthe origin of symbols will soon see that 666 is our number because we havefive fingers on each hand: had we had but four, our mystic number wouldhave been expressed by 555, and would have stood for our present 365. Had nbeen the number on each hand, the great number would have been

 (n + 1) (4n^2 + 2n + 1)

With no finger on each hand, the number would have been 1: with one fingerless than none at all on each hand, it would have been 0. But what doesthis mean? Here is a question for an algebraical paradoxer! So soon as wehave found out how many fingers the inhabitants of any one planet have oneach hand, we have the means of knowing their number of the Beast, andthence all about them. Very much struck with this hint of discovery, Iturned my attention to the means of developing it. The first point was toclear my vision of all the old cataracts. I propose the followingexperiment, subject of course to the consent of parties. Let Dr. ThornDouble-Vahu Mr. James Smith, and Thau Mr. Reddie: if either be deparadoxedby the treatment, I will consent to undergo it myself. Provided always thatthe temperature required be not so high as the Doctor hints at: if theTurkish Baths will do for this world, I am content. 

The three paradoxers last named and myself have a pentasyllable convention, under which, though we go far beyond civility, we keep within civilization. Though Mr. James Smith pronounced that I must be dishonest if I did not seehis argument, which he knew I should not do [to say nothing of recentaccusation]; though Dr. Thorn declared me a competitor for fire andbrimstone--and my wife, too, which doubles the joke: though Mr. Reddie{362} was certain I had garbled him, evidently on purpose to make falsehoodappear truth; yet all three profess respect for me as to everything butpower to see truth, or candor to admit it. And on the other hand, thoughthese were the modes of opening communication with me, and though I have nodoubt that all three are proper persons of whom to inquire whether I shouldgo up-stairs or down-stairs, etc. , yet I am satisfied they are thoroughlyrespectable men, as to everything but reasoning. And I dare say our severalprofessions are far more true in extent than in many which are made undermore parliamentary form. We find excuses for each other: they makeallowances for my being hoodwinked by Aristotle, by Newton, by the Devil;and I permit them to feel, for I know they cannot get on without it, thattheir reasons are such as none but a knave or a sinner can resist. But_they_ are content with cutting a slice each out of my character: neitherof them is more than an uncle, a Bone-a-part; I now come to a dreadfulnephew, Bone-the-whole. 

I will not give the name of the poor fellow who has fallen so far belowboth the _honestum_ and the _utile_, to say nothing of the _decorum_ or the_dulce_. [662] He is the fourth who has taken elaborate notice of me; and myadvice to him would be, _Nec quarta loqui persona laboret_. [663] Accordingto him, I scorn humanity, scandalize learning, and disgrace the press; itadmits of no manner of doubt that my object is to mislead the public andsilence truth, at the expense of the interests of science, the wealth ofthe nation, and the lives of my fellow men. The only thing left to besettled is, whether this is due to ignorance, natural distaste for truth, personal malice, a wish to curry favor with the Astronomer Royal, or meretoadyism. The only accusation which has truth in it is, that I have mademyself a "public scavenger of science": the assertion, which is the {363}most false of all is, that the results of my broom and spade are "shotright in between the columns of" the _Athenæum_. I declare I never in mylife inserted a word between the columns of the _Athenæum_: I feel huffedand miffed at the very supposition. I _have_ made myself a publicscavenger; and why not? Is the mud never to be collected into a heap? Ilook down upon the other scavengers, of whom there have been a few--merehistorical drudges; Montucla, Hutton, etc. --as not fit to compete with me. I say of them what one crossing-sweeper said of the rest: "They are wellenough for the common thing; but put them to a bit of fancy-work, such assweeping round a post, and see what a mess they make of it!" Who can touchme at sweeping round a paradoxer? If I complete my design of publishing aseparate work, an old copy will be fished up from a stall two hundred yearshence by the coming man, and will be described in an article which will endby his comparing our century with his own, and sighing out in the best NewZealand pronunciation--

 "Dans ces tems-là C'était déjà comme ça!"[664]

ORTHODOX PARADOXERS. 

And pray, Sir! I have been asked by more than one--do your orthodox neverfall into mistake, nor rise into absurdity? They not only do both, but theyadmit it of each other very freely; individually, they are convinced ofsin, but not of any particular sin. There is not a syndoxer among them allbut draws his line in such a way as to include among paradoxers a greatmany whom I should exclude altogether from this work. My worst specimensare but exaggerations of what may be found, occasionally, in the thoughtsof sagacious investigators. At the end of the {364} glorious dream, welearn that there is a way to Hell from the gates of Heaven, as well as fromthe City of Destruction: and that this is true of other things besidesChristian pilgrimage is affirmed at the end of the Budget of Paradoxes. IfD'Alembert[665] had produced _enough_ of a quality to match his celebratedmistake on the chance of throwing head in two throws, he would have been inmy list. If Newton had produced _enough_ to match his reception of thestory that Nausicaa, Homer's Phæacian princess, invented the celestialsphere, followed by his serious surmise that she got it from theArgonauts, --then Newton himself would have had an appearance entered forhim, in spite of the _Principia_. In illustration, I may cite a few wordsfrom _Tristram Shandy_:

"'A soldier, ' cried my uncle Toby, interrupting the Corporal, 'is no moreexempt from saying a foolish thing, Trim, than a man of letters. '--'But notso often, an' please your honor, ' replied the Corporal. My uncle Toby gavea nod. "

I now proceed to die out. Some prefatory remarks will follow in time. [666]I shall have occasion to insist that all is not barren: I think I shallfind, on casting up, that two out of five of my paradoxers are not to beutterly condemned. Among the better lot will be found all gradations ofmerit; at the same time, as was remarked on quite a different subject, there may be little to choose between the last of the saved and the firstof the lost. The higher and better class is worthy of blame; the lower andworse class is worthy of praise. The higher men are to be reproved for nottaking up things in which they could do some good: the lower men are to becommended for taking up things in which they can do no great harm. Thecircle problem is like Peter Peebles's lawsuit:

{365}

"'But, Sir, I should really spoil any cause thrust on me so hastily. '--'Yecannot spoil it, Alan, ' said my father, 'that is the very cream of thebusiness, man, --... The case is come to that pass that Stair or Arnistoncould not mend it, and I don't think even you, Alan, can do it much harm. '"

I am strongly reminded of the monks in the darker part of the Middle Ages. To a certain proportion of them, perhaps two out of five, we are indebtedfor the preservation of literature, and their contemporaries for goodteaching and mitigation of socials evils. But the remaining three were thefleas and flies and thistles and briars with whom the satirist lumps them, about a century before the Reformation:

 "Flen, flyys, and freris, populum domini male cædunt; Thystlis and breris crescentia gramina lædunt. Christe nolens guerras qui cuncta pace tueris, Destrue per terras breris, flen, flyys, and freris. Flen, flyys, and freris, foul falle hem thys fyften yeris, For non that her is lovit flen, flyys ne freris. "[667]

I should not be quite so savage with my second class. Taken together, theymay be made to give useful warning to those who are engaged in learningunder better auspices: aye, even useful hints; for bad things are veryoften only good things spoiled or misused. My plan is that of a predecessorin the time of Edward the Second:

 "Meum est propositum genti imperitæ Artes frugi reddere melioris vitæ. "[668]

To this end I have spoken with freedom of books as books, of opinions asopinions, of ignorance as ignorance, of {366} presumption as presumption;and of writers as I judge may be fairly inferred from what they havewritten. Some--to whom I am therefore under great obligation--havepermitted me to enlarge my plan by assaults to which I have alluded;assaults which allow a privilege of retort, of which I have often availedmyself; assaults which give my readers a right of partnership in theamusement which I myself have received. 

For the present I cut and run: a Catiline, pursued by a chorus of Ciceros, with _Quousque tandem? Quamdiu nos? Nihil ne te?_[669] ending with, _In teconferri pestem istam jam pridem oportebat, quam tu in nos omnes jamdiumachinaris!_ I carry with me the reflection that I have furnished to thosewho need it such a magazine of warnings as they will not find elsewhere; _asignatis cavetote_:[670] and I throw back at my pursuers--_Valete, doctoressine doctrina; facite ut proxima congressu vos salvos corporibus et sanosmentibus videamus. _[671] Here ends the Budget of Paradoxes. 

{367}

 * * * * *

APPENDIX. 

I think it right to give the proof that the ratio of the circumference tothe diameter is incommensurable. This method of proof was given byLambert, [672] in the _Berlin Memoirs_ for 1761, and has been also given inthe notes to Legendre's[673] Geometry, and to the English translation ofthe same. Though not elementary algebra, it is within the reach of astudent of ordinary books. [674]

Let a continued fraction, such as

 a ----- b + c ----- d + e - f + etc. , be abbreviated into a/b+ c/d+ e/f+ etc. : each fraction being understood asfalling down to the side of the preceding sign +. In every such fraction wemay suppose b, d, f, etc. {368} positive; a, c, e, &c. Being as required:and all are supposed integers. If this succession be continued adinfinitum, and if a/b, c/d, e/f, etc. All lie between -1 and +1, exclusive, the limit of the fraction must be incommensurable with unity; that is, cannot be A/B, where A and B are integers. 

First, whatever this limit may be, it lies between -1 and +1. This isobviously the case with any fraction p/(q + [omega]), where [omega] isbetween ±1: for, p/q, being < 1, and p and q integer, cannot be brought upto 1, by the value of [omega]. Hence, if we take any of the fractions

 a/b, a/b+ c/d, a/b+ c/d+ e/f, etc. 

say a/b+ c/d+ e/f+ g/h we have, g/h being between ±1, so is e/f+ g/h, sotherefore is c/d+ e/f+ g/h; and so therefore is a/b+ c/d+ e/f+ g/h. 

Now, if possible, let a/b+ c/d+ etc. Be A/B at the limit; A and B beingintegers. Let

 P = A c/d+ e/f+ etc. , Q = P e/f+ g/h+ etc. , R = Q g/h + i/k + etc. 

P, Q, R, etc. Being integer or fractional, as may be. It is easily shownthat all must be integer: for

{369}

 A/B = a/b+ P/A, or, P = aB - bA

 P/A = c/d+ Q/P, or, Q = cA - dP

 Q/P = e/f+ R/Q, or, R = eP - fQ

etc. , etc. Now, since a, B, b, A, are integers, so also is P; and thence Q;and thence R, etc. But since A/B, P/A, Q/P, R/Q, etc. Are all between -1and +1, it follows that the unlimited succession of integers P, Q, R, areeach less in numerical value than the preceding. Now there can be no such_unlimited_ succession of _descending_ integers: consequently, it isimpossible that a/b+ c/d+, etc. Can have a commensurable limit. 

It easily follows that the continued fraction is incommensurable if a/b, c/d, etc. , being at first greater than unity, become and continue less thanunity after some one point. Say that i/k, l/m, ... Are all less than unity. Then the fraction i/k+ l/m+ ... Is incommensurable, as proved: let it be[kappa]. Then g/(h + [kappa]) is incommensurable, say [lambda]; e/(f +[lambda]) is the same, say [mu]; also c/(d + [mu]), say [nu], and a/(b +[nu]), say [rho]. But [rho] is the fraction a/b+ c/d+ ... Itself; which istherefore incommensurable. 

Let [phi]z represent

 a a^2 a^3 1 + - + ------- + -------------- + .... Z 2z(z+1) 2·3·z(z+1)(z+2)

{370} Let z be positive: this series is convergent for all values of a, andapproaches without limit to unity as z increases without limit. Change zinto z + 1, and form [phi]z - [phi](z+1): the following equation willresult--

 a [phi]z-[phi](z+1) = ------([phi](z+2)) z(z+1)

 a [phi](z+1) a [phi](z+1) a [phi](z+2) or a = - ---------- · z + - ---------- · --- ---------- z [phi]z z [phi]z z+1 [phi](z+1)

 a = [psi]z(z+[psi](z+1))

[psi]z being (a/z)([phi](z+1)/[phi]z); of which observe that it diminisheswithout limit as z increases without limit. Accordingly, we have

 [psi]z = a/z+ [psi](z+1) = a/z+ a/(z+1)+ [psi](z+2) = a/z+ a/(z+1)+ a/(z+2)+ [psi](z+3), etc. 

And, [psi](z + n) diminishing without limit, we have

 a/z · [phi](z+1)/[phi]z = (a/z+) (a/(z+1)+) (a/(z+2)+) (a/((z+3)+ ... ))

Let z = ½; and let 4a = -x^2. Then (a/z)[phi](z+1) is -(x^2/2) ( 1 -x^2/(2·3) + x^4/(2·3·4·5... )) or -(x/2) sin x. Again [phi]z is 1 - x^2/2 +x^4/(2·3·4) or cos x: and the continued fraction is

 (¼)x^2/(½)+ (¼)x^2/(3/2)+ (¼)x^2/(5/2)+ ... Or -x/2 x/1+ -x^2/3+ -x^2/5+ ... 

{371} whence tan x = x/1+ -x^2/3+ -x^2/5+ -x^2/7+ ... 

Or, as written in the usual way, tan x = x ------- 1 - x^2 ------- 3 - x^2 ------- 5 - x^2 ------- 7 - ... 

This result may be proved in various ways: it may also be verified bycalculation. To do this, remember that if

 a_1/b_1+ a_2/b_2+ a_3/b_3+ ... A_n/b_n = P_n/Q_n; then

 P_1=a_1, P_2=b_2 P_1, P_3=b_3 P_2+a_3 P_1, P_4=b_4 P_3+a_4 P_2, etc. Q_1=b_1, Q_2=b_2 Q_1+a_2, Q_3=b_3 Q_2+a_3 Q_1, Q_4=b_4 Q_3+a_4 Q_2, etc. 

in the case before us we have

 a_1=x, a_2=-x^2, a_3=-x^2, a_4=-x^2, a_5=-x^2, etc. B_1=1, b_2=3, b_3=5, b_4=7, b_5=9, etc. 

 P_1=x Q_1=1 P_2=3x Q_2=3-x^2 P_3=15x-x^3 Q_3=15-6x^2 P_4=105x-10x^3 Q_4=105-45x^2+x^4 P_5=945x-105x^3+x^5 Q_5=945-420x^2+15x^4 P_6=10395x-1260x^3+21x^5 Q_6=10395-4725x^2+210x^4-x^6

We can use this algebraically, or arithmetically. If we divide P_n by Q_n, we shall find a series agreeing with the known series for tan x, _as faras_ n _terms_. That series is

 x + x^3/3 + 2x^5/15 + 17x^7/315 + 62x^9/2835 + ... 

{372} Take P_5, and divide it by Q_5 in the common way, and the first fiveterms will be as here written. Now take _x_ = . 1, which means that theangle is to be one tenth of the actual unit, or, in degrees 5°. 729578. Wefind that when x = . 1, P_6 = 1038. 24021, Q_6 = 10347. 770999; whence P_6divided by Q_6 gives . 1003346711. Now 5°. 729578 is 5°43'46½"; and from theold tables of Rheticus[675]--no modern tables carry the tangents sofar--the tangent of this angle is . 1003347670. 

Now let x = ¼[pi]; in which case tan x = 1. If ¼[pi] be commensurable withthe unit, let it be (m/n), m and n being integers: we know that ¼[pi] < 1. We have then

 1=(m/n)/1- (m^2/n^2)/3- (m^2/n^2)/5- ... = m/n- m^2/3n- m^2/5n- m^2/7n- ... 

Now it is clear that m^2/3n, m^2/5n, m^2/7n, etc. Must at last become andcontinue severally less than unity. The continued fraction is thereforeincommensurable, and cannot be unity. Consequently [pi]^2 cannot becommensurable: that is, [pi] is an incommensurable quantity, and so also is[pi]^2. 

I thought I should end with a grave bit of appendix, deeply mathematical:but paradox follows me wherever I go. The foregoing is--in my ownlanguage--from Dr. (now Sir David) Brewster's[676] English edition ofLegendre's Geometry, (Edinburgh, 1824, 8vo. ) translated by some one who isnot named. I picked up a notion, which others had at Cambridge in 1825, that the translator was the late Mr. Galbraith, [677] then known atEdinburgh as a writer and teacher. 

{373} But it turns out that it was by a very different person, and onedestined to shine in quite another walk; it was a young man named ThomasCarlyle. [678] He prefixed, from his own pen, a thoughtful and ingeniousessay on Proportion, as good a substitute for the fifth Book of Euclid ascould have been given in the space; and quite enough to show that he wouldhave been a distinguished teacher and thinker on first principles. But heleft the field immediately. 

 * * * * *

(The following is the passage referred to at Vol. II, page 54. )

Michael Stifelius[679] edited, in 1554, a second edition of the Algebra(_Die Coss. _), of Christopher Rudolff. [680] This is one of the earliestworks in which + and - are used. 

Stifelius was a queer man. He has introduced into this very work of Rudolffhis own interpretation of the number of the Beast. He determined to fix thecharacter of Pope Leo: so he picked the numeral letters from LEODECIMVS, and by taking in X from LEO X. And striking out M as standing for_mysterium_, he hit the number exactly. This discovery completed hisconversion to Luther, and his determination to throw off his monastic vows. Luther dealt with him as straight-forwardly as with Melanchthon about hisastrology: he accepted the conclusions, but told him to clear his mind ofall the premises about the Beast. Stifelius {374} did not take the advice, and proceeded to settle the end of the world out of the prophet Daniel: hefixed on October, 1533. The parishioners of some cure which he held, havingfull faith, began to spend their savings in all kinds of good eating anddrinking; we may charitably hope this was not the way of preparing for theevent which their pastor pointed out. They succeeded in making themselvesas fit for Heaven as Lazarus, so far as beggary went: but when the timecame, and the world lasted on, they wanted to kill their deceiver, andwould have done so but for the interference of Luther. {375}

 * * * * *

INDEX. 

Pages denoted by numerals of this kind (_287_) refer to biographical notes, chiefly by the editor. Numerals like 426 refer to books discussed by DeMorgan, or to leading topics in the text. Numerals like 126 indicate minorreferences. 

 Abbott, Justice, I, _181_. Abernethy, J. , II, _219_. Aboriginal Britons, a poem, II, 270. Academy of Sciences, French, I, 163. Adair, J. , I, _223_. Adam, M. , I, _65_. Adams, J. C. , I, _43_, 82, 385, 388; II, 131, 135, 140, 303. Ady, Joseph, II, 42, _42_. Agnew, H. C. , I, 328. Agricola, J. , I, 394. Agricultural Laborer's letter, II, 16. Agrippa, H. C. , I, _48_, 48. Ainsworth, W. H. , II, _132_. Airy, I, _85_, 88, 152, 242; II, 85, 140, 150, 303, 347. Alchemy, I, 125. Alfonso X (El Sabio), II, _269_. Alford, H. , II, _221_. Alfred, King, Ballad of, II, 22. Algebra, I, 121. Algebraic symbols, I, 121. Almanac, I, 300; II, 147, 148, 207. (_See Easter. _) Aloysius Lilius, I, 362. Alsted, J. H. , II, _282_. Ameen Bey, II, 15. Amicable Society, I, 347. Ampère, I, 86. Amphisbæna serpent, I, 31. Anagrams, De Morgan, I, 138. Anaxagoras, II, _59_. Angherà, II, 60, _60_, 61, 279. Annuities, Fallacies of, I, 157. Antichrist, I, 130. Antimony, I, 125. Antinewtonism, I, 162. Antinomians, I, 394. Antiphon, II, _59_. Antonie, F. , I, _126_, 126. Apollonius, I, 41, 107. Apparitions, II, 47. Arago, I, _243_, 390. Aratus, II, _167_. Arbuthnot, I, _133_, 134. Archer, H. , II, 90. Archimedes, I, 5, 11, 42, 83, 107. Archytas, I, _53_. Argoli, I, _104_. Aristocrat, as a scientist, I, 131. Aristotle, I, 5, 331. Arnobius, II, _73_. Arson, P. J. , II, _207_. Ashton, R. , II, _99_. Astrology, I, 118, 127, 128, 350; II, 43. Astronomer's Drinking Song, I, 380. Astronomical Aphorisms, I, 398. Paradox, I, 394. Police Report, I, 390. Society. (_See Royal Astronomical Society. _) Astronomy, Bailly's exaggerated view of, I, 166. Astunica, Didacas, I, 90. Athanasian Creed, I, 371. Atheists, Philosophical, I, 1. Atoms, II, 191. {376} Attraction, I, 136, 151, 155. Augustine, St. , II, _23_. Aurora borealis, I, 134. Austen, Jane, I, 191. Auzout, A. , II, _300_. Aviation, Early ideas of, II, 8. 

 Babbage, C. , I, _207_, 290, 291; II, 181. Bachet, de Méziriac, I, _161_. Bacon, F. , I, 5, _75_, 75, 76, 79, 89, 145, 331. Bacon, R. , I, 5, _126_, 126, 360; II, 94. Baconian controversy, I, 141. Baden Powell, II, _267_. Bailly, J. S. , I, 166, _166_, 308. Baily, F. , I, 308, _309_; II, 16, 143, 188. Baily, R. , II, 16. Baker, T. , II, _302_. Bakewell, F. C. , II, 156, _156_. Banks, J. , I, 28. Barberini, M. , II, _267_. Barker, C. , II, _262_. Baronius, I, 33, 35; II, _62_. Barrême, I, _42_. Barrett, G. , II, _188_. Barrow, I. , I, _160_; II, 302. Baruel, de, I, 165. Bassano, Duc de, II, 3, 339. Baxter, T. , I, 146. Bayle, P. , II, _73_. Beaufort, F. , II, _267_. Beaugrand, I, 119, _121_. Beaulieu, I, _119_, 119, 121. Beaune, de, II, _59_. Bécourt, R. , II, 277. Bedford, Duke of, (6th), I, _182_. Behmen, I, _168_, 254; II, 317. Bellenden, W. , I, _175_. Bentley, I, _110_. Berkeley, G. , II, 346. Bernard, E. , II, _297_, 300. Bernardus Trevisanus, I, _126_, 126. Bernoullis, I, 130, _150_, _335_, 336. Bertius, P. , II, _300_. Bèse, I, _66_. Bessel, I, _384_; II, 2. Bethune, I, _99_, 279, 291. Bettesworth, I, 19. Beza. (_See Bèse. _) Bickersteth, E. H. , I, _238_. Bidder, I, _86_. Biden, J. , II, 158, _160_. Bidle, (Biddle), I, 239. Biot, I, _85_. Birch, T. , I, _108_; II, 304, 313. Birks, T. R. , II, 158, _158_. Bishop, G. , I, _386_. Bishops as Paradoxers, I, 226. Boccaccio, I, 38. Boethius, I, _42_, 45. Böhme. (_See Behmen. _) Boncompagni, I, _298_. Boniface, St. , I, 32. Bonnycastle, J. , II, _16_. Booker, I, 115. Boole, G. , I, _261_, 332; II, 75, 79. --A tribute to, II, 79. Borelli, G. A. , II, _300_. Borello, I, _69_. Boreman, I, 113. Borron, Mrs. , II, 7. Boscovich, I, _156_, 164. Bouguer, II, _301_. Bouillaud, I, _87_; II, 295. Bouvard, A. , I, _327_. Bovillus, I, _44_; II, 324. --Epitome of, I, 44. Bowdler, H. M. , I, _194_. Bowring, J. , I, _352_; II, 256. Boyle, R. , I, 24, _125_; II, 300. Bradley, I, 24. Bradwardine, I, _227_, 228, 229. Brahe. (_See Tycho B. _) Brancker, I, 107; II, _300_. Brenan, J. , I, 330, _330_. Brewster, D. , I, 39, _137_, 140; II, 214, 288, 372. Briggs, I, _69_; II, 299, 302. Bright, J. , II, _235_. Brinkley, J. , I, _311_. Britannicus, D. , II, 8. British Museum library, I, 151. Brothers, R. , I, _315_; II, 97. Brougham, Henry, Lord, I, _191_. Brouncker (Brounker), I, _132_; II, 302. Brown, W. , II, _168_. Browne, T. , I, 31. Brucker, I, _61_. Brunet, I, _402_. Brünnow, I, _386_. Bruno, I, _59_, 59. Bryson, II, _59_. Bürgi, I, 52. Buffon, I, _282_. Bulstrode, II, 84. Bungus, I, _55_, 55, 57. Buoncompagno, U. , I, _362_. {377} Burgon, J. W. , II, _30_. Buridan, I, _37_. --Questiones morales, I, 37. Buridan's Ass, I, 37. Burke, E. , I, _173_. Burlesque, Frend's, I, 208. Burnet, G. , I, _107_. Burney, Frances, I, _190_. Burton, Frances B. , I, 374. Busby, R. , II, _313_. Buteo, I, _51_. Butler, G. , I, _199_. Butler, S. , II, _218_. Buxton, J. , I, _86_. Byrgius. (_See Bürgi. _) Byrne, O. , I, _329_; II, 186, 190. Byron, I, 186; II, 270, 273. 

 Cabbala, I, 272. Calculating Boys, I, 86. Calculus, I, 129. Calendar. (_See Easter. _) Cambridge Poets, II, 269. Campanus, I, 42, _43_. Canning, Geo. , II, _145_. Carcavi, I, _106_. Cardanus, II, _59_. Carlile, R. , I, _271_. Carlyle, T. , II, _373_. Carnot, I, 107. Caroline tables, I, 124. Casaubon, I, _111_. Case, J. , I, _128_, 128. Cassini, J. , I, _172_. Castel, I, _148_, 148. Castiglioni, I, _139_. Castlereagh, I, 185, _186_. Cataldi, I, _69_, 69. Catcott, A. , I, _237_. Causans, de, I, _298_. Cavalieri, I, _106_. Cavendish, C. , I, _106_; II, 299, 312. Cavendish, W. , I, _290_. Caxton, W. , II, _281_. Cayley, A. , II, _292_. Cecil, R. (1st Earl of Salisbury), II, _330_. Centrifugal force, II, 268. Ceulen. (_See Van Ceulen. _) Challis, J. , I, _390_; II, 141. Chalmers, I, _102_; II, 219. Chambers, E. , II, _282_. Chambers, R. , I, _344_, 344. Charles IX, II, 94. Charles X, II, 1. Chasles, I, _39_, 229. Chesterfield, Earl of (4th), II, _298_. Chiffinch, W. , II, _50_. Ch'in Chiu-shang, II, 66. Chitty, J. , II, _323_. Chiu-chang, Suan-shu, II, 67. Christian Evidence Society, I, 270. Christie, I, 27. Christmann, I, 272, _272_. Church question, I, 62. Church, The word, II, 30. Circle squarers. (_See Squaring the Circle. _) Circulating media of mathematics, I, 107. Ciruelo. (_See Sanchez. _) Clairaut, I, _219_, 382. Clarence, Duke of, I, 179. Clarke, R. , I, _255_. Clavius, I, 11, _69_, 111, 112, 335, 362, 363, 372; II, 59. Clayton. , Geo. , II, _98_. Cluvier, D. , II, 332, _332_. Cobb, Mary, II, 117. Cobbett, W. , I, _177_, 200, 399. Cobden, R. , II, 217. Cocker, I, _42_; II, 64, 173, 251, 307. Cody, P. , II, 208. Coke, E. , II, _331_. Colburn, Z. , I, _86_. Colenso, I, _63_, 247; II, 191. Collins, J. , I, _107_; II, 297, 300, 302, 313. Colvill, W. H. , II, 68. Cometic astrology, I, 128. Comets, I, 128; II, 68, 83. Cominale, C. , I, _162_, 162. Compton, S. J. A. , II, _19_. Computation, Paradoxes of, II, 251, 267. Condamine, C. M. De la, II, _301_. Conduitt, John, I, _397_. Conduitt, Mrs. , I, _136_. Congregation of the Index, I, 90. Converse propositions, I, 295. Convocation at Oxford, I, 96. Cooke, Margaret, I, _310_. Cooper, A. A. (Shaftsbury), II, _181_. Copernicus, I, 5, 6, _76_, 90, 121, 172, 380; II, 165, 335. Copley, J. S. , I, _198_. Cormouls, I, 225. Cosmology, I, 172. {378} Cotes, R. , II, _301_. Cottle, Mrs. , II, 97, _97_, 161. Craig, J. , I, _129_, 129. Creed, Mathematics of a, I, 329. Cribb, T. , I, _314_. Crotus, J. , I, 318. Cruickshank, G. , I, _186_. Cube, Duplication of, I, 349. Cumyns, Eliza, I, 299. Cunningham, I, 172, _172_. Curabelle, I, _221_. Curious Calculations, II, 66. Curll, E. , II, _279_. Cusa, I, _44_, 47, 360. Custom, II, 324. Cyclometry, II, 208. (_See Squaring of the Circle. _) Cyclopædias, Review of, II, 280. 

 D'Alembert, I, _382_; II, 283, 364. Dalgarno, I, 116, _117_. Dalton, J. , I, _255_. D'Arblay, Mme. , I, _190_. Darwin, E. , II, _8_. Darwinism, Primitive, I, 344. Dary, M. , II, _305_. Daval, P. , II, _298_. Davies, T. S. , II, 151, _151_, 188. Day, A. , I, 295, _295_. De Baruel, I, 165. De Beaune. (_See Beaune. _) De Becourt, II, 277, _277_. Debenham, J. , I, _393_. De Causans. (_See Causans. _) Dechales. (_See de Challes. _) De Challes, I, _45_. Decimal coinage, II, 80, 168, 169. Decimals run riot, II, 80. Dee, J. , II, _302_. De Fauré, I, 149. De la Leu, I, _297_. Delambre, I, _160_, 167, 354; II, 165. Democritus, II, _34_. De Moivre, I, 24, _376_; II, 298. De Molières, I, _220_. De Molina, I, _297_. Demonville, I, 291, 293. De Morgan, A. , I, 191, 383; II, 194. --Refusal of LL. D. , I, 191. De Morgan, G. C. , I, 383. De Morgan, Mrs. , I, _196_; II, 194. Denison, J. , I, 348, 353. Desaguliers, I, _153_, 156, 157. Desargues, I, _119_, 221. Descartes, I, 5, 59, 105, 132, 165, 204, 220; II, 94. De Serres, II, _60_. De Sluse. (_See Sluse. _) De Thou, I, 51, _111_, 113; II, 295. De Vausenville, I, 12. Devonshire, Duke of (7th), I, _290_. Diamandi, I, 86. Didacus Astunica, I, 90. Diderot, II, _4_, 283, 339. Digby, K. , I, _108_. Digges, T. , and L. , II, _302_. Dionysius Exiguus, I, _360_. Dircks, H. , II, 138, _138_. Discoverers and discoveries, II, 206. Discovery, Basis of, I, 85. D'Israeli, I. , I, _115_, 118, 136, 188, 227. Ditton, I, _133_, 133. Division, Nature of, II, 248. Dobson, J. , I, _234_, 234. Dodson, J. , II, _312_. Dodt, I, _52_. Doggerel verse, I, 341. Dolland, I, _377_. Double Vahu Process, II, 360. Douglas, G. , I, _232_. Drach, S. M. , II, _317_. Drayson, G. A. W. , II, _132_, 132. Dryden, II, _71_. Dual arithmetic, II, 186. Duchesne, I, _52_. Dumortier, I, 313. Duncan, A. , I, _179_. Dunkin, E. , II, _349_. Duodecimal scale, II, 68. Duplication Problem, I, 349. Dupuy, J. And P. , II, _295_. Dutens, L. , II, _90_. Dyer, G. , I, _178_. 

 Earth, Figure of, II, 54. Easter, I, 359. Easter Day Paradoxes, I, 353. Ebrington, Thos. , I, _247_. Edgeworth, Maria, I, _191_. Editorial System, I, 15. Edleston, I, _140_; II, 296. Edwards, J. , I, _144_. Edwards, T. , I, _112_. Eirenæus Philalethes, I, _125_, 125, 126. Eldon, Lord (1st), II, _197_. Elephant story, I, 58. Elizabeth, Queen, I, 128. Ellenborough, Baron, I, _181_. {379} Ellicot, I, 24. Ellis, I, _76_, 82. Engel, I, 230. English language, Origin of, I, 215. Enriques, F. , II, _367_. Epps, J. , I, _153_; II, 143. Equation of fifth degree, I, 250, 373. Erasmus, I, 110. Erastus, I, _65_. Erichsen, I, 163. Ersch, II, _193_, 282. Erskine, T. , II, _127_. Esperanto, Forerunner of, I, 116. Euclid, I, 5, 43; II, 118, 151. --Without Axioms, I, 287. Eudoxus, II, _164_. Euler, I, 221, _382_; II, 3, 4, 303, 331, 339. Eusebius, II, _220_. Eustace, J. C. , II, _46_. Eutocius, I, _41_; II, 60. Evelyn, J. , I, _108_. Everett, J. , I, _346_. Evidence, I, 57, 58. 

 Faber. (_See Stapulensis. _) Fairfax, Mary, I, _242_. Falco, I, 53. Faraday, M. , II, _351_. Fauré, de, I, 149; II, 238. Fawcett, H. , II, _249_. Ferguson, J. , II, _20_. Fermat, I, 122, 221; II, _300_. Ferrari, S. , II, 68. Fiction, New era in, I, 189. Fienus, I, _74_, 74. Filopanti, Q. B. , II, _93_. Finæus, I, _50_, 50, 113. Finleyson, J. , I, 314, _314_. Flamsteed, I, _87_, 309; II, 45, 143, 302, 306. Fletcher, I, 378. Fludd, II, _318_. Fly-leaf Paradox, II, 264. Folkes, M. , I, _136_; II, 301. Fontenelle, I, _103_. Forbes, D. , I, _237_. Forman, W. , I, 296, _296_, 306. Forster, T. I. M. , I, 320, _320_. Foscarini, I, _90_. Foster, S. , II, _310_. Fourier, II, _66_. Fox, G. , I, _397_. Francis, P. , II, _96_. Francoeur, I, _365_. Frankland, W. B. , I, 230, 287. Franklin, J. , II, _265_. Freedom of opinion, Growth of, I, 265. Freher, A. , II, 319. French academy on circle squaring, I, 163. Frend, W. , I, _196_, 196, 206, 208, 252. Fresnel, II, _48_. Fromondus, I, _74_, 74, 99. Frost, I. And J. , I, 394. Fry, Elizabeth, I, 224. Fuller, T. , I, _86_. Fulton, R. , I, _148_. 

 Gadbury, J. , I, _115_, 115. Galbraith, J. A. , II, 372. Galileo, I, 5, 6, 32, _76_, 82, 83, 96, 122, 381. Galle, J. G. , I, _386_; II, 7. Galloway, I, _56_, 57; II, 143. Gamblers, I, 280. Garrick, I, 21. Gascoigne, W. , II, _299_. Gassendi, I, _107_. Gauss, I, 86, 107, 310. Gemistus, G. , I, _188_. Gentleman's Monthly, Miscellany, I, 208. Gephryander. (_See Salicetus. _) Gergonne, I, _336_. Ghetaldi, I, _83_; II, 59. Ghost paradox, II, 47. Giddy (Gilbert), II, _174_. Gilbert, Davies, II, 66, _174_. Gilbert, William, I, 6, _68_, 68, 76. Gillot, II, _315_. Glazier (Glazion), II, 7. Godwin, F. , I, _103_. Godwin, W. , I, _174_. Golius, I, _106_. Gompertz, B. , I, _378_. Goulburn, I, _288_. Goulden, S. , II, 88. Graham, I, 24. Grandamicus, I, _104_, 104. Granger, J. , I, _156_. Grant, A. R. , II, 131. Grant, R. , I, _392_; II, 131. Grassi, O. , I, _262_. Grassini, I, _231_. Graunt, J. , I, 113, _114_, 154. Gravity, I, 151, 244, 348, 353. {380} --Newton's apple, I, 136. Greek numerals, II, _77_. Greene, R. , I, _135_, 135. Greenhill, Sir G. , I, 136. Greenwich observatory, I, 87. Gregg, T. D. , II, 75, _75_. Gregorian Calendar, I, 363. Gregory, D. , I, _66_; II, 301. Gregory, J. , I, _118_, 118, 207; II, 302. Gregory O. , II, _71_. Gregory, Pope, I, 362. Grevil, I, _202_. Grey, C. , (2d Earl), I, _315_; II, 247. Grosart, I, _141_, 141, 145. Grove, W. R. , II, _320_. Gruber, II, _193_, 282. Gruenberger, I, _70_. Grynaeus, I, _66_. Guaricus, I, _43_. Guillim, J. , II, _28_. Guldin, I, _83_. Gumpach, Von, II, 137, _137_. Gunning, H. , I, _198_. Gurney. (_See Fry, E. _) Guthrie, W. , I, _395_. 

 Hailes, J. D. , II, 135, _135_. Hailesean system of astronomy, II, 135. Hale, M. , I, _123_, 123. Hales, S. , I, _123_. Hall, B. , II, _181_. Hallam, I, 159. Halley, I, 24, _124_, 311; II, 301, 332. Halliwell-Phillips, II, _148_, 296. Hamilton, W. , I, _112_, 117, 331, 335, 339, 341, 342; II, 52, 53, 111. Hamilton, W. Rowan, I, _332_; II, 104, 256, 343. Hanover, King of, I, 201. Hardy, C. , I, _298_. Hardy, T. , I, _178_. Harriot, T. , II, _302_. Harvey, I, _76_, 78; II, 201. Hauff, I, _230_. Haughton, S. , II, 372. Hauksbee, F. , I, _156_. Hayes, C. , I, _132_, 132. Heath, D. D. , I, _76_. Heinfetter, H. , II, 94, _94_. Helmont, J. B. Van, I, _125_. Henson, II, 8. Heraclitus, II, _34_. Herbart, J. F. , I, 253, _253_; II, 78. Hérigone, II, _59_. Herschel, J. , I, _80_, 299, 326, 383, 386; II, 88, 95, 181, 255, 261, 262. Herschel, W. , I, _81_, 151, 192, 225, 233, 299; II, 288, 348. Heywood, F. , II, _49_. Hicks, J. P. , II, _67_. Higgins, G. , I, _257_, 274. Hilarius, Pope, I, _359_. Hill, J. , I, 21, 22, 23, 24. Hill, Rev. R. , I, _192_. Hill, Sir R. , I, _165_, 232. Hind, J. R. , I, _384_. Hippocrates, II, _59_. Hoax, An interesting, I, 163. --Lunar Caustic, I, 288. --Moon (Herschel), I, 326; II, 131. Hobbes, I, _105_, 109, 143, 144; II, 80. Hobhouse, J. C. , II, _126_. Hodder, J. , II, _265_. Hodge, C. B. , I, 114. Hodges, W. , I, 237. Hoffmann, J. J. , II, _282_. Hoffmann, J. J. I. Von, I, _230_. Holloway, B. , I, _237_. Holmes, O. W. , I, 109. Holyoake, G. J. , I, _399_, 399. Hone, W. , I, 124, 180, 184, 185. Hook, T. E. , II, _261_. Hooke, I, _77_; II, 300. Hooker, R. , II, _201_. Hopkins, J. , II, 41. Horace, I, 40. Horne, G. , I, _152_, 152, 154, 155, 236. Horne, J. , I, _178_. Horner, L. , I, _176_. Horner, W. G. , II, _66_, 151, 187. Houlston, W. , II, 156, _156_. Howard, E. , I, 131. Howison, W. , I, 256, _256_. Howitt, W. , II, 193, _193_. Howley, I, 63. Hulls, I, _147_, 147; II, 8. Hume, J. , I, _352_; II, 174. Husaín Rifki, II, 16. Hussein Effendi, II, 15. Hutchinson, J. , I, 154, _154_. Hutton, C. , I, _153_, 161; II, 303, 340. Huyghens, I, _100_, 133; II, 300. 

 Imaginary numbers, II, 186. Impalement by request, II, 133. Inaudi, I, 86. Index Expurgatorius, I, 90. {381} Infant prodigies, I, 86. Inglis, J. B. , II, _52_. Inglis, R. H. , I, _352_. Ingliz Selim Effendi, II, 15. Innocent I. , I, _359_. Irving, E. , II, _54_. Ivory, J. , II, 142, _142_. 

 Jabir ben Aflah, II, _59_. Jack, R. , I, 149. Jacotot, J. , I, 278, _278_. Jameson, Mrs. , II, _63_. Jeffreys, G. , I, _183_. Jenner, E. , II, _205_. Jesuit contributions, I, 164. Johnson, H. C. , I, 350. Johnson, S. , I, 20, _190_, 259; II, 117. Johnston, W. H. , II, 67. Jombert, I, _161_. Jonchère, I, _146_, 146. Jones, W. , I, _135_; II, 298, 301. Jones, Rev. W. , I, _237_. Jonson, B. , I, 13. Journals, Three classes of, II, 144. 

 Kantesian Jeweler, I, 258. Karsten, I, _230_. Kästner, I, _43_, 110, 112. Kater, I, 11. Keckermann, I, 3. Keill, J. , II, _302_. Kepler, I, 52, _76_, 82, 381; II, 166. Kerigan, T. , I, _308_, 353. Keroualle, De, II, 50. Kersey, I, 107. King, Wm. , I, _246_. Kircher, Adolphe, I, 229. Kircher, Athanasius, I, _103_. Kirkringius, T. , I, 125, _125_. Kittle, I, 236. Klein, F. , II, 367. Knight, C. , II, _109_, 280, 289. Knight, G. , I, 151, _151_. Knight, R. P. , II, _274_. Knight, Wm. , I, _97_. Koenig, S. , I, _150_. 

 Lacomme, I, 46. La Condamine, II, _301_. Lacroix, I, 41, 159, 207. Lactantius, I, 33, _96_. Lagrange, I, 221, _288_, 313, 382; II, 86. Laing, F. H. , II, _186_, 186. Lalande, I, _159_. Lamb, C. , I, 178; II, 270. Lambert, J. H. , I, _336_; II, 214, 367. Lambert, John, II, _309_. Language, Test of, II, 327. Lansbergius, I, _70_, 70. Laplace, I, 24, _255_, 382; II, 1, 340. Lardner, D. , II, _253_. Lardner, N. , I, 14; II, _221_. Laud, I, _145_. Lauder, W. , I, 297. Laurent, P. , I, _309_, 309. Laurie, J. , II, 4. Laurie, P. , II, _42_. Laurus, I, 381. Law, Edmund, I, _181_. Law, Edward, I, _181_. Law, W. , I, 168, _254_; II, 317. Le Coq, I, _86_. Lee, R. , I, _66_. Lee, S. , I, _131_. Lee, W. , I, 157. Legate, I, _59_. Legendre, I, _229_; II, 215, 367. Legh, P. , II, _68_, 83. Leibnitz, I, 5, 7; II, 46. Leo, St. , I, _359_. Leverrier, I, _43_, 82, 383, 386, 388, 390; II, 7, 135, 140, 303. Lewis, G. C. , II, 162, _162_. Libri, I, _40_, 62; II, 295. Lilius, Aloysius, I, 362. Lilly, I, _115_; II, 302. Lipen, M. , I, _298_. Little, J. , I, 206. Livingston, R. , I, _148_. Locke, J. , I, _142_, 142, 144; --and Socinianism, I, 142. Locke, R. , I, 146. Locke, R. A. , I, _326_; II, 86, 131. Logan, W. E. , I, _337_. Logic, Formal, I, 158; II, 75. --Has no paradoxes, I, 330. London Mathematical Society, I, 383. London, University of, I, 259; II, 71. Long, G. , II, _290_. Long, J. St. J. , II. _38_, 205. Longitude problems, I, 132, 146, 249. Longley, C. T. , I, _325_. Longomontanus, I, _105_, 105. Lottery, I. 281. Lovett, R. , I, 165, _166_. Lowe, R. , II, _169_. Lowndes, W. T. , I. _402_. Lubbock, J. , I, _279_; II, 148. {382} Lucas, F. , II, _28_. Lucian, I, _102_. Lunar Caustic Joke, I, 288. Lunn, J. R. , II, _66_. Lydiat, T. , II, _302_. Lyndhurst, I, _198_. 

 Macclesfield, Earls of, I, 7; II, _296_, 301. Macclesfield, Letters, II, 296. MacElshender, II, 87. Machin, J. , II, _301_. Mackey, John, I, 349. Mackey, S. A. , I, _256_. Maclear, T. , II, _181_. Macleod, H. D. , II, 184, _184_. Magic, I, 118. Magna Charta, I, 25. Magnus, I, 42. Maitland, S. , I, _63_, 163. Malacarne, I, 119. Malden, H. , II, _162_. Malius, II, _342_. Mallemens, II, _333_. Mankind gullible, I, 115. Manning, H. E. , (Card. ), II, _233_. Mansel, H. L. , II, _162_. Marcelis, J. , I, _129_, 129. Maret, II, _3_. Margarita Philosophica. (_See Reisch. _) Marryat, Capt. , II, _87_. Marsh, H. , I, _199_, 271. Martin, B. , I, _152_, 153. Martin, R. , II, _236_. Maseres, F. , I, _197_, 203. Mason, M. , II, _132_. Mathematical Illustrations of Doctrine, II, 70; --Psychology, I, 253; --Society, I, 374, 376, 382; --Theology, I, 149. Mathematics, Condensed history of, II, 58. Matter to Spirit, II, 194. Maty, I, 23. Maupertuis, II, _301_. Maurice, F. D. , II, _101_. Maurolycus, I, _121_. Maxwell, A. , I, _102_. Meadley, G. W. , I, _223_. Mechanics Magazine, II, 141, 145. Medici, Cosmo de, II, _295_. Medicine, Status of, I, 266. Melanchthon, II, _323_. Menestrier, I, _127_, 127. Mercator, G. , II, _92_. Mercator's projection, I, 84. Mersenne, I, _106_, 107; II, 295, 297. Meslier, J. , II, _195_. Meteorologist, An early, I, 320. Meteorology, I, 327. Metius, A. And P. , I, _52_, _99_, 99. Meton, II, 167. Metric System, Forerunner of, I, 171. Méziriac, I, 161. Milbanke, A. I. , I, 225. Mill, Jas. , I, 260. Miller, Joe, I, _182_. Miller, S. , I, 167. Mills, Elizabeth, W. , II, 7. Milne, J. , I, _286_. Milner, I. , I, _251_, 251. Milner, J. , II, _23_. Milner's lamp, I, 252. Milward, II, _250_. Miracles _vs_. Nature, II, 6. Mitchell, J. , I, _242_. Molière, I, _232_. Molina, A. C. De, I, _297_. Mollendorff, von, II, _3_, 338. Mondeux, I, 86. Montague, C. , II, _311_. Montmort, P. R. De, II, _301_. Montucla, I, 40, 45, 54, 65, 117, 120, 159, 163; II, 60. Moon Hoax, I, 326; II, 131. Moon, Nature of, II, 84; --Rotation of, II, 4, 19, 84, 87. More, Hannah, I, _189_, 192. More, Henry, I, 123. Moore, Dr. John, I, _190_. Moore, Sir John, I, _190_. Morgan, S. , I, 6. Morgan, T. , I, 191. Morgan, W. , I, _223_, 224. Morhof, I, _61_. Morin, I, _99_, 99. Morinus, J. B. , I, 149. Morland, S. , II, _302_. Mormonism, II, 69. Morrison, R. J. , I, _321_; II, 43. Mose, H. , II, _266_. Mottelay, I, 68. Motti, II, 60. Mouton, I, _172_; II, 300. Muggleton, I, 394, _395_. Multiplication, Nature of, II, 251. Murchison, R. I. , I, _384_. Murhard, I, _43_, 298. {383} Murphy, A. , II, _308_. Murphy, J. L. , II, 54, _54_. Murphy, P. , I, _327_, 398. Murphy, R. , I, 349, _349_. Murray, J. , I, _186_; II, 145. Murray, L. , II, _326_. Murray, Mungo, II, 310. Musgrave, T. , I, _324_. Mydorge, I, _298_. Mystrom, J. W. , II, 182. Mythological paradoxes, I, 256. 

 Names of Religious Bodies, II, 22. Napier, J. , I, 5, 66, _67_, 82. Napoleon, Doubts as to, I, 246. Nautical Almanac, I, 300; II, 147. Neal, I, _98_. Negative numbers, I, 196, 203. Neile, W. , II, _190_. Neptune, Discovery of, I, 387; II, 140. (_See Adams, Leverrier. _) Nesse, C, I, _128_, 128. Newcomb, S. , I, 162. Newcomen, T. , I, _147_. Newton, Sir Isaac, I, 5, 6, 8, 24, 39, 84, 88, 130, 136, 139, 144, 145, 148, 152, 154, 155, 162, 165, 167, 197, 225, 237, 242, 257, 282, 296, 297, 309, 311, 382, 394, 395, 396, 397; II, 2, 70, 184, 297, _302_, 305. Newton, John, II, _305_. Nicene Creed, I, 371. Nichol, J. P. , II, _289_. Nicholas. (_See Cusa. _) Nichols, J. , I, _175_. Nicolas, N. H. , I, _354_. Nicollet, I, _326_; II, 131. Nicolson, W. , I, _201_. Nieuwentijt, II, _333_. Nizzoli, M. , II, 275. Non-Euclidean geometry, II, 83. Northampton, Marquis of (2d), II, _19_. Novum Organum Moralium, II, 74. Number, Mystery of, I, 55, 56, 169. Number of the Beast (666), I, 55, 130, 272, 298, 352; II, 77, 159, 217, 218, 361, 373. Numeral system, II, 68. Nursery rhymes, II, 150. 

 Occam, Wm. Of, II, _40_. Odgers, N. , II, 191, _191_. Oinopides of Chios, II, _59_. Oldenburgh, H. , II, _300_, 302. Orthodox Paradoxes, II, 363. Orthography, Paradoxes of, II, 267. Ortwinus, I, 319. Oughtred, W. , II, _298_, 303. Owenson, I, _191_. Ozanam, I, _161_, 312. 

 Pagi, I, 32. Paine, T. , I, 173, _173_, 271. Paley, W. , I, _222_; II, 226. Palmer, C. , I, 225. Palmer, H. , I, _141_, 141, 145. Palmer, J. , II, _253_. Palmer, T. F. , II, _254_. Palmer, W. , II, _37_. Palmerston, Viscount (3d), I, _290_, 352. Palmézeaux, I, 167. Panizzi, A. , I, _151_. Papist, II, 26. Paracelsus, II, 322. Paradox defined, I, 2, 31. Paradox, religious, I, 236. Paradoxers in general, II, 352. Parallels, Theory of, I, 229, 344. Pardies, I. G. , II, _300_. Park, Mungo, II, _91_, 132. Parker, F. , II, _94_. Parker, G. (Earl of Macclesfield), II, _296_. Parr, S. , I, 173, _173_, 175, 176, 184. Parsey, I, 293, _293_. Partridge, J. , I, _305_. Pasbergius, I, 381. Pascal, I, 39, 119, 122, _220_, _221_; II, 73. Pascal's Hexagram, I, 221. Passot, I, 279, _279_. Passover, I, 358, 372. Patriotic paradox, I, 231. Paucton, I, _172_. Paulian, I, 165, _165_. Peacock, Geo. , I, _196_, 350. Peacock, T. L. , I, _190_, 340. Pearce, A. J. , II, 43. Pearne, T. , I, _239_. Peel, Sir R. , I, 290, _352_. Peel, W. Y. , I, _290_. Pèlerin, J. , II, _324_. Pell, J. , I, _105_, 105, 107; II, 300, 302, 312. Pepys, I, _113_, 114. Perigal, H. , II, 19, _20_. {384} Perpetual motion, I, 118, 348; II, 55, 138. Perspective, New theory of, I, 293. Peters, W. , II, 11, 315. Petitioning Comet, I, 127. Petrie, W. M. F. , I, _328_. Petty, I, _114_; II, 300. Philalethes, Eirenaeus, I, _125_, 125, 126. Philalethes, Eugenius, I, _255_. Phillips, R. , I, 242, _242_, 245. Philo of Gadara, I, 40, _40_. Philosopher's stone, I, 118, 125. Philosophical atheists, II, 1. Philosophy and Religion, II, 37. Phonetic spelling, II, 81. [pi], values of, I, 11, 43, 45, 46, 52, 69, 100, 110, 129, 135, 146, 245, 283, 284, 294, 347, 348, 349, 350; II, 60, 63, 105, 110, 118, 135, 156, 209, 279, 315, 316. Pighius, I, _372_. Pike, S. , I, 236, _236_. Pindar, P. , II, _272_. Piozzi, Mrs. , I, _235_; II, 272. Piscator, B. , II, 25. Pitman, F. , II, 81, _81_. Place, F. , I, _199_. Planets inhabitable, I, 100, 102. Plato, I, 5. Platt, H. , I, _126_, 126. Playfair, J. , I, _233_. Pletho, G. , I, _188_. Pliny, II, 280. Ploucquet, I, _336_, 337. Poe, E. A. , II, 132. Poincaré, I, 136. Poisson, I, _292_; II, 2. Pollock, J. F. , II, _174_. Pons, II, _45_. Pope, Wm. , I, 277, _277_. Porta, I, _68_, 68, 83. Porteus, B. , I, _193_, 203. Porteus, H. F. A. , II, 157, _157_. Porus, I, 44. Powell, Baden, II, _267_. Powell, W. S. , I, _222_. Pratt, H. F. A. , II, 157, _157_. Pratt, O. , II, _69_. Predaval, Count de, I, _348_. Prescot, B. , I, 270, _270_, 278. Prester John, I, 70, _71_, 152. Price, R. , I, _223_. Probability, Discourse on, I, 279. Proclus, I, 188, _188_. Prodigies, Youthful, I, 219, 332. Pronunciation, II, 330. Protestant and Papal Christendom, II, 33. Protimalethes, II, 6. Ptolemy, I, 5, 33, _380_. Pullicino, II, 61, _61_. Pusey, I, _64_. Pyramids, The, I, 328; II, 95, 136. Pythagoras, II, _59_. 

 Quadrature problem. (_See Squaring the circle. _) Quarles, F. , II, _277_. Quintilian, II, 280. Quotem, C. , I, 399. 

 Rabelais, I, _102_. Rainbow Paradox, II, 334. Ramachandra, Y. , I, _374_. Ramchundra, I, 374. Ramus, I, 5. Recalcati, II, 208, 314. Recorde, R. , II, _328_. Reddie, Jas. , II, 183, _183_, 344, 360. Reeve, J. , I, _395_. Regiomontanus, I, _48_, 360. Reisch, I, _45_; II, 281. Religion and Philosophy, II, 37. Religious bodies, Names of, II, 22; --customs, Attacks on, I, 177; --Insurance, I, 345; --Paradox, I, 236; --Tract society, I, 192. Remigius, I, _50_. Reuchlin, J. , II, _323_. Revelations, Napier on, I, 66. Revilo, (O. Byrne), I, 241, 329, _329_. Reyneau, C. R. , II, _301_. Rheticus, I, _69_; II, 372. Rhonius, II, 300. Ribadeneira, P. De, II, _62_. Riccioli, I, _96_. Richards, G. , II, _270_. Rigaud, J. , II, _299_. Rigaud, S. J. , II, _299_. Rigaud, S. P. , I, _140_; II, 298, 313. Ringelbergh, J. S. , II, _281_. Ripley, G. , I, _126_, 126. Ritchie, W. , I, 295, _295_. Ritterhusius, I, _60_. Rive, J. -J. , I, _160_. Robertson, Jas. , I, _237_. Roberval, I, _105_, 122. {385} Robinson, B. , I, _148_, 148. Robinson, H. C. , I, _314_; II, 52, 275. Robinson, R. , I, _177_. Robinson, T. R. , II, _181_. Roblin, J. , II, 136. Rogers, S. , II, _260_. Roget, P. M. , I, _398_. Roomen, A. Van, I, _110_. Ross, J. C. , I, _303_. Rosse, I, 26. Rossi, G. , I, 231, _231_. Rotation of the Moon, II, 4, 19. Rough, W. , I, 198. Rowning, J. , I, _155_. Royal Astronomical Society, I, 27; --Forerunner of, I, 374. Royal Society, I, 21, 22, 24-30, 56, 57, 136, 153, 163, 164, 165. Rudio, I, 159; II, 367. Rudolff, C. , II, 373. Russell, Earl (1st), I, _296_. Rutherford, W. , II, _109_. 

 Sabatíer, A. , II, _267_. Sabellius, I, _241_. Sacrobosco, I, _360_. Sadler, T. , I, _238_, 241. Saint-Martin, I, 167, _168_, 206. St. -Mesmin, M. De. , I, 280. St. Vincent, G. De. , I, _110_, 117. St. Vitus, Patron of Cyclometers, II, 60. Sales, de, I, 167. Salicetus, I, 64. Salisbury, Earl of (1st), II, _330_. Salmasius, Claudius, II, _168_. Salusbury, Hester, I, _235_. Sanchez, Petro, I, 229, _229_. Sanders, W. , I, 207. Sanderson, R. , I, _135_. Sara, R. , I, _297_. Saunderson, N. , I, _377_; II, 301. Scaliger, I, _44_, 110, 111, 112, 113; II, 238. Scévole de St. Marthe, I, 113. Schooten, Van, II, _59_. Schopp, I, _60_. Schott, I, _64_; II, 64. Schumacher, H. C. , I, _107_; II, 297. Schwab, I, _230_. Scientific paradoxes, I, 232. Scott, Michael, I, _38_. Scott's Devils, I, 38. Scott, W. , I, 20, 27, 38, 39, 155, _191_. Scripture and Science, II, 261. Search, John, I, 247. Selden, J. , II, _250_. Sénarmont, II, _48_. Serres, De, II, _60_. Shaftesbury, Earl of, II, _181_. Shakespeare, I, 13. Shanks, II, _63_, 65, 109. Shaw, P. , I, _142_. Sheepshanks, J. , I, _147_. Sheepshanks, R. , I, _290_. Shelley, I, 174. Shepherd, S. , I, _124_. Sherburne, E. , II, _295_. Sheridan, R. B. , I, _175_. Sheridan, T. , I, _175_. Shoberl, F. , II, _270_. Shrewsbury, I, _108_. Siddons, Mrs. , I, 189. Simms, W. , I, _152_. Simplicius, II, _164_. Simpson, T. , I, 377; II, 304. Simson, R. , I, _197_, 202, 233. Sinclair, G. , I, _207_. Slander Paradoxes, II, 138. Sloane, I, 24. Sluse, R. De, I, _118_, 118; II, 300. Smith, Adam, II, _112_. Smith, Jas. , I, _46_; II, 103, _103_, 154, 236, 237, 238, 241, 336, 360. Smith, Jas. , II, _217_. Smith, Jas. (Shepherd), II, _55_, 193. Smith, Joseph, II, _69_. Smith, Richarda, I, _242_. Smith, Thomas, I, 346, _346_. Smith, Wm. , II, _152_. Smyth, C. P. , I, _328_; II, 65. Snell, I, _75_, 75. Socinianism, I, 142, 143. Socinus, I, 3, _143_. Socrates Scholasticus, I, _358_. Sohncke, L. A. , II, _131_. Somerville, Mrs. , I, _242_. South, J. , II, _181_. Southcott, Joanna, II, _58_, 97, 239. Spearman, R. , I, _237_. Speculative thought in England, I, 374. Spedding, I, _76_, 82, 142. Speed, J. , I, _201_. Speke, I, _70_. Spelling, phonetic, II, 81. Spence, W. , I, _231_, 231. Spencer, Earl (3d), II, _9_. {386} Spinoza, I, 3, _37_. Spiritualism, II, 47, 55, 207. Spurius Cassius Viscellinus, II, _342_. Spurius Maelius, II, _342_. Squaring the circle, I, 8, 42, 44, 46, 47, 50, 52, 53, 69, 70, 75, 109, 117, 119, 129, 135, 146, 149, 159, 163, 164, 347, 348, 374; II, 10, 11, 60, 105, 154, 156, 208, 278, 314. Stäckel, I, 230. Stanhope, P. D. , (Earl of Chesterfield), II, _298_. Stapulensis, I, _44_; II, 324. Star polygons, I, 229. Starkie, G. , I, _126_, 126. Statter, D. , II, 80. Steamship suggested, I, 147. Steel, Jas. , II, 68. Stenography, II, 81. Stephens, I, _44_; II, 324. Stephenson, G. , II, _138_. Stephenson, R. , II, _138_. Stevin, I, _83_, 313; II, 59. Stewart, D. , II, _53_. Stewart, R. , I, _186_. Stifel, M. , II, _373_. Strafford, Earl of, I, _240_. Stratford, W. S. , I, _300_. Street, T. , I, _124_. Stukely, W. , I, _236_. Suffield, G. , II, _66_. Suidas, II, _29_. Sumner, C. R. , I, _324_. Sumner, J. B. , I, _324_. Sun as an electric space, II, 41. Supernatural, The, II, 193. Suvaroff, II, _85_. Swastika, II, 231. Swedenborg, E. , I, _255_. Swift, I, 19, 133. Sylvester, J. J. , II, _336_. Symington, W. , I, _148_. Symons, II, 4, 5, 20, 84, 85. Sympathetic powder, I, 108. Synesius, I, 125. 

 Talbot, G. , I, _22_, _108_. Talbot's powder, I, 108. Tartaglia, II, 59. Tasse, I, _106_. Tate, J. , I, _199_. Tauler, J. , II, _322_. Taylor, Brook, II, _301_. Taylor, John, I, _352_; II, 95. Taylor, Robt. , I, _270_. Taylor, T. , I, 188, _188_. Teissier, I, _113_. Temple, H. J. , I, 290. Tenterden, Chief Justice, I, _181_. Thales, II, _59_, 83. Theism independent of Revelation, I, 399. Thelwall, J. , I, _178_. Theodoretus, I, _358_. Theological Paradoxes, I, 316. Theology, Mathematical, I, 129, 149. Theophrastus, II, _167_. Thiébault, II, _3_, 338. Thom, D. , II, _226_, 240. Thom, J. H. , II, _226_. Thompson, P. , I, _7_. Thompson, T. P. , I, _252_, 287, 344; II, 83, 185. Thomson, Dr. , I, 21. Thomson, W. , I, _325_. Thorn, W. , II, 158, _158_, 360. Thorndike, H. , II, _313_. Thrale, Mrs. , I, _235_. Thurlow, Baron, I, _222_. Thyræus, I, 50. Tides, New theory of, I, 393. Tombstones of mathematicians, I, 106. Tonal System, II, 182. Tooke, H. , I, _178_. Torriano, E. , I, 250. Towneley, II, 300. Townley, C. , II, _300_. Trisection problem, I, 118; II, 10, 12, 13, 15. Troughton, I, _152_. Turnor, E. , I, _137_. Tycho Brahe, I, 5, _76_, 381; II, 302, 335. 

 Upton, W. , II, 12, _12_, 15. Ursus, I, _52_. 

 Valentine, B. , I, _125_, 125. Van Ceulen, I, 52, 70, 100. Van de Weyer, I, _313_. Van Etten, I, _161_. Van Helmont, I, _125_, 125. Van Roomen, I, _110_. Van Schooten, II, _59_. Vaughan, T. , I, _255_. Victorinus, I, _359_. Viète, I, _51_; II, 210, 295. Virgil, St. , I, 32, _33_, 34, 99. {387} Virginia, University of, I, 233. Viscellinus, II, _342_. Vitruvius, II, _281_. Vivian, T. , I, 172, _172_. Vogel, A. F. , I, 373. Voltaire, I, 103, 165, 166, 167, 168, 248; II, 268. Von Gumpach, II, 137, _137_. Von Hutten, I, 318. Von Wolzogen. (_See Wolzogen. _) Vyse, R. W. H. , I, _328_. 

 Walker, W. E. , II, _316_. Walkingame, F. , II, _173_. Wallich, N. , II, _14_. Wallis, J. , I, 107, 109, 110; II, 299, 313. Walpole, I, _23_, 131. Walsh, John, I, 260, _260_; II, 157. Wapshare, J. , II, _230_. Warburton, H. , I, _349_. Warburton, Wm. , I, _55_, 112; II, 174. Ward, S. , II, _299_. Waring, E. , I, _203_, 222. Warner, W. , II, _302_, 312. Warren, S. , II, _340_. Watkins, J. , II, _270_. Watson, Bp. , I, _223_. Watt, R. , I, _102_, 402. Watts, I. , II, 18. Weddle, T. , II, _187_. Wentworth, Thos. , I, _240_. Wharton, I, 115. Whately, R. , I, 246, _246_, 324. Whately's Paradox, I, 246. Whewell, I, _101_, 101, 273, 314, 380; II, 104, 246, 247. Whigs, II, 22. Whiston, J. , I, _147_. Whiston, W. , I, 133, _133_, 146, 156, 311. White, H. K. , II, _271_. White, J. B. , I, 248. White, R. , I, 11. Whitford, I, 105. Whitworth, W. A. , II, _344_. Whizgig, On the, I, 254. Wightman, I, 59. Wilberforce, W. , II, _236_. Wilkins, J. , I, 96, _100_, 116, 226. Williams, J. B. , I, _378_. Williams, T. , I, 171, _171_. Wilson, Sir J. , I, _221_. Wilson, J. M. , II, _344_. Wilson, R. , II, 7, _7_. Wilson's Theorem, I, _222_. Wingate, E. , II, _308_. Winter, I, 46. Wirgman, T. , I, 258, _258_. Wiseman, N. P. S. , II, _26_, 61, 294. Wolcot, J. (Peter Pindar), II, _272_. Wollstonecraft, I, 173, _173_. Wolzogen, I, _106_. Wood, A. , I, _98_. Wood, John, I, _233_. Wood, Wm. , I, 246, _246_. Woodley, W. , I, 307, _307_. Wordsworth, II, 273. Wright, E. , I, 84. Wright, T. , I, 151, _151_, 152, 153. Wright, W. , II, 9. Wronski, I, 249, _250_. Wrottesley, J. (Baron), II, _181_. 

 Young, B. , II, _69_. Young, J. W. A. , II, 367. Young, T. , I, 24, 30, _250_. Youthful Prodigies, I, 219. Yvon, I, _297_. 

 Zach, von, II, _45_, 196. Zachary, Pope, I, 32, 34. Zadkiel, I, _321_; II, 43. Zetetic Astronomy, II, 88. Zodiac, II, 136. Zytphen, II, _335_. 

 * * * * *

Notes

Transcriber's note: References to Notes in Volume I are shown as in theprinted book, with the resequenced footnote numbers in the ProjectGutenberg Edition (EText-No. 23100) added thus {123}. 

[1] See Vol. I, page 255, note 6 {584}. 

[2] "I have no need for this hypothesis. "

[3] "Ah, it is a beautiful hypothesis; it explains many things. "

[4] "What we know is very slight; what we don't know is immense. "

[5] Brewster relates (_Life of Sir Isaac Newton_, Vol. II, p. 407) that, ashort time before his death, Newton remarked: "I do not know what I mayappear to the world, but to myself I seem to have been only like a boyplaying on the seashore, and diverting myself in now and then finding asmoother pebble or a prettier shell than ordinary, whilst the great oceanof truth lay all undiscovered before me. "

[6] See Vol. I, p. 292, note 1 {632}. 

[7] "What is all that!"

[8] "I have some good news to tell you: at the Bureau of Longitudes theyhave just received a letter from Germany announcing that M. Bessel hasverified by observation your theoretical discoveries on the satellites ofJupiter. "

[9] "Man follows only phantoms. "

[10] See Vol. I, page 382, note 13 {786}. 

[11] Dieudonné Thiébault (1733-1807) was a Jesuit in his early life, but heleft the order and took up the study of law. In 1765 he went to Prussia andbecame a favorite of Frederick the Great. He returned to France in 1785 andbecame head of the Lycée at Versailles. 

[12] _Memories of Twenty Years of Residence in Berlin. _ There was a secondFrench and an English edition in 1805. 

[13] Richard Joachim Heinrich von Mollendorff (1724-1816) began his careeras a page of Frederick the Great (1740) and became field marshal (1793) andcommander of the Prussian army on the Rhine (1794). 

[14] Hugues Bernard Maret (1763-1839) was not Duc de Bassano in 1807, thistitle not being conferred upon him until 1809. He was ambassador to Englandin 1792 and to Naples in 1793. Napoleon made him head of the cabinet andhis special confidant. The Bourbons exiled him in 1816. 

[15] Denis Diderot (1713-1784), whose _Lettre sur les aveugles_ (1749)introduced him to the world as a philosopher, and whose work on the_Encyclopédie_ is so well known. 

[16] "Sir, (a + b^{n}) / n = x, whence God exists; answer!"

[17] This was one James Laurie of Musselburgh. 

[18] Jelinger Cookson Symons (1809-1860) was an office-holder with adecided leaning towards the improvement of education and social conditions. He wrote _A Plea for Schools_ (1847), _The Industrial Capacities of SouthWales_ (1855), and _Lunar Motion_ (1856), to which last work the criticprobably refers. 

[19] "Protimalethes" followed this by another work along the same line thefollowing year, _The Independence of the Testimony of St. Matthew and St. John tested and vindicated by the theory of chances_. 

[20] Wilson had already taken up the lance against science in his_Strictures on Geology and Astronomy, in reference to a supposed want ofharmony between these sciences and some parts of Divine Revelation_, Glasgow, 1843. He had also ventured upon poetry in his _Pleasures ofPiety_, Glasgow, 1837. 

[21] Mrs. Borron was Elizabeth Willesford Mills before her marriage. Shemade an attempt at literature in her _Sibyl's Leaves_, London (printed atDevonport), 1826. 

[22] See Vol. I, page 386, note 10 {801}. 

[23] See Vol. I, page 43, notes 7 {32} and 8 {33}. 

[24] His flying machine, designed in 1843, was one of the earliest attemptsat aviation on any extensive scale. 

[25] Erasmus Darwin (1731-1802) was the grandfather of Charles Darwin. Thework here mentioned had great influence, being translated into French, Portuguese, and Italian. Canning parodied it in his _Loves of theTriangles_. 

[26] See Vol. I, page 147, note 1 {312}. 

[27] The notes on this page were written on the day of the funeral ofWilbur Wright, June 1, 1912, the man who realized all of these prophecies, and then died a victim of municipal crime, --of typhoid fever. 

[28] John Charles, third Earl Spencer (1782-1845), to whose efforts theReform Bill was greatly indebted for its final success. 

[29] This was published in London in 1851 instead of 1848. 

[30] This appeared in 1846. 

[31] This was done in _The Circle Squared_, published at Brighton in 1865. 

[32] It first appeared in 1847, under the title, _The Scriptural Calendarand Chronological Reformer, 1848. Including a review of tracts by Dr. Wardlaw and others on the Sabbath question. By W. H. Black. _ The one abovementioned, for 1849, was printed in 1848, and was also by Black(1808-1872). He was pastor of the Seventh Day Baptists and was interestedin archeology and in books. He catalogued the manuscripts of the AshmoleanMuseum at Oxford. 

[33] William Upton, a Trinity College man, Dublin. He also wrote _Upton'sPhysioglyphics_, London, 1844; _Pars prima. Geometria vindicata;antiquorumque Problematum, ad hoc tempus desperatorum, Trisectionis Anguli, Circulique Quadraturae, Solutio, per Eucliden effecta, London_ (printed atSouthampton), 1847; _The Uptonian Trisection_, London, 1866; and _TheCircle Squared_, London, 1872. 

[34] For example, if [theta] = 90° we should have 3 cos 30° = 1 + [root](4- sin^2 90°), or 3. ½ [root]3 = 1 + [root]3, or ½ [root]3 = 1. 

[35] Nathaniel Wallich (1786-1854) was surgeon at the Danish settlement atSerampore when the East India Company took over the control in 1807. Heentered the British medical service and was invalided to England in 1828. His _Plantae Asiaticae Rariores_ (3 vols. , London, 1830-1832) wasrecognized as a standard. He became vice-president of the Linnean Society, F. R. S. , and fellow of the Royal Asiatic Society. 

[36] But if [theta] = 90° this asserts that

 cos 30° = (sin 270° . Cos 225° + sin^2 90° . Sin 225°) / [root](sin^2 270° . Cos^2 225° + sin^{4} 90° + sin 270° . Sin 450° . Sin^2 90°), or that

 ½ [root]3 = (-1 . (-1 /[root]2) + 1 . (-1/[root]2) / [root]1 . ½ + 1 - 1 . 1 . 1) = 0 / [root]½, so that De Morgan must have made some error in copying. 

[37] John Bonnycastle (died in 1821) was professor of mathematics atWoolwich. His edition of Bossut's _History of Mathematics_ (1803), and hisworks on elementary mathematics were well known. 

[38] The bibliographies give Husaín Rifki as the translator, a practicalgeometry as the work, and 1802 as the date. 

[39] See Vol. I, page 309, note 2 {670}. 

[40] Probably in _The Improvement of the Mind_ which Isaac Watts(1674-1748) published in 1741. His _Horae Lyricae_ appeared in 1706, andthe _Hymns_, by which he is still well known, in 1707. 

[41] Spencer Joshua Alwyne Compton, second Marquis of Northampton(1790-1851), was a poet, a scientist, and a statesman. He was president ofthe Royal Society from 1838 to 1849. 

[42] Besides the writings here mentioned Perigal published a work on_Geometric Maps_ (London, 1853), and _Graphic Demonstrations of GeometricProblems_ (1891). 

[43] See Vol. II, page 5, note 18. 

[44] James Ferguson (1710-1776) was a portrait painter, an astronomer, anda popular writer and lecturer on various subjects. 

[45] In the old ballad of King Alfred and the Shepherd, when the latter istempting the disguised king into his service, he says:

 "Of whig and whey we have good store, And keep good pease-straw fire. "

_Whig_ is then a preparation of milk. But another commonly cited derivationmay be suspected from the word _whiggamor_ being used before _whig_, asapplied to the political party; _whig_ may be a contraction. Perhaps bothderivations conspired: the word _whiggamor_, said to be a word of commandto the horses, might contract into _whig_, and the contraction might bewelcomed for its own native meaning. --A. De M. 

[46] This was p. 147 in the first edition. 

[47] St. Augustine (354-430) was bishop of Hippo. His _Confessiones_, in 13books, was written in 397, and his _De Civitate Dei_ in 426. 

[48] "He was wont to indulge in certain Punic subtleties lest he shouldweary the reader by much speaking. "

[49] John Milner (1751-1826), bishop of Castabala, a well-knownantiquarian. 

[50] It will be said that when the final happiness is spoken of in "sureand certain hope, " it is _the_ Resurrection, generally; but when afterwardsapplication is made to the individual, simple "hope" is all that ispredicated which merely means "wish?" I know it: but just before thegeneral declaration, it is declared that it _has_ pleased God of his greatmercy to _take unto Himself_, the soul of our dear brother: and between the"hopes" hearty thanks are given that it _has_ pleased God to deliver ourdear brother out of the miseries of this wicked world, with an additionalprayer that the number of the elect may shortly be accomplished. All whichmeans, that our dear brother is declared to be taken to God, to be in aplace not so miserable as this world--a description which excludes the"wicked place"--and to be of the elect. Yes, but it will be said again! doyou not know that when this Liturgy was framed, all who were not in theroad to Heaven were excommunicated burial service read over them. Supposingthe fact to have been true in old time, which is a very spicy supposition, how does that excuse the present practice? Have you a right _always_ to saywhat you believe _cannot always_ be true, because you think it was once_always_ true? Yes, but, choose whom you please, you cannot be _certain_ Heis _not_ gone to Heaven. True, and choose which Bishop you please, youcannot be demonstratively _certain_, he is _not_ a concealed unbeliever:may I therefore say of the whole bench, _singulatim et seriatim_, that they_are_ unbelievers? No! No! The voice of common sense, of which common logicis a part, is slowly opening the eyes of the multitude to the unprincipledreasoning of theologians. Remember 1819. What chance had ParliamentaryReform when the House of Commons thanked the Manchester sabre-men? If youdo not reform your Liturgy, it will be reformed for you, and sooner thanyou think! The dishonest interpretations, by defence of which even theminds of children are corrupted, and which throw their shoots intoliterature and commerce, will be sent to the place whence they came: andover the door of the established organization for teaching religion will beposted the following notice:

 "Shift and Subterfuge, Shuffle and Dodge, No longer here allowed to lodge!"

All this ought to be written by some one who belongs to the Establishment:in him, it would be quite prudent and proper; in me, it is kind andcharitable. --A. De M. 

[51] But few do have access to it, for the work is not at all common, andthis Piscator is rarely mentioned. 

[52] This derivation has been omitted. --S. E. De M. 

[53] A blow for a blow. Roland and Oliver were two of the paladins ofCharlemagne whose exploits were so alike that each was constantly receivingcredit for what the other did. Finally they met and fought for five days onan island in the Rhine, but even at the end of that period it was merely adrawn battle. 

[54] "In the name of the church. "

[55] "From the chair, " officially. 

[56] Nicholas Patrick Stephen Wiseman (1802-1865), whose elevation to thearchbishopric of Westminster and the cardinalate (1850) led to the actprohibiting Roman Catholics from assuming episcopal titles in England, alaw that was never enforced. 

[57] He was born in 1812 and was converted to Catholicism in 1839. Hefounded the _Tablet_ in London in 1840, removing its office to Dublin in1849. He became M. P. In 1852, and at the time of his death (1855) he waspreparing a memorial to the Pope asking him to annul the proclamation of anIrish bishop prohibiting his priests from taking part in politics. 

[58] John Guillim (1565-1621) was the first to systematize and illustratethe whole science of heraldry. He published _A display of Heraldrie:manifesting a more easie accesse to the knowledge thereof_ in 1610. 

[59] "Faith. "

[60] "Faithful. "

[61] "For the faith vindicated. "

[62] The words are of the same root, and hence our word _fiddle_. Somesuppose this root means a _rope_, which, as that to which you trust, becomes, in one divergence, confidence itself--just as a _rock_, and otherwords, come to mean reliance--and in another, a little string. --A. De M. 

[63] The Greek lexicographer, a Christian, living after 1000 A. D. Hislexicon was first printed at Milan in 1499. 

[64] _Skindapsos. _

[65] This was John William Burgon (1813-1888), Gresham professor oftheology (1867) and dean of Chichester. He was an ultra-conservative, opposing the revised version of the New Testament, and saying of theadmission of women to the university examinations that it was "a thinginexpedient and immodest. "

[66] _Ekklesia_, or _ecclesia_. 

[67] _Ennomos ekklesia. _

[68] "Without doubt I shall perish forever. "

[69] "Every man is an animal. " "Sortes is a man. " "Sortes is an animal. "

[70] "For a special purpose. "

[71] Heraclitus of Ephesus, the weeping philosopher, 6th century B. C. 

[72] Democritus, the laughing philosopher, founder of the atomistic theory, 5th century B. C. 

[73] "Ends to which. "

[74] "Ends from which. "

[75] "In just as many syllables, " "With just as many letters, " "In just asmany words. "

[76] "I shall make a way, " "I shall find a way. "

[77] The notion that the Evil Spirit is a functionary liable to bedismissed for not attending to his duty, is, so far as my reading goes, utterly unknown in theology. My first wrinkle on the subject was the remarkof the Somersetshire farmer upon Palmer the poisoner-- "Well! if the Devildon't take he, he didn't ought to be allowed to be devil no longer. "--A. DeM. 

William Palmer (1824-1856) was a member of the Royal College of Surgeonsand practised medicine at London. He was hanged in 1856 for having poisoneda friend and was also suspected of having poisoned his wife and brother fortheir insurance money, besides being guilty of numerous other murders. Histrial was very much in the public attention at the time. 

[78] Advantages and dangers. 

[79] The old priory of St. Mary of Bethlehem in London, was used as anasylum for the insane. The name was corrupted to Bedlam. 

[80] Referring to the common English pronunciation of St. John, almostSinjin. John St. John Long (1798-1834), an Irishman by birth, practisedmedicine in London. He claimed to have found a specific for rheumatism andtuberculosis, but upon the death of one of his patients in 1830 he wastried for manslaughter. He died of tuberculosis four years later, refusingto take his own treatment. 

[81] William of Occam (d. 1349), so called from his birthplace, Ockham, inSurrey. He was a Franciscan, and lectured on philosophy in the Sorbonne. 

[82] He signs himself "James Hopkins, schoolmaster, " and this seems to havebeen his only published effort. 

[83] Joseph Ady (1770-1852) was a famous swindler. One of his best-knownschemes was to send out letters informing the recipients that they wouldlearn something to their advantage on payment of a certain sum. He spentsome time in prison. 

[84] Sir Peter Laurie (c. 1779-1861) was worth referring to, for he wasprominent as a magistrate and was honored because of his interest in allsocial reforms. He made a fortune as a contractor, became sheriff of Londonin 1823, and was knighted in the following year. He became Lord Mayor ofLondon in 1832. 

[85] See Vol. I, page 321, note 2 {691}. The _Astronomy in a nutshell_appeared in 1860. _The Herald of Astrology_ was first published in Londonin 1831, "by Zadkiel the Seer. " It was continued as _The AstrologicalAlmanac_ (London, 1834), as _Zadkiel's Almanac and Herald of Astrology_(_ibid. _, 1835, edited by R. J. Morrison, and subsequently by A. J. Pearce), and as _Raphael's Prophetic Almanac_ (1840-1855). 

[86] See Vol. I, page 172, note 3 {382}. 

[87] See Vol. I, page 87, note 4 {133}. 

[88] Franz Xaver, Freiherr von Zach (1754-1832) was director of theobservatory at Seeberge near Gotha. He wrote the _Tabulae specialesaberrationis et mutationis_ (1806-7), _Novae et correctae tabulae solis_(1792), and _L'attraction des montagnes et ses effets sur le fil à plomb_(1814). 

[89] Jean Louis Pons (1761-1831) was connected with the observatory atMarseilles for thirty years (1789-1819). He later became director of theobservatory at Marlia, near Lucca, and subsequently filled the same officeat Florence. He was an indefatigable searcher for comets, discovering 37between 1801 and 1827, among them being the one that bears Encke's name. 

[90] This hypothesis has now become an established fact. 

[91] John Chetwode Eustace (c. 1762-1815) was born in Ireland. Although aRoman Catholic priest he lived for a time at Cambridge where he did sometutoring. His _Classical Tour_ appeared in 1813 and went through severaleditions. 

[92] "Crimes should be exposed when they are punished, but disgraceful actsshould be hidden. "

[93] Henri Hureau de Sénarmont (1808-1862) was professor of mineralogy atthe _Ecole des mines_ and examiner at the _Ecole polytechnique_ at Paris. 

[94] Augustin Jean Fresnel (1788-1827), "Ingenieur des ponts et chaussées, "gave the first experimental proofs of the wave theory of light. He studiedthe questions of interference and polarization, and determined theapproximate velocity of light. 

[95] "As is my custom. "

[96] Francis Heywood (1796-1858) made the first English translation ofKant's _Critick of Pure Reason_ (1838, reprinted in 1848). The _Analysis_came out, as here stated, in 1844. 

[97] Louise Renée de Keroualle, Duchess of Portsmouth and Aubigny(1649-1734), was a favorite of Charles II. She used her influence to keephim under the control of Louis XIV. 

[98] William Chiffinch (c. 1602-1688) was page of the king's bed-chamberand keeper of the private closet to Charles II. He was one of the king'sintimates and was an unscrupulous henchman. 

[99] "Well devised. "

[100] "John Bellingham Inglis. His _Philobiblion_ "translated from thefirst edition (of Ricardus d'Aungervile, Bishop of Durham), 1473, " appearedat London in 1832. It was republished in America (Albany, N. Y. ) in 1864. 

[101] "What are you laughing at?"

[102] See Vol. I, page 314, note 4 {681}. 

[103] See Vol. I, page 112, note 7 {211}. 

[104] Referring to Hamilton's edition of the _Collected Works of DugaldStewart_, 10 volumes, Edinburgh, 1854-58. It is not commonly rememberedthat Stewart (1753-1828) taught mathematics at the University of Edinburghbefore he took up philosophy. 

[105] This was Hamilton's edition of the _Works of Thomas Reid_ (2 vols. , Edinburgh, 1846-1863). Reid (1710-1796) included mathematics in his work inphilosophy at Aberdeen. In 1764 he succeeded Adam Smith at Glasgow. 

[106] Edward Irving (1792-1834), the famous preacher. At first he assistedDr. Chalmers at Glasgow, but in 1822 he went to London where he met withgreat success. A few years later he became mentally unbalanced and wasfinally expelled from his church (1832) for heresy. He was a great friendof Carlyle. 

[107] He also wrote a number of other paradoxes, including _An Essaytowards a Science of Consciousness_ (1838), _Instinctive Natural Religion_(1858), _Popular Treatise on the structure, diseases, and treatment of thehuman teeth_ (1837), and _On Headache_ (1859). 

[108] James Smith (1801-1857), known as Shepherd Smith, was a socialist anda mystic, with a philosophy that was wittily described as "Orientalpantheism translated into Scotch. " He was editor of several journals. 

[109] Joanna Southcott (1750-1814) was known for her rhyming prophecies inwhich she announced herself as the woman spoken of in Revelations xii. Shehad at one time as many as 100, 000 disciples, and she established a sectthat long survived her. 

[110] Thales, c. 640-548 B. C. 

[111] Pythagoras, 580-501 B. C. 

[112] Anaxagoras, 499-428 B. C. , the last of the Ionian school, teacher ofEuripides and Pericles. Plutarch speaks of him as having squared thecircle. 

[113] Oinopides of Chios, contemporary of Anaxagoras. Proclus tells us thatOinopides was the first to show how to let fall a perpendicular to a linefrom an external point. 

[114] Bryson and Antiphon, contemporaries of Socrates, invented theso-called method of exhaustions, one of the forerunners of the calculus. 

[115] He wrote, c. 440 B. C. , the first elementary textbook on mathematicsin the Greek language. The "lunes of Hippocrates" are well known ingeometry. 

[116] Jabir ben Aflah. He lived c. 1085, at Seville, and wrote on astronomyand spherical trigonometry. The _Gebri filii Affla Hispalensis deastronomia libri_ IX was published at Nuremberg in 1533. 

[117] Hieronymus Cardanus, or Girolamo Cardano (1501-1576), the greatalgebraist. His _Artis magnae sive de regulis Algebrae_ was published atNuremberg in 1545. 

[118] Nicolo Tartaglia (c. 1500-1557), the great rival of Cardan. 

[119] See note 5 {98}, Vol. I, page 69. 

[120] See note 10 {124}, Vol. I. , page 83. 

[121] See note 9 {123}, Vol. I, page 83. 

[122] Pierre Hérigone lived in Paris the first half of the 17th century. His _Cours mathématique_ (6 vols. , 1634-1644) had some standing but was notat all original. 

[123] Franciscus van Schooten (died in 1661) was professor of mathematicsat Leyden. He edited Descartes's _La Géométrie_. 

[124] Florimond de Beaune (1601-1652) was the first Frenchman to write acommentary on Descartes's _La Géométrie_. He did some noteworthy work inthe theory of curves. 

[125] See note 3 {23}, Vol. I, page 41. 

[126] Olivier de Serres (b. In 1539) was a writer on agriculture. Montuclaspeaks of him in his _Quadrature du cercle_ (page 227) as having assertedthat the circle is twice the inscribed equilateral triangle, although, asDe Morgan points out, this did not fairly interpret his position. 

[127] Angherà wrote not only the three works here mentioned, but also the_Problemi del più alto interesse scientifico, geometricamente risoluti edimostrati_, Naples, 1861. His quadrature was defended by Giovanni Motti ina work entitled _Matematica Vera. Falsità del sistema ciclometricod'Archimede, quadratura del cerchio d'Angherà, ricerca algebraica dei latidi qualunque poligono regolare inscritto in un circolo_, Voghera, 1877. The_Problemi_ of 1861 contains Angherà's portrait, and states that he lived atMalta from 1849 to 1861. It further states that the Malta publications arein part reproduced in this work. 

[128] This was his friend Paolo Pullicino whose _Elogio_ was pronounced byL. Farrugia at Malta in 1890. He wrote a work _La Santa Effegie della BlataVergine Maria_, published at Valetta in 1868. 

[129] St. Vitus, St. Modestus, and St. Crescentia were all martyred thesame day, being torn limb from limb after lions and molten lead had provedof no avail. At least so the story runs. 

[130] The reference is to Cardinal Wiseman. See Vol. II, page 26, note 56. 

[131] "Worthy of esteem. "

[132] Pedro de Ribadeneira (Ribadeneyra, Rivadeneira), was born at Toledoin 1526 and died in 1611. He held high position in the Jesuit order. Thework referred to is the _Flos Sanctorum o libro de las vidas de lossantos_, of which there was an edition at Barcelona in 1643. His life ofLoyola (1572) and _Historia ecclesiástica del Cisma del reino deInglaterra_ were well known. 

[133] Cæsar Baronius (1538-1607) was made a cardinal in 1595 and becamelibrarian at the Vatican in 1597. The work referred to appeared at Rome in1589. 

[134] Mrs. Jameson's (1794-1860) works were very popular half a centuryago, and still have some circulation among art lovers. The first edition ofthe work mentioned appeared in 1848. 

[135] The barnyard cock. 

[136] Shanks did nothing but computing. The title should, of course, read"to 607 Places of _Decimals_. " He later carried the computation to 707decimal places. (_Proc. Roy. Society_, XXI, p. 319. ) He also prepared atable of prime numbers up to 60, 000. (_Proc. Roy. Society_, XXII, p. 200. )

[137] See Vol. I, page 42, note 4 {24}. 

[138] See Vol. I, page 64, note 1 {78}. 

[139] See Vol. I, page 328, note 1 {704}. 

[140] George Suffield published _Synthetic Division in Arithmetic_, towhich reference is made, in 1863. 

[141] John Robert Lunn wrote chiefly on Church matters, although hepublished a work on motion in 1859. 

[142] Jean Baptiste Joseph, Baron Fourier (1768-1830), sometime professorin the Military School at Paris, and later at the _Ecole polytechnique_. Heis best known by his _Théorie analytique de la chaleur_ (Paris, 1822), inwhich the Fourier series is used. The work here referred to is the _Analysedes équations déterminées_ (Paris, 1831). 

[143] William George Horner (1786-1837) acquired a name for himself inmathematics in a curious manner. He was not a university man nor was he amathematician of any standing. He taught school near Bristol and at Bath, and seems to have stumbled upon his ingenious method for finding theapproximate roots of numerical higher equations, including as a specialcase the extracting of the various roots of numbers. Davies Gilbertpresented the method to the Royal Society in 1819, and it was reprinted inthe _Ladies' Diary_ for 1838, and in the _Mathematician_ in 1843. Themethod was original as far as Horner was concerned, but it is practicallyidentical with the one used by the Chinese algebraist Ch'in Chiu-shang, inhis _Su-shu Chiu-chang_ of 1247. But even Ch'in Chiu-shang can hardly becalled the discoverer of the method since it is merely the extension of aprocess for root extracting that appeared in the _Chiu-chang Suan-shu_ ofthe second century B. C. 

[144] He afterwards edited Loftus's _Inland Revenue Officers' Manual_(London, 1865). The two equations mentioned were x^3 - 2x = 5 and y^3 -90y^2 + 2500y - 16, 000 = 0, in which y = 30 - 10x. Hence each place of y isthe complement of the following place of x with respect to 9. 

[145] Probably the John Power Hicks who wrote a memoir on T. H. Key, London, 1893. 

[146] Possibly the one who wrote on the quadrature of the circle in 1881. 

[147] As it is. But what a pity that we have not 12 fingers, withduodecimal fractions instead of decimals! We should then have 0. 6 for ½, 0. 4 for 1/3, 0. 8 for 2/3, 0. 3 for ¼, 0. 9 for ¾, and 0. 16 for 1/8, insteadof 0. 5, 0. 333+, 0. 666+, 0. 25, 0. 75, and 0. 125 as we now have with ourdecimal system. In other words, the most frequently used fractions inbusiness would be much more easily represented on the duodecimal scale thanon the decimal scale that we now use. 

[148] He wrote Hints for an _Essay on Anemology and Ombrology_ (London, 1839-40) and _The Music of the Eye_ (London, 1831). 

[149] Brigham Young (1801-1877) was born at Whitingham, Vermont, andentered the Mormon church in 1832. In 1840 he was sent as a missionary toEngland. After the death of Joseph Smith he became president of the Mormons(1847), leading the church to Salt Lake City (1848). 

[150] Joseph Smith (1805-1844) was also born in Vermont, and was four yearsthe junior of Brigham Young. The _Book of Mormon_ appeared in 1827, and thechurch was founded in 1830. He was murdered in 1844. 

[151] Orson Pratt (1811-1881) was one of the twelve apostles of the MormonChurch (1835), and made several missionary journeys to England. He wasprofessor of mathematics in the University of Deseret (the Mormon name forUtah). Besides the paper mentioned Pratt wrote the _Divine Authenticity ofthe Book of Mormon_ (1849), _Cubic and Biquadratic Equations_ (1866), and a_Key to the Universe_ (1866). 

[152] "It does not follow. "

[153] Dryden (1631-1700) published his _Religio Laici_ in 1682. The use ofthe word "proportion" in the sense of ratio was common before his time, buthe uses it in the sense of having four terms; that is, that price is toprice as offence is to offence. 

[154] Olinthus Gilbert Gregory (1774-1841) succeeded Hutton as professor ofmathematics at Woolwich. He was, with De Morgan, much interested infounding the University of London. He wrote on astronomy (1793), mechanics(1806), practical mathematics (1825), and Christian evidences (1811). 

[155] See Vol. I, page 220, note 6 {482}. The _Pensées_ appearedposthumously in 1670. 

[156] "The right thing to do is not to wager at all. " "Yes, but you oughtto wager; you have started out; and not to wager at all that God exists isto wager that he does not exist. "

[157] He lived about 300 A. D. , in Africa, and wrote _Libri septem adversusGentes_. This was printed at Rome in 1542-3. 

[158] Pierre Bayle (1647-1706) was professor of philosophy at theProstestant University at Sedan from 1675 until its dissolution in 1681. Hethen became professor at Rotterdam (1681-1693). In 1684 he began thepublication of his journal of literary criticism _Nouvelles de laRépublique des Lettres_. He is best known for his erudite _Dictionnairehistorique et critique_ (1697). 

[159] "But Christ himself does not prove what he promises. It is true. For, as I have said, there cannot be any absolute proof of future events. Therefore since it is a condition of future events that they cannot begrasped or comprehended by any efforts of anticipation, is it not morereasonable, out of two alternatives that are uncertain and that are hangingin doubtful expectation, to give credence to the one that gives some hoperather than to the one that offers none at all? For in the former casethere is no danger if, as is said to threaten, it becomes empty and void;while in the latter case the danger is greatest, that is, the loss ofsalvation, if when the time comes it is found that it was not a falsehood. "

[160] Gregg wrote several other paradoxes, including the following: _TheAuthentic Report of the extraordinary case of Tresham Dames Gregg ... Hiscommittal to Bridewell for refusing to give his recognizance_ (Dublin, 1841), _An Appeal to Public Opinion upon a Case of Injury and Wrong ... Inthe case of a question of prerogative that arose between_ [R. Whately] _... Archbishop of Dublin and the author_ (London, 1861), _The Cosmology of SirIsaac Newton proved to be in accordance with the Bible_ (London, 1871), _The Steam Locomotive as revealed in the Bible_ (London 1863) and _On theSacred Law of 1866, conferring perpetual life with immunity from decay anddisease. A cento of decisive scriptural oracles strangely discovered_(London and Dublin, 1875). These titles will help the reader to understandthe man whom De Morgan so pleasantly satirizes. 

[161] See Vol. I, page 261, note 2 {592}. 

[162] "They have found it. "

[163] The late Greeks used the letters of their alphabet as numerals, adding three early alphabetic characters. The letter [chi] represented 600, [xi] represented 60, and [digamma] stood for 6. This gives 666, the numberof the Beast given in the Revelation. 

[164] "Allowing for necessary exceptions. "

[165] Mr. Gregg is not alone in his efforts to use the calculus in originallines, as any one who has read Herbart's application of the subject topsychology will recall. 

[166] See Vol. I, page 105, note 4 {188}; page 109, note 1 {197}. 

[167] The full title shows the plan, --_The Decimal System as a whole, inits relation to time, measure, weight, capacity, and money, in unison witheach other. _ But why is this so much worse than the French plan of which wehave only the metric system and the decimal division of the angle left?

[168] One of the brothers of Sir Isaac Pitman (1813-1897), the inventor ofmodern stenography. Of these brothers, Benjamin taught the art in America, Jacob in Australia, and Joseph, Henry, and Frederick in England. 

[169] For example, _The Phonographic Lecturer_ (London, 1871 etc. ), _ThePhonographic Student_ (1867, etc. ), and _The Shorthand Magazine_ (1866, etc. ). 

[170] See Vol. II, page 68, note 148. 

[171] It involves the theory of non-Euclidean geometry, Euclid's postulateof parallels being used in proving this theorem. 

[172] Referring to the fact that none of the works of Thales is extant. 

[173] The author was one B. Bulstrode. Parts 4 and 5 were printed atCalcutta. 

[174] See Vol. II, page 5, note 18. 

[175] See Vol. I, page 85, note 2 {129}. 

[176] Alexander Vasilievich Suvaroff (1729-1800), a Russian general whofought against the Turks, in the Polish wars, and in the early Napoleoniccampaigns. When he took Ismail in 1790 he sent this couplet to EmpressCatherine. 

[177] "Newton hath determined rightly, " "Newton hath not determinedrightly. "

[178] See Vol. I, page 288, note 3 {621}. 

[179] See Vol. I, page 326, note 1 {700}. 

[180] "With great honor. "

[181] Apparently unknown to biographers. He seems to have written nothingelse. 

[182] Captain Marryat (1792-1848) published _Snarley-yow, or the Dog Fiend_in 1837. 

[183] He is not known to biographers, and published nothing else under thisname. 

[184] See Vol. I, page 80, note 5 {119}. 

[185] He published a _Family and Commercial Illustrated Almanack and YearBook ... For 1861_ (Bath, 1860). 

[186] Louis Dutens (1730-1812) was born at Tours, but went to England as ayoung man. He made the first collection of the works of Leibnitz, againstthe advice of Voltaire, who wrote to him: "Les écrits de Leibnitz sontépars comme les feuilles de la Sybille, et aussi obscurs que les écrits decette vieille. " The work appeared at Geneva, in six volumes, in 1769. 

[187] Mungo Park (1771-1806), the first European to explore the Niger(1795-6). His _Travels in the Interior of Africa_ appeared in 1799. He diedin Africa. 

[188] Gerhard Mercator (1512-1594) the well-known map maker of Louvain. The"Mercator's Projection" was probably made as early as 1550, but theprinciple of its construction was first set forth by Edward Wright (London, 1599). 

[189] Quirico Barilli Filopanti wrote a number of works and monographs. Hesucceeded in getting his _Cesare al Rubicone_ and _Degli_ _usi idraulicidella Tela_ in the _Memoria letta ... All' Accademia delle Scienze inBologna_ (1847, 1866). He also wrote _Dio esiste_ (1881), _Dio Liberale_(1880), and _Sunto della memoria sulle geuranie ossia di alcune singolarirelazioni cosmiche della terra e del cielo_ (1862). 

[190] The periods of disembodiment may be interesting. They will be seenfrom the following dates: Descartes (1596-1650), William III (1650-1702);Roger Bacon (1214 to c. 1294), Boccaccio (1313-1375). Charles IX was bornin 1550 and died in 1574. 

[191] His real name was Frederick Parker, and he wrote several works on theGreek language and on religion. Among these were a translation of the NewTestament from the Vatican MS. (1864), _The Revealed History of Man_(1854), _An Enquiry respecting the Punctuation of Ancient Greek_ (1841), and _Rules for Ascertaining the sense conveyed in Ancient GreekManuscripts_ (1848, the seventh edition appearing in 1862). 

[192] See Vol. I, page 352, second note 1 {736}. 

The literature on the subject of the Great Pyramid, considered from thestandpoint of metrology, is extensive. 

[193] See Vol. I, page 80, note 5 {119}. 

[194] Sir Philip Francis (1740-1818) was a Whig politician. The evidencethat he was the author of the _Letters of Junius_ (1769-1772) is purelycircumstantial. He was clerk in the war office at the time of theirpublication. In 1774 he was made a member of the Supreme Council of Bengal, and was a vigorous opponent of Warren Hastings, the two fighting a duel in1780. He entered parliament in 1784 and was among the leaders in theagitation for parliamentary reform. 

[195] Mrs. Cottle published a number of letters that attracted attention atthe time. Among these were letters to the emperor of France and king ofSardinia (1859) relating to the prophecies of the war between France andAustria; to G. C. Lavis and Her Majesty's Ministers (1859) relating to herclaims as a prophetess; and to the "Crowned Heads" at St. James, the Kingof Prussia, and others (1860), relating to certain passages of Scripture. She also wrote _The Lamb's Book of Life for the New Jerusalem Church andKingdom, interpreted for all nations_ (1861). 

[196] See Vol. I, page 315, note 2 {685}, and Vol. II, page 58, note 109. 

[197] A Congregational minister, who published a number of sermons, chieflyobituaries, between 1804 and 1851. His _Frailty of Human Life_, two sermonsdelivered on the occasion of the death of Princess Charlotte, went throughat least three editions. 

[198] He was secretary of the Congregational Board and editor of the_Congregational Year Book_ (from 1846) and the _Congregational Manual_. 

[199] Frederick Denison Maurice (1805-1872) began his preaching as aUnitarian but entered the Established Church in 1831, being ordained in1834. He was professor of English and History at King's College, London, from 1840 to 1853. He was one of the founders of Queen's College for women, and was the first principal of the Working Men's College, London. Thesubject referred to by De Morgan is his expression of opinion in his_Theological Essays_ (1853) that future punishment is not eternal. As aresult of this expression he lost his professorship at King's College. In1866 he was made Knightbridge Professor of Casuistry, Moral Theology, andMoral Philosophy at Cambridge. 

[200] See Vol. I, page 46, note 1 {42}. Besides the books mentioned inthis list he wrote _The Ratio between Diameter and Circumferencedemonstrated by angles, and Euclid's Theorem, Proposition 32, Book I, proved to be fallacious_ (Liverpool, 1870). This is the theorem whichasserts that the exterior angle of a triangle is equal to the sum of thetwo opposite interior angles, and that the sum of the interior anglesequals two right angles. He also published his _Curiosities of Mathematics_in 1870, a work containing an extensive correspondence with every one whowould pay any attention to him. De Morgan was then too feeble to show anyinterest in the final effort of the subject of some of his keenest satire. 

[201] See Vol. I, page 332, note 4 {709}. 

[202] See Vol. I, page 101, note 4 {174}. 

[203] "The circle-squaring disease"; literally, "the circle-measuringdisease. "

[204] See Vol. II, page 63, note 136. 

[205] William Rutherford (c. 1798-1871), teacher of mathematics atWoolwich, secretary of the Royal Astronomical Society, editor of _TheMathematician_, and author of various textbooks. _The Extension of [pi] to440 places_, appeared in the _Proceedings_ of the Royal Society in 1853 (p. 274). 

[206] Charles Knight (1791-1873) was associated with De Morgan for manyyears. After 1828 he superintended the publications of the Society for theDiffusion of Useful Knowledge, to which De Morgan contributed, and heedited the _Penny Cyclopedia_ (1833-1844) for which De Morgan wrote thearticles on mathematics. 

[207] Sir William Hamilton. See Vol. I, page 112, note 7 {211}. 

[208] Adam Smith (1723-1790) was not only known for his _Wealth of Nations_(1776), but for his _Theory of Moral Sentiments_ (1759), published while hewas professor of moral philosophy at Glasgow (1752-1764). He was LordRector of the university in 1787. 

[209] See Vol. I, page 332, note 4 {709}. 

[210] "Whip. "

[211] "Terrible lash. "

[212] "An accomplished fact [an accomplished fault]. "

[213] See _Extracts from the Diary and Letters of Mrs. Mary Cobb_, London, 1805. 

[214] "Gentle in manner. "

[215] "Brave in action. " The motto of Earl Newborough was "Suaviter inmodo, fortiter in re. "

[216] "Reduction to an absurdity, " a method of proof occasionally used ingeometry and in logic. 

[217] "He has lost the right of being moved (struck) by evidence. "

[218] For _radix quadratus_. The usual root sign is supposed to be derivedfrom _r_ (for radix), and at one time _q_ was commonly used for square, asin Viète's style of writing Aq for A^2. 

[219] The Garde Douloureuse was a castle in the marches of Wales andreceived its name because of its exposure to attacks by the Welsh. 

[220] "Out of the fight. "

[221] "Hidden. "

[222] John Cam Hobhouse (1786-1869), Baron Broughton, was committed toNewgate for two months in 1819 for his anonymous pamphlet, _A TriflingMistake_. This was a great advertisement for him, and upon his release hewas at once elected to parliament for Westminster. He was a strongsupporter of all reform measures, and was Secretary for War in 1832. He wascreated Baron Broughton de Gyfford in 1851. 

[223] Thomas Erskine (1750-1823), the famous orator. He became LordChancellor in 1806, but sat in the House of Commons most of his life. 

[224] The above is explained in the MS. By a paragraph referring to someanagrams, in one of which, by help of the orthography suggested, adesignation for this cyclometer was obtained from the letters of hisname. --S. E. De M. 

[225] "A personal verb agrees with its subject. "

[226] See Vol. I, page 326, note 1 {700}. 

[227] See Vol. I, page 326, note 2 {701}. 

[228] Apparently unknown to biographers. 

[229] The _Bibliotheca Mathematica_ of Ludwig Adolph Sohncke (1807-1853), professor of mathematics at Königsberg and Halle, covered the period from1830 to 1854, being completed by W. Engelmann. It appeared in 1854. 

[230] See Vol. I, page 392, note 2 {805}. 

[231] See Vol. I, page 43, note 7 {32}. 

[232] See Vol. II, page 91, note 187. 

[233] Mason made a notable balloon trip from London to Weilburg, in theDuchy of Nassau, in November, 1836, covering 500 miles in 18 hours. Hepublished an account of this trip in 1837, and a work entitled_Aeronautica_ in 1838. 

[234] William Harrison Ainsworth (1805-1885) the novelist. 

[235] On this question see Vol. I, page 326, note 2 {701}. 

[236] Major General Alfred Wilks Drayson, author of various works ongeology, astronomy, military surveying, and adventure. 

[237] Hailes also wrote several other paradoxes on astronomy and circlesquaring during the period 1843-1872. 

[238] See Vol. I, page 43, note 8 {33}. 

[239] See Vol. I, page 43, note 7 {32}. 

[240] "Very small errors are not to be condemned. "

[241] He seems to have written nothing else. 

[242] Besides the paradoxes here mentioned by De Morgan he wrote severalother works, including the following: _Abriss der Babylonisch-AssyrischenGeschichte_ (Mannheim, 1854), _A Popular Inquiry into the Moon's rotationon her axis_ (London, 1856), _Practical Tables for the reduction of theMahometan dates to the Christian kalendar_ (London, 1856), _Grundzüge einerneuen Weltlehre_ (Munich, 1860), and _On the historical Antiquity of thePeople of Egypt_ (London, 1863). 

[243] Dircks (1806-1873) was a civil engineer of prominence, and a memberof the British Association and the Royal Society of Edinburgh. He wrote(1863) on "Pepper's Ghost, " an ingenious optical illusion invented by him. There was a second edition of the _Perpetuum Mobile_ in 1870. 

[244] George Stephenson (1781-1848), the inventor of the first successfulsteam locomotive. His first engine was tried in 1814. 

[245] Robert Stephenson (1803-1859), the only son of George. Most of theearly improvements in locomotive manufacture were due to him. He was alsowell known for his construction of great bridges. 

[246] "In its proper place. "

[247] "A fool always finds a bigger fool to admire him. "

[248] See Vol. I, page 43, note 7 {32}. 

[249] See Vol. I, page 43, note 8 {33}. 

[250] See Vol. I, page 85, note 2 {129}. 

[251] See Vol. I, page 390, note 1 {390}. 

[252] From 1823 to 1852 it was edited by I. C. Robertson; from 1852 to 1857by R. A. Brooman; and from 1857 to 1863 by Brooman and E. J. Reed. 

[253] Sir James Ivory (1765-1842) was, as a young man, manager of a flaxmill in Scotland. In 1804 he was made professor of mathematics at the RoyalMilitary College, then at Marlow and later at Sandhurst. He was deeplyinterested in mathematical physics, and there is a theorem on theattraction of ellipsoids that bears his name. He was awarded three medalsof the Royal Society, and was knighted together with Herschel and Brewster, in 1831. 

[254] See Vol. I, page 56, note 1 {64}. 

[255] See Vol. I, page 153, note 5 {338}. 

[256] See Vol. I, page 309, note 2 {670}. 

[257] See Vol. I, page 87, note 4 {133}. 

[258] George Canning (1770-1857), the Tory statesman and friend of Scott, was much interested in founding the _Quarterly Review_ (1808) and was acontributor to its pages. 

[259] See Vol. I, page 186, note 14 {418}. 

[260] See Vol. II, page 141, note 252. 

[261] De Morgan had a number of excellent articles in this publication. 

[262] See Vol. I, page 279, note 1 {611}. 

[263] James Orchard Halliwell (1820-1889), afterwards Halliwell-Phillips, came into prominence as a writer at an early age. When he was seventeen hewrote a series of lives of mathematicians for the _Parthenon_. His _RaraMathematica_ appeared when he was but nineteen. He was a great bibliophileand an enthusiastic student of Shakespeare. 

[264] This was written at the age of twenty-two. 

[265] The subject of this criticism is of long past date, and as it hasonly been introduced by the author as an instance of faulty editorship, Ihave omitted the name of the writer of the libel, and a few lines offurther detail. --S. E. De M. 

[266] "Condemned souls. "

[267] The editor of the _Mechanics' Magazine_ died soon after the above waswritten. --S. E. De M. 

[268] Thomas Stephens Davies (1795-1851) was mathematical master atWoolwich and F. R. S. He contributed a series of "Geometrical Notes" to the_Mechanics' Magazine_ and edited the _Mathematician_. He also published anumber of text-books. 

[269] See Vol. II, page 66, note 143. 

[270] The _Dictionary of Greek and Roman Biography_ (1849), edited by SirWilliam Smith (1813-1893), whose other dictionaries on classical andbiblical matters are well known. 

[271] "O J. S. ! This is the worst! the greatest possible injury!"

[272] See Vol. I, page 44, note 9 {34} and page 110, note 5 {201}. 

[273]

 "If there's a man whom the judge's pitiless sentence awaiteth, His head condemned to penalties and tribulations, Let neither penitentiaries tire him with laborer's burdens Nor let his stiffened hands be harrassed by work in the mines. He must square the circle! For what else do I care?--all Known punishments this one task hath surely included. "

[274] Houlston was in the customs service. He also published _Inklings ofAreal Autometry_, London, 1874. 

[275] This is Frederick C. Bakewell. He had already published _NaturalEvidence of a Future Life_ (London, 1835), _Philosophical Conversations_(London, 1833, with other editions), and _Electric Science_ (London, 1853, with other editions). 

[276] Henry F. A. Pratt had already published _A Dissertation on the powerof the intercepted pressure of the Atmosphere_ (London, 1844) and _TheGenealogy of Creation_ (1861). Later he published a work _On OrbitalMotion_ (1863), and _Astronomical Investigations_ (1865). 

[277] See Vol. I, page 260, note 1 {591}. 

[278] Thomas Rawson Birks (1810-1883), a theologian and controversialist, fellow of Trinity College, Cambridge, and (1872) professor of moralphilosophy in that university. He wrote _Modern Rationalism_ (1853), _TheBible and Modern Thought_ (1861), _The First Principles of Moral Science_(1873), and _Modern Physical Fatalism and the Doctrine of Evolution_(1876), the last being an attack on Herbert Spencer's _First Principles_. 

[279] Pseudonym for William Thorn. In the following year (1863) hepublished a second work, _The Thorn-Tree: being a History of ThornWorship_, a reply to Bishop Colenso's work entitled _The Pentateuch and theBook of Joshua critically examined_. 

[280] Besides _The Pestilence_ (1866) he published _The True Church_(1851), _The Church and her destinies_ (1855), _Religious reformationimperatively demanded_ (1864), and _The Bible plan unfolded_ (secondedition, 1872). 

[281] See Vol. II, page 97, note 195. 

[282] Sir George Cornewall Lewis (1806-1863) also wrote an _Essay on theOrigin and Formation of the Romance Languages_ (1835), an _Essay on theGovernment of Dependencies_ (1841), and an _Essay on Foreign Jurisdictionand the Extradition of Criminals_ (1859). He was Chancellor of theExchequer in 1855 and Home Secretary in 1859. 

[283] Henry Malden (1800-1876), a classical scholar, fellow of TrinityCollege, Cambridge, and professor of Greek at University College(1831-1876), then (1831) the University of London. He wrote a _History ofRome to 390 B. C. _ (1830), and _On the Origin of Universities andAcademical Degrees_ (1835). 

[284] Henry Longueville Mansel (1820-1871), theologian and metaphysician, reader in theology at Magdalen College, Oxford (1855), and professor ofecclesiastical history and Dean of St. Paul's (1866). He wrote onmetaphysics, and his Bampton Lectures (1858) were reprinted several times. 

[285] "Hejus gave freely, gave freely. God is propitious, God is favorableto him who gives freely. God is honored with a banquet of eggs at the crossroads, the god of the world. God, with benignant spirit, desired insacrifice a goat, a bull to be carried within the precincts of the holyplace. God, twice propitiated, blesses the pit of the sacred libation. "

[286] Eudoxus of Cnidus (408-355 B. C. ) had much to do with the earlyscientific astronomy of the Greeks. The fifth book of Euclid is generallyattributed to him. His astronomical works are known chiefly through thepoetical version of Aratus mentioned in note 13, page 167. 

[287] Simplicius, a native of Cilicia, lived in the 6th century of our era. He was driven from Athens by Justinian and went to Persia (531), but hereturned later and had some fame as a teacher. 

[288] See Vol. I, page 160, note 3 {348}. 

[289] See Vol. I, page 76, note 3 {112}. 

[290] "Through right and wrong. "

[291] "It is therefore to arrive at this parallelism, or to preserve it, that Copernicus feared to be obliged to have recourse to this equal andopposite movement which destroys the effect which he attributed so freelyto the first, of deranging the parallelism. "

[292] A contemporary of Plato and a disciple of Aristotle. 

[293] Meton's solstice, the beginning of the Metonic cycles, has beenplaced at 432 B. C. Ptolemy states that he made the length of the year 365¼+ 1/72 days. 

[294] Aratus lived about 270 B. C. , at the court of Antigonus of Macedonia, and probably practiced medicine there. He was the author of twoastronomical poems, the [Greek: Phainomena], apparently based on the lostwork of Eudoxus, and the [Greek: Diosêeia] based on Aristotle's_Meteorologica_ and _De Signis Ventorum_ of Theophrastus. 

[295] "The nineteen (-year) cycle of the shining sun. "

[296] Claudius Salmasius (1588-1653), or Claude Saumaise, was adistinguished classicist, and professor at the University of Leyden. Theword [Greek: êleioio] means Elian, thus making the phrase refer to thebrilliant one of Elis. 

[297] Sir William Brown (1784-1864). In 1800 the family moved to Baltimore, and there the father, Alexander Brown, became prominent in the linen trade. William went to Liverpool where he acquired great wealth as a merchant andbanker. He was made a baronet in 1863. 

[298] Robert Lowe (1811-1892), viscount Sherbrooke, was a fellow ofMagdalen College, Oxford (1835). He went to Australia in 1842 and was verysuccessful at the bar. He returned to England in 1850 and became leaderwriter on the _Times_. He was many years in parliament, and in 1880 wasraised to the peerage. 

[299] See Vol. I, page 42, note 4 {24}. 

[300] Francis Walkingame (fl. About 1751-1785), whose _Tutor's Assistant_went through many editions from 1751-1854. 

[301] Davies Gilbert (1767-1839). His family name was Giddy, but he assumedhis wife's name. He sat in parliament from 1806 to 1832. In 1819 he securedthe establishment of the Cape of Good Hope observatory. He was Treasurer(1820-1827) and President (1827-1830) of the Royal Society. 

[302] See Vol. I, page 55, note 2 {63}. 

[303] Sir Jonathan Frederick Pollock (1783-1870) entered parliament in 1831and was knighted in 1834. 

[304] Joseph Hume (1777-1855) entered parliament in 1812 and for thirtyyears was leader of the Radical party. 

[305] "What! when I say, 'Nicole, bring me my slippers, ' is that prose?"

[306] Captain Basil Hall (1788-1844), a naval officer, carried on a seriesof pendulum observations in 1820-1822, while on a cruise of the west coastof North America. The results were published in 1823 in the _PhilosophicalTransactions_. He also wrote two popular works on travel that went throughnumerous editions. 

[307] Anthony Ashley Cooper (1801-1885), Earl of Shaftesbury. His name isconnected with philanthropic work and factory legislation. 

[308] See Vol. I, page 207, note 12 {469}. 

[309] See Vol. I, page 80, note 5 {119}. 

[310] Sir Thomas Maclear (1794-1879), an Irishman by birth, becameAstronomer Royal at the Cape of Good Hope in 1833. He was an indefatigableobserver. He was knighted in 1860. 

[311] Thomas Romney Robinson (1792-1882), another Irish astronomer ofprominence. He was a deputy professor at Trinity College, Dublin, but tookcharge of the Armagh observatory in 1823 and remained there until hisdeath. 

[312] Sir James South (1785-1867) was in early life a surgeon, but gave uphis practice in 1816 and fitted up a private observatory. He contributed tothe science of astronomy, particularly with respect to the study of doublestars. 

[313] Sir John Wrottesley (1798-1867), second Baron Wrottesley. Like SirJames South, he took up the study of astronomy after a professionalcareer, --in his case in law. He built a private observatory in 1829 andmade a long series of observations, publishing three star catalogues. Hewas president of the Astronomical Society from 1841 to 1843, and of theRoyal Society from 1854 to 1857. 

[314] He seems to have written nothing else. 

[315] See Vol. II, page 68, note 147. 

[316] "The wills are free, and I wish neither the one nor the other. "

[317] "The force of inertia conquered. "

[318] Reddie also wrote _The Mechanics of the Heavens_, referred to laterin this work. He must not be confused with Judge James Reddie (1773-1852), of Glasgow, who wrote on international law, although this is done in theprinted edition of the British Museum catalogue, for he is mentioned by DeMorgan somewhat later as alive in 1862. 

[319] Henry Dunning Macleod (1821-1902), a lawyer and writer on politicaleconomy, was a Scotchman by birth. He wrote on economical questions, andlectured on banking at Cambridge (1877) and at King's College, London(1878). He was a free lance in his field, and was not considered orthodoxby the majority of economists of his time. He was an unsuccessful candidatefor the chairs of political economy at Cambridge (1863), Edinburgh (1871), and Oxford (1888). 

[320] See Vol. I, page 252, note 2 {576}. 

[321] Francis Henry Laing (1816-1889) was a graduate of Queen's College, Cambridge, and a clergyman in the Church of England until 1846, when heentered the Church of Rome. He taught in various Jesuit colleges until1862, when his eccentricity was too marked to warrant the Church inallowing him to continue. He published various controversial writingsduring his later years. Of course if he had known the works of Wessel, Gaus, Buée, Argand, and others, he would not have made such a sorryexhibition of his ignorance of mathematics. 

[322] See Vol. I, page 329, note 1 {705}. The book went into a secondedition in 1864. 

[323] Thomas Weddle (1817-1853) was, at the time of publishing this paper, a teacher in a private school. In 1851 he became professor of mathematicsat Sandhurst. He contributed several papers to the _Cambridge and DublinMathematical Journal_, chiefly on geometry. 

[324] See Vol. II, page 109, note 205. 

[325] See Vol. II, page 66, note 143. 

[326] See Vol. II, page 151, note 268. 

[327] George Barrett (1752-1821) worked from 1786 to 1811 on a set of lifeinsurance and annuity tables. He invented a plan known as the "columnarmethod" for the construction of such tables, and as De Morgan states, thiswas published by Francis Baily, appearing in the appendix to his work onannuities, in the edition of 1813. Some of his tables were used inBabbage's _Comparative View of the various Institutions for the Assuranceof Lives_ (1826). 

[328] See Vol. I, page 309, note 2 {670}. 

[329] This was his _Practical short and direct Method of Calculating theLogarithm of any given Number, and the Number corresponding to any givenLogarithm_ (1849). 

[330] This is William Neile (1637-1670), grandson of Richard Neile (notNeal), Archbishop of York. At the age of 19, in 1657, he gave the firstrectification of the semicubical parabola. Although he communicated it toBrouncker, Wren, and others, it was not published until 1639, when itappeared in John Wallis's _De Cycloide_. 

[331] I myself "was a considerable part. "

[332] He also wrote _A Glance at the Universe_ ("2d thousand" in 1862), and_The Resurrection Body_ (1869). 

[333] See Vol. I, page 63, note 1 {74}. 

[334] As Swift gave it in his _Poetry. A Rhapsody_, it is as follows:

 "So, naturalists observe, a flea Has smaller fleas that on him prey; And these have smaller still to bite 'em. And so proceed _ad infinitum_. "

[335] Perhaps 1, 600, 000, 000 years, if Boltwood's recent computations basedon radium disintegration stand the test. This would mean, according toMacCurdy's estimate, 60, 000, 000 years since life first appeared on theearth. 

[336] De Morgan wrote better than he knew, for this work, the _AllgemeineEncyclopädie der Wissenschaften und Künste_, begun at Leipsic in 1818, isstill (1913) unfinished. Section I, A-G, consists of 99 parts in 56volumes; Section II, H-N, consists of 43 volumes and is not yet completed;and Section III, O-Z, consists of 25 volumes thus far, with most of thework still to be done. Johann Samuel Ersch (1766-1828), the founder, washead librarian at Halle. Johann Gottfried Gruber (1774-1851), hisassociate, was professor of philosophy at the same university. 

[337] William Howitt (1792-1879) was a poet, a spiritualist, and amiscellaneous writer. He and his wife became spiritualists about 1850. Hewrote numerous popular works on travel, nature and history. 

[338] See Vol. II, page 55, note 108. 

[339] As will be inferred from the text, C. D. Was Mrs. De Morgan, andA. B. Was De Morgan. 

[340] Jean Meslier (1678-1733), curé of Estrepigny, in Champagne, was askeptic, but preached only strict orthodoxy to his people. It was only inhis manuscript, _Mon Testament_, that was published after his death, andthat caused a great sensation in France, that his antagonism toChristianity became known. 

[341] Baron Zach relates that a friend of his, in a writing intended forpublication, said _Un esprit doit se frotter contre un autre_. The censorsstruck it out. The Austrian police have a keen eye for consequences. --A. DeM. 

"One mind must rub against another. " On Baron Zach, see Vol. II, page 45, note 4. 

[342] Referring to the first Lord Eldon (1751-1838), who was LordChancellor from 1799 to 1827, with the exception of one year. 

[343] "Sleeping power. "

[344] "Causes sleep. "

[345] Richard Hooker (c. 1554-1600), a theologian, "the ablest livingadvocate of the Church of England as by law established. "

[346] See Vol. I, page 76, note 3 {112}. 

[347] "Other I, "--other self. 

[348] This "utter rejection" has been repeated (1872) by the samewriter. --S. E. De M. 

[349] Edward Jenner (1749-1823) was a physician and biologist. His firstexperiments in vaccination were made in 1796, and his discovery waspublished in 1798. 

[350] See Vol. II, page 38, note 80. 

[351] "You will go most safely in the middle (way). "

[352] Pierre Joseph Arson was known early in the 19th century for hiscontroversy with Hoëné Wronski the mathematician, whom he attacked in his_Document pour l'histoire des grands fourbes qui ont figuré sur la terre_(1817-1818). 

[353] "We enter the course by night and are consumed by fire. "

[354] See Vol. I, page 51, note 3 {51}. 

[355] See Vol. I, page 336, note 8 {713}. 

[356] See Vol. I, page 137, note 8 {286}. 

[357] See Vol. I, page 229, note 2 {515}. 

[358] Richard Cobden (1804-1865), the cotton manufacturer and statesman whowas prominent in his advocacy of the repeal of the Corn Laws. 

[359] James Smith (1775-1839), solicitor to the Board of Ordnance. With hisbrother Horatio he wrote numerous satires. His _Horace in London_ (1813)imitated the Roman poet. His works were collected and published in 1840. 

[360] Samuel Butler (1612-1680), the poet and satirist, author of_Hudibras_ (1663-1678). 

[361] "Is it not fine to be sure of one's action when entering in a combatwith another? There, push me a little in order to see. NICOLE. Well! what'sthe matter? M. JOURDAIN. Slowly. Ho there! Ho! gently. Deuce take therascal! NICOLE. You told me to push. M. JOURDAIN. Yes, but you pushed me_en tierce_, before you pushed _en quarte_, and you did not give me time toparry. "

[362] John Abernethy (1764-1831), the famous physician and surgeon. 

[363] See Vol. I, page 102, note 5 {175}. 

[364] "With what measure ye mete, it shall be measured to you again. "

[365] Eusebius of Cæsarea (c. 260-340), leader of the moderate party at theCouncil of Nicæa, and author of a _History of the Christian Church_ in tenbooks (c. 324 A. D. ). 

[366] Nathaniel Lardner (1684-1768), a non-conformist minister and one ofthe first to advocate the scientific study of early Christian literature. 

[367] Henry Alford (1810-1871) Dean of Canterbury (1857-1871) and editor ofthe Greek Testament (1849-1861). 

[368] The work was _The Number and Names of the Apocalyptic Beasts: with anexplanation and application. Part I. _ London, 1848, as mentioned below. Thom also wrote _The Assurance of Faith, or Calvinism identified withUniversalism_ (London, 1833), and various other religious works. 

[369] See Vol. I, page 222, note 14 {490}. 

[370] John Hamilton Thom (1808-1894) was converted to Unitarianism and waslong a minister in that church, preaching in the Renshaw Street Chapel from1831 to 1866. De Morgan refers to the Liverpool Unitarian controversyconducted by James Martineau and Henry Giles in response to a challenge bythirteen Anglican Clergy. In 1839 Thom contributed four lectures and aletter to this controversy. Among his religious works were a _Life ofBlanco White_ (1845) and _Hymns, Chants, and Anthems_ (1854). 

[371] The spelling of these names is occasionally changed to meet thecondition that the numerical value of the letters shall be 666, "the numberof the beast" of Revelations. The names include Julius Cæsar; ValeriusJovius Diocletianus (249-313), emperor from 287 to 305, persecutor of theChristians; Louis, presumably Louis XIV; Gerbert (940-1003), who reigned asPope Sylvester II from 999 to 1003, known to mathematicians for his abacusand his interest in geometry, and accused by his opponents as being inleague with the devil; Linus, the second Bishop of Rome, the successor ofPeter; Camillo Borghese (1552-1621), who reigned as Pope Paul V from 1605to 1621, and who excommunicated all Venice in 1606 for its claim to tryecclesiastics before lay tribunals, thus taking a position which he wasforced to abandon; Luther, Calvin; Laud (see Vol. I, page 145, note 7{307}); Genseric (c. 406-477), king of the Vandals, who sacked Rome in 455and persecuted the orthodox Christians in Africa; Boniface III, who waspope for nine months in 606; Beza (see Vol. I, page 66, note 6 {83});Mohammed; [Greek: braski], who was Giovanni Angelo Braschi (1717-1799), andwho reigned as Pope Pius VI from 1775 to 1799, dying in captivity becausehe declined to resign his temporal power to Napoleon; Bonaparte; and, under[Greek: Ion Paune], possibly Pope John XIV, who reigned in 983 and 984during the absence of Boniface VII in Constantinople. 

[372] The Greek words and names are also occasionally misspelled so as tofit them to the number 666. They are [Greek: Lateinos] (Latin), [Greek: hêlatinê basileia] (the Latin kingdom), [Greek: ekklêsia italika] (theItalian Church), [Greek: euanthas] (blooming), [Greek: teitan] (Titan), [Greek: arnoume] (renounce), [Greek: lampetis] (the lustrous), [Greek: honikêtês] (conqueror), [Greek: kakos hodêgos] (bad guide), [Greek: alêthêsblaberos] (truthful harmful one), [Greek: palai baskanos] (a slanderer ofold), [Greek: amnos adikos] (unmanageable lamb), [Greek: antemos](Antemos), [Greek: gensêrikos] (Genseric), [Greek: euinas] (with stoutfibers), [Greek: Benediktos] (Benedict), [Greek: Bonibazios g. Papa x. ê. E. E. A. ] (Boniface III, pope 68, bishop of bishops I), [Greek: oulpios](baneful), [Greek: dios eimi hê hêras] (I, a god, am the), [Greek: hê missahê papikê] (the papal brief), [Greek: loutherana] (Lutheran), [Greek:saxoneios] (Saxon), [Greek: Bezza antitheos] (Beza antigod), [Greek: hêalazoneia biou] (the illusion of life), [Greek: Maometis] (Mahomet);[Greek: Maometês b. ] (Mahomet II), [Greek: theos eimi epi gaiês] (I am lordover the earth), [Greek: iapetos] (Iapetos, father of Atlas), [Greek:papeiskos] (Papeiskos), [Greek: dioklasianos] (Diocletian), [Greek: cheina](Cheina = Cain? China?), [Greek: braski] (Braschi, as explained in note10), [Greek: Ion Paune] (Paunian violet, but see note 10), [Greek: koupoks](cowpox), [Greek: Bonnepartê] (Bonneparte), [Greek: N. Bonêparte] (N. Boneparte), [Greek: euporia] (facility), [Greek: paradosis] (surrender), [Greek: to megathêrion] (the megathereum, the beast). 

[373] James Wapshare, whose _Harmony of the Word of God in Spirit and inTruth_ appeared in 1849. 

[374] The literature relating to the _Swastika_ is too extended to permitof any adequate summary in these notes. 

[375] Henry Edward Manning (1808-1892), at first an Anglican clergyman, hebecame a Roman Catholic priest in 1851, and became Cardinal in 1875. Hesucceeded Cardinal Wiseman as Archbishop of Westminster in 1865. He wrote anumber of religious works. 

[376] John Bright (1811-1889), Quaker, cotton manufacturer, and statesman. He worked with Cobden for free trade, peace, and reform of the electorate. 

[377] "The fallacy of many questions. "

[378] William Wilberforce (1759-1833), best known for his long fight forthe abolition of the slave trade. 

[379] Richard Martin (1754-1834), high sheriff of County Galway and ownerof a large estate in Connemara. Curiously enough, he was known both for hisreadiness in duelling and for his love for animals. He was known as"Humanity Martin, " and in 1822 secured the passage of an act "to preventthe cruel and improper treatment of cattle. " He was one of the founders(1824) of the Royal Society for the Prevention of Cruelty to Animals. He isusually considered the original of Godfrey O'Malley in Lever's novel, _Charles O'Malley_. 

[380] See Vol. I, page 149, note 1 {323}, also text on same page. 

[381] See Vol. I, page 44, note 9 {34}, also text, Vol. I, page 110. 

[382] "Penitential seat. "

[383] "Well placed upon the cushion. "

[384] See Vol. II, page 58, note 109. 

[385] "He has lost the right of being influenced by evidence. "

[386] "Hung up. "

[387] "A few things to the wise, nothing to the unlettered. "

[388] The fallacy results from dividing both members of an equation by 0, x- 1 being the same as 1 - 1, and calling the quotients finite. 

[389] "If you order him to the sky he will go. "

[390] _Similia similibus curanter_, "Like cures like, " the homeopathicmotto. 

[391] "Without harm to the proprieties. "

[392] "What are you doing? I am standing here. "

[393] Lors feist l'Anglois tel signe. La main gausche toute ouverte il levahault en l'aer, puis ferma au poing les quatres doigtz d'icelle et lepoulce estendu assit sus la pinne du nez. Soubdain après leva la dextretoute ouverte, et toute ouverte la baissa, joignant la poulce au lieu quefermait le petit doigt de la gausche, et les quatre doigtz d'icelle mouvoitlentement en l'aer. Puis au rebours feit de la dextre ce qu'il avoit faictde la gausche, et de la gausche ce que avoit faict de la dextre. --A. De M. 

[394] _Suaviter in modo, fortiter in re_, "Gentle in manners, firm inaction. "

[395] See Vol. I, page 101, note 4 {174}. 

[396] See Vol. I, page 315, note 3 {686}. 

[397] Henry Fawcett (1833-1884) became totally blind in 1858, but in spiteof this handicap he became professor of political economy at Cambridge andsat in parliament for a number of years. He championed the cause of reformand in particular he was prominent in the protection of the nativeinterests of India. The establishing of the parcels post (1882) took placewhile he was postmaster general (1880-1884). 

[398] Of course the whole thing depends upon what definition of division istaken. We can multiply 2 ft. By 3 ft. If we define multiplication so as toallow it, or 2 ft. By 3 lb, getting foot-pounds, as is done in physics. 

[399] Richard Milward (1609-1680), for so the name is usually given, wasrector of Great Braxted (Essex) and canon of Windsor. He was long theamanuensis of John Selden, and the _Table Talk_ was published nine yearsafter Milward's death, from notes that he left. Some doubt has been castupon the authenticity of the work owing to many of the opinions that itascribes to Selden. 

[400] John Selden (1584-1654) was a jurist, legal antiquary, and Orientalscholar. He sat in the Long Parliament, and while an advocate of reform hewas not an extremist. He was sent to the Tower for his support of theresolution against "tonnage and poundage, " in 1629. His _History of Tythes_(1618) was suppressed at the demand of the bishops. His _De Diis Syriis_(1617) is still esteemed a classic on Semitic mythology. 

[401] See Vol. I, page 42, note 4 {24}. 

[402] See Vol. II, page 249, note 398. 

[403] John Palmer (1742-1818) was a theatrical manager. In 1782 he setforth a plan for forwarding the mails by stage coaches instead of bypostmen. Pitt adopted the plan in 1784. Palmer was made comptroller-generalof the post office in 1786 and was dismissed six years later forarbitrarily suspending a deputy. He had been verbally promised 2½% on theincreased revenue, but Pitt gave him only a pension of £3000. In 1813 hewas awarded £50, 000 in addition to his pension. 

[404] Dionysius Lardner (1793-1859), professor of natural philosophy inLondon University (now University College). His _Cabinet Cyclopædia_(1829-1849) contained 133 volumes. De Morgan wrote on probabilities, andLardner on various branches of mathematics, and there were many otherwell-known contributors. Lardner is said to have made $200, 000 on a lecturetour in America. 

[405] Thomas Fysche Palmer (1747-1802) joined the Unitarians in 1783, andin 1785 took a charge in Dundee. He was arrested for sedition because of anaddress that it was falsely alleged that he gave before a society known asthe "Friends of Liberty. " As a matter of fact the address was given by anuneducated weaver, and Palmer was merely asked to revise it, declining todo even this. Nevertheless he was sentenced to Botany Bay (1794) for sevenyears. The trial aroused great indignation. 

[406] See Vol. I, page 80, note 5 {119}. 

[407] See Vol. II, page 244, note 394. 

[408] See Vol. I, page 352, note 1 {731}. 

[409] See Vol. I, page 332, note 4 {709}. 

[410] "The lawyers are brought into court; let them accuse each other. "

[411] Samuel Rogers (1763-1855), the poet and art connoisseur. He declinedthe laureateship on the death of Wordsworth (1850). Byron, his pretendedfriend, wrote a lampoon (1818) ridiculing his cadaverous appearance. 

[412] Theodore Edward Hook (1788-1841), the well-known wit. He is satirizedas Mr. Wagg in _Vanity Fair_. The _John Bull_ was founded in 1820 and Hookwas made editor. 

[413] "On pitying the heretic. "

[414] A term of medieval logic. Barbara: All M is P, all S is M, hence allS is P. Celarent: No M is P, all S is M, hence no S is P. 

[415] "Simply, " "According to which, " "It does not follow. "

[416]

 "O sweet soul, what good shall I declare That heretofore was thine, since such are thy remains!"

[417] "Stupid fellow!"

[418] Christopher Barker (c. 1529-1599), also called Barkar, was theQueen's printer. He began to publish books in 1569, but did no actualprinting until 1576. In 1575 the Geneva Bible was first printed in England, the work being done for Barker. He published 38 partial or completeeditions of the Bible from 1575 to 1588, and 34 were published by hisdeputies (1588-1599). 

[419] James Franklin (1697-1735) was born in Boston, Mass. , and was sent toLondon to learn the printer's trade. He returned in 1717 and started aprinting house. Benjamin, his brother, was apprenticed to him but ran away(1723). James published the _New England Courant_ (1721-1727), and Benjaminis said to have begun his literary career by writing for it. 

[420] James Hodder was a writing master in Tokenhouse Yard, Lothbury, in1661, and later kept a boarding school in Bromley-by-Bow. His famousarithmetic appeared at London in 1661 and went through many editions. Itwas the basis of Cocker's work. (See Vol. I, page 42, note 4 {24}. ) It waslong thought to have been the first arithmetic published in America, and itwas the first English one. There was, however, an arithmetic published muchearlier than this, in Mexico, the _Sumario compendioso ... Con algunasreglas tocantes al Aritmética_, by "Juan Diaz Freyle, " in 1556. 

[421] Henry Mose, Hodder's successor, kept a school in Sherborne Lane, London. 

[422] Rear Admiral Sir Francis Beaufort (1774-1857), F. R. S. , washydrographer to the Navy from 1829 to 1855. He prepared an atlas that wasprinted by the Society for the Diffusion of Useful Knowledge. 

[423] Antoine Sabatier (1742-1817), born at Castres, was known as the Abbébut was really nothing more than a "clerc tonsuré. " He lived at Court andwas pensioned to write against the philosophers of the Voltaire group. Heposed as the defender of morality, a commodity of which he seems to havepossessed not the slightest trace. 

[424] Maffeo Barberini was pope, as Urban VIII, from 1623 to 1644. It wasduring his ambitious reign that Galileo was summoned to Rome to make hisrecantation (1633), the exact nature of which is still a matter of dispute. 

[425] This Baden Powell (1796-1860) was the Savilian professor of geometry(1827-1860) at Oxford. 

[426] "Memoirs of the famous bishop of Chiapa, by which it appears that hehad butchered or burned or drowned ten million infidels in America in orderto convert them. I believe that this bishop exaggerated; but if we shouldreduce these sacrifices to five million victims, this would still beadmirable. "

[427] Alfonso X (1221-1284), known as El Sabio (the Wise), was interestedin astronomy and caused the Alphonsine Tables to be prepared. These tablewere used by astronomers for a long time. It is said that when thePtolemaic system of the universe was explained to him he remarked that ifhe had been present at the Creation he could have shown how to arrangethings in a much simpler fashion. 

[428] George Richards (c. 1767-1837), fellow of Oriel (1790-1796), Bamptonlecturer (1800), Vicar of St. Martin's-in-the-Fields, Westminster (1824), and a poet of no mean ability. 

[429] The "Aboriginal Britons, " an excellent poem, by Richards. (Note byByron. )--A. De M. 

[430] John Watkins (d. After 1831), a teacher and miscellaneous writer. 

[431] Frederic Shoberl (1775-1853), a miscellaneous writer. 

[432] He wrote, besides the _Aboriginal Britons_, _Songs of the AboriginalBards_ (1792), _Modern France: a Poem_ (1793), _Odin, a drama_ (1804), _Emma, a drama on the model of the Greek theatre_ (1804), _Poems_ (2volumes, 1804), and a _Monody on the Death of Lord Nelson_ (1806). 

[433] Henry Kirke White (1785-1806), published his first volume of poems atthe age of 18. Southey and William Wilberforce became interested in him andprocured for him a sizarship at St. John's College, Cambridge. He at onceshowed great brilliancy, but he died of tuberculosis at the age of 21. 

[434] John Wolcot, known as Peter Pindar (1738-1819), was a Londonphysician. He wrote numerous satirical poems. His _Bozzy and Piozzi, or theBritish Biographers_, appeared in 1786, and reached the 9th edition in1788. 

[435] See Vol. I, page 235, note 8 {532}. 

[436] Richard Payne Knight (1750-1824) was a collector of bronzes, gems, and coins, many of his pieces being now in the British Museum. He sat inparliament for twenty-six years (1780-1806), but took no active part inlegislation. He opposed the acquisition of the Elgin Marbles, holding themto be of little importance. His _Analytical Inquiry into the Principles ofTaste_ appeared in 1808. 

[437] Mario Nizzoli (1498-1566), a well-known student of Cicero, was for atime professor at the University of Parma. His _Observationes in M. TulliumCiceronem_ appeared at Pratalboino in 1535. It was revised by his nephewunder the title _Thesaurus Ciceronianus_ (Venice, 1570). 

[438] See Vol. I, page 314, note 4 {681}. 

[439]

 "Like the geometer, who bends all his powers To measure the circle, and does not succeed, Thinking what principle he needs. "

[440] Francis Quarles (1592-1644), a religious poet. He wrote paraphrasesof the Bible and numerous elegies. In the early days of the revolutionarystruggle he sided with the Royalists. One of his most popular works was the_Emblems_ (1635), with illustrations by William Marshall. 

[441] Regnault de Bécourt wrote _La Création du monde, ou Systèmed'organisation primitive suivi de l'interprétation des principauxphénomènes et accidents que se sont opérés dans la nature depuis l'originede univers jusqu'à nos jours_ (1816). This may be the work translated byDalmas. 

[442] "Because it lacks a holy prophet. "

[443] Angherà. See Vol. II, page 60, note 127. 

[444] Edmund Curll (1675-1747), a well-known bookseller, publisher, andpamphleteer. He was for a time at "The Peacock without Temple Bar, " andlater at "The Dial and Bible against St. Dunstan's Church. " He was finedrepeatedly for publishing immoral works, and once stood in the pillory forit. He is ridiculed in the _Dunciad_ for having been tossed in a blanket bythe boys of Westminster School because of an oration that displeased them. 

[445] See Vol. II, page 109, note 206. 

[446] Encyclopædia. 

[447] Author of the _Historia Naturalis_ (77 A. D. )

[448] Author of the _De Institutione Oratorio Libri_ XII (c. 91 A. D. )

[449] His _De Architectures Libri_ X was not merely a work on architectureand building, but on the education of the architect. 

[450] Cyclophoria. 

[451] William Caxton (c. 1422-c. 1492), sometime Governor of the Company ofMerchant Adventurers in Bruges (between 1449 and 1470). He learned the artof printing either at Bruges or Cologne, and between 1471 and 1477 set up apress at Westminster. Tradition says that the first book printed in Englandwas his _Game and Playe of Chesse_ (1474). The _Myrrour of the Worlde andth'ymage of the same_ appeared in 1480. It contains a brief statement onarithmetic, the first mathematics to appear in print in England. 

[452] See Vol. I, page 45, note 6 {40}. De Morgan is wrong as to the dateof the _Margarita Philosophica_. The first edition appeared at Freiburg in1503. 

[453] Reisch was confessor to Maximilian I (1459-1519), King of the Romans(1486) and Emperor (1493-1519). 

[454] Joachim Sterck Ringelbergh (c. 1499-c. 1536), teacher of philosophyand mathematics in various cities of France and Germany. His _Institutionumastronomicarum libri III_ appeared at Basel in 1528, his _Cosmographia_ atParis in 1529, and his _Opera_ at Leyden in 1531. 

[455] Johannes Heinrich Alsted (1588-1638) was professor of philosophy andtheology at his birthplace, Herborn, in Nassau, and later at Weissenberg. He published several works, including the _Elementale mathematicum_ (1611), _Systema physicae harmonicae_ (1612), _Methodus admirandorummathematicorum_ (1613), _Encyclopædia septem tomis distincta_ (1630), andthe work mentioned above. 

[456] Johann Jakob Hoffmann (1635-1706), professor of Greek and history athis birthplace, Basel. He also wrote the _Epitome metrica historiæuniversalis civilis et sacræ ab orbe condito_ (1686). 

[457] Ephraim Chambers (c. 1680-1740), a crotchety, penurious, butkind-hearted freethinker. His _Cyclopædia, or an Universal Dictionary_ wastranslated into French and is said to have suggested the great_Encyclopédie_. 

[458] _Encyclopédie, ou Dictionnaire raisonné des sciences, des arts et desmétiers, par un société de gens de lettres. Mis en ordre et publié par M. Diderot, et quant à la partie mathématique, par M. D'Alembert. _ Paris, 1751-1780, 35 volumes. 

[459] "From the egg" (state). 

[460] See Vol. I, page 382, note 12 {785}. 

[461] See Vol. II, page 4, note 15. 

[462] "In morals nothing should serve man as a model but God; in the arts, nothing but nature. "

[463] _Encyclopédie Méthodique, ou par ordre de matières. _ Paris, 1782-1832, 166½ volumes. 

[464] See Vol. II, page 193, note 336. 

[465] _Encyclopædia Metropolitana; or, Universal Dictionary of Knowledge. _London, 1845, 29 volumes. A second edition came out in 1848-1858 in 40volumes. 

[466] See Vol. I, page 137, note 8 {286}. 

[467] See Vol. I, page 80, note 5 {119}. 

[468] De Morgan should be alive to satirize some of the statements on thehistory of mathematics in the eleventh edition. 

[469] John Pringle Nichol (1804-1859), Regius professor of astronomy atGlasgow and a popular lecturer on the subject. He lectured in the UnitedStates in 1848-1849. His _Views of the Architecture of the Heavens_ (1838)was a very popular work, and his _Planetary System_ (1848, 1850) containsthe first suggestion for the study of sun spots by the aid of photography. 

[470] See Vol. II, page 109, note 206. 

[471] George Long (1800-1879), a native of Poulton, in Lancashire, wascalled to the University of Virginia when he was only twenty-four years oldas professor of ancient languages. He returned to England in 1828 to becomeprofessor of Greek at London University. From 1833 to 1849 he edited thetwenty-nine volumes of the _Penny Cyclopædia_. He was an authority on Romanlaw. 

[472] A legal phrase, "Qui tam pro domina regina, quam pro se ipsosequitur, "--"Who sues as much on the Queen's account as on his own. "

[473] Arthur Cayley (1821-1895) was a fellow of Trinity College, Cambridge(1842-1846) and was afterwards a lawyer (1849-1863). During his fourteenyears at the bar he published some two hundred mathematical papers. In 1863he became professor of mathematics at Cambridge, and so remained until hisdeath. His collected papers, nine hundred in number, were published by theCambridge Press in 13 volumes (1889-1898). He contributed extensively tothe theory of invariants and covariants. De Morgan's reference to hiscoining of new names is justified, although his contemporary, ProfessorSylvester, so far surpassed him in this respect as to have been dubbed "themathematical Adam. "

[474] See Vol. II, page 26, note 56. 

[475] See Vol. I, page 111, note 3 {207}. 

[476] See Vol. I, page 87, note 6 {135}. 

[477] Pierre Dupuy (1582-1651) was a friend and relative of De Thou. Withthe collaboration of his brother and Nicolas Rigault he published the 1620and 1626 editions of De Thou's History. He also wrote on law and history. His younger brother, Jacques (died in 1656), edited his works. The two hada valuable collection of books and manuscripts which they bequeathed to theRoyal Library at Paris. 

[478] See Vol. I, page 51, note 3 {51}. 

[479] It was Cosmo de' Medici (1590-1621) who was the patron of Galileo. 

[480] See Vol. I, page 40, note 4 {20}. 

[481] See Vol. I, page 106, note 4 {188}. 

[482] Sir Edward Sherburne (1618-1702), a scholar of considerablereputation. The reference by De Morgan is to _The Sphere of MarcusManilius_, in the appendix to which is a _Catalogue of Astronomers, ancientand modern_. 

[483] George Parker, second Earl of Macclesfield (1697-1764). He erected anobservatory at Shirburn Castle, Oxfordshire, in 1739, and fitted it outwith the best equipment then available. He was President of the RoyalSociety in 1752. 

[484] See Vol. II, page 148, note 263. 

[485] See Vol. I, page 140, note 7 {296}. 

[486] See Vol. I, page 106, note 4 {188}. 

[487] Edward Bernard (1638-1696), although Savilian professor of astronomyat Oxford, was chiefly interested in archeology. 

[488] See Vol. I, page 107, note 1 {190}. 

[489] See Vol. I, page 107, note 1 {190}. 

[490] See Vol. I, page 135, note 3 {281}. 

[491] Philip Dormer Stanhope, fourth Earl of Chesterfield (1694-1773), wellknown for the letters written to his son which were published posthumously(1774). 

[492] Peter Daval (died in 1763), Vice-President of the Royal Society, andan astronomer of some ability. 

[493] See Vol. I, page 376, note 1 {766}. 

[494] William Oughtred (c. 1573-1660), a fellow of King's College, Cambridge, and afterwards vicar of Aldbury, Surrey, wrote the best-knownarithmetic and trigonometry of his time. His _Arithmeticæ in Numero &Speciebus Institutio ... Quasi Clavis Mathematicæ est_ (1631) went throughmany editions and appeared in English as _The Key to the Mathematicks newforged and filed_ in 1647. 

[495] See Vol. I, page 140, note 5 {294}. 

[496] Stephen Jordan Rigaud (1816-1859) was senior assistant master ofWestminster School (1846) and head master of Queen Elizabeth's School atIpswich (1850). He was made Bishop of Antigua in 1858 and died of yellowfever the following year. 

[497] He also wrote a memoir of his father, privately printed at Oxford in1883. 

[498] See Vol. I, page 69, note 3 {96}. 

[499] See Vol. I, page 106, note 4 {188}. 

[500] William Gascoigne was born at Middleton before 1612 and was killed inthe battle of Marston Moor in 1644. He was an astronomer and invented themicrometer with movable threads (before 1639). 

[501] Seth Ward (1617-1689) was deprived of his fellowship at Cambridge forrefusing to sign the covenant. He became professor of astronomy at Oxford(1649), Bishop of Exeter (1662), Bishop of Salisbury (1667), and Chancellorof the Garter (1671). He is best known for his solution of Kepler's problemto approximate a planet's orbit, which appeared in his _Astronomiageometrica_ in 1656. 

[502] See Vol. I, page 110, note 2 {198}. 

[503] See Vol. I, page 100, note 2 {172}. 

[504] See Vol. I, page 107, note 1 {190}. 

[505] See Vol. I page 114, note 6 {220}. 

[506] See Vol. I, page 77, note 4 {118}. 

[507] See Vol. I, page 125, note 3 {253}. 

[508] See Vol. I, page 105, note 2 {186}. 

[509] Heinrich Oldenburgh (1626-1678) was consul in England for the City ofBremen, his birthplace, and afterwards became a private teacher in London. He became secretary of the Royal Society and contributed on physics andastronomy to the _Philosophical Transactions_. 

[510] Thomas Brancker, or Branker (1636-1676) wrote the _Doctrinæ sphæricæadumbratio et usus globorum artificialium_ (1662) and translated thealgebra of Rhonius with the help of Pell. The latter work appeared underthe title of _An Introduction to Algebra_ (1668), and is noteworthy ashaving brought before English mathematicians the symbol ÷ for division. Thesymbol never had any standing on the Continent for this purpose, butthereafter became so popular in England that it is still used in all theEnglish-speaking world. 

[511] See Vol. I, page 118, note 1 {230}. 

[512] Pierre Bertius (1565-1629) was a native of Flanders and was educatedat London and Leyden. He became a professor at Leyden, and later held thechair of mathematics at the Collège de France. He wrote chiefly ongeography. 

[513] See Vol. II, page 297, note 487. 

[514] Giovanni Alphonso Borelli (1608-1679) was professor of mathematics atMessina (1646-1656) and at Pisa (1656-1657), after which he taught in Romeat the Convent of St. Panteleon. He wrote several works on geometry, astronomy, and physics. 

[515] See Vol. I, page 172, note 2 {381}. 

[516] Ignace Gaston Pardies (c. 1636-1673), a Jesuit, professor of ancientlanguages and later of mathematics and physics at the Collège of Pau, andafterwards professor of rhetoric at the Collège Louis-le-Grand at Paris. Hewrote on geometry, astronomy and physics. 

[517] Pierre Fermat was born in 1608 (or possibly in 1595) near Toulouse, and died in 1665. Although connected with the parliament of Toulouse, hissignificant work was in mathematics. He was one of the world's geniuses inthe theory of numbers, and was one of the founders of the theory ofprobabilities and of analytic geometry. After his death his son publishedhis edition of Diophantus (1670) and his _Varia opera mathematica_ (1679). 

[518] This may be Christopher Townley (1603-1674) the antiquary, or hisnephew, Richard, who improved the micrometer already invented by Gascoigne. 

[519] Adrien Auzout a native of Rouen, who died at Rome in 1691. Heinvented a screw micrometer with movable threads (1666) and made manyimprovements in astronomical instruments. 

[520] See Vol. I, page 66, note 9 {86}. 

[521] See Vol. I, page 124, note 7 {248}. 

[522] John Machin (d. 1751) was professor of astronomy at Gresham College(1713-1751) and secretary of the Royal Society. He translated Newton's_Principia_ into English. His computation of [pi] to 100 places is given inWilliam Jones's _Synopsis palmariorum matheseos_ (1706). 

[523] Pierre Rémond de Montmort (1678-1719) was canon of Notre Dame untilhis marriage. He was a gentleman of leisure and devoted himself to thestudy of mathematics, especially of probabilities. 

[524] Roger Cotes (1682-1716), first Plumian professor of astronomy andphysics at Cambridge, and editor of the second edition of Newton's_Principia_. His posthumous _Harmonia Mensurarum_ (1722) contains "Cotes'sTheorem" on the binomial equation. Newton said of him, "If Mr. Cotes hadlived we had known something. "

[525] See Vol. I, page 135, note 3 {281}. 

[526] See Vol. I, page 377, note 4 {769}. 

[527] Charles Réné Reyneau (1656-1728) was professor of mathematics atAngers. His _Analyse démontrée, ou Manière de resoudre les problèmes demathématiques_ (1708) was a successful attempt to popularize the theoriesof men like Descartes, Newton, Leibnitz, and the Bernoullis. 

[528] Brook Taylor (1685-1731), secretary of the Royal Society, and studentof mathematics and physics. His _Methodus incrementorum directa et inversa_(1715) was the first treatise on the calculus of finite differences. Itcontained the well-known theorem that bears his name. 

[529] Pierre Louis Moreau de Maupertuis (1698-1759) was sent with Clairaut(1735) to measure an arc of a meridian in Lapland. He was head of thephysics department in the Berlin Academy from 1745 until 1753. He wrote_Sur la figure de la terre_ (1738) and on geography and astronomy. 

[530] Pierre Bouguer (1698-1758) was professor of hydrography at Paris, andwas one of those sent by the Academy of Sciences to measure an arc of ameridian in Peru (1735). The object of this and the work of Maupertuis wasto determine the shape of the earth and see if Newton's theory wassupported. 

[531] Charles Marie de la Condamine (1701-1774) was a member of the ParisAcademy of Sciences and was sent with Bouguer to Peru, for the purposementioned in the preceding note. He wrote on the figure of the earth, butwas not a scientist of high rank. 

[532] See Vol. I, page 136, note 5 {283}. 

[533] See Vol. II, page 296, note 483. 

[534] Thomas Baker (c. 1625-1689) gave a geometric solution of thebiquadratic in his _Geometrical Key, or Gate of Equations unlocked_ (1684). 

[535] See Vol. I, page 160, note 5 {350}. 

[536] See Vol. I, page 87, note 4 {133}. 

[537] See Vol. I, page 132, note 2 {272}. 

[538] See Vol. I, page 118, second note 1 {231}. 

[539] The name of Newton is so well known that no note seems necessary. Hewas born at Woolsthorpe, Lincolnshire, in 1642, and died at Kensington in1727. 

[540] John Keill (1671-1721), professor of astronomy at Oxford from 1710, is said to have been the first to teach the Newtonian physics by directexperiment, the apparatus being invented by him for the purpose. He wroteon astronomy and physics. His _Epistola de legibus virium centripetarum_, in the Philosophical Transactions for 1708, accused Leibnitz of havingobtained his ideas of the calculus from Newton, thus starting the prioritycontroversy. 

[541] Thomas Digges (d. In 1595) wrote _An Arithmeticall Militare Treatise, named Stratioticos_ (1579), and completed _A geometrical practise, namedPantometria_ (1571) that had been begun by his father, Leonard Digges. 

[542] John Dee (1527-1608), the most famous astrologer of his day, andsomething of a mathematician, wrote a preface to Billingsley's translationof Euclid into English (1570). 

[543] See Vol. I, page 76, note 3 {112}. 

[544] Thomas Harriot (1560-1621) was tutor in mathematics to Sir WalterRaleigh, who sent him to survey Virginia (1585). He was one of the bestEnglish algebraists of his time, but his _Artis Analyticæ Praxis adAequationes Algebraicas resolvendas_ (1631) did not appear until ten yearsafter his death. 

[545] Thomas Lydiat (1572-1626), rector of Alkerton, devoted his lifechiefly to the study of chronology, writing upon the subject and takingissue with Scaliger (1601). 

[546] See Vol. I, page 69, note 3 {96}. 

[547] Walter Warner edited Harriot's _Artis Analyticae Praxis_ (1631). Tarporley is not known in mathematics. 

[548] See Vol. I, page 105, note 2 {186}. 

[549] See Vol. I, page 115, note 3 {224}. 

[550] See Vol. II, page 300, note 509. 

[551] See Vol. I, page 107, note 1 {190}. 

[552] Sir Samuel Morland (1625-1695) was a diplomat and inventor. For someyears he was assistant to John Pell, then ambassador to Switzerland. Hewrote on arithmetical instruments invented by him (1673), on hydrostatics(1697) and on church history (1658). 

[553] See Vol. I, page 153, note 4 {337}. 

[554] See Vol. I, page 85, note 2 {129}. 

[555] See Vol. I, page 43, note 8 {33}. 

[556] See Vol. I, page 43, note 7 {32}. 

[557] See Vol. I, page 382, note 13 {786}. The history of the subject maybe followed in Braunmühl's _Geschichte der Trigonometrie_. 

[558] See Vol. I, page 377, note 3 {768}. 

[559] See Vol. I, page 108, note 2 {192}. 

[560] Michael Dary wrote _Dary's Miscellanies_ (1669), _Gauging epitomised_(1669), and _The general Doctrine of Equation_ (1664). 

[561] John Newton (1622-1678), canon of Hereford (1673), educationalreformer, and writer on elementary mathematics and astronomy. 

[562] See Vol. I, page 87, note 4 {133}. 

[563] "The average of the two equal altitudes of the sun before and afterdinner. "

[564] See Vol. I, page 42, note 4 {24}. 

[565] London, 1678. It went though many editions. 

[566] "This I who once ... "

[567] Arthur Murphy (1727-1805) worked in a banking house until 1754. Hethen went on the stage and met with some success at Covent Garden. Hisfirst comedy, _The Apprentice_ (1756) was so successful that he left thestage and took to play writing. His translation of Tacitus appeared in1793, in four volumes. 

[568] Edmund Wingate (1596-1656) went to Paris in 1624 as tutor to PrincessHenrietta Maria and remained there several years. He wrote _L'usage de larègle de proportion_ (Paris, 1624, with an English translation in 1626), _Arithmétique Logarithmétique_ (Paris, 1626, with an English translation in1635), and _Of Natural and Artificial Arithmetick_ (London, 1630, revisedin 1650-1652), part I of which was one of the most popular textbooks everproduced in England. 

[569] John Lambert (1619-1694) was Major-General during the Revolution andhelped to draw up the request for Cromwell to assume the protectorate. Hewas imprisoned in the Tower by the Rump Parliament. He was confined inGuernsey until the clandestine marriage of his daughter Mary to CharlesHatton, son of the governor, after which he was removed (1667) to St. Nicholas in Plymouth Sound. 

[570] Samuel Foster (d. In 1652) was made professor of astronomy at GreshamCollege in March, 1636, but resigned in November of that year, beingsucceeded by Mungo Murray. Murray vacated his chair by marriage in 1641 andFoster succeeded him. He wrote on dialling and made a number ofimprovements in geometric instruments. 

[571] "Twice of the word a minister, " that is, twice a minister of theGospel. 

[572] This is _The Lives of the Professors of Gresham College to which isprefixed the Life of the Founder, Sir Thomas Gresham_, London, 1740. It waswritten by John Ward (c. 1679-1758), professor of rhetoric (1720) atGresham College and vice-president (1752) of the Royal Society. 

[573] Charles Montagu (1661-1715), first Earl of Halifax, was Lord of theTreasury in 1692, and was created Baron Halifax in 1700 and ViscountSunbury and Earl of Halifax in 1714. He introduced the bill establishingthe Bank of England, the bill becoming a law in 1694. He had troubles ofhis own, without considering Newton, for he was impeached in 1701, and wasthe subject of a damaging resolution of censure as auditor of the exchequerin 1703. Although nothing came of either of these attacks, he was out ofoffice during much of Queen Anne's reign. 

[574] See Vol. II, page 302, note 547. 

[575] See Vol. I, page 105, note 2 {186}. 

[576] James Dodson (d. 1757) was master of the Royal Mathematical School, Christ's Hospital. He was De Morgan's great-grandfather. The_Anti-Logarithmic Canon_ was published in 1742. 

[577] See Vol. I, page 106, note 4 {188}. 

[578] See Vol. I, page 110, note 2 {198}. 

[579] Richard Busby, (1606-1695), master of Westminster School (1640) hadamong his pupils Dryden and Locke. 

[580] See Vol. I, page 107, note 1 {190}. 

[581] Herbert Thorndike (1598-1672), fellow of Trinity College, Cambridge(1620-1646), and Prebend of Westminster (1661), was a well-knowntheological writer of the time. 

[582] See Vol. I, page 140, note 5 {294}. 

[583] See Vol. I, page 108, note 2 {192}. 

[584] "Labor performed returns in a circle. "

[585] See Vol. II, page 208. 

[586] "Whatever objections one may make to the above arguments, one alwaysfalls into an absurdity. "

[587] See Vol. II. Page 11, note 29. _The Circle Squared; and the solutionof the problem adapted to explain the difference between square andsuperficial measurement_ appeared at Brighton in 1865. 

[588] "And beyond that nothing. "

[589] Gillott (1759-1873) was the pioneer maker of steel pens by machinery, reducing the price from 1s. Each to 4d. A gross. He was a great collectorof paintings and old violins. 

[590] William Edward Walker wrote five works on circle squaring (1853, 1854, 1857, 1862, 1864), mostly and perhaps all published at Birmingham. 

[591] Solomon M. Drach wrote _An easy Rule for formulizing all EpicyclicalCurves_ (London, 1849), _On the Circle area and Heptagon-chord_ (London, 1864), _An easy general Rule for filling up all Magic Squares_ (London, 1873), and _Hebrew Almanack-Signs_ (London, 1877), besides numerousarticles in journals. 

[592] See Vol. I, page 168, note 3 {371}. 

[593] See Vol. I, page 254, note 2 {580}. 

[594] See Vol. I, page 98, note 6 {163}. 

[595] Robert Fludd or Flud (1574-1637) was a physician with a large Londonpractice. He denied the diurnal rotation of the earth, and was attacked byKepler and Mersenne, and accused of magic by Gassendi. His _ApologiaCompendiania, Fraternitatem de Rosea Cruce suspicionis ... Maculisaspersam, veritatis quasi Fluctibus abluens_ (Leyden, 1616) is one of alarge number of works of the mystic type. 

[596] Consult _To the Christianity of the Age. Notes ... Comprising anelucidation of the scope and contents of the writings ... Of DionysiusAndreas Freher_ (1854). 

[597] Sir William Robert Grove (1811-1896), although called to the bar(1835) and to the bench (1853), is best known for his work as a physicist. He was professor of experimental philosophy (1840-1847) at the LondonInstitution, and invented a battery (1839) known by his name. His_Correlation of Physical Forces_ (1846) went through six editions and wastranslated into French. 

[598] Johann Tauler (c. 1300-1361), a Dominican monk of Strassburg, amystic, closely in touch with the Gottesfreunde of Basel. His _Sermons_first appeared in print at Leipsic in 1498. 

[599] Paracelsus (c. 1490-1541). His real name was Theophrastes Bombast vonHohenheim, and he took the name by which he is generally known because heheld himself superior to Celsus. He was a famous physician and pharmacist, but was also a mystic and neo-Platonist. He lectured in German on medicineat Basel, but lost his position through the opposition of the orthodoxphysicians and apothecaries. 

[600] See Vol. I, page 256, note 2 {588}. 

[601] Philip Schwarzerd (1497-1560) was professor of Greek at Wittenberg. He helped Luther with his translation of the Bible. 

[602] Johann Reuchlin (1455-1522), the first great German humanist, wasvery influential in establishing the study of Greek and Hebrew in Germany. His lectures were mostly delivered privately in Heidelberg and Stuttgart. Unlike Melanchthon, he remained in the Catholic Church. 

[603] Joseph Chitty (1776-1841) published his _Precedents of Pleading_ in1808 and his _Reports of Cases on Practice and Pleading_ in 1820-23 (2volumes). 

[604] See Vol. I, page 44, note 1 {35}. 

[605] See Vol. I, page 44, note 4 {38}. 

[606] Jean Pèlerin, also known as Viator, who wrote on perspective. Hiswork appeared in 1505, with editions in 1509 and 1521. 

[607] Henry Stephens. See Vol. I, page 44, note 3 {37}. 

[608] The well-known grammarian (1745-1826). He was born at Swatara, inPennsylvania, and practised law in New York until 1784, after which heresided in England. His grammar (1795) went through 50 editions, and theabridgment (1818) through 120 editions. Murray's friend Dalton, thechemist, said that "of all the contrivances invented by human ingenuity forpuzzling the brains of the young, Lindley Murray's grammar was the worst. "

[609] Robert Recorde (c. 1510-1558) read and probably taught mathematicsand medicine at Cambridge up to 1545. After that he taught mathematics atOxford and practised medicine in London. His _Grounde of Artes_, publishedabout 1540, was the first arithmetic published in English that had anyinfluence. It went through many editions. The _Castle of Knowledge_appeared in 1551. It was a textbook on astronomy and the first to set forththe Copernican theory in England. Like Recorde's other works it was writtenon the catechism plan. His _Whetstone of Witte ... Containying thextractionof Rootes: The Cosike practise, with the rule of Equation: and the woorkesof Surde Nombres_ appeared in 1557, and it is in this work that the modernsign of equality first appears in print. The word "Cosike" is an adjectivethat was used for a long time in Germany as equivalent to algebraic, beingderived from the Italian _cosa_, which stood for the unknown quantity. 

[610] Robert Cecil (c. 1563-1612), first Earl of Salisbury, Secretary ofState under Elizabeth (1596-1603) and under James I (1603-1612). 

[611] In America the German pronunciation is at present universal amongmathematicians, as in the case of most other German names. This is due, nodoubt, to the great influence that Germany has had on American education inthe last fifty years. 

[612] The latest transliteration is substantially K'ung-fu-tz[vu]. 

[613] The tendency seems to be, however, to adopt the forms used ofindividuals or places as rapidly as the mass of people comes to be preparedfor it. Thus the spelling Leipzig, instead of Leipsic, is coming to be verycommon in America. 

[614] Sir Edward Coke (1552-1634), the celebrated jurist. 

[615] Dethlef Cluvier or Clüver (d. 1708 at Hamburg) was a nephew, not agrandson, of Philippe Cluvier, or Philipp Clüver (1580-c. 1623). Dethleftraveled in France and Italy and then taught mathematics in London. Hewrote on astronomy and philosophy and also published in the _ActaEruditorum_ (1686) his _Schediasma geometricum de nova infinitorumscientia_. _Quadratura circuli infinitis modis demonstrata_, and his_Monitum ad geometras_ (1687). Philippe was geographer of the Academy ofLeyden. His _Introductionis in universam geographiam tam veterem quam novamlibri sex_ appeared at Leyden in 1624, about the time of his death. 

[616] See Vol. I, page 124, note 7 {248}. 

[617] Bernard Nieuwentijt (1654-1718), a physician and burgomaster atPurmerend. His _Considerationes circa Analyseos ad quantitates infiniteparvas applicatæ Principia et Calculi Differentialis usum_ (Amsterdam, 1694) was attacked by Leibnitz. He replied in his _Considerationes secundæ_(1694), and also wrote the _Analysis Infinitorum, seu CurvilineorumProprietates ex Polygonorum Natura deductæ_ (1695). His most famous workwas on the existence of God, _Het Regt Gebruik der Werelt Beschouwingen_(1718). 

[618] "From a given line to construct" etc. 

[619] "Pirates do not fight one another. "

[620] Claude Mallemens (Mallement) de Messanges (1653-1723) was professorof philosophy at the Collège du Plessis, in Paris, for 34 years. The workto which De Morgan refers is probably the _Fameux Problème de la quadraturedu cercle, résolu géometriquement par le cercle et a ligne droite_ thatappeared in 1683. 

[621] On Tycho Brahe see Vol. I, page 76, note 3 {112}. 

[622] Wilhelm Frederik von Zytphen also published the _Tidens Ström_, achronological table, in 1840. The work to which De Morgan refers, the_Solens Bevægelse i Verdensrummet_, appeared first in 1861. De Morgan seemsto have missed his _Nogl Ord om Cirkelens Quadratur_ which appeared in1865, at Copenhagen. 

[623] James Joseph Sylvester (1814-1897), professor of natural philosophyat University College, London (1837-1841), professor of mathematics at theUniversity of Virginia (1841-1845), actuary in London (1845-1855), professor of mathematics at Woolwich (1877-1884) and at Johns HopkinsUniversity, Baltimore (1877-1884), and Savilian professor of geometry atOxford (1884-1894). 

[624] See Vol. I, page 76, note 3 {112}. 

[625] See Vol. II, page 205, note 349. 

[626] See Vol. I, page 76, note 3 {112}. 

[627] See Vol. I, page 46, note 1 {42}. 

[628] See Vol. II, page 183, note 318. 

[629] See Vol. I, page 321, note 2 {691}. 

[630] James Mill, born 1773, died 1836. 

[631] See Vol. II, page 3, note 11. 

[632] See Vol. II, page 3, note 13. 

[633] See Vol. II, page 3, note 14. 

[634] This anecdote is printed at page 4 (Vol. II); but as it is used inillustration here, and is given more in detail, I have not omittedit. --S. E. De M. 

[635] See Vol. II, page 4, note 15. 

[636] See Vol. I, page 382, note 13 {786}. 

[637] "Monsieur, (a + b^{n})/n = x, whence God exists; answer that!"

[638] "Monsieur, you know very well that your argument requires thedevelopment of x according to integral powers of n. "

[639] See Vol. I, page 153, note 4 {337}. 

[640] Thomas Love Peacock (1785-1866) an English novelist and poet. 

[641] Perhaps Dr. Samuel Warren (1807-1877), the author of _Ten Thousand aYear_ (serially in Blackwood's in 1839; London, 1841). 

[642] See Vol. I, page 255, note 6 {584}. 

[643] "From many, one; much in little; Ultima Thule (the most remoteregion); without which not. "

[644] Spurius Mælius (fl. 440 B. C. ), who distributed corn freely among thepoor in the famine of 440 B. C. And was assassinated by the patricians. 

[645] Spurius Cassius Viscellinus, Roman consul in 502, 493, and 486 B. C. Put to death in 485. 

[646] "O what a fine bearing, he said, that has no brain. "

[647] Sir William Rowan Hamilton. See Vol. I, page 332, note 4 {709}. 

[648] William Allen Whitworth, the author of the well-known _Choice andChance_ (Cambridge, 1867), and other works. 

[649] James Maurice Wilson, whose _Elementary Geometry_ appeared in 1868and went through several editions. 

[650] See Vol. II, page 183, note 315. 

[651] "Force of inertia conquered, " and "Victory in the whole heavens. "

[652] "With all his might. "

[653] George Berkeley (1685-1753), Bishop of Cloyne, the idealisticphilosopher and author of the _Principles of Human Knowledge_ (1710), _TheAnalyst, or a Discourse addressed to an Infidel Mathematician_ (1734), and_A Defense of Freethinking in Mathematics_ (1735). He asserted that spaceinvolves the idea of movement without the sensation of resistance. Spacesensation less than the "minima sensibilia" is, therefore, impossible. Fromthis he argues that infinitesimals are impossible concepts. 

[654] See Vol. I, page 85, note 2 {129}. 

[655] See Vol. I, page 81, note 6 {120}. 

[656] Edwin Dunkin revised Lardner's _Handbook of Astronomy_ (1869) andMilner's _The Heavens and the Earth_ (1873) and wrote _The Midnight Sky_(1869). 

[657] Michael Faraday (1791-1867) the celebrated physicist and chemist. Hewas an assistant to Sir Humphrey Davy (1813) and became professor ofchemistry at the Royal Institution, London, in 1827. 

[658] "If you teach a fool he shows no joyous countenance; he cordiallyhates you; he wishes you buried. "

[659] "Every man is an animal, Sortes is a man, therefore Sortes is ananimal. "

[660]

 "May some choice patron bless each grey goose quill; May every Bavius have his Bufo still. "--POPE, _Prologue to the Satires. _

Bavius has become proverbial as a bad poet from the lines in Vergil's_Eclogues_ (III, 90):

 "Qui Bavium non odit, amet tua carmina, Maevi, Atque idem jungat vulpes, et mulgeat hircos. "

"He who does not hate Bavius shall love thy verses, O Maevius; and the sameshall yoke foxes and shall milk he-goats. "

Bavius and Maevius were the worst of Latin poets, condemned by Horace aswell as Vergil. 

[661] See Vol. II, page 158, note 279. 

[662] "Honest, " "useful, " "handsome, " "sweet. "

[663] "Let not the fourth man attempt to speak. "

[664]

 "In those old times, --ah 'Twas just like this, ah!"

[665] See Vol. I, page 382, note 12 {785}. 

[666] These remarks were never written. --S. E. De M. 

[667]

 "Fleas, flies, and friars, are masters who sadly the people abuse, And thistles and briars are sure growing grains to abuse. O Christ, who hatest strife and slayst all things in peace, Destroy where'er are rife, briars, friars, flies and fleas. Fleas, flies, and friars foul fall them these fifteen years For none that there is loveth fleas, flies, nor freres. "

[668] "It is my plan to restore to an unskilled race the worthy arts of abetter life. "

[669] The first sentences of the first oration of Cicero against Catiline:"Quo usque tandem abutere, Catilina, patientia nostra?" (How long, OCatiline, will you abuse our patience?) "Quamdiu etiam furor iste tuus noseludet?" (How long will this your madness baffle us?) "Nihilne te nocturnumpraesidium Palati, ... Nihil horum ora voltusque moverunt?" (Does the nightwatch of the Palatium, ... Do the faces and expressions of all these menfail to move you?) "In te conferri ... " (This plague should have beeninflicted upon you long ago, which you have plotted against us so long. )

[670] "Beware of the things that are marked. "

[671] "Farewell, ye teachers without learning! See to it that at our nextmeeting we may find you strong in body and sound in mind. "

[672] See Vol. I, page 336, note 8 {713}. 

[673] See Vol. I, page 229, note 2 {515}. 

[674] This proof, although capable of improvement, is left as in theoriginal. Those who may be interested in the mathematics of the question, may consult F. Enriques, _Fragen der Elementargeometrie_ (German byFleischer), Leipsic, 1907, Part II, p. 267; F. Rudio, _Archimedes_, _Huygens_, _Lambert_, _Legendre_. _Vier Abhandlungen über dieKreismessung_, Leipsic, 1892; F. Klein, _Famous Problems of ElementaryGeometry_ (English by Beman and Smith), Boston, 1895; J. W. A. Young, _Monographs on Modern Mathematics_, New York, 1911, Chap. IX (by the editorof the present edition of De Morgan. )

[675] See Vol. I, page 69, note 2 {95}. 

[676] See Vol. I, page 137, note 8 {286}. 

[677] Joseph Allen Galbraith who, with Samuel Haughton, wrote the Galbraithand Haughton's _Scientific Manuals_. (Euclid, 1856; Algebra, 1860;Trigonometry, 1854; Optics, 1854, and others. )

[678] This note on Carlyle (1795-1881) is interesting. The translation ofLegendre appeared in the same year (1824) as his translation of Goethe's_Wilhelm Meister_. 

[679] Michael Stifel (1487-1567), also known as Stiefel, Styfel, andStifelius, was an Augustine monk but became a convert to Lutheranism. Hewas professor of mathematics at Jena (1559-1567). His edition of the _Coss_appeared at Königsberg in 1553, the first edition having been published in1525. The + and - signs first appeared in print in Widman's arithmetic of1489, but for purposes of algebra this book was one of the first to makethem known. 

[680] Christoff Rudolff was born about 1500 and died between 1540 and 1552. _Die Coss_ appeared in 1525 and his arithmetic in 1526. 

 * * * * *

Corrections made to printed original. 

Page 9, "long-fostered prejudice": 'perjudice' in original. 

Page 73, "Pensées, ch. 7": 'Pansées' in original. 

Page 127, "and pulled out a plum": 'und' in original. 

Page 147, "did not come forward": 'forword' in original. 

Page 172, "come into general circulation": 'circulalation' in original. 

Ibid. , "the more difficult fractions which we have got": 'he have got' inoriginal. 

Page 192, "it has been stated": 'started' in original. 

Page 216, "the obsolete word tetch of the same meaning": 'meaing' inoriginal. 

Page 228, "[Greek: dioklasianos]": 'dioklalasianos' in original. 

Page 233. After `Henry E. Manning' were printed two paragraphs `Shillingversus Franc. ' and `Teutonic Long Hundred 120 versus 100 or the Decimalquestion. ' These appear to have been set in error, there is no applicablecontext. 

Page 316, "in a manner depending upon the difference": 'maner' in original. 

Page 322, "neither what Newton did, nor what was done before him. ": 'not'(for 'nor') in original. 

Page 344, "Victoria toto coelo": 'tolo' in original. 

Page 368, "cannot be brought up to 1": 'up to ±' in original. 

Page 371, "Q_2=b_2 Q_1+a_2": 'Q_2=b_2 Q_1-a_2' in original. 

Note 50, "all who were not in the road to Heaven were excommunicated":'excomunicated' in original. 

Note 372, "[Greek: hê alazoneia biou]": 'alaxoneia' in original. 

Ibid. , "Iapetos": 'Ispetos' in original. 

Ibid. , "Papeiskos": 'Paspeisoks' in original. 

Ibid. , "[Greek: dioklasianos]": 'dioklalasianos' in original.