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 * * * * * * * * * * * * * *

 THE ELEMENTS OF GEOMETRIE

 of the most auncient Philosopher _EVCLIDE_ of Megara. 

 _Faithfully (now first) translated into the Englishe toung, by _H. Billingsley_, Citizen of London_. 

 _Whereunto are annexed certaine Scholies, Annotations, and Inuentions, of the best Mathematiciens, both of time past, and in this our age. _

 _With a very fruitfull Præface made by _M. I. Dee_, specifying the chiefe Mathematicall Sciẽces, what they are, and wherunto commodious: where, also, are disclosed certaine new Secrets Mathematicall and Mechanicall, vntill these our daies, greatly missed. _

 Imprinted at London by _Iohn Daye_. 

❧ The Translator to the Reader. 

_There is (gentle Reader) nothing (the word of God onely setapart) which so much beautifieth and adorneth the soule andminde of mã, as doth the knowledge of good artes and sciences:as the knowledge of naturall and morall Philosophie. The onesetteth before our eyes, the creatures of God, both in theheauens aboue, and in the earth beneath: in which as in aglasse, we beholde the exceding maiestie and wisedome of God, in adorning and beautifying them as we see: in geuing vnto themsuch wonderfull and manifolde proprieties, and naturallworkinges, and that so diuersly and in such varietie: farther inmaintaining and conseruing them continually, whereby to praiseand adore him, as by S. Paule we are taught. The other teachethvs rules and preceptes of vertue, how, in common life amongestmen, we ought to walke vprightly: what dueties pertaine to ourselues, what pertaine to the gouernment or good order both of anhousholde, and also of a citie or common wealth. The readinglikewise of histories, conduceth not a litle, to the adorning ofthe soule & minde of man, a studie of all men cõmended: by itare seene and knowen the artes and doinges of infinite wise mengone before vs. In histories are contained infinite examples ofheroicall vertues to be of vs followed, and horrible examples ofvices to be of vs eschewed. Many other artes also there arewhich beautifie the minde of man: but of all other none do moregarnishe & beautifie it, then those artes which are calledMathematicall. Unto the knowledge of which no man can attaine, without the perfecte knowledge and instruction of theprinciples, groundes, and Elementes of Geometrie. But perfectlyto be instructed in them, requireth diligent studie and readingof olde auncient authors. Amongest which, none for a beginner isto be preferred before the most auncient Philosopher _Euclide_of _Megara_. For of all others he hath in a true methode andiuste order, gathered together whatsoeuer any before him had ofthese Elementes written: inuenting also and adding many thingesof his owne: wherby he hath in due forme accomplished the arte:first geuing definitions, principles, & groundes, wherof hededuceth his Propositions or conclusions, in such wonderfullwise, that that which goeth before, is of necessitie required tothe proufe of that which followeth. So that without the diligentstudie of _Euclides_ Elementes, it is impossible to attaine vntothe perfecte knowledge of Geometrie, and consequently of any ofthe other Mathematicall sciences. Wherefore considering the want& lacke of such good authors hitherto in our Englishe tounge, lamenting also the negligence, and lacke of zeale to theircountrey in those of our nation, to whom God hath geuen bothknowledge, & also abilitie to translate into our tounge, andto publishe abroad such good authors, and bookes (the chiefeinstrumentes of all learninges): seing moreouer that many goodwittes both of gentlemen and of others of all degrees, muchdesirous and studious of these artes, and seeking for them asmuch as they can, sparing no paines, and yet frustrate of theirintent, by no meanes attaining to that which they seeke: I hauefor their sakes, with some charge & great trauaile, faithfullytranslated into our vulgare toũge, & set abroad in Print, thisbooke of _Euclide_. Whereunto I haue added easie and plainedeclarations and examples by figures, of the definitions. Inwhich booke also ye shall in due place finde manifoldeadditions, Scholies, Annotations, and Inuentions: which I hauegathered out of many of the most famous & chiefe Mathematiciẽs, both of old time, and in our age: as by diligent reading it incourse, ye shall well perceaue. The fruite and gaine which Irequire for these my paines and trauaile, shall be nothing els, but onely that thou gentle reader, will gratefully accept thesame: and that thou mayest thereby receaue some profite: andmoreouer to excite and stirre vp others learned, to do the like, & to take paines in that behalfe. By meanes wherof, our Englishetounge shall no lesse be enriched with good Authors, then areother straunge tounges: as the Dutch, French, Italian, andSpanishe: in which are red all good authors in a maner, foundamongest the Grekes or Latines. Which is the chiefest cause, that amongest thẽ do florishe so many cunning and skilfull men, in the inuentions of straunge and wonderfull thinges, as inthese our daies we see there do. Which fruite and gaine if Iattaine vnto, it shall encourage me hereafter, in such like sortto translate, and set abroad some other good authors, bothpertaining to religion (as partly I haue already done) and alsopertaining to the Mathematicall Artes. Thus gentlereader farewell. _ (?¿)

 [Decoration]

 ❧ TO THE VNFAINED LOVERS of truthe, and constant Studentes of Noble _Sciences, _IOHN DEE_ of London, hartily_ wisheth grace from heauen, and most prosperous _successe in all their honest attemptes and_ exercises. 

Diuine _Plato_, the great Master of many worthy Philosophers, and theconstant auoucher, and pithy perswader of _Vnum_, _Bonum_, and _Ens_: inhis Schole and Academie, sundry times (besides his ordinary Scholers)was visited of a certaine kinde of men, allured by the noble fame of_Plato_, and the great commendation of hys profound and profitabledoctrine. But when such Hearers, after long harkening to him, perceaued, that the drift of his discourses issued out, to conclude, this _Vnum_, _Bonum_, and _Ens_, to be Spirituall, Infinite, Æternall, Omnipotent, &c. Nothyng beyng alledged or expressed, How, worldly goods: how, worldly dignitie: how, health, Strẽgth or lustines of body: nor yet themeanes, how a merueilous sensible and bodyly blysse and felicitiehereafter, might be atteyned: Straightway, the fantasies of thosehearers, were dampt: their opinion of _Plato_, was clene chaunged: yeahis doctrine was by them despised: and his schole, no more of themvisited. Which thing, his Scholer, _Aristotle_, narrowly cõsidering, founde the cause therof, to be, “For that they had no forwarnyng andinformation, in generall, ” whereto his doctrine tended. For, so, mightthey haue had occasion, either to haue forborne his schole hauntyng: (ifthey, then, had misliked his Scope and purpose) or constantly to hauecontinued therin: to their full satisfaction: if such his finall scope &intent, had ben to their desire. Wherfore, _Aristotle_, euer, afterthat, vsed in brief, to forewarne his owne Scholers and hearers, “bothof what matter, and also to what ende, he tooke in hand to speake, orteach. ” While I consider the diuerse trades of these two excellentPhilosophers (and am most sure, both, that _Plato_ right well, otherwisecould teach: and that _Aristotle_ mought boldely, with his hearers, hauedealt in like sorte as _Plato_ did) I am in no little pang ofperplexitie: Bycause, that, which I mislike, is most easy for me toperforme (and to haue _Plato_ for my exãple. ) And that, which I know tobe most commendable: and (in this first bringyng, into common handling, the _Artes Mathematicall_) to be most necessary: is full of greatdifficultie and sundry daungers. Yet, neither do I think it mete, for sostraunge matter (as now is ment to be published) and to so straunge anaudience, to be bluntly, at first, put forth, without a peculiarPreface: Nor (Imitatyng _Aristotle_) well can I hope, that accordyng tothe amplenes and dignitie of the _State Mathematicall_, I am able, either playnly to prescribe the materiall boundes: or precisely toexpresse the chief purposes, and most wonderfull applications therof. And though I am sure, that such as did shrinke from _Plato_ his schole, after they had perceiued his finall conclusion, would in these thingeshaue ben his most diligent hearers (so infinitely mought their desires, in fine and at length, by our _Artes Mathematicall_ be satisfied) yet, by this my Præface & forewarnyng, Aswell all such, may (to their greatbehofe) the soner, hither be allured: as also the _Pythagoricall_, and_Platonicall_ perfect scholer, and the constant profound Philosopher, with more ease and spede, may (like the Bee, ) gather, hereby, both waxand hony. 

 [The intent of this Preface. ]

Wherfore, seyng I finde great occasion (for the causes alleged, andfarder, in respect of my _Art Mathematike generall_) to vse “a certaineforewarnyng and Præface, whose content shalbe, that mighty, mostplesaunt, and frutefull _Mathematicall Tree_, with his chief armes andsecond (grifted) braunches: Both, what euery one is, and also, whatcommodity, in generall, is to be looked for, aswell of griff as stocke:And forasmuch as this enterprise is so great, that, to this our tyme, itneuer was (to my knowledge) by any achieued: And also it is most hard, in these our drery dayes, to such rare and straunge Artes, to wyn dueand common credit:” Neuertheles, if, for my sincere endeuour to satisfieyour honest expectation, you will but lend me your thãkefull mynde awhile: and, to such matter as, for this time, my penne (with spede) ishable to deliuer, apply your eye or eare attentifely: perchaunce, atonce, and for the first salutyng, this Preface you will finde a lessonlong enough. And either you will, for a second (by this) be made muchthe apter: or shortly become, well hable your selues, of the lyons claw, to coniecture his royall symmetrie, and farder propertie. Now then, gentle, my frendes, and countrey men, Turne your eyes, and bend yourmyndes to that doctrine, which for our present purpose, my simple talentis hable to yeld you. 

All thinges which are, & haue beyng, are found vnder a triple diuersitiegenerall. For, either, they are demed Supernaturall, Naturall, or, of athird being. Thinges Supernaturall, are immateriall, simple, indiuisible, incorruptible, & vnchangeable. Things Naturall, aremateriall, compounded, diuisible, corruptible, and chaungeable. ThingesSupernaturall, are, of the minde onely, comprehended: Things Naturall, of the sense exterior, ar hable to be perceiued. In thinges Naturall, probabilitie and coniecture hath place: But in things Supernaturall, chief demõstration, & most sure Science is to be had. By whichproperties & comparasons of these two, more easily may be described, thestate, condition, nature and property of those thinges, which, we beforetermed of a third being: which, by a peculier name also, are called_Thynges Mathematicall_. For, these, beyng (in a maner) middle, betwenethinges supernaturall and naturall: are not so absolute and excellent, as thinges supernatural: Nor yet so base and grosse, as things naturall:But are thinges immateriall: and neuerthelesse, by materiall thingshable somewhat to be signified. And though their particular Images, byArt, are aggregable and diuisible: yet the generall _Formes_, notwithstandyng, are constant, vnchaungeable, vntrãsformable, andincorruptible. Neither of the sense, can they, at any tyme, be perceiuedor iudged. Nor yet, for all that, in the royall mynde of man, firstconceiued. But, surmountyng the imperfectiõ of coniecture, weenyng andopinion: and commyng short of high intellectuall cõceptiõ, are theMercurial fruite of _Dianœticall_ discourse, in perfect imaginationsubsistyng. A meruaylous newtralitie haue these thinges _Mathematicall_, and also a straunge participatiõ betwene thinges supernaturall, immortall, intellectual, simple and indiuisible: and thynges naturall, mortall, sensible, compounded and diuisible. Probabilitie and sensibleprose, may well serue in thinges naturall: and is commendable: InMathematicall reasoninges, a probable Argument, is nothyng regarded: noryet the testimony of sense, any whit credited: But onely a perfectdemonstration, of truthes certaine, necessary, and inuincible:vniuersally and necessaryly concluded: is allowed as sufficient for “anArgument exactly and purely Mathematical. ”

 [Note the worde, Vnit, to expresse the Greke Monas, & not Vnitie: as we haue all, commonly, till now, vsed. ]

Of _Mathematicall_ thinges, are two principall kindes: namely, _Number_, and _Magnitude_. 

 [Number. ]

_Number_, we define, to be, a certayne Mathematicall Sũme, of _Vnits_. And, an _Vnit_, is that thing Mathematicall, Indiuisible, byparticipation of some likenes of whose property, any thing, which is indeede, or is counted One, may resonably be called One. We account an_Vnit_, a thing _Mathematicall_, though it be no Number, and alsoindiuisible: because, of it, materially, Number doth consist: which, principally, is a thing _Mathematicall_. 

 [Magnitude. ]

_Magnitude_ is a thing _Mathematicall_, by participation of some likenesof whose nature, any thing is iudged long, broade, or thicke. “A thicke_Magnitude_ we call a _Solide_, or a _Body_. What _Magnitude_ so euer, is Solide or Thicke, is also broade, & long. A broade magnitude, we calla _Superficies_ or a Plaine. Euery playne magnitude, hath also length. A long magnitude, we terme a _Line_. A _Line_ is neither thicke norbroade, but onely long: Euery certayne Line, hath two endes:

 [A point. ]

The endes of a line, are _Pointes_ called. A _Point_, is a thing_Mathematicall_, indiuisible, which may haue a certayne determinedsituation. ” If a Poynt moue from a determined situation, the way whereinit moued, is also a _Line_: mathematically produced, whereupon, of theauncient Mathematiciens, [A Line. ]

a _Line_ is called the race or course of a _Point_. A Poynt we define, by the name of a thing Mathematicall: though it be no Magnitude, andindiuisible: because it is the propre ende, and bound of a Line: whichis a true _Magnitude_. 

 [Magnitude. ]

And _Magnitude_ we may define to be that thing _Mathematicall_, which isdiuisible for euer, in partes diuisible, long, broade or thicke. Therefore though a Poynt be no _Magnitude_, yet _Terminatiuely_, werecken it a thing _Mathematicall_ (as I sayd) by reason it is properlythe end, and bound of a line. Neither _Number_, nor _Magnitude_, haueany Materialitie. First, we will consider of _Number_, and of theScience _Mathematicall_, to it appropriate, called _Arithmetike_: andafterward of _Magnitude_, and his Science, called _Geometrie_. But thatname contenteth me not: whereof a word or two hereafter shall be sayd. How Immateriall and free from all matter, _Number_ is, who doth notperceaue? yea, who doth not wonderfully wõder at it? For, neither pure_Element_, nor _Aristoteles, Quinta Essentia_, is hable to serue forNumber, as his propre matter. Nor yet the puritie and simplenes ofSubstance Spirituall or Angelicall, will be found propre enough thereto. And therefore the great & godly Philosopher _Anitius Boetius_, sayd:_Omnia quæcun[que] a primæua rerum natura constructa sunt, Numerorumvidentur ratione formata. Hoc enim fuit principale in animo ConditorisExemplar_. That is: +_All thinges (which from the very first originallbeing of thinges, haue bene framed and made) do appeare to be Formed bythe reason of Numbers. For this was the principall example or patternein the minde of the Creator_. + O comfortable allurement, O rauishingperswasion, to deale with a Science, whose Subiect, is so Auncient, sopure, so excellent, so surmounting all creatures, so vsed of theAlmighty and incomprehensible wisdome of the Creator, in the distinctcreation of all creatures: in all their distinct partes, properties, natures, and vertues, by order, and most absolute number, brought, from_Nothing_, to the _Formalitie_ of their being and state. By _Numbers_propertie therefore, of vs, by all possible meanes, (to the perfectionof the Science) learned, we may both winde and draw our selues into theinward and deepe search and vew, of all creatures distinct vertues, natures, properties, and _Formes_: And also, farder, arise, clime, ascend, and mount vp (with Speculatiue winges) in spirit, to behold inthe Glas of Creation, the _Forme of Formes_, the _Exemplar Number_ ofall thinges _Numerable_: both visible and inuisible, mortall andimmortall, Corporall and Spirituall. Part of this profound and diuineScience, had _Ioachim_ the Prophesier atteyned vnto: by _NumbersFormall, Naturall_, and _Rationall_, forseyng, concludyng, andforshewyng great particular euents, long before their comming. Hisbookes yet remainyng, hereof, are good profe: And the noble Earle of_Mirandula_, (besides that, ) a sufficient witnesse: that _Ioachim, inhis prophesies, proceded by no other way, then by Numbers Formall_. Andthis Earle hym selfe, in Rome, [Ano. 1488. ]

* set vp 900. Conclusions, in all kinde of Sciences, openly to bedisputed of: and among the rest, in his Conclusions _Mathematicall_, (inthe eleuenth Conclusion) hath in Latin, this English sentence. _ByNumbers, a way is had, to the searchyng out, and vnderstandyng of euerythyng, hable to be knowen. For the verifying of which Conclusion, I promise to aunswere to the 74. Questions, vnder written, by the way ofNumbers_. Which Cõclusions, I omit here to rehearse: aswell auoidyngsuperfluous prolixitie: as, bycause _Ioannes Picus, workes_, arecommonly had. But, in any case, I would wish that those Conclusions werered diligently, and perceiued of such, as are earnest Obseruers andConsiderers of the constant law of nũbers: which is planted in thyngsNaturall and Supernaturall: and is prescribed to all Creatures, inuiolably to be kept. For, so, besides many other thinges, in thoseConclusions to be marked, it would apeare, how sincerely, & within myboundes, I disclose the wonderfull mysteries, by numbers, to be atteynedvnto. 

Of my former wordes, easy it is to be gathered, that _Number_ hath atreble state: One, in the Creator: an other in euery Creature (inrespect of his complete constitution:) and the third, in Spirituall andAngelicall Myndes, and in the Soule of mã. In the first and third state, _Number_, is termed _Number Numbryng_. But in all Creatures, otherwise, _Number_, is termed _Nũber Numbred_. And in our Soule, Nũber bearethsuch a swaye, and hath such an affinitie therwith: that some of the old_Philosophers_ taught, _Mans Soule, to be a Number mouyng it selfe_. Andin dede, in vs, though it be a very Accident: yet such an Accident itis, that before all Creatures it had perfect beyng, in the Creator, Sempiternally. _Number Numbryng_ therfore, is the discretion discerning, and distincting of thinges. But in God the Creator, This discretion, inthe beginnyng, produced orderly and distinctly all thinges. For his_Numbryng_, then, was his Creatyng of all thinges. And his Continuall_Numbryng_, of all thinges, is the Conseruation of them in being: And, where and when he will lacke an _Vnit_: there and then, that particularthyng shalbe _Discreated_. Here I stay. But our Seuerallyng, distinctyng, and _Numbryng_, createth nothyng: but of Multitudeconsidered, maketh certaine and distinct determination. And albeit thesethynges be waighty and truthes of great importance, yet (by the infinitegoodnes of the Almighty _Ternarie_, ) Artificiall Methods and easy wayesare made, by which the zelous Philosopher, may wyn nere this Riuerish_Ida_, this Mountayne of Contemplation: and more then Contemplation. Andalso, though _Number_, be a thyng so Immateriall, so diuine, andæternall: yet by degrees, by litle and litle, stretchyng forth, andapplying some likenes of it, as first, to thinges Spirituall: and then, bryngyng it lower, to thynges sensibly perceiued: as of a momentanyesounde iterated: then to the least thynges that may be seen, numerable:And at length, (most grossely, ) to a multitude of any corporall thyngesseen, or felt: and so, of these grosse and sensible thynges, we aretrayned to learne a certaine Image or likenes of numbers: and to vseArte in them to our pleasure and proffit. So grosse is our conuersation, and dull is our apprehension: while mortall Sense, in vs, ruleth thecommon wealth of our litle world. Hereby we say, Three Lyons, are three:or a _Ternarie_. Three Egles, are three, or a _Ternarie_. 

 [☞]

Which * _Ternaries_, are eche, the _Vnion_, _knot_, and _Vniformitie_, of three discrete and distinct _Vnits_. That is, we may in eche_Ternarie_, thrise, seuerally pointe, and shew a part, _One_, _One_, and_One_. Where, in Numbryng, we say One, two, Three. But how farre, thesevisible Ones, do differre from our Indiuisible Vnits (in pure_Arithmetike_, principally considered) no man is ignorant. Yet fromthese grosse and materiall thynges, may we be led vpward, by degrees, so, informyng our rude Imagination, toward the cõceiuyng of _Numbers_, absolutely (:Not supposing, nor admixtyng any thyng created, Corporallor Spirituall, to support, conteyne, or represent those _Numbers_imagined:) that at length, we may be hable, to finde the number of ourowne name, gloriously exemplified and registred in the booke of the_Trinitie_ most blessed and æternall. 

But farder vnderstand, that vulgar Practisers, haue Numbers, otherwise, in sundry Considerations: and extend their name farder, then to Numbers, whose least part is an _Vnit_. For the common Logist, Reckenmaster, orArithmeticien, in hys vsing of Numbers: of an Vnit, imagineth lessepartes: and calleth them _Fractions_. As of an _Vnit_, he maketh anhalfe, and thus noteth it, ½. And so of other, (infinitely diuerse)partes of an _Vnit_. Yea and farder, hath, _Fractions of Fractions. &c_. And, forasmuch, as, _Addition_, _Substraction_, _Multiplication_, _Diuision_ and _Extraction of Rotes_, are the chief, and sufficientpartes of _Arithmetike_:

 [Arithmetike. ]

which is, the _Science that demonstrateth the properties, of Numbers, and all operatiõs, in numbers to be performed_:

 [Note. ]

“How often, therfore, these fiue sundry sortes of Operations, do, forthe most part, of their execution, differre from the fiue operations oflike generall property and name, in our Whole numbers practisable, Sooften, (for a more distinct doctrine) we, vulgarly account and name it, an other kynde of _Arithmetike_. ” And by this reason:

 [1. ]

the Consideration, doctrine, and working, in whole numbers onely: where, of an _Vnit_, is no lesse part to be allowed: is named (as it were) an_Arithmetike_ by it selfe. And so of the _Arithmetike of Fractions_. 

 [2. ]

In lyke sorte, the necessary, wonderfull and Secret doctrine ofProportion, and proportionalytie hath purchased vnto it selfe a peculiermaner of handlyng and workyng: and so may seme an other forme of_Arithmetike_. 

 [3. ]

Moreouer, the _Astronomers_, for spede and more commodious calculation, haue deuised a peculier maner of orderyng nũbers, about theyr circularmotions, by Sexagenes, and Sexagesmes. By Signes, Degrees and Minutes&c. Which commonly is called the _Arithmetike_ of _Astronomical_ or_Phisicall Fractions_. That, haue I briefly noted, by the name of_Arithmetike Circular_. Bycause it is also vsed in circles, not_Astronomicall. &c. _

 [4. ]

Practise hath led _Numbers_ farder, and hath framed them, to take vponthem, the shew of _Magnitudes_ propertie: Which is _Incommensurabilitie_and _Irrationalitie_. (For in pure _Arithmetike_, an _Vnit_, is thecommon Measure of all Numbers. ) And, here, Nũbers are become, as Lynes, Playnes and Solides: some tymes _Rationall_, some tymes _Irrationall_. And haue propre and peculier characters, (as ²√. ³√. And so of other. Which is to signifie _Rote Square, Rote Cubik: and so forth_:) & propreand peculier fashions in the fiue principall partes: Wherfore thepractiser, estemeth this, a diuerse _Arithmetike_ from the other. Practise bryngeth in, here, diuerse compoundyng of Numbers: as sometyme, two, three, foure (or more) _Radicall_ nũbers, diuersly knit, bysignes, of More & Lesse: as thus ²√12 + ³√15. Or thus ⁴√19 + ³√12 - ²√2. &c. And some tyme with whole numbers, or fractions of whole Number, amõgthem: as 20 + ²√24. ³√16 + 33 - ²√10. ⁴√44 + 12¼ + ³√9. And so, infinitely, may hap the varietie. After this: Both the one and the otherhath fractions incident: and so is this _Arithmetike_ greately enlarged, by diuerse exhibityng and vse of Compositions and mixtynges. Considerhow, I (beyng desirous to deliuer the student from error andCauillation) do giue to this _Practise_, the name of the _Arithmetike ofRadicall numbers_: Not, of _Irrationall_ or _Surd Numbers_: which otherwhile, are Rationall: though they haue the Signe of a Rote before them, which, _Arithmetike_ of whole Numbers most vsuall, would say they had nosuch Roote: and so account them _Surd Numbers_: which, generally spokẽ, is vntrue: as _Euclides_ tenth booke may teach you. Therfore to callthem, generally, _Radicall Numbers_, (by reason of the signe √. Prefixed, ) is a sure way: and a sufficient generall distinction from allother ordryng and vsing of Numbers: And yet (beside all this) Consider:the infinite desire of knowledge, and incredible power of mans Searchand Capacitye: how, they, ioyntly haue waded farder (by mixtyng ofspeculation and practise) and haue found out, and atteyned to the verychief perfection (almost) of _Numbers_ Practicall vse. Which thing, iswell to be perceiued in that great Arithmeticall Arte of _Æquation_:commonly called the _Rule of Coss. _ or _Algebra_. The Latines termed it, _Regulam Rei & Census_, that is, the +_Rule of the thyng and hisvalue_+. With an apt name: comprehendyng the first and last pointes ofthe worke. And the vulgar names, both in Italian, Frenche and Spanish, depend (in namyng it, ) vpon the signification of the Latin word, _Res_:+_A thing_+: vnleast they vse the name of _Algebra_. And therin(commonly) is a dubble error. The one, of them, which thinke it to be of_Geber_ his inuentyng: the other of such as call it _Algebra_. For, first, though _Geber_ for his great skill in Numbers, Geometry, Astronomy, and other maruailous Artes, mought haue semed hable to hauefirst deuised the sayd Rule: and also the name carryeth with it a verynere likenes of _Geber_ his name: yet true it is, that a _Greke_Philosopher and Mathematicien, named _Diophantus_, before _Geber_ histyme, wrote 13. Bookes therof (of which, six are yet extant: and I hadthem to *vse, [* Anno. 1550. ]

of the famous Mathematicien, and my great frende, _Petrus Montaureus_:)And secondly, the very name, is _Algiebar_, and not _Algebra_: as by theArabien _Auicen_, may be proued: who hath these precise wordes inLatine, by _Andreas Alpagus_ (most perfect in the Arabik tung) sotranslated. _Scientia faciendi Algiebar & Almachabel. I. Scientiainueniendi numerum ignotum, per additionem Numeri, & diuisionem &æquationem_. Which is to say: +_The Science of workyng Algiebar andAlmachabel_+, that is, the +_Science of findyng an vnknowen number, byAddyng of a Number, & Diuision & æquation_+. Here haue you the name: andalso the principall partes of the Rule, touched. To name it, _The rule, or Art of Æquation_, doth signifie the middle part and the State of theRule. This Rule, hath his peculier Characters:

 [5. ]

and the principal partes of _Arithmetike_, to it appertayning, dodifferre from the other _Arithmeticall operations_. This _Arithmetike, hath Nũbers_ Simple, Cõpound, Mixt: and Fractions, accordingly. ThisRule, and _Arithmetike of Algiebar_, is so profound, so generall and so(in maner) conteyneth the whole power of Numbers Application practicall:that mans witt, can deale with nothyng, more proffitable about numbers:nor match, with a thyng, more mete for the diuine force of the Soule, (in humane Studies, affaires, or exercises) to be tryed in. Perchaunceyou looked for, (long ere now, ) to haue had some particular profe, oreuident testimony of the vse, proffit and Commodity of Arithmetikevulgar, in the Common lyfe and trade of men. Therto, then, I will nowframe my selfe: But herein great care I haue, least length of sundryprofes, might make you deme, that either I did misdoute your zelousmynde to vertues schole: or els mistrust your hable witts, by some, togesse much more. A profe then, foure, fiue, or six, such, will I bryng, as any reasonable man, therwith may be persuaded, to loue & honor, yealearne and exercise the excellent Science of _Arithmetike_. 

And first: who, nerer at hand, can be a better witnesse of the frutereceiued by _Arithmetike_, then all kynde of Marchants? Though not all, alike, either nede it, or vse it. How could they forbeare the vse andhelpe of the Rule, called the Golden Rule? Simple and Compounde: bothforward and backward? How might they misse _Arithmeticall_ helpe in theRules of Felowshyp: either without tyme, or with tyme? and betwene theMarchant & his Factor? The Rules of Bartering in wares onely: or part inwares, and part in money, would they gladly want? Our Marchantventurers, and Trauaylers ouer Sea, how could they order their doyngesiustly and without losse, vnleast certaine and generall Rules forExchaũge of money, and Rechaunge, were, for their vse, deuised? The Ruleof Alligation, in how sundry cases, doth it conclude for them, suchprecise verities, as neither by naturall witt, nor other experience, they, were hable, els, to know? And (with the Marchant then to make anend) how ample & wonderfull is the Rule of False positions? especiallyas it is now, by two excellent Mathematiciens (of my familieracquayntance in their life time) enlarged? I meane _Gemma Frisius_, and_Simon Iacob_. Who can either in brief conclude, the generall andCapitall Rules? or who can Imagine the Myriades of sundry Cases, andparticular examples, in Act and earnest, continually wrought, tried andconcluded by the forenamed Rules, onely? How sundry other _Arithmeticallpractises_, are commonly in Marchantes handes, and knowledge: They themselues, can, at large, testifie. 

The Mintmaster, and Goldsmith, in their Mixture of Metals, either ofdiuerse kindes, or diuerse values: how are they, or may they, exactly bedirected, and meruailously pleasured, if _Arithmetike_ be their guide?And the honorable Phisiciãs, will gladly confesse them selues, muchbeholding to the Science of _Arithmetike_, and that sundry wayes: Butchiefly in their Art of Graduation, and compounde Medicines. And though_Galenus_, _Auerrois_, _Arnoldus_, _Lullus_, and other haue publishedtheir positions, aswell in the quantities of the Degrees aboueTemperament, as in the Rules, concluding the new _Forme_ resulting: yeta more precise, commodious, and easy _Method_, is extant: by aCountreyman of ours

 [R. B. ]

(aboue 200. Yeares ago) inuented. And forasmuch as I am vncertaine, whohath the same: or when that litle Latin treatise, (as the Author writit, ) shall come to be Printed: (Both to declare the desire I haue topleasure my Countrey, wherin I may: and also, for very good profe ofNumbers vse, in this most subtile and frutefull, PhilosophicallConclusion, ) I entend in the meane while, most briefly, and with myfarder helpe, to communicate the pith therof vnto you. 

First describe a circle: whose diameter let be an inch. Diuide theCircumference into foure equall partes. Frõ the Center, by those 4. Sections, extend 4. Right lines: eche of 4. Inches and a halfe long: orof as many as you liste, aboue 4. Without the circumference of thecircle: So that they shall be of 4. Inches long (at the least) withoutthe Circle. Make good euident markes, at euery inches end. If you list, you may subdiuide the inches againe into 10. Or 12. Smaller partes, equall. At the endes of the lines, write the names of the 4. Principallelementall Qualities. _Hote_ and _Colde_, one against the other. Andlikewise _Moyst_ and _Dry_, one against the other. And in the Circlewrite _Temperate_. Which _Temperature_ hath a good Latitude: asappeareth by the Complexion of man. And therefore we haue allowed vntoit, the foresayd Circle: and not a point Mathematicall or Physicall. 

 [* Take some part of Lullus counsayle in his booke de Q. Essentia. ]

Now, when you haue two thinges Miscible, whose degrees are * truelyknowen: Of necessitie, either they are of one Quantitie and waight, orof diuerse. If they be of one Quantitie and waight: whether theirformes, be Contrary Qualities, or of one kinde (but of diuerseintentions and degrees) or a _Temperate_, and a Contrary, _The formeresulting of their Mixture, is in the Middle betwene the degrees of theformes mixt_. As for example, let _A_, be _Moist_ in the first degree:and _B_, _Dry_ in the third degree. Adde 1. And 3. That maketh 4: thehalfe or middle of 4. Is 2. This 2. Is the middle, equally distant from_A_ and _B_

 [* Note. ]

(for the * _Temperament_ is counted none. And for it, you must put aCiphre, if at any time, it be in mixture). 

 HOTE +C | | + | | + | | +E | MOIST A TEMPERATE B DRYE +------+------+------+------+------+------+------+------+ |D | + | | + | | + | | + COLD

Counting then from _B_, 2. Degrees, toward _A_: you finde it to be _Dry_in the first degree: So is the _Forme resulting_ of the Mixture of _A_, and _B_, in our example. I will geue you an other example. Suppose, youhaue two thinges, as _C_, and _D_: and of _C_, the Heate to be in the 4. Degree: and of _D_, the Colde, to be remisse, euen vnto the_Temperament_. Now, for _C_, you take 4: and for _D_, you take a Ciphre:which, added vnto 4, yeldeth onely 4. The middle, or halfe, whereof, is2. Wherefore the _Forme resulting_ of _C_, and _D_, is Hote in thesecond degree: for, 2. Degrees, accounted from _C_, toward _D_, endeiuste in the 2. Degree of heate. Of the third maner, I will geue also anexample: which let be this:

 [Note. ]

I haue a liquid Medicine whose Qualitie of heate is in the 4. Degreeexalted: as was _C_, in the example foregoing: and an other liquidMedicine I haue: whose Qualitie, is heate, in the first degree. Of echeof these, I mixt a like quantitie: Subtract here, the lesse frõ themore: and the residue diuide into two equall partes: whereof, the onepart, either added to the lesse, or subtracted from the higher degree, doth produce the degree of the Forme resulting, by this mixture of _C_, and _E_. As, if from 4. Ye abate 1. There resteth 3. The halfe of 3. Is1½: Adde to 1. This 1½: you haue 2½. Or subtract from 4. This 1½: youhaue likewise 2½ remayning. Which declareth, the _Forme resulting_, tobe _Heate_, in the middle of the third degree. 

 [The Second Rule. ]

“But if the Quantities of two thinges Commixt, be diuerse, and theIntensions (of their Formes Miscible) be in diuerse degrees, andheigthes. (Whether those Formes be of one kinde, or of Contrary kindes, or of a Temperate and a Contrary, _What proportion is of the lessequantitie to the greater, the same shall be of the difference, which isbetwene the degree of the Forme resulting, and the degree of the greaterquantitie of the thing miscible, to the difference, which is betwene thesame degree of the Forme resulting, and the degree of the lessequantitie_. As for example. Let two pound of Liquor be geuen, hote inthe 4. Degree: & one pound of Liquor be geuen, hote in the thirddegree. ” I would gladly know the Forme resulting, in the Mixture ofthese two Liquors. Set downe your nũbers in order, thus. ___________________________ | | | | {P}. _2. _ | _Hote. 4. _ | | | | | {P}. _1. _ | _Hote. 3. _ | |____________|______________|

Now by the rule of Algiebar, haue I deuised a very easie, briefe, andgenerall maner of working in this case. Let vs first, suppose that_Middle Forme resulting_, to be 1{x}: as that Rule teacheth. And because(by our Rule, here geuen) as the waight of 1. Is to 2: So is thedifference betwene 4. (the degree of the greater quantitie) and 1{x}: tothe difference betwene 1{x} and 3: (the degree of the thing, in lessequãtitie. And with all, 1{x}, being alwayes in a certaine middell, betwene the two heigthes or degrees). For the first difference, I set4-1{x}: and for the second, I set 1{x}-3. And, now againe, I say, as 1. Is to 2. So is 4-1{x} to 1{x}-3. Wherfore, of these foure proportionallnumbers, the first and the fourth Multiplied, one by the other, do makeas much, as the second and the third Multiplied the one by the other. Let these Multiplications be made accordingly. And of the first and thefourth, we haue 1{x}-3. And of the second & the third, 8-2{x}. Wherfore, our Æquation is betwene 1{x}-3: and 8-2{x}. Which may be reduced, according to the Arte of Algiebar: as, here, adding 3. To eche part, geueth the Æquation, thus, 1{x}=11-2{x}. And yet againe, contracting, orReducing it: Adde to eche part, 2{x}: Then haue you 3{x} æquall to 11:thus represented 3{x}=11. Wherefore, diuiding 11. By 3: the Quotient is3⅔: the _Valew_ of our 1{x}, _Coss_, or _Thing_, first supposed. Andthat is the heigth, or Intension of the _Forme resulting:_ which is, _Heate_, in two thirdes of the fourth degree: And here I set the shew ofthe worke in conclusion, thus. The proufe hereof is easie: bysubtracting 3. From 3⅔, resteth ⅔. Subtracte the same heigth of theForme resulting, (which is 3⅔) frõ 4: then resteth ⅓: You see, that ⅔ isdouble to ⅓: as 2. {P}. Is double to 1. {P}. So should it be: by the rulehere geuen. Note. As you added to eche part of the Æquation, 3: so if yefirst added to eche part 2{x}, it would stand, 3{x}-3=8. And now addingto eche part 3: you haue (as afore) 3{x}=11. _________________________ | | | _ | {P}. _2. _ | _Hote. 4. _ | ⅓ _ _The forme_ | | | _ _3⅔ resulting. _ | {P}. _1. _ | _Hote. 3. _ | _ ⅔ |___________|_____________|

And though I, here, speake onely of two thyngs Miscible: and mostcommonly mo then three, foure, fiue or six, (&c. ) are to be Mixed: (andin one Compound to be reduced: & the Forme resultyng of the same, toserue the turne) yet these Rules are sufficient: duely repeated anditerated. 

 [Note. ]

In procedyng first, with any two: and then, with the Forme Resulting, and an other: & so forth: For, the last worke, concludeth the Formeresultyng of them all: I nede nothing to speake, of the Mixture (heresupposed) what it is. Common Philosophie hath defined it, saying, _Mixtio est miscibilium, alteratorum, per minima coniunctorum, Vnio_. Euery word in the definition, is of great importance. I nede not alsospend any time, to shew, how, the other manner of distributing ofdegrees, doth agree to these Rules. Neither nede I of the farder vsebelonging to the Crosse of Graduation (before described) in this placedeclare, vnto such as are capable of that, which I haue all ready sayd. Neither yet with examples specifie the Manifold varieties, by theforesayd two generall Rules, to be ordered. The witty and Studious, here, haue sufficient: And they which are not hable to atteine to this, without liuely teaching, and more in particular: would haue largerdiscoursing, then is mete in this place to be dealt withall: And other(perchaunce) with a proude snuffe will disdaine this litle: and would bevnthankefull for much more. I, therfore conclude: and wish such as hauemodest and earnest Philosophicall mindes, to laude God highly for this:and to Meruayle, that the profoundest and subtilest point, concerning_Mixture of Formes and Qualities Naturall_, is so Matcht and maryed withthe most simple, easie, and short way of the noble Rule of _Algiebar_. Who can remaine, therfore vnpersuaded, to loue, alow, and honor theexcellent Science of _Arithmetike_? For, here, you may perceiue that thelitle finger of _Arithmetike_, is of more might and contriuing, then ahunderd thousand mens wittes, of the middle sorte, are hable toperfourme, or truely to conclude, with out helpe thereof. 

Now will we farder, by the wise and valiant Capitaine, be certified, what helpe he hath, by the Rules of _Arithmetike_: in one of the Artesto him appertaining: And of the Grekes named

 [Τακτικὴ. ]

Τακτικὴ. “That is, the Skill of Ordring Souldiers in Battell ray afterthe best maner to all purposes. ” This Art so much dependeth vpponNumbers vse, and the Mathematicals, that _Ælianus_ (the best writertherof, ) in his worke, to the _Emperour Hadrianus_, by his perfection, in the Mathematicals, (beyng greater, then other before him had, )thinketh his booke to passe all other the excellent workes, written ofthat Art, vnto his dayes. For, of it, had written _Æneas_: _Cyneas_ of_Thessaly_: _Pyrrhus Epirota_: and _Alexander_ his sonne: _Clearchus_:_Pausanias_: _Euangelus_: _Polybius_, familier frende to _Scipio_:_Eupolemus_: _Iphicrates_, _Possidonius_: and very many other worthyCapitaines, Philosophers and Princes of Immortall fame and memory: Whosefayrest floure of their garland (in this feat) was _Arithmetike_: and alitle perceiuerance, in _Geometricall_ Figures. But in many other casesdoth _Arithmetike_ stand the Capitaine in great stede. As inproportionyng of vittayles, for the Army, either remaining at a stay: orsuddenly to be encreased with a certaine number of Souldiers: and for acertain tyme. Or by good Art to diminish his company, to make thevictuals, longer to serue the remanent, & for a certaine determinedtyme: if nede so require. And so in sundry his other accountes, Reckeninges, Measurynges, and proportionynges, the wise, expert, andCircumspect Capitaine will affirme the Science of _Arithmetike_, to beone of his chief Counsaylors, directers and aiders. Which thing (by goodmeanes) was euident to the Noble, the Couragious, the loyall, andCurteous

 [☞]

_Iohn_, late Earle of Warwicke. Who was a yong Gentleman, throughlyknowne to very few. Albeit his lusty valiantnes, force, and Skill inChiualrous feates and exercises: his humblenes, and frendelynes to allmen, were thinges, openly, of the world perceiued. But what rotes(otherwise, ) vertue had fastened in his brest, what Rules of godly andhonorable life he had framed to him selfe: what vices, (in some thenliuing) notable, he tooke great care to eschew: what manly vertues, inother noble men, (florishing before his eyes, ) he Sythingly aspiredafter: what prowesses he purposed and ment to achieue: with what featsand Artes, he began to furnish and fraught him selfe, for the betterseruice of his Kyng and Countrey, both in peace & warre. These (I say)his Heroicall Meditations, forecastinges and determinations, no twayne, (I thinke) beside my selfe, can so perfectly, and truely report. Andtherfore, in Conscience, I count it my part, for the honor, preferment, & procuring of vertue (thus, briefly) to haue put his Name, in theRegister of _Fame Immortall_. 

To our purpose. This _Iohn_, by one of his actes (besides many other:both in England and Fraunce, by me, in him noted. ) did disclose hisharty loue to vertuous Sciences: and his noble intent, to excell inMartiall prowesse: When he, with humble request, and instantSolliciting: got the best Rules (either in time past by Greke orRomaine, or in our time vsed: and new Stratagemes therin deuised) forordring of all Companies, summes and Numbers of mẽ, (Many, or few) withone kinde of weapon, or mo, appointed: with Artillery, or without: onhorsebacke, or on fote: to giue, or take onset: to seem many, being few:to seem few, being many. To marche in battaile or Iornay: with many suchfeates, to Foughten field, Skarmoush, or Ambushe appartaining:

 [This noble Earle, dyed Anno. 1554. Skarse of 24. Yeares of age: hauing no issue by his wife: Daughter to the Duke of Somerset. ]

And of all these, liuely designementes (most curiously) to be in velameparchement described: with Notes & peculier markes, as the Arterequireth: and all these Rules, and descriptions Arithmeticall, inclosedin a riche Case of Gold, he vsed to weare about his necke: as his Iuellmost precious, and Counsaylour most trusty. Thus, _Arithmetike_, of him, was shryned in gold: Of _Numbers_ frute, he had good hope. Now, Numberstherfore innumerable, in _Numbers_ prayse, his shryne shall finde. 

What nede I, (for farder profe to you) of the Scholemasters of Iustice, to require testimony: how nedefull, how frutefull, how skillfull a thing_Arithmetike_ is? I meane, the Lawyers of all sortes. Vndoubtedly, theCiuilians, can meruaylously declare: how, neither the Auncient Romainelawes, without good knowledge of _Numbers art_, can be perceiued: Nor(Iustice in infinite Cases) without due proportion, (narrowlyconsidered, ) is hable to be executed. How Iustly, & with great knowledgeof Arte, did _Papinianus_ institute a law of partition, and allowance, betwene man and wife after a diuorce? But how _Accursius_, _Baldus_, _Bartolus_, _Iason_, _Alexander_, and finally _Alciatus_, (beingotherwise, notably well learned) do iumble, gesse, and erre, from theæquity, art and Intent of the lawmaker: _Arithmetike_ can detect, andconuince: and clerely, make the truth to shine. Good _Bartolus_, tyredin the examining & proportioning of the matter: and with _Accursius_Glosse, much cumbred: burst out, and sayd: _Nulla est in toto libro, hacglossa difficilior: Cuius computationem nec Scholastici nec Doctoresintelligunt. &c. _ That is: +_In the whole booke, there is no Glosseharder then this: Whose accoumpt or reckenyng, neither the Scholers, northe Doctours vnderstand. &c. _+ What can they say of _Iulianus_ law, _Siita Scriptum. &c. _ Of the Testators will iustly performing, betwene thewife, Sonne and daughter? How can they perceiue the æquitie of_Aphricanus_, _Arithmeticall_ Reckening, where he treateth of _LexFalcidia_? How can they deliuer him, from his Reprouers: and theirmaintainers: as _Ioannes_, _Accursius Hypolitus_ and _Alciatus_? HowIustly and artificially, was _Africanus_ reckening made? Proportionatingto the Sommes bequeathed, the Contributions of eche part? Namely, forthe hundred presently receiued, 17-1/7. And for the hundred, receiuedafter ten monethes, 12-6/7: which make the 30: which were to becõtributed by the legataries to the heire. For, what proportion, 100hath to 75: the same hath 17-1/7 to 12-6/7: Which is Sesquitertia: thatis, as 4, to 3. Which make 7. Wonderfull many places, in the Ciuile law, require an expert _Arithmeticien_, to vnderstand the deepe Iudgemẽt, &Iust determinatiõ of the Auncient Romaine Lawmakers. But much moreexpert ought he to be, who should be hable, to decide with æquitie, theinfinite varietie of Cases, which do, or may happen, vnder euery one ofthose lawes and ordinances Ciuile. Hereby, easely, ye may nowconiecture: that in the Canon law: and in the lawes of the Realme (whichwith vs, beare the chief Authoritie), Iustice and equity might begreately preferred, and skilfully executed, through due skill ofArithmetike, and proportions appertainyng. The worthy Philosophers, andprudent lawmakers (who haue written many bookes _De Republica:_ How thebest state of Common wealthes might be procured and mainteined, ) hauevery well determined of Iustice: (which, not onely, is the Base andfoundacion of Common weales: but also the totall perfection of all ourworkes, words, and thoughtes:) defining it, [Iustice. ]

“to be that vertue, by which, to euery one, is rendred, that to himappertaineth. ” God challengeth this at our handes, to be honored as God:to be loued, as a father: to be feared as a Lord & master. Ourneighbours proportiõ, is also prescribed of the Almighty lawmaker: whichis, to do to other, euen as we would be done vnto. These proportions, are in Iustice necessary: in duety, commendable: and of Common wealthes, the life, strength, stay and florishing. _Aristotle_ in his _Ethikes_(to fatch the sede of Iustice, and light of direction, to vse andexecute the same) was fayne to fly to the perfection, and power ofNumbers: for proportions Arithmeticall and Geometricall. _Plato_ in hisbooke called _Epinomis_ (which boke, is the Threasury of all hisdoctrine) where, his purpose is, to seke a Science, which, when a manhad it, perfectly: he might seme, and so be, in dede, _Wise_. He, briefly, of other Sciences discoursing, findeth them, not hable to bringit to passe: But of the Science of Numbers, he sayth. _Illa, quæ numerummortalium generi dedit, id profecto efficiet. Deum autem aliquem, magisquam fortunam, ad salutem nostram, hoc munus nobis arbitror contulisse. &c. Nam ipsum bonorum omnium Authorem, cur non maximi boni, Prudentiædico, causam arbitramur? +That Science, verely, which hath taughtmankynde number, shall be able to bryng it to passe. And, I thinke, a certaine God, rather then fortune, to haue giuen vs this gift, for ourblisse. For, why should we not Iudge him, who is the Author of all goodthings, to be also the cause of the greatest good thyng, namely, Wisedome?+_ There, at length, he proueth _Wisedome_ to be atteyned, bygood Skill of _Numbers_. With which great Testimony, and the manifoldprofes, and reasons, before expressed, you may be sufficiently and fullypersuaded: of the perfect Science of _Arithmetike_, to make thisaccounte: That

 [☞]

of all Sciences, next to _Theologie_, it is most diuine, most pure, mostample and generall, most profounde, most subtile, most commodious andmost necessary. Whose next Sister, is the Absolute Science of_Magnitudes_: of which (by the Direction and aide of him, whose_Magnitude_ is Infinite, and of vs Incomprehensible) I now entend, so towrite, that both with the _Multitude_, and also with the _Magnitude_ ofMeruaylous and frutefull verities, you (my frendes and Countreymen) maybe stird vp, and awaked, to behold what certaine Artes and Sciences, (toour vnspeakable behofe) our heauenly father, hath for vs prepared, andreuealed, by sundry _Philosophers_ and _Mathematiciens_. 

Both, _Number_ and _Magnitude_, haue a certaine Originall sede, (as itwere) of an incredible property: and of man, neuer hable, Fully, to bedeclared. Of _Number_, an Vnit, and of _Magnitude_, a Poynte, doo seemeto be much like Originall causes: But the diuersitie neuerthelesse, isgreat. We defined an _Vnit_, to be a thing Mathematicall Indiuisible:A Point, likewise, we sayd to be a Mathematicall thing Indiuisible. Andfarder, that a Point may haue a certaine determined Situation: that is, that we may assigne, and prescribe a Point, to be here, there, yonder. &c. Herein, (behold) our Vnit is free, and can abyde no bondage, or tobe tyed to any place, or seat: diuisible or indiuisible. Agayne, byreason, a Point may haue a Situation limited to him: a certaine motion, therfore (to a place, and from a place) is to a Point incident andappertainyng. But an _Vnit_, can not be imagined to haue any motion. A Point, by his motion, produceth, Mathematically, a line: (as we saydbefore) which is the first kinde of Magnitudes, and most simple: An_Vnit_, can not produce any number. A Line, though it be produced of aPoint moued, yet, it doth not consist of pointes: Number, though it benot produced of an _Vnit_, yet doth it Consist of vnits, as a materiallcause. But formally, [Number. ]

Number, is the Vnion, and Vnitie of Vnits. Which vnyting and knitting, is the workemanship of our minde: which, of distinct and discrete Vnits, maketh a Number: by vniformitie, resulting of a certaine multitude ofVnits. And so, euery number, may haue his least part, giuen: namely, anVnit: But not of a Magnitude, (no, not of a Lyne, ) the least part can begiuẽ: by cause, infinitly, diuision therof, may be conceiued. AllMagnitude, is either a Line, a Plaine, or a Solid. Which Line, Plaine, or Solid, of no Sense, can be perceiued, nor exactly by hãd (any way)represented: nor of Nature produced: But, as (by degrees) Number didcome to our perceiuerance: So, by visible formes, we are holpen toimagine, what our Line Mathematicall, is. What our Point, is. Soprecise, are our Magnitudes, that one Line is no broader then an other:for they haue no bredth: Nor our Plaines haue any thicknes. Nor yet ourBodies, any weight: be they neuer so large of dimensiõ. Our Bodyes, wecan haue Smaller, then either Arte or Nature can produce any: andGreater also, then all the world can comprehend. Our least Magnitudes, can be diuided into so many partes, as the greatest. As, a Line of aninch long, (with vs) may be diuided into as many partes, as may thediameter of the whole world, from East to West: or any way extended:What priuiledges, aboue all manual Arte, and Natures might, haue our twoSciences Mathematicall? to exhibite, and to deale with thinges of suchpower, liberty, simplicity, puritie, and perfection? And in them, socertainly, so orderly, so precisely to procede: as, excellent is thatworkemã Mechanicall Iudged, who nerest can approche to the representingof workes, Mathematically demonstrated?

 [☞]

And our two Sciences, remaining pure, and absolute, in their propertermes, and in their owne Matter: to haue, and allowe, onely suchDemonstrations, as are plaine, certaine, vniuersall, and of an æternallveritye?

 [Geometrie. ]

This Science of _Magnitude_, his properties, conditions, andappertenances: commonly, now is, and from the beginnyng, hath of allPhilosophers, ben called _Geometrie_. But, veryly, with a name to baseand scant, for a Science of such dignitie and amplenes. And, perchaunce, that name, by cõmon and secret consent, of all wisemen, hitherto hathben suffred to remayne: that it might carry with it a perpetuallmemorye, of the first and notablest benefite, by that Science, to commonpeople shewed: Which was, when Boundes and meres of land and ground werelost, and confounded (as in _Egypt_, yearely, with the ouerflowyng of_Nilus_, the greatest and longest riuer in the world) or, that groundbequeathed, were to be assigned: or, ground sold, were to be layd out:or (when disorder preuailed) that Commõs were distributed intoseueralties. For, where, vpon these & such like occasiõs, Some byignorãce, some by negligẽce, Some by fraude, and some by violence, didwrongfully limite, measure, encroach, or challenge (by pretence of iustcontent, and measure) those landes and groundes: great losse, disquietnes, murder, and warre did (full oft) ensue: Till, by Godsmercy, and mans Industrie, The perfect Science of Lines, Plaines, andSolides (like a diuine Iusticier, ) gaue vnto euery man, his owne. Thepeople then, by this art pleasured, and greatly relieued, in theirlandes iust measuring: & other Philosophers, writing Rules for landmeasuring: betwene them both, thus, confirmed the name of _Geometria_, that is, (according to the very etimologie of the word) Land measuring. Wherin, the people knew no farder, of Magnitudes vse, but in Plaines:and the Philosophers, of thẽ, had no feet hearers, or Scholers: farderto disclose vnto, then of flat, plaine _Geometrie_. And though, thesePhilosophers, knew of farder vse, and best vnderstode the etymologye ofthe worde, yet this name _Geometria_, was of them applyed generally toall sortes of Magnitudes: vnleast, otherwhile, of _Plato_, and_Pythagoras_: When they would precisely declare their owne doctrine. Then, was

 [* Plato. 7. De Rep. ]

* _Geometria_, with them, _Studium quod circa planum versatur_. But, well you may perceiue by _Euclides Elementes_, that more ample is ourScience, then to measure Plaines: and nothyng lesse therin is tought (ofpurpose) then how to measure Land. An other name, therfore, must nedesbe had, for our Mathematicall Science of Magnitudes: which regardethneither clod, nor turff: neither hill, nor dale: neither earth norheauen: but is absolute _Megethologia_: not creping on ground, anddasseling the eye, with pole perche, rod or lyne: but “liftyng the hartaboue the heauens, by inuisible lines, and

 [☞]

immortall beames meteth with the reflexions, of the lightincomprehensible: and so procureth Ioye, and perfection vnspeakable. ” Ofwhich true vse of our _Megethica_, or _Megethologia_, _Diuine Plato_seemed to haue good taste, and iudgement: and (by the name of_Geometrie_) so noted it: and warned his Scholers therof: as, in hysseuenth _Dialog_, of the Common wealth, may euidently be sene. Where (inLatin) thus it is: right well translated: _Profecto, nobis hoc nonnegabunt, Quicun[que] vel paululum quid Geometriæ gustârunt, quin hæcScientia, contrà, omnino se habeat, quàm de ea loquuntur, qui in ipsaversantur. _ In English, thus. +_Verely_+ (sayth _Plato_) +_whosoeuerhaue, (but euen very litle) tasted of Geometrie, will not denye vnto vs, this: but that this Science, is of an other condicion, quite contrary tothat, which they that are exercised in it, do speake of it. _+ And thereit followeth, of our _Geometrie_, _Quòd quæritur cognoscendi illiusgratia, quod semper est, non & eius quod oritur quando[que] & interit. Geometria, eius quod est semper, Cognitio est. Attollet igitur(ô Generose vir) ad Veritatem, animum: at[que] ita, ad Philosophandumpreparabit cogitationem, vt ad supera conuertamus: quæ, nunc, contraquàm decet, ad inferiora deijcimus. &c. Quàm maximè igitur præcipiendumest, vt qui præclarissimam hanc habitãt Civitatem, nullo modo, Geometriam spernant. Nam & quæ præter ipsius propositum, quodam modoesse videntur, haud exigua sunt. &c. _ It must nedes be confessed (saith_Plato_) +_That =[Geometrie]= is learned, for the knowyng of that, whichis euer: and not of that, which, in tyme, both is bred and is brought toan ende. &c. Geometrie is the knowledge of that which is euerlastyng. Itwill lift vp therfore (O Gentle Syr) our mynde to the Veritie: and bythat meanes, it will prepare the Thought, to the Philosophicall loue ofwisdome: that we may turne or conuert, toward heauenly thinges =[bothmynde and thought]= which now, otherwise then becommeth vs, we cast downon base or inferior things. &c. Chiefly, therfore, Commaundement must begiuen, that such as do inhabit this most honorable Citie, by no meanes, despise Geometrie. For euen those thinges =[done by it]= which, inmanner, seame to be, beside the purpose of Geometrie: are of no smallimportance. &c. _+ And besides the manifold vses of _Geometrie_, inmatters appertainyng to warre, he addeth more, of second vnpurposedfrute, and commoditye, arrising by _Geometrie_: saying: _Scimus quinetiam, ad Disciplinas omnes facilius per discendas, interesse omnino, attigerit ne Geometriam aliquis, an non. &c. Hanc ergo Doctrinam, secundo loco discendam Iuuenibus statuamus. _ That is. +_But, also, weknow, that for the more easy learnyng of all Artes, it importeth much, whether one haue any knowledge in Geometrie, or no. &c. Let vs therforemake an ordinance or decree, that this Science, of young men shall belearned in the second place. _+ This was _Diuine Plato_ his Iudgement, both of the purposed, chief, and perfect vse of _Geometrie_: and of hissecond, dependyng, deriuatiue commodities. And for vs, Christen men, a thousand thousand mo occasions are, to haue nede of the helpe of *

 [I. D. * Herein, I would gladly shake of, the earthly name, of Geometrie. ]

_Megethologicall_ Contemplations: wherby, to trayne our Imaginations andMyndes, by litle and litle, to forsake and abandon, the grosse andcorruptible Obiectes, of our vtward senses: and to apprehend, by suredoctrine demonstratiue, Things Mathematicall. And by them, readily to beholpen and conducted to conceiue, discourse, and conclude of thingsIntellectual, Spirituall, æternall, and such as concerne our Blisseeuerlasting: which, otherwise (without Speciall priuiledge ofIllumination, or Reuelation frõ heauen) No mortall mans wyt (naturally)is hable to reach vnto, or to Compasse. And, veryly, by my small Talent(from aboue) I am hable to proue and testifie, that the litterall Text, and order of our diuine Law, Oracles, and Mysteries, require more skillin Numbers, and Magnitudes: then (commonly) the expositors haue vttered:but rather onely (at the most) so warned: & shewed their own wanttherin. (To name any, is nedeles: and to note the places, is, here, noplace: But if I be duely asked, my answere is ready. ) And without thelitterall, Grammaticall, Mathematicall or Naturall verities of suchplaces, by good and certaine Arte, perceiued, no Spirituall sense(propre to those places, by Absolute _Theologie_) will thereon depend. 

 [☞]

“No man, therfore, can doute, but toward the atteyning of knowledgeincomparable, and Heauenly Wisedome: Mathematicall Speculations, both ofNumbers and Magnitudes: are meanes, aydes, and guides: ready, certaine, and necessary. ” From henceforth, in this my Preface, will I frame mytalke, to _Plato_ his fugitiue Scholers: or, rather, to such, who wellcan, (and also wil, ) vse their vtward senses, to the glory of God, thebenefite of their Countrey, and their owne secret contentation, orhonest preferment, on this earthly Scaffold. To them, I will orderlyrecite, describe & declare a great Number of Artes, from our twoMathematicall fountaines, deriued into the fieldes of _Nature_. Wherby, such Sedes, and Rotes, as lye depe hyd in the groũd of _Nature_, arerefreshed, quickened, and prouoked to grow, shote vp, floure, and giuefrute, infinite, and incredible. And these Artes, shalbe such, as vponMagnitudes properties do depende, more, then vpon Number. And by goodreason we may call them Artes, and Artes Mathematicall Deriuatiue: for(at this tyme) I Define

 [An Arte. ]

+An Arte, to be a Methodicall cõplete Doctrine, hauing abundancy ofsufficient, and peculier matter to deale with, by the allowance of theMetaphisicall Philosopher: the knowledge whereof, to humaine state isnecessarye. + And that I account, [Art Mathematicall Deriuatiue. ]

+An Art Mathematicall deriuatiue, which by Mathematicall demonstratiueMethod, in Nũbers, or Magnitudes, ordreth and confirmeth his doctrine, as much & as perfectly, as the matter subiect will admit. + And for that, I entend to vse the name and propertie of a

 [A Mechanitien. ]

_Mechanicien_, otherwise, then (hitherto) it hath ben vsed, I thinke itgood, (for distinction sake) to giue you also a brief description, whatI meane therby. +A Mechanicien, or a Mechanicall workman is he, whoseskill is, without knowledge of Mathematicall demonstration, perfectly toworke and finishe any sensible worke, by the Mathematicien principall orderiuatiue, demonstrated or demonstrable. + Full well I know, that hewhich inuenteth, or maketh these demonstrations, is generally called _Aspeculatiue Mechanicien_: which differreth nothyng from a _MechanicallMathematicien_. So, in respect of diuerse actions, one man may haue thename of sundry artes: as, some tyme, of a Logicien, some tymes (in thesame matter otherwise handled) of a Rethoricien. Of these trifles, I make, (as now, in respect of my Preface, ) small account: to fyle thẽfor the fine handlyng of subtile curious disputers. In other places, they may commaunde me, to giue good reason: and yet, here, I will not bevnreasonable. 

 [+1. +]

First, then, from the puritie, absolutenes, and Immaterialitie ofPrincipall _Geometrie_, is that kinde of _Geometrie_ deriued, whichvulgarly is counted _Geometrie_: and is the +Arte of Measuring sensiblemagnitudes, their iust quãtities and contentes. +

 [Geometrie vulgar. ]

This, teacheth to measure, either at hand: and the practiser, to be bythe thing Measured: and so, by due applying of Cumpase, Rule, Squire, Yarde, Ell, Perch, Pole, Line, Gaging rod, (or such like instrument) tothe Length, Plaine, or Solide measured, [1. ]

* to be certified, either of the length, perimetry, or distance lineall:and this is called, _Mecometrie_. Or

 [2. ]

* to be certified of the content of any plaine Superficies: whether itbe in ground Surueyed, Borde, or Glasse measured, or such like thing:which measuring, is named _Embadometrie_. 

 [3. ]

* Or els to vnderstand the Soliditie, and content of any bodily thing:as of Tymber and Stone, or the content of Pits, Pondes, Wells, Vessels, small & great, of all fashions. Where, of Wine, Oyle, Beere, or Alevessells, &c, the Measuring, commonly, hath a peculier name: and iscalled _Gaging_. And the generall name of these Solide measures, is_Stereometrie_. 

 [+2. +]

Or els, this _vulgar Geometrie_, hath consideration to teach thepractiser, how to measure things, with good distance betwene him and thething measured: and to vnderstand thereby, either

 [1. ]

* how Farre, a thing seene (on land or water) is from the measurer: andthis may be called _Apomecometrie_:

 [2. ]

Or, how High or depe, aboue or vnder the leuel of the measurers stãding, any thing is, which is sene on land or water, called _Hypsometrie_. 

 [3. ]

* Or, it informeth the measurer, how Broad any thing is, which is in themeasurers vew: so it be on Land or Water, situated: and may be called_Platometrie_. Though I vse here to condition, the thing measured, to beon Land, or Water Situated:

 [Note. ]

yet, know for certaine, that the sundry heigthe of Cloudes, blasingStarres, and of the Mone, may (by these meanes) haue their distancesfrom the earth: and, of the blasing Starres and Mone, the Soliditie(aswell as distances) to be measured: But because, neither these thingsare vulgarly taught: nor of a common practiser so ready to be executed:I, rather, let such measures be reckened incident to some of our otherArtes, dealing with thinges on high, more purposely, then this vulgarLand measuring Geometrie doth: as in _Perspectiue_ and _Astronomie, &c. _

Of these Feates (farther applied) is Sprong the Feate of _Geodesie_, orLand Measuring: more cunningly to measure & Suruey Land, Woods, andWaters, a farre of. More cunningly, I say: But God knoweth (hitherto) inthese Realmes of England and Ireland (whether through ignorance orfraude, I can not tell, in euery particular)

 [Note. ]

how great wrong and iniurie hath (in my time) bene committed by vntruemeasuring and surueying of Land or Woods, any way. And, this I am sure:that the Value of the difference, betwene the truth and such Surueyes, would haue bene hable to haue foũd (for euer) in eche of our twoVniuersities, an excellent Mathematicall Reader: to eche, allowing(yearly) a hundred Markes of lawfull money of this realme: which, indede, would seme requisit, here, to be had (though by other wayesprouided for) as well, as, the famous Vniuersitie of Paris, hath twoMathematicall Readers: and eche, two hundreth French Crownes yearly, ofthe French Kinges magnificent liberalitie onely. Now, againe, to ourpurpose returning: Moreouer, of the former knowledge Geometricall, aregrowen the Skills of _Geographie_, _Chorographie_, _Hydrographie_, and_Stratarithmetrie_. 

“+‡Geographie‡+ teacheth wayes, by which, in sũdry formes, (as_Sphærike_, _Plaine_ or other), the Situation of Cities, Townes, Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, &such other things, vpõ the outface of the earthly Globe (either in thewhole, or in some principall mẽber and portion therof cõtayned) may bedescribed and designed, in cõmensurations Analogicall to Nature andveritie: and most aptly to our vew, may be represented. ” Of this Artehow great pleasure, and how manifolde commodities do come vnto vs, dailyand hourely: of most men, is perceaued. While, some, to beautifie theirHalls, Parlers, Chambers, Galeries, Studies, or Libraries with: othersome, for thinges past, as battels fought, earthquakes, heauenlyfyringes, & such occurentes, in histories mentioned: therby liuely, asit were, to vewe the place, the region adioyning, the distance from vs:and such other circumstances. Some other, presently to vewe the largedominion of the Turke: the wide Empire of the Moschouite: and the litlemorsell of ground, where Christendome (by profession) is certainlyknowen. Litle, I say, in respecte of the rest. &c. Some, either fortheir owne iorneyes directing into farre landes: or to vnderstand ofother mens trauailes. To conclude, some, for one purpose: and some, foran other, liketh, loueth, getteth, and vseth, Mappes, Chartes, &Geographicall Globes. Of whose vse, to speake sufficiently, wouldrequire a booke peculier. 

+‡Chorographie‡+ seemeth to be an vnderling, and a twig, of_Geographie_: and yet neuerthelesse, is in practise manifolde, and invse very ample. “This teacheth Analogically to describe a small portionor circuite of ground, with the contentes: not regarding whatcommensuration it hath to the whole, or any parcell, without it, contained. But in the territory or parcell of ground which it taketh inhand to make description of, it leaueth out (or vndescribed) no notable, or odde thing, aboue the ground visible. Yea and sometimes, of thingesvnder ground, geueth some peculier marke: or warning: as of Mettallmines, Cole pittes, Stone quarries. &c. ” Thus, a Dukedome, a Shiere, a Lordship, or lesse, may be described distinctly. But marueilouspleasant, and profitable it is, in the exhibiting to our eye, andcommensuration, the plat of a Citie, Towne, Forte, or Pallace, in trueSymmetry: not approching to any of them: and out of Gunne shot. &c. Hereby, the _Architect_ may furnishe him selfe, with store of whatpatterns he liketh: to his great instruction: euen in those thingeswhich outwardly are proportioned: either simply in them selues: orrespectiuely, to Hilles, Riuers, Hauens, and Woods adioyning. Some also, terme this particular description of places, _Topographie_. 

“+‡Hydrographie‡+, deliuereth to our knowledge, on Globe or in Plaine, the perfect Analogicall description of the Ocean Sea coastes, throughthe whole world: or in the chiefe and principall partes thereof:” withthe Iles and chiefe particular places of daungers, conteyned within theboundes, and Sea coastes described: as, of Quicksandes, Bankes, Pittes, Rockes, Races, Countertides, Whorlepooles. &c. This, dealeth with theElement of the water chiefly: as _Geographie_ did principally take theElement of the Earthes description (with his appertenances) to taske. And besides thys, _Hydrographie_, requireth a particular Register ofcertaine Landmarkes (where markes may be had) from the sea, well hableto be skried, in what point of the Seacumpase they appeare, and whatapparent forme, Situation, and bignes they haue, in respecte of anydaungerous place in the sea, or nere vnto it, assigned: And in allCoastes, what Mone, maketh full Sea: and what way, the Tides and Ebbes, come and go, the _Hydrographer_ ought to recorde. The Soundingeslikewise: and the Chanels wayes: their number, and depthes ordinarily, at ebbe and flud, ought the _Hydrographer_, by obseruation and diligenceof _Measuring_, to haue certainly knowen. And many other pointes, arebelonging to perfecte _Hydrographie_, and for to make a _Rutter_, by: ofwhich, I nede not here speake: as of the describing, in any place, vponGlobe or Plaine, the 32. Pointes of the Compase, truely: (wherof, scarsly foure, in England, haue right knowledge: bycause, the linestherof, are no straight lines, nor Circles. ) Of making due proiection ofa Sphere in plaine. Of the Variacion of the Compas, from true Northe:And such like matters (of great importance, all) I leaue to speake of, in this place: bycause, I may seame (al ready) to haue enlarged theboundes, and duety of an Hydographer, much more, then any man (to thisday) hath noted, or prescribed. Yet am I well hable to proue, all thesethinges, to appertaine, and also to be proper to the Hydrographer. Thechief vse and ende of this Art, is the Art of Nauigation: but it hathother diuerse vses: euen by them to be enioyed, that neuer lacke sightof land. 

+‡Stratarithmetrie‡+, is the Skill, (appertainyng to the warre, ) bywhich a man can set in figure, analogicall to any _Geometricall_ figureappointed, any certaine number or summe of men: of such a figurecapable: (by reason of the vsuall spaces betwene Souldiers allowed: andfor that, of men, can be made no Fractions. Yet, neuertheles, he canorder the giuen summe of men, for the greatest such figure, that ofthem, cã be ordred) and certifie, of the ouerplus: (if any be) and ofthe next certaine summe, which, with the ouerplus, will admit a figureexactly proportionall to the figure assigned. By which Skill, also, ofany army or company of men: (the figure & sides of whose orderlystanding, or array, is knowen) he is able to expresse the iust number ofmen, within that figure conteined: or (orderly) able to be conteined. 

 [* Note. ]

* And this figure, and sides therof, he is hable to know: either beyngby, and at hand: or a farre of. Thus farre, stretcheth the descriptionand property of _Stratarithmetrie_: sufficient for this tyme and place. 

 [The difference betwene Stratarithmetrie and Tacticie. ]

“It differreth from the Feate _Tacticall_, _De aciebus instruendis. _bycause, there, is necessary the wisedome and foresight, to what purposehe so ordreth the men: and Skillfull hability, also, for any occasion, or purpose, to deuise and vse the aptest and most necessary order, arrayand figure of his Company and Summe of men. ” By figure, I meane: as, either of a _Perfect Square_, _Triangle_, _Circle_, _Ouale_, _longsquare_, (of the Grekes it is called _Eteromekes_) _Rhombe_, _Rhomboïd_, _Lunular_, _Ryng_, _Serpentine_, and such other Geometricall figures:Which, in warres, haue ben, and are to be vsed: for commodiousnes, necessity, and auauntage &c. And no small skill ought he to haue, thatshould make true report, or nere the truth, of the numbers and Summes, of footemen or horsemen, in the Enemyes ordring. A farre of, to make anestimate, betwene nere termes of More and Lesse, is not a thyng veryrife, among those that gladly would do it. 

 [I. D. Frende, you will finde it hard, to performe my description of this Feate. But by Chorographie, you may helpe your selfe some what: where the Figures knowne (in Sides and Angles) are not Regular: And where, Resolution into Triangles can serue. &c. And yet you will finde it strange to deale thus generally with Arithmeticall figures: and, that for Battayle ray. Their contentes, differ so much from like Geometricall Figures. ]

Great pollicy may be vsed of the Capitaines, (at tymes fete, and inplaces conuenient) as to vse Figures, which make greatest shew, of somany as he hath: and vsing the aduauntage of the three kindes of vsuallspaces: (betwene footemen or horsemen) to take the largest: or when hewould seme to haue few, (beyng many:) contrarywise, in Figure, andspace. The Herald, Purseuant, Sergeant Royall, Capitaine, or who soeueris carefull to come nere the truth herein, besides the Iudgement of hisexpert eye, his skill of Ordering _Tacticall_, the helpe of hisGeometricall instrument: Ring, or Staffe Astronomicall: (commodiouslyframed for cariage and vse) He may wonderfully helpe him selfe, byperspectiue Glasses. In which, (I trust) our posterity will proue moreskillfull and expert, and to greater purposes, then in these dayes, can(almost) be credited to be possible. 

Thus haue I lightly passed ouer the Artificiall Feates, chieflydependyng vpon vulgar _Geometrie_: & commonly and generally reckenedvnder the name of _Geometrie_. But there are other (very many)_Methodicall Artes_, which, declyning from the purity, simplicitie, andImmateriality, of our Principall Science of _Magnitudes_: do yetneuertheles vse the great ayde, direction, and Method of the saydprincipall Science, and haue propre names, and distinct: both from theScience of _Geometrie_, (from which they are deriued) and one from theother. As +Perspectiue, Astronomie, Musike, Cosmographie, Astrologie, Statike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie, Menadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Architecture, Nauigation, Thaumaturgike+ and +Archemastrie+. I thinke it necessary, orderly, of these to giue some peculier descriptions: and withall, totouch some of their commodious vses, and so to make this Preface, to bea little swete, pleasant Nosegaye for you: to comfort your Spirites, beyng almost out of courage, and in despayre, (through brutish brute)Weenyng that _Geometrie_, had but serued for buildyng of an house, or acurious bridge, or the roufe of Westminster hall, or some witty prettydeuise, or engyn, appropriate to a Carpenter, or a Ioyner &c. That thething is farre otherwise, then the world, (commonly) to this day, hathdemed, by worde and worke, good profe wilbe made. 

Among these Artes, by good reason, +‡Perspectiue‡+ ought to be had, ereof _Astronomicall Apparences_, perfect knowledge can be atteyned. Andbycause of the prerogatiue of _Light_, beyng the first of _GodsCreatures_: and the eye, the light of our body, and his Sense mostmighty, and his organ most Artificiall and _Geometricall_: At_Perspectiue_, we will begyn therfore. +Perspectiue, is an ArtMathematicall, which demonstrateth the maner, and properties, of allRadiations Direct, Broken, and Reflected. + This Description, orNotation, is brief: but it reacheth so farre, as the world is wyde. Itconcerneth all Creatures, all Actions, and passions, by Emanation ofbeames perfourmed. Beames, or naturall lines, (here) I meane, not oflight onely, or of colour (though they, to eye, giue shew, witnes, andprofe, wherby to ground the Arte vpon) but also of other _Formes_, both_Substantiall_, and _Accidentall_, the certaine and determined actiueRadiall emanations. By this Art (omitting to speake of the highestpointes) we may vse our eyes, and the light, with greater pleasure: andperfecter Iudgement: both of things, in light seen, & of other: which bylike order of Lightes Radiations, worke and produce their effectes. Wemay be ashamed to be ignorant of the cause, why so sundry wayes our eyeis deceiued, and abused: as, while the eye weeneth a roũd Globe orSphere (beyng farre of) to be a flat and plaine Circle, and so likewiseiudgeth a plaine Square, to be roũd: supposeth walles parallels, toapproche, a farre of: rofe and floure parallels, the one to benddownward, the other to rise vpward, at a little distance from you. Againe, of thinges being in like swiftnes of mouing, to thinke thenerer, to moue faster: and the farder, much slower. Nay, of two thinges, wherof the one (incomparably) doth moue swifter then the other, to demethe slower to moue very swift, & the other to stand: what an error isthis, of our eye? Of the Raynbow, both of his Colours, of the order ofthe colours, of the bignes of it, the place and heith of it, (&c) toknow the causes demonstratiue, is it not pleasant, is it not necessary?of two or three Sonnes appearing: of Blasing Sterres: and such likethinges: by naturall causes, brought to passe, (and yet neuertheles, offarder matter, Significatiue) is it not commodious for man to know thevery true cause, & occasion Naturall? Yea, rather, is it not, greatly, against the Souerainty of Mans nature, to be so ouershot and abused, with thinges (at hand) before his eyes? as with a Pecockes tayle, and aDoues necke: or a whole ore, in water, holden, to seme broken. Thynges, farre of, to seeme nere: and nere, to seme farre of. Small thinges, toseme great: and great, to seme small. One man, to seme an Army. Or a manto be curstly affrayed of his owne shaddow. Yea, so much, to feare, that, if you, being (alone) nere a certaine glasse, and proffer, withdagger or sword, to foyne at the glasse, you shall suddenly be moued togiue backe (in maner) by reason of an Image, [☞ A marueilous Glasse. ]

appearing in the ayre, betwene you & the glasse, with like hand, swordor dagger, & with like quicknes, foyning at your very eye, likewise asyou do at the Glasse. Straunge, this is, to heare of: but moremeruailous to behold, then these my wordes can signifie. Andneuerthelesse by demonstration Opticall, the order and cause therof, iscertified: euen so, as the effect is consequent. Yea, thus much more, dare I take vpon me, toward the satisfying of the noble courrage, thatlongeth ardently for the wisedome of Causes Naturall: as to let himvnderstand, that, in London, he may with his owne eyes, haue profe ofthat, which I haue sayd herein. A Gentleman, (which, for his goodseruice, done to his Countrey, is famous and honorable:

 [S. W. P. ]

and for skill in the Mathematicall Sciences, and Languages, is the Odman of this land. &c. ) euen he, is hable: and (I am sure) will, verywillingly, let the Glasse, and profe be sene: and so I (here) requesthim: for the encrease of wisedome, in the honorable: and for thestopping of the mouthes malicious: and repressing the arrogancy of theignorant. Ye may easily gesse, what I meane. This Art of _Perspectiue_, is of that excellency, and may be led, to the certifying, and executingof such thinges, as no man would easily beleue: without Actuall profeperceiued. I speake nothing of _Naturall Philosophie_, which, without_Perspectiue_, can not be fully vnderstanded, nor perfectly atteinedvnto. Nor, of _Astronomie_: which, without _Perspectiue_, can not wellbe grounded: Nor _Astrologie_, naturally Verified, and auouched. Thatpart hereof, which dealeth with Glasses (which name, Glasse, is agenerall name, in this Arte, for any thing, from which, a Beamereboundeth) is called _Catoptrike_: and hath so many vses, bothmerueilous, and proffitable: that, both, it would hold me to long, tonote therin the principall conclusions, all ready knowne: And also(perchaunce) some thinges, might lacke due credite with you: And I, therby, to leese my labor: and you, to slip into light Iudgement, [* ☞]

Before you haue learned sufficiently the powre of Nature and Arte. 

Now, to procede: +‡Astronomie‡, is an Arte Mathematicall, whichdemonstrateth the distance, magnitudes, and all naturall motions, apparences, and passions propre to the Planets and fixed Sterres: forany time past, present and to come: in respect of a certaine Horizon, orwithout respect of any Horizon. + By this Arte we are certified of thedistance of the Starry Skye, and of eche _Planete_ from the Centre ofthe Earth: and of the greatnes of any Fixed starre sene, or _Planete_, in respect of the Earthes greatnes. As, we are sure (by this Arte) thatthe Solidity, Massines and Body of the _Sonne_, conteineth the quantitieof the whole Earth and Sea, a hundred thre score and two times, lesse by⅛ one eight parte of the earth. But the Body of the whole earthly globeand Sea, is bigger then the body of the Mone, three and forty timeslesse by ⅛ of the Mone. Wherfore the _Sonne_ is bigger then the _Mone_, 7000 times, lesse, by 59 39/64 that is, precisely 6940 25/64 bigger thenthe _Mone_. And yet the vnskillfull man, would iudge them a like bigge. Wherfore, of Necessity, the one is much farder from vs, then the other. The _Sonne_, when he is fardest from the earth (which, now, in our age, is, when he is in the 8. Degree, of Cancer) is, 1179 Semidiameters ofthe Earth, distante. And the _Mone_ when she is fardest from the earth, is 68 Semidiameters of the earth and ⅓ The nerest, that the _Mone_commeth to the earth, is Semidiameters 52¼ The distance of the StarrySkye is, frõ vs, in Semidiameters of the earth 20081½ Twenty thousandfourescore, one, and almost a halfe. Subtract from this, the _Mones_nerest distance, from the Earth: and therof remaineth Semidiameters ofthe earth 20029¼ Twenty thousand nine and twenty and a quarter. 

 [Note. ]

So thicke is the heauenly Palace, that the _Planetes_ haue all theirexercise in, and most meruailously perfourme the Commaũdement and Chargeto them giuen by the omnipotent Maiestie of the king of kings. This isthat, which in _Genesis_ is called _Ha Rakia_. Consider it well. TheSemidiameter of the earth, cõteineth of our common miles 3436 4/11 threethousand, foure hundred thirty six and foure eleuenth partes of onemyle: Such as the whole earth and Sea, round about, is 21600. One andtwenty thousand six hundred of our myles. Allowyng for euery degree ofthe greatest circle, thre score myles. Now if you way well with yourselfe but this litle parcell of frute _Astronomicall_, as concerning thebignesse, Distances of _Sonne_, _Mone_, _Sterry Sky_, and the hugemassines of _Ha Rakia_, will you not finde your Consciences moued, withthe kingly Prophet, to sing the confession of Gods Glory, and say, +_TheHeauens declare the glory of God, and the Firmament =[Ha Rakia]= shewethforth the workes of his handes_+. And so forth, for those fiue firststaues, of that kingly Psalme. Well, well, It is time for some to layhold on wisedome, and to Iudge truly of thinges: and notso to expoundthe Holy word, all by Allegories: as to Neglect the wisedome, powre andGoodnes of God, in, and by his Creatures, and Creation to be seen andlearned. By parables and Analogies of whose natures and properties, thecourse of the Holy Scripture, also, declareth to vs very many Mysteries. The whole Frame of Gods Creatures, (which is the whole world, ) is to vs, a bright glasse: from which, by reflexion, reboundeth to our knowledgeand perceiuerance, Beames, and Radiations: representing the Image of hisInfinite goodnes, Omnipotẽcy, and wisedome. And we therby, are taughtand persuaded to Glorifie our Creator, as God: and be thankefulltherfore. Could the Heathenistes finde these vses, of these most pure, beawtifull, and Mighty Corporall Creatures: and shall we, after that thetrue _Sonne_ of rightwisenesse is risen aboue the _Horizon_, of ourtemporall _Hemisphærie_, and hath so abundantly streamed into ourhartes, the direct beames of his goodnes, mercy, and grace: Whose heatAll Creatures feele: Spirituall and Corporall: Visible and Inuisible. Shall we (I say) looke vpon the _Heauen_, _Sterres_, and _Planets_, asan Oxe and an Asse doth: no furder carefull or inquisitiue, what theyare: why were they Created, How do they execute that they were Createdfor? Seing, All Creatures, were for our sake created: and both we, andthey, Created, chiefly to glorifie the Almighty Creator: and that, byall meanes, to vs possible. _Nolite ignorare_ (saith _Plato inEpinomis_) _Astronomiam, Sapientissimũ quiddam esse. _ +_Be ye notignorant, Astronomie to be a thyng of excellent wisedome. _+_Astronomie_, was to vs, from the beginning commended, and in manercommaunded by God him selfe. In asmuch as he made the _Sonne_, _Mone_, and _Sterres_, to be to vs, for _Signes_, and knowledge of Seasons, andfor Distinctions of Dayes, and yeares. Many wordes nede not. But I wish, euery man should way this word, _Signes_. And besides that, conferre italso with the tenth Chapter of _Hieremie_. And though Some thinke, thatthere, they haue found a rod: Yet Modest Reason, will be indifferentIudge, who ought to be beaten therwith, in respect of our purpose. Leauing that: I pray you vnderstand this: that without great diligenceof Obseruation, examination and Calculation, their periods and courses(wherby _Distinction_ of Seasons, yeares, and New Mones might preciselybe knowne) could not exactely be certified. Which thing to performe, isthat _Art_, which we here haue Defined to be _Astronomie_. Wherby, wemay haue the distinct Course of Times, dayes, yeares, and Ages: aswellfor Consideratiõ of Sacred Prophesies, accomplished in due time, foretold: as for high Mysticall Solemnities holding: And for all otherhumaine affaires, Conditions, and couenantes, vpon certaine time, betwene man and man: with many other great vses: Wherin, (verely), wouldbe great incertainty, Confusion, vntruth, and brutish Barbarousnes:without the wonderfull diligence and skill of this Arte: continuallylearning, and determining Times, and periodes of Time, by the Record ofthe heauenly booke, wherin all times are written: and to be read with an_Astronomicall staffe_, in stede of a festue. 

+‡Musike‡+, of Motion, hath his Originall cause: Therfore, after themotions most swift, and most Slow, which are in the Firmament, of Natureperformed: and vnder the _Astronomers Consideration_: now I will Speakeof an other kinde of _Motion_, producing sound, audible, and of Mannumerable. _Musike_ I call here that _Science_, which of the Grekes iscalled _Harmonice_. Not medling with the Controuersie betwene theauncient _Harmonistes_, and _Canonistes_. +Musike is a MathematicallScience, which teacheth, by sense and reason, perfectly to iudge, andorder the diuersities of soundes, hye and low. + _Astronomie_ and_Musike_ are Sisters, saith _Plato_. As, for _Astronomie_, the eyes: So, for _Harmonious Motion_, the eares were made. But as _Astronomie_ hath amore diuine Contemplation, and cõmodity, then mortall eye can perceiue:So, is _Musike_ to be considered, [1. ]

that the * Minde may be preferred, before the eare. And from audiblesound, we ought to ascende, to the examination: which numbers are_Harmonious_, and which not. And why, either, the one are: or the otherare not. I could at large, [2. ]

in the heauenly * motions and distances, describe a meruailous Harmonie, of _Pythagoras_ Harpe

 [3. ]

with eight stringes. Also, somwhat might be sayd of _Mercurius_ * twoHarpes, [4. ]

eche of foure Stringes Elementall. And very straunge matter, might bealledged of the _Harmonie_, [5. ]

to our * Spirituall part appropriate. As in _Ptolomaus_ third boke, inthe fourth and sixth Chapters may appeare. *

 [6. ]

And what is the cause of the apt bonde, and frendly felowship, of theIntellectuall and Mentall part of vs, with our grosse & corruptiblebody: but a certaine Meane, and _Harmonious Spiritualitie, with bothparticipatyng, & of both (in a maner) resultynge In

 [7. ]

the * Tune of Mans voyce, and also

 [8. ]

* the sound of Instrument_, what might be sayd, of _Harmonie_: No commonMusicien would lightly beleue. 

 [I. D. Read in Aristotle his 8. Booke of Politikes: the 5, 6, and 7. Chapters. Where you shall haue some occasion farder to thinke of Musike, than commonly is thought. ]

But of the sundry Mixture (as I may terme it) and concurse, diuersecollation, and Application of these _Harmonies_: as of thre, foure, fiue, or mo: Maruailous haue the effectes ben: and yet may be founde, and produced the like: with some proportionall consideration for ourtime, and being: in respect of the State, of the thinges then: in which, and by which, the wondrous effectes were wrought. _Democritus_ and_Theophrastus_ affirmed, that, by _Musike_, griefes and diseases of theMinde, and body might be cured, or inferred. And we finde in Recorde, that _Terpander_, _Arion_, _Ismenias_, _Orpheus_, _Amphion_, _Dauid_, _Pythagoras_, _Empedocles_, _Asclepiades_ and _Timotheus_, by_Harmonicall_ Consonãcy, haue done, and brought to pas, thinges, morethen meruailous, to here of. Of them then, making no farder discourse, in this place: Sure I am, that Common _Musike_, commonly vsed, is foundto the _Musiciens_ and Hearers, to be so Commodious and pleasant, Thatif I would say and dispute, but thus much: That it were to be otherwisevsed, then it is, I should finde more repreeuers, then I could findepriuy, or skilfull of my meaning. In thinges therfore euident, andbetter knowen, then I can expresse: and so allowed and liked of, (as Iwould wish, some other thinges, had the like hap) I will spare toenlarge my lines any farder, but consequently follow my purpose. 

+‡Of Cosmographie‡+, I appointed briefly in this place, to geue you someintelligence. +Cosmographie, is the whole and perfect description of theheauenly, and also elementall parte of the world, and their homologallapplication, and mutuall collation necessarie. + This Art, requireth_Astronomie_, _Geographie_, _Hydrographie_ and _Musike_. Therfore, it isno small Arte, nor so simple, as in common practise, it is (slightly)considered. This matcheth Heauen, and the Earth, in one frame, and aptlyapplieth parts Correspõdent: So, as, the Heauenly Globe, may (inpractise) be duely described vpon the Geographicall, and HydrographicallGlobe. And there, for vs to consider an _Æquonoctiall Circle_, _anEcliptike line_, _Colures_, _Poles_, _Sterres_ in their true Longitudes, Latitudes, Declinations, and Verticalitie: also Climes, and Parallels:and by an _Horizon_ annexed, and reuolution of the earthly Globe (as theHeauen, is, by the _Primouant_, caried about in 24. æquall Houres) tolearne the Risinges and Settinges of Sterres (of _Virgill_ in his_Georgikes_: of _Hesiod_: of _Hippocrates_ in his _Medicinall Sphære_, to Perdicca King of the Macedonians: of _Diocles_, to King _Antigonus_, and of other famous _Philosophers_ prescribed) a thing necessary, fordue manuring of the earth, for _Nauigation_, for the Alteration of mansbody: being, whole, Sicke, wounded, or brused. By the Reuolution, also, or mouing of the Globe Cosmographicall, the Rising and Setting of theSonne: the Lengthes, of dayes and nightes: the Houres and times (bothnight and day) are knowne: with very many other pleasant and necessaryvses: Wherof, some are knowne: but better remaine, for such to know andvse:

 [☞]

who of a sparke of true fire, can make a wonderfull bonfire, by applyingof due matter, duely. 

+‡Of Astrologie‡+, here I make an Arte, seuerall from _Astronomie_: notby new deuise, but by good reason and authoritie: for, +Astrologie, isan Arte Mathematicall, which reasonably demonstrateth the operations andeffectes, of the naturall beames, of light, and secrete influence: ofthe Sterres and Planets: in euery element and elementall body: at alltimes, in any Horizon assigned. + This Arte is furnished with many othergreat Artes and experiences: As with perfecte _Perspectiue_, _Astronomie_, _Cosmographie_, _Naturall Philosophie_ of the 4. Elementes, the Arte of Graduation, and some good vnderstãding in_Musike_: and yet moreouer, with an other great Arte, hereafterfollowing, though I, here, set this before, for some considerations memouing. Sufficient (you see) is the stuffe, to make this rare andsecrete Arte, of: and hard enough to frame to the ConclusionSyllogisticall. Yet both the manifolde and continuall trauailes of themost auncient and wise Philosophers, for the atteyning of this Arte: andby examples of effectes, to confirme the same: hath left vnto vssufficient proufe and witnesse: and we, also, daily may perceaue, Thatmans body, and all other Elementall bodies, are altered, disposed, ordred, pleasured, and displeasured, by the Influentiall working of the_Sunne_, _Mone_, and the other Starres and Planets. And therfore, sayth_Aristotle_, in the first of his _Meteorologicall_ bookes, in the secondChapter: _Est autem necessariò Mundus iste, supernis lationibus ferècontinuus. Vt, inde, vis eius vniuersa regatur. Ea siquidem Causà primaputanda omnibus est, vnde motus principium existit. _ That is: +_This=[Elementall]= World is of necessitie, almost, next adioyning, to theheauenly motions: That, from thence, all his vertue or force may begouerned. For, that is to be thought the first Cause vnto all: fromwhich, the beginning of motion, is. _+ And againe, in the tenth Chapter. _Oportet igitur & horum principia sumamus, & causas omnium similiter. Principium igitur vt mouens, præcipuum[que] & omnium primum, Circulusille est, in quo manifeste Solis latio, &c. _ And so forth. His_Meteorologicall_ bookes, are full of argumentes, and effectualldemonstrations, of the vertue, operation, and power of the heauenlybodies, in and vpon the fower Elementes, and other bodies, of them(either perfectly, or vnperfectly) composed. And in his second booke, _De Generatione & Corruptione_, in the tenth Chapter. _Quocirca & primalatio, Ortus & Interitus causa non est: Sed obliqui Circuli latio: eanam[que] & continua est, & duobus motibus fit:_ In Englishe, thus. +_Wherefore the vppermost motion, is not the cause of Generation andCorruption, but the motion of the Zodiake: for, that, both, iscontinuall, and is caused of two mouinges. _+ And in his second booke, and second Chapter of hys _Physikes_. _Homo nam[que] generat hominem, at[que] Sol. _ +_For Man (sayth he) and the Sonne, are cause of mansgeneration. _+ Authorities may be brought, very many: both of 1000. 2000. Yea and 3000. Yeares Antiquitie: of great _Philosophers_, _Expert_, _Wise_, and godly men, for that Conclusion: which, daily and hourely, wemen, may discerne and perceaue by sense and reason: All beastes dofeele, and simply shew, by their actions and passions, outward andinward: All Plants, Herbes, Trees, Flowers, and Fruites. And finally, the Elementes, and all thinges of the Elementes composed, do geueTestimonie (as _Aristotle_ sayd) that theyr +_Whole Dispositions, vertues, and naturall motions, depend of the Actiuitie of the heauenlymotions and Influences. Whereby, beside the specificall order and forme, due to euery seede: and beside the Nature, propre to the IndiuiduallMatrix, of the thing produced: What shall be the heauenly Impression, the perfect and circumspecte Astrologien hath to Conclude. _+ Not onely(by _Apotelesmes_) τὸ ὁτὶ]. But by Naturall and Mathematicalldemonstration τὸ διότι. Whereunto, what Sciences are requisite (withoutexception) I partly haue here warned: And in my _Propædeumes_ (besidesother matter there disclosed) I haue Mathematically furnished vp thewhole Method: To this our age, not so carefully handled by any, thateuer I saw, or heard of. I was, [* Anno. 1548 and 1549. In Louayn. ]

(for * 21. Yeares ago) by certaine earnest disputations, of the Learned_Gerardus Mercator_, and _Antonius Gogaua_, (and other, ) therto soprouoked: and (by my constant and inuincible zeale to the veritie) inobseruations of Heauenly Influencies (to the Minute of time, ) than, sodiligent: And chiefly by the Supernaturall influence, from the Starre ofIacob, so directed: That any Modest and Sober Student, carefully anddiligently seking for the Truth, will both finde & cõfesse, therin, tobe the Veritie, of these my wordes: And also become a ReasonableReformer, of three Sortes of people: about these InfluentiallOperations, greatly erring from the truth. 

 [Note. ]

Wherof, the one, is +Light Beleuers+, the other, +Light Despisers+, andthe third +Light Practisers+. The first, & most cõmon Sort, thinke theHeauen and Sterres, to be answerable to any their doutes or desires:

 [1. ]

which is not so: and, in dede, they, to much, ouer reache. The Secondsorte thinke no Influentiall vertue (frõ the heauenly bodies) to beareany Sway in Generation

 [2. ]

and Corruption, in this Elementall world. And to the _Sunne_, _Mone_ and_Sterres_ (being so many, so pure, so bright, so wonderfull bigge, sofarre in distance, so manifold in their motions, so constant in theirperiodes. &c. ) they assigne a sleight, simple office or two, and soallow vnto thẽ (according to their capacities) as much vertue, and powerInfluentiall, as to the Signe of the _Sunne_, _Mone_, and seuen Sterres, hanged vp (for Signes) in London, for distinction of houses, & suchgrosse helpes, in our worldly affaires: And they vnderstand not (or willnot vnderstand) of the other workinges, and vertues of the Heauenly_Sunne_, _Mone_, and _Sterres_: not so much, as the Mariner, or Husbandman: no, not so much, as the _Elephant_ doth, as the _Cynocephalus_, asthe Porpentine doth: nor will allow these perfect, and incorruptiblemighty bodies, so much vertuall Radiation, & Force, as they see in alitle peece of a _Magnes stone_: which, at great distance, sheweth hisoperation. And perchaunce they thinke, the Sea & Riuers (as the Thames)to be some quicke thing, and so to ebbe, and flow, run in and out, ofthem selues, at their owne fantasies. God helpe, God helpe. Surely, these men, come to short: and either are to dull: or willfully blind:or, perhaps, to malicious. The third man, is the common and vulgare_Astrologien_, or Practiser: who, being not duely, artificially, andperfectly

 [3. ]

furnished: yet, either for vaine glory, or gayne: or like a simple dolt, & blinde Bayard, both in matter and maner, erreth: to the discredit ofthe _Wary_, and modest _Astrologien_: and to the robbing of those mostnoble corporall Creatures, of their Naturall Vertue: being most mighty:most beneficiall to all elementall Generation, Corruption and theappartenances: and most Harmonious in their Monarchie: For whichthinges, being knowen, and modestly vsed: we might highly, andcontinually glorifie God, with the princely Prophet, saying. +_TheHeauens declare the Glorie of God: who made the Heauẽs in his wisedome:who made the Sonne, for to haue dominion of the day: the Mone andSterres to haue dominion of the nyght: whereby, Day to day vtterethtalke: and night, to night declareth knowledge. Prayse him, all yeSterres, and Light. Amen. _+

In order, now foloweth, of +‡Statike‡+, somewhat to say, what we meaneby that name: and what commodity, doth, on such Art, depend. +Statike, is an Arte Mathematicall, which demonstrateth the causes of heauynes, and lightnes of all thynges: and of motions and properties, to heauynesand lightnes, belonging. + And for asmuch as, by the Bilanx, or Balance(as the chief sensible Instrument, ) Experience of these demonstrationsmay be had: we call this Art, _Statike:_ that is, _the Experimentes ofthe Balance_. Oh, that men wist, what proffit, (all maner of wayes) bythis Arte might grow, to the hable examiner, and diligent practiser. “Thou onely, knowest all thinges precisely (O God) who hast made weightand Balance, thy Iudgement: who hast created all thinges in _Number, Waight, and Measure_: and hast wayed the mountaines and hils in aBalance: who hast peysed in thy hand, both Heauen and earth. We therforewarned by the Sacred word, to Consider thy Creatures: and by thatconsideration, to wynne a glyms (as it were, ) or shaddow ofperceiuerance, that thy wisedome, might, and goodnes is infinite, andvnspeakable, in thy Creatures declared: And being farder aduertised, bythy mercifull goodnes, that, three principall wayes, were, of the, vsedin Creation of all thy Creatures, namely, _Number_, _Waight_ and_Measure_, And for as much as, of _Number_ and _Measure_, the two Artes(auncient, famous, and to humaine vses most necessary, ) are, all ready, sufficiently knowen and extant: This third key, we beseche thee (throughthy accustomed goodnes, ) that it may come to the nedefull and sufficientknowledge, of such thy Seruauntes, as in thy workemanship, would gladlyfinde, thy true occasions (purposely of the vsed) whereby we shouldglorifie thy name, and shew forth (to the weaklinges in faith) thywondrous wisedome and Goodnes. Amen. ”

Meruaile nothing at this pang (godly frend, you Gentle and zelousStudent. ) An other day, perchaunce, you will perceiue, what occasionmoued me. Here, as now, I will giue you some ground, and withall someshew, of certaine commodities, by this Arte arising. And bycause thisArte is rare, my wordes and practises might be to darke: vnleast you hadsome light, holden before the matter: and that, best will be, in giuingyou, out of _Archimedes_ demonstrations, a few principal Conclusions, asfoloweth. 

 +1. +

 +The Superficies of euery Liquor, by it selfe consistyng, and in quyet, is Sphæricall: the centre whereof, is the same, which is the centre of the Earth. +

 +2. +

 +If Solide Magnitudes, being of the same bignes, or quãtitie, that any Liquor is, and hauyng also the same Waight: be let downe into the same Liquor, they will settle downeward, so, that no parte of them, shall be aboue the Superficies of the Liquor: and yet neuertheles, they will not sinke vtterly downe, or drowne. +

 +3. +

 +If any Solide Magnitude beyng Lighter then a Liquor, be let downe into the same Liquor, it will settle downe, so farre into the same Liquor, that so great a quantitie of that Liquor, as is the parte of the Solid Magnitude, settled downe into the same Liquor: is in Waight, æquall, to the waight of the whole Solid Magnitude. +

 +4. +

 +Any Solide Magnitude, Lighter then a Liquor, forced downe into the same Liquor, will moue vpward, with so great a power, by how much, the Liquor hauyng æquall quantitie to the whole Magnitude, is heauyer then the same Magnitude. +

 +5. +

 +Any Solid Magnitude, heauyer then a Liquor, beyng let downe into the same Liquor, will sinke downe vtterly: And wilbe in that Liquor, Lighter by so much, as is the waight or heauynes of the Liquor, hauing bygnes or quantitie, æquall to the Solid Magnitude. +

 +6. +

 [I. D. The Cutting of a Sphære according to any proportion assigned may by this proposition be done Mechanically by tempering Liquor to a certayne waight in respect of the waight of the Sphære therein Swymming. ]

 +If any Solide Magnitude, Lighter then a Liquor, be let downe into the same Liquor, the waight of the same Magnitude, will be, to the Waight of the Liquor. (Which is æquall in quantitie to the whole Magnitude, ) in that proportion, that the parte, of the Magnitude settled downe, is to the whole Magnitude. +

By these verities, great Errors may be reformed, in Opinion of theNaturall Motion of thinges, Light and Heauy. Which errors, are inNaturall Philosophie (almost) of all mẽ allowed: to much trusting toAuthority: and false Suppositions. As, +Of any two bodyes, the heauyer, to moue downward faster then the lighter. +

 [A common error, noted. ]

This error, is not first by me, Noted: but by one _Iohn Baptist deBenedictis_. The chief of his propositions, is this: which seemeth aParadox. 

+If there be two bodyes of one forme, and of one kynde, æquall inquantitie or vnæquall, [A paradox. ]

they will moue by æquall space, in æquall tyme: So that both theyrmouynges be in ayre, or both in water: or in any one Middle. +

Hereupon, in the feate of +Gunnyng+, [N. T. ]

certaine good discourses (otherwise) may receiue great amendement, andfurderance. 

 [The wonderfull vse of these Propositions. ]

In the entended purpose, also, allowing somwhat to the imperfection ofNature: not aunswerable to the precisenes of demonstration. Moreouer, bythe foresaid propositions (wisely vsed. ) The Ayre, the water, the Earth, the Fire, may be nerely, knowen, how light or heauy they are (Naturally)in their assigned partes: or in the whole. And then, to thingesElementall, turning your practise: you may deale for the proportion ofthe Elementes, in the thinges Compounded. Then, to the proportions ofthe Humours in Man: their waightes: and the waight of his bones, andflesh. &c. Than, by waight, to haue consideration of the Force of man, any maner of way: in whole or in part. Then, may you, of Ships waterdrawing, diuersly, in the Sea and in fresh water, haue pleasantconsideration: and of waying vp of any thing, sonken in Sea or in freshwater &c. And (to lift vp your head a loft:) by waight, you may, asprecisely, as by any instrument els, measure the Diameters of _Sonne_and _Mone. &c. _ Frende, I pray you, way these thinges, with the iustBalance of Reason. And you will finde Meruailes vpon Meruailes: Andesteme one Drop of Truth (yea in Naturall Philosophie) more worth, thenwhole Libraries of Opinions, vndemonstrated: or not aunswering toNatures Law, and your experience. Leauing these thinges, thus: I willgiue you two or three, light practises, to great purpose: and so finishmy Annotation _Staticall_. In Mathematicall matters, by the Mechaniciensayde, we will behold, here, the Commodity of waight. 

 [The practise Staticall, to know the proportion, betwene the Cube, and the Sphære. ]

Make a Cube, of any one Vniforme: and through like heauy stuffe: of thesame Stuffe, make a Sphære or Globe, precisely, of a Diameter æquall tothe Radicall side of the Cube. Your stuffe, may be wood, Copper, Tinne, Lead, Siluer. &c. (being, as I sayd, of like nature, condition, and likewaight throughout. ) And you may, by Say Balance, haue prepared a greatnumber of the smallest waightes: which, by those Balance can bediscerned or tryed: and so, haue proceded to make you a perfect Pyle, company & Number of waightes: to the waight of six, eight, or tweluepound waight: most diligently tryed, all. And of euery one, the Contentknowen, in your least waight, that is wayable. [They that can not hauethese waightes of precisenes: may, by Sand, Vniforme, and well dusted, make them a number of waightes, somewhat nere precisenes: by halfingeuer the Sand: they shall, at length, come to a least common waight. Therein, I leaue the farder matter, to their discretion, whom nede shallpinche. ] The _Venetians_ consideration of waight, may seme preciseenough: by eight descentes progressionall, * halfing, from a grayne. 

 [I. D. * For, so, haue you . 256. Partes of a Graine. ]

Your Cube, Sphære, apt Balance, and conuenient waightes, being ready:fall to worke. ❉. First, way your Cube. Note the Number of the waight. Way, after that, your Sphære. Note likewise, the Nũber of the waight. Ifyou now find the waight of your Cube, to be to the waight of the Sphære, as 21. Is to 11: Then you see, how the Mechanicien and _Experimenter_, without Geometrie and Demonstration, are (as nerely in effect) toughtthe proportion of the Cube to the Sphere: as I haue demonstrated it, inthe end of the twelfth boke of _Euclide_. Often, try with the same Cubeand Sphære. Then, chaunge, your Sphære and Cube, to an other matter: orto an other bignes: till you haue made a perfect vniuersall Experienceof it. Possible it is, that you shall wynne to nerer termes, in theproportion. 

When you haue found this one certaine Drop of Naturall veritie, procedeon, to Inferre, and duely to make assay, of matter depending. As, bycause it is well demonstrated, that a Cylinder, whose heith, andDiameter of his base, is æquall to the Diameter of the Sphære, isSesquialter to the same Sphære (that is, as 3. To 2:) To the number ofthe waight of the Sphære, adde halfe so much, as it is: and so haue youthe number of the waight of that Cylinder. Which is also Comprehended ofour former Cube: So, that the base of that Cylinder, is a Circledescribed in the Square, which is the base of our Cube. But the Cube andthe Cylinder, being both of one heith, haue their Bases in the sameproportion, in the which, they are, one to an other, in their Massinesor Soliditie. But, before, we haue two numbers, expressing theirMassines, Solidities, and Quantities, by waight: wherfore, [* =The proportion of the Square to the Circle inscribed. =]

we haue * the proportion of the Square, to the Circle, inscribed in thesame Square. And so are we fallen into the knowledge sensible, andExperimentall of _Archimedes_ great Secret: of him, by great trauaile ofminde, sought and found. Wherfore, to any Circle giuen, you can giue aSquare æquall:

 [* =The Squaring of the Circle, Mechanically. =]

* as I haue taught, in my Annotation, vpon the first proposition of thetwelfth boke, And likewise, to any Square giuen, you may giue a Circleæquall:

 [* =To any Square geuen, to geue a Circle, equall. =]

* If you describe a Circle, which shall be in that proportion, to yourCircle inscribed, as the Square is to the same Circle: This, you may do, by my Annotations, vpon the second proposition of the twelfth boke of_Euclide_, in my third Probleme there. Your diligence may come to aproportion, of the Square to the Circle inscribed, nerer the truth, thenis the proportion of 14. To 11. And consider, that you may begyn at theCircle and Square, and so come to conclude of the Sphære, & the Cube, what their proportion is: as now, you came from the Sphære to theCircle. For, of Siluer, or Gold, or Latton Lamyns or plates (thoroughone hole drawẽ, as the maner is) if you make a Square figure & way it:and then, describing theron, the Circle inscribed: & cut of, & fileaway, precisely (to the Circle) the ouerplus of the Square: you shallthen, waying your Circle, see, whether the waight of the Square, be toyour Circle, as 14. To 11. As I haue Noted, in the beginning of_Euclides_ twelfth boke. &c. After this resort to my last proposition, vpon the last of the twelfth. And there, helpe your selfe, to the end. And, here, Note this, by the way. 

 [Note Squaring of the Circle without knowledge of the proportion betwene Circumference and Diameter. ]

That we may Square the Circle, without hauing knowledge of theproportion, of the Circumference to the Diameter: as you haue hereperceiued. And otherwayes also, I can demonstrate it. So that, many hauecumberd them selues superfluously, by trauailing in that point first, which was not of necessitie, first: and also very intricate. And easily, you may, (and that diuersly) come to the knowledge of the Circumference:the Circles Quantitie, being first knowen. Which thing, I leaue to yourconsideration: making hast to despatch an other Magistrall Probleme: andto bring it, nerer to your knowledge, and readier dealing with, then theworld (before this day, ) had it for you, that I can tell of. And thatis, _A Mechanicall Dubblyng of the Cube: &c. _ Which may, thus, be done:

 [To Dubble the Cube redily: by Art Mechanicall: depending vppon Demonstration Mathematicall. ]

+Make of Copper plates, or Tyn plates, a foursquare vpright Pyramis, ora Cone: perfectly fashioned in the holow, within. Wherin, let greatdiligence be vsed, to approche (as nere as may be) to the Mathematicallperfection of those figures. At their bases, let them be all open: euerywhere, els, most close, and iust to. From the vertex, to theCircumference of the base of the Cone: & to the sides of the base of thePyramis:+

 [=I. D. = =The 4. Sides of this Pyramis must be 4. Isosceles Triangles alike and æquall. =]

+Let 4. Straight lines be drawen, in the inside of the Cone and Pyramis:makyng at their fall, on the perimeters of the bases, equall angles onboth sides them selues, with the sayd perimeters. These 4. Lines (in thePyramis: and as many, in the Cone) diuide: one, in 12. æquall partes:and an other, in 24. An other, in 60, and an other, in 100. (reckenyngvp from the vertex. ) Or vse other numbers of diuision, as experienceshall teach you. +

 [=I. D. = =* In all workinges with this Pyramis or Cone, Let their Situations be in all Pointes and Conditions, alike, or all one: while you are about one Worke. Els you will erre. =]

+Then, * set your Cone or Pyramis, with the vertex downward, perpendicularly, in respect of the Base. (Though it be otherwayes, ithindreth nothyng. ) So let thẽ most stedily be stayed. + Now, if there bea Cube, which you wold haue Dubbled. Make you a prety Cube of Copper, Siluer, Lead, Tynne, Wood, Stone, or Bone. Or els make a hollow Cube, orCubik coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may soproportiõ in respect of your Pyramis or Cone, that the Pyramis or Cone, will be hable to conteine the waight of them, in water, 3. Or 4. Times:at the least: what stuff so euer they be made of. Let not your Solidangle, at the vertex, be to sharpe: but that the water may come withease, to the very vertex, of your hollow Cone or Pyramis. Put one ofyour Solid Cubes in a Balance apt: take the waight therof exactly inwater. Powre that water, (without losse) into the hollow Pyramis orCone, quietly. Marke in your lines, what numbers the water Cutteth: Takethe waight of the same Cube againe: in the same kinde of water, whichyou had before:

 [=I. D. = =* Consider well whan you must put your waters togyther: and whan, you must empty your first water, out of your Pyramis or Cone. Els you will erre. =]

put that* also, into the Pyramis or Cone, where you did put the first. Marke now againe, in what number or place of the lines, the waterCutteth them. Two wayes you may conclude your purpose: it is to wete, either by numbers or lines. By numbers: as, if you diuide the side ofyour Fundamentall Cube into so many æquall partes, as it is capable of, conueniently, with your ease, and precisenes of the diuision. For, asthe number of your first and lesse line (in your hollow Pyramis orCone, ) is to the second or greater (both being counted from the vertex)so shall the number of the side of your Fundamentall Cube, be to thenũber belonging to the Radicall side, of the Cube, dubble to yourFundamentall Cube: Which being multiplied Cubik wise, will sone shew itselfe, whether it be dubble or no, to the Cubik number of yourFundamentall Cube. By lines, thus: As your lesse and first line, (inyour hollow Pyramis or Cone, ) is to the second or greater, so let theRadical side of your Fundamẽtall Cube, be to a fourth proportionallline, by the 12. Proposition, of the sixth boke of _Euclide_. Whichfourth line, shall be the Rote Cubik, or Radicall side of the Cube, dubble to your Fundamentall Cube: which is the thing we desired. 

 [☞ God be thanked for this Inuention, & the fruite ensuing. ]

For this, may I (with ioy) say, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ: thanking theholy and glorious Trinity: hauing greater cause therto, then

 [* Vitruuius. Lib. 9. Cap. 3. ]

* _Archimedes_ had (for finding the fraude vsed in the Kinges Crowne, ofGold): as all men may easily Iudge: by the diuersitie of the frutefollowing of the one, and the other. Where I spake before, of a hollowCubik Coffen: the like vse, is of it: and without waight. Thus. Fill itwith water, precisely full, and poure that water into your Pyramis orCone. And here note the lines cutting in your Pyramis or Cone. Againe, fill your coffen, like as you did before. Put that Water, also, to thefirst. Marke the second cutting of your lines. Now, as you procededbefore, so must you here procede. 

 [* Note. ]

* And if the Cube, which you should Double, be neuer so great: you haue, thus, the proportion (in small) betwene your two litle Cubes: And then, the side, of that great Cube (to be doubled) being the third, will hauethe fourth, found, to it proportionall: by the 12. Of the sixth ofEuclide. 

 [Note, as concerning the Sphæricall Superficies of the Water. ]

Note, that all this while, I forget not my first Proposition Staticall, here rehearsed: that, the Superficies of the water, is Sphæricall. Wherein, vse your discretion: to the first line, adding a small hearebreadth, more: and to the second, halfe a heare breadth more, to hislength. For, you will easily perceaue, that the difference can be nogreater, in any Pyramis or Cone, of you to be handled. Which you shallthus trye. _For finding the swelling of the water aboue leuell. _

 [☞]

“Square the Semidiameter, from the Centre of the earth, to your firstWaters Superficies. Square then, halfe the Subtendent of that watrySuperficies (which Subtendent must haue the equall partes of hismeasure, all one, with those of the Semidiameter of the earth to yourwatry Superficies): Subtracte this square, from the first: Of theresidue, take the Rote Square. That Rote, Subtracte from your firstSemidiameter of the earth to your watry Superficies: that, whichremaineth, is the heith of the water, in the middle, aboue the leuell. ”Which, you will finde, to be a thing insensible. And though it weregreatly sensible, *

 [* Note. ]

yet, by helpe of my sixt Theoreme vpon the last Proposition of Euclidestwelfth booke, noted: you may reduce all, to a true Leuell. But, fartherdiligence, of you is to be vsed, against accidentall causes of thewaters swelling: as by hauing (somwhat) with a moyst Sponge, before, made moyst your hollow Pyramis or Cone, will preuent an accidentallcause of Swelling, &c. Experience will teach you abundantly: with greatease, pleasure, and cõmoditie. 

Thus, may you Double the Cube Mechanically, Treble it, and so forth, inany proportion. 

 [Note this Abridgement of Dubbling the Cube. &c. ]

Now will I Abridge your paine, cost, and Care herein. Without allpreparing of your Fundamentall Cubes: you may (alike) worke thisConclusion. For, that, was rather a kinde of Experimentall demõstration, then the shortest way: and all, vpon one Mathematicall Demonstrationdepending. “Take water (as much as conueniently will serue your turne:as I warned before of your Fundamentall Cubes bignes) Way it precisely. Put that water, into your Pyramis or Cone. Of the same kinde of water, then take againe, the same waight you had before: put that likewise intothe Pyramis or Cone. For, in eche time, your marking of the lines, howthe Water doth cut them, shall geue you the proportion betwen theRadicall sides, of any two Cubes, wherof the one is Double to the other:working as before I haue taught you:

 [* ☞ Note. ]

* sauing that for you Fundamentall Cube his Radicall side: here, you maytake a right line, at pleasure. ”

Yet farther proceding with our droppe of Naturall truth:

 [To giue Cubes one to the other in any proportion, Rationall or Irrationall. ]

+you may (now) geue Cubes, one to the other, in any proportiõ geuẽ:Rationall or Irrationall+: on this maner. Make a hollow Parallelipipedonof Copper or Tinne: with one Base wãting, or open: as in our CubikeCoffen. Frõ the bottome of that Parallelipipedon, raise vp, manyperpendiculars, in euery of his fower sides. Now if any proportion beassigned you, in right lines: Cut one of your perpendiculars (or a lineequall to it, or lesse then it) likewise: by the 10. Of the sixth ofEuclide. And those two partes, set in two sundry lines of thoseperpendiculars (or you may set them both, in one line) making theirbeginninges, to be, at the base: and so their lengthes to extend vpward. Now, set your hollow Parallelipipedon, vpright, perpendicularly, steadie. Poure in water, handsomly, to the heith of your shorter line. Poure that water, into the hollow Pyramis or Cone. Marke the place ofthe rising. Settle your hollow Parallelipipedon againe. Poure water intoit: vnto the heith of the second line, exactly. 

 [* Emptying the first. ]

Poure that water * duely into the hollow Pyramis or Cone: Marke nowagaine, where the water cutteth the same line which you marked before. For, there, as the first marked line, is to the second: So shall the twoRadicall sides be, one to the other, of any two Cubes: which, in theirSoliditie, shall haue the same proportion, which was at the firstassigned: were it Rationall or Irrationall. 

Thus, in sundry waies you may furnishe your selfe with such straunge andprofitable matter: which, long hath bene wished for. And though it beNaturally done and Mechanically: yet hath it a good DemonstrationMathematicall. 

 [=The demonstrations of this Dubbling of the Cube, and of therest. =]

Which is this: Alwaies, you haue two Like Pyramids: or two Like Cones, in the proportions assigned: and like Pyramids or Cones, are inproportion, one to the other, in the proportion of their Homologallsides (or lines) tripled. Wherefore, if to the first, and second lines, found in your hollow Pyramis or Cone, you ioyne a third and a fourth, incontinuall proportion: that fourth line, shall be to the first, as thegreater Pyramis or Cone, is to the lesse: by the 33. Of the eleuenth ofEuclide. If Pyramis to Pyramis, or Cone to Cone, be double, [I. D. = * Hereby, helpe your self to become a præcise practiser. And so consider, how, nothing at all, you are hindred (sensibly) by the Conuexitie of the water. =]

then shall * Line to Line, be also double, &c. But, as our first line, is to the second, so is the Radicall side of our Fundamentall Cube, tothe Radicall side of the Cube to be made, or to be doubled: andtherefore, to those twaine also, a third and a fourth line, incontinuall proportion, ioyned: will geue the fourth line in thatproportion to the first, as our fourth Pyramidall, or Conike line, wasto his first: but that was double, or treble, &c. As the Pyramids orCones were, one to an other (as we haue proued) therfore, this fourth, shalbe also double or treble to the first, as the Pyramids or Cones wereone to an other: But our made Cube, is described of the second inproportion, of the fower proportionall lines:

 [= * By the 33. Of the eleuenth booke of Euclide. =]

therfore * as the fourth line, is to the first, so is that Cube, to thefirst Cube: and we haue proued the fourth line, to be to the first, asthe Pyramis or Cone, is to the Pyramis or Cone: Wherefore the Cube is tothe Cube, as Pyramis is to Pyramis, or Cone is to Cone. 

 [I. D. = * And your diligence in practise, can so (in waight of water) performe it: Therefore, now, you are able to geue good reason of your whole doing. =]

But we * Suppose Pyramis to Pyramis, or Cone to Cone, to be double ortreble. &c. Therfore Cube, is to Cube, double, or treble, &c. Which wasto be demonstrated. And of the Parallelipipedõ, it is euidẽt, that thewater Solide Parallelipipedons, are one to the other, as their heithesare, seing they haue one base. Wherfore the Pyramids or Cones, made ofthose water Parallelipipedons, are one to the other, as the lines are(one to the other) betwene which, our proportion was assigned. But theCubes made of lines, after the proportiõ of the Pyramidal or Conik_homologall_ lines, are one to the other, as the Pyramides or Cones are, one to the other (as we before did proue) therfore, the Cubes made, shalbe one to the other, as the lines assigned, are one to the other:Which was to be demonstrated. Note. 

 [* _Note this Corollary. _]

* This, my Demonstratiõ is more generall, then onely in Square Pyramisor Cone: Consider well. Thus, haue I, both Mathematically andMechanically, ben very long in wordes: yet (I trust) nothing tedious tothem, who, to these thinges, are well affected. And verily I am forced(auoiding prolixitie) to omit sundry such things, easie to be practised:which to the Mathematicien, would be a great Threasure: and to theMechanicien, no small gaine. 

 [* The great Commodities following of these new Inuentions. ]

* Now may you, +Betwene two lines giuen, finde two middle proportionals, in Continuall proportion: by the hollow Parallelipipedon, and the hollowPyramis, or Cone. + Now, any Parallelipipedon rectangle being giuen: threright lines may be found, proportionall in any proportion assigned, ofwhich, shal be produced a Parallelipipedon, æquall to theParallelipipedon giuen. Hereof, I noted somwhat, vpon the 36. Proposition, of the 11. Boke of _Euclide_. Now, all those thinges, which_Vitruuius_ in his Architecture, specified hable to be done, by dubblingof the Cube: Or, by finding of two middle proportionall lines, betwenetwo lines giuen, may easely be performed. Now, that Probleme, which Inoted vnto you, in the end of my Addition, vpon the 34. Of the 11. Bokeof _Euclide_, is proued possible. Now, may any regular body, beTransformed into an other, &c. Now, any regular body: any Sphere, yeaany Mixt Solid: and (that more is) Irregular Solides, may be made (inany proportiõ assigned) like vnto the body, first giuen. Thus, of a_Manneken_, (as the _Dutch_ Painters terme it) in the same _Symmetrie_, may a Giant be made: and that, with any gesture, by the Manneken vsed:and contrarywise. Now, may you, of any Mould, or Modell of a Ship, makeone, of the same Mould (in any assigned proportion) bigger or lesser. 

 [* ☞]

Now, may you, of any * Gunne, or little peece of ordinaũce, make another, with the same _Symmetrie_ (in all pointes) as great, and aslittle, as you will. Marke that: and thinke on it. Infinitely, +may youapply this, so long sought for, and now so easily concluded: andwithall, so willingly and frankly communicated to such, as faithfullydeale with vertuous studies. +

 [Such is the Fruite of the Mathematicall Sciences and Artes. ]

Thus, can the Mathematicall minde, deale Speculatiuely in his own Arte:and by good meanes, Mount aboue the cloudes and sterres: And thirdly, hecan, by order, Descend, to frame Naturall thinges, to wonderfull vses:and when he list, retire home into his owne Centre: and there, preparemore Meanes, to Ascend or Descend by: and, all, to the glory of God, andour honest delectation in earth. 

Although, the Printer, hath looked for this Præface, a day or two, yetcould I not bring my pen from the paper, before I had giuen youcomfortable warning, and brief instructions, of some of the Commodities, by _Statike_, hable to be reaped: In the rest, I will therfore, be asbrief, as it is possible: and with all, describing them, somwhataccordingly. And that, you shall perceiue, by this, which in ordercommeth next. For, wheras, it is so ample and wonderfull, that, an wholeyeare long, one might finde fruitfull matter therin, to speake of: andalso in practise, is a Threasure endeles: yet will I glanse ouer it, with wordes very few. 

This do I call +‡Anthropographie‡+. Which is an Art restored, and of mypreferment to your Seruice. I pray you, thinke of it, as of one of thechief pointes, of Humane knowledge. Although it be, but now, firstCõfirmed, with this new name: yet the matter, hath from the beginning, ben in consideration of all perfect Philosophers. +Anthropographie, isthe description of the Number, Measure, Waight, figure, Situation, andcolour of euery diuerse thing, conteyned in the perfect body of MAN:with certain knowledge of the Symmetrie, figure, waight, Characterization, and due locall motion, of any parcell of the saydbody, assigned: and of Nũbers, to the sayd parcell appertainyng. + This, is the one part of the Definition, mete for this place: Sufficient tonotifie, the particularitie, and excellency of the Arte: and why it is, here, ascribed to the Mathematicals. Yf the description of the heauenlypart of the world, had a peculier Art, called _Astronomie:_ If thedescription of the earthly Globe, hath his peculier arte, called_Geographie_. If the Matching of both, hath his peculier Arte, called_Cosmographie:_ Which is the Descriptiõ of the whole, and vniuersallframe of the world: Why should not the description of

 [MAN is the Lesse World. ]

him, who is the Lesse world: and, frõ the beginning, called_Microcosmus_ (that is. _The Lesse World. _) And for whose sake, andseruice, all bodily creatures els, were created: Who, also, participateth with Spirites, and Angels: and is made to the Image andsimilitude of _God_: haue his peculier Art? and be called the _Arte ofArtes_: rather, then, either to want a name, or to haue to base andimpropre a name? You must of sundry professions, borow or challengehome, peculier partes hereof: and farder procede: as, God, Nature, Reason and Experience shall informe you. The Anatomistes will restore toyou, some part: The Physiognomistes, some: The Chyromantistes some. TheMetaposcopistes, some: The excellent, _Albert Durer_, a good part: theArte of Perspectiue, will somwhat, for the Eye, helpe forward:_Pythagoras_, _Hipocrates_, _Plato_, _Galenus_, _Meletius_, & many other(in certaine thinges) will be Contributaries. And farder, the Heauen, the Earth, and all other Creatures, will eche shew, and offer theirHarmonious seruice, to fill vp, that, which wanteth hereof: and withyour own Experience, concluding: you may Methodically register thewhole, for the posteritie: Whereby, good profe will be had, of ourHarmonious, and

 [Micro Cosmus. ]

Microcosmicall constitution. 

 [* ☞]

The outward Image, and vew hereof: to the Art of _Zographie_ andPainting, to Sculpture, and Architecture: (for Church, House, Fort, orShip) is most necessary and profitable: for that, it is the chiefe baseand foundation of them. 

 [* Lib. 3. Cap. 1. ]

Looke in * _Vitruuius_, whether I deale sincerely for your behoufe, orno. Looke in _Albertus Durerus_, _De Symmetria humani Corporis_. Lookein the 27. And 28. Chapters, of the second booke, _De occultaPhilosophia_. Consider the _Arke_ of _Noe_. And by that, wade farther. Remember the _Delphicall Oracle NOSCE TEIPSVM_ +_(Knowe thy selfe)_+ solong agoe pronounced: of so many a Philosopher repeated: and of the_Wisest_ attempted: And then, you will perceaue, how long agoe, you hauebene called to the Schole, where this Arte might be learned. Well. I amnothing affrayde, of the disdayne of some such, as thinke Sciences andArtes, to be but Seuen. Perhaps, those Such, may, with ignorance, andshame enough, come short of them Seuen also: and yet neuerthelesse theycan not prescribe a certaine number of Artes: and in eche, certainevnpassable boundes, to God, Nature, and mans Industrie. New Artes, daylyrise vp: and there was no such order taken, that, [☞]

All Artes, should in one age, or in one land, or of one man, be madeknowen to the world. Let vs embrace the giftes of God, and wayes towisedome, in this time of grace, from aboue, continually bestowed onthem, who thankefully will receiue them: _Et bonis Omnia Cooperabunturin bonum. _

+‡Trochilike, ‡ is that Art Mathematicall, which demonstrateth theproperties of all Circular motions, Simple and Compounde. + And bycausethe frute hereof, vulgarly receiued, is in Wheles, it hath the name of_Trochilike:_ as a man would say, _Whele Art_. By this art, a Whele maybe geuen which shall moue ones about, in any tyme assigned. Two Whelesmay be giuen, whose turnynges about in one and the same tyme, (or equalltymes), shall haue, one to the other, any proportion appointed. ByWheles, may a straight line be described: Likewise, a Spirall line inplaine, Conicall Section lines, and other Irregular lines, at pleasure, may be drawen. These, and such like, are principall Conclusions of thisArte: and helpe forward many pleasant and profitable Mechanicall workes:

 [Saw Milles. ]

As Milles, to Saw great and very long Deale bordes, no man being by. Such haue I seene in Germany: and in the Citie of Prage: in the kingdomeof Bohemia: Coyning Milles, Hand Milles for Corne grinding: And allmaner of Milles, and Whele worke: By Winde, Smoke, Water, Waight, Spring, Man or Beast, moued. Take in your hand, _Agricola De reMetallica:_ and then shall you (in all Mines) perceaue, how great nedeis, of Whele worke. By Wheles, straunge workes and incredible, are done:as will, in other Artes hereafter, appeare. A wonderfull example offarther possibilitie, and present commoditie, was sene in my time, in acertaine Instrument: which by the Inuenter and Artificer (before) wassolde for xx. Talentes of Golde: and then had (by misfortune) receauedsome iniurie and hurt: And one _Ianellus_ of _Cremona_ did mend thesame, and presented it vnto the Emperour _Charles_ the fifth. _Hieronymus Cardanus_, can be my witnesse, that therein, was one Whele, which moued, and that, in such rate, that, in 7000. Yeares onely, hisowne periode should be finished. A thing almost incredible: But howfarre, I keepe me within my boundes: very many men (yet aliue) can tell. 

+‡Helicosophie‡+, is nere Sister to _Trochilike:_ and is, +An ArteMathematicall, which demonstrateth the designing of all Spirall lines inPlaine, on Cylinder, Cone, Sphære, Conoid, and Sphæroid, and theirproperties appertayning. + The vse hereof, in _Architecture_, and diuerseInstrumentes and Engines, is most necessary. For, in many thinges, theSkrue worketh the feate, which, els, could not be performed. By helpehereof, [* Atheneus Lib. 5. Cap. 8. ]

it is * recorded, that, where all the power of the Citie of Syracusa, was not hable to moue a certaine Ship (being on ground) mightie_Archimedes_, setting to, his Skruish Engine, caused _Hiero_ the king, by him self, at ease, to remoue her, as he would. 

 [Proclus. Pag. 18. ]

Wherat, the King wondring: Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδειλέγοντι πιϛευτεόν. _From this day, forward_ (said the King) _Creditought to be giuen to Archimedes, what soeuer he sayth. _

+‡Pneumatithmie‡ demonstrateth by close hollow Geometricall Figures, (regular and irregular) the straunge properties (in motion or stay) ofthe Water, Ayre, Smoke, and Fire, in theyr cõtinuitie, and as they areioyned to the Elementes next them. + This Arte, to the NaturallPhilosopher, is very proffitable: to proue, that _Vacuum_, or _Emptines_is not in the world. And that, all Nature, abhorreth it so much: that, contrary to ordinary law, the Elementes will moue or stand. As, Water toascend: rather then betwene him and Ayre, Space or place should be left, more then (naturally) that quãtitie of Ayre requireth, or can fill. Againe, Water to hang, and not descend: rather then by descending, toleaue Emptines at his backe. The like, is of Fire and Ayre: they willdescend: when, either, their Cõtinuitie should be dissolued: or theirnext Element forced from them. And as they will not be extended, todiscontinuitie: So, will they not, nor yet of mans force, can be prestor pent, in space, not sufficient and aunswerable to their bodilysubstance. Great force and violence will they vse, to enioy theirnaturall right and libertie. 

 [To go to the bottom of the Sea without daunger. ]

Hereupon, two or three men together, by keping Ayre vnder a greatCauldron, and forcyng the same downe, orderly, may without harme descendto the Sea bottome: and continue there a tyme &c. Where, Note, how thethicker Element (as the Water) giueth place to the thynner (as, is theayre:) and receiueth violence of the thinner, in maner. &c. Pumps andall maner of Bellowes, haue their ground of this Art: and many otherstraunge deuises. As, _Hydraulica_, Organes goyng by water. &c. Of thisFeat, (called commonly _Pneumatica_, ) goodly workes are extant, both inGreke, and Latin. With old and learned Schole men, it is called_Scientia de pleno & vacuo. _

+‡Menadrie‡, is an Arte Mathematicall, which demonstrateth, how, aboueNatures vertue and power simple: Vertue and force may be multiplied: andso, to direct, to lift, to pull to, and to put or cast fro, anymultiplied or simple, determined Vertue, Waight or Force: naturally, not, so, directible or moueable. + Very much is this Art furdred by otherArtes: as, in some pointes, by _Perspectiue_: in some, by _Statike_: insome, by _Trochilike_: and in other, by _Helicosophie_: and_Pneumatithmie_. By this Art, all Cranes, Gybbettes, & Ingines to liftvp, or to force any thing, any maner way, are ordred: and the certainecause of their force, is knowne: As, the force which one man hath withthe Duche waghen Racke: therwith, to set vp agayne, a mighty waghenladen, being ouerthrowne. The force of the Crossebow Racke, iscertainly, here, demonstrated. The reason, why one mã, doth with aleauer, lift that, which Sixe men, with their handes onely, could not, so easily do. By this Arte, in our common Cranes in London, where powreis to Crane vp, the waight of 2000. Pound: by two Wheles more (by goodorder added) Arte concludeth, that there may be Craned vp 200000. Poundwaight &c. So well knew _Archimedes_ this Arte: that he alone, with hisdeuises and engynes, (twise or thrise) spoyled and discomfited the wholeArmy and Hoste of the Romaines, besieging _Syracusa_, [=Plutarchus in Marco Marcello. =]

_Marcus Marcellus the Consul_, being their Generall Capitaine. 

 [=Synesius in Epistolis. =]

Such huge Stones, so many, with such force, and so farre, did he withhis engynes hayle among them, out of the Citie. 

 [=Polybius. =]

 [=Plinius. =]

 [=Quintilianus. =]

 [=T. Liuius. =]

And by Sea likewise: though their Ships might come to the walls of_Syracusa_, yet hee vtterly confounded the Romaine Nauye. What with hismighty Stones hurlyng:

 [=* Athenæus. =]

what with Pikes of * 18 fote long, made like shaftes: which he forcedalmost a quarter of a myle. What, with his catchyng hold of their Shyps, and hoysing them vp aboue the water, and suddenly letting them fall intothe Sea againe:

 [= * Galenus. =]

 [=Anthemius. =]

what with his * Burning Glasses: by which he fired their other Shippes afar-of: what, with his other pollicies, deuises, and engines, he somanfully acquit him selfe: that all the Force, courage, and pollicie ofthe Romaines (for a great season) could nothing preuaile, for thewinning of Syracusa. Wherupon, the Romanes named _Archimedes_, _Briareus_, and _Centimanus_. _Zonaras_ maketh mention of one _Proclus_, who so well had perceiued _Archimedes_ Arte of _Menadrie_, and had sowell inuented of his owne, that with his Burning Glasses, [Burning Glasses. ]

being placed vpon the walles of Bysance, he multiplied so the heate ofthe Sunne, and directed the beames of the same against his enemies Nauiewith such force, and so sodeinly (like lightening) that he burned anddestroyed both man and ship. And _Dion_specifieth of _Priscus_, a _Geometricien_ in Bysance, who inuented and vsed sondry Engins, ofForce multiplied: Which was cause, that the _Emperour Seuerus_ pardonedhim, his life, after he had wonne Bysance: Bycause he honored the Arte, wytt, and rare industrie of _Priscus_. But nothing inferior to theinuention of these engines of Force, was the inuention of Gunnes. 

 [Gunnes. ]

Which, from an English man, had the occasion and order of firstinuenting: though in an other land, and by other men, it was firstexecuted. And they that should see the record, where the occasion andorder generall, of Gunning, is first discoursed of, would thinke: that, “small thinges, slight, and cõmon: comming to wise mens consideration, and industrious mens handling, may grow to be of force incredible. ”

+‡Hypogeiodie‡, is an Arte Mathematicall, demonstratyng, how, vnder theSphæricall Superficies of the earth, at any depth, to any perpendicularline assigned (whose distance from the perpendicular of the entrance:and the Azimuth, likewise, in respect of the said entrance, is knowen)certaine way may be præscribed and gone: And how, any way aboue theSuperficies of the earth designed, may vnder earth, at any depthlimited, be kept: goyng alwayes, perpendicularly, vnder the way, onearth designed: And, contrarywise, Any way, (straight or croked, ) vnderthe earth, beyng giuen: vppon the vtface, or Superficies of the earth, to Lyne out the same: So, as, from the Centre of the earth, perpendiculars drawen to the Sphæricall Superficies of the earth, shallprecisely fall in the Correspondent pointes of those two wayes. This, with all other Cases and circumstances herein, and appertenances, thisArte demonstrateth. + This Arte, is very ample in varietie ofConclusions: and very profitable sundry wayes to the Common Wealth. Theoccasion of my Inuenting this Arte, was at the request of two Gentlemen, who had a certaine worke (of gaine) vnder ground: and their groundes didioyne ouer the worke: and by reason of the crokednes, diuers depthes, and heithes of the way vnder ground, they were in doubt, and atcontrouersie, vnder whose ground, as then, the worke was. The name onely(before this) was of me published, _De Itinere Subterraneo_: The rest, be at Gods will. For Pioners, Miners, Diggers for Mettalls, Stone, Cole, and for secrete passages vnder ground, betwene place and place (as thisland hath diuerse) and for other purposes, any man may easily perceaue, both the great fruite of this Arte, and also in this Arte, the greataide of Geometrie. 

+‡Hydragogie‡, demonstrateth the possible leading of Water, by Natureslawe, and by artificiall helpe, from any head (being a Spring, standing, or running Water) to any other place assigned. + Long, hath this Artebene in vse: and much thereof written: and very marueilous workestherein, performed: as may yet appeare, in Italy: by the Ruynesremaining of the Aqueductes. In other places, of Riuers leading throughthe Maine land, Nauigable many a Mile. And in other places, of themarueilous forcinges of Water to Ascend. Which all, declare the greatskill, to be required of him, who should in this Arte be perfecte, forall occasions of waters possible leading. To speake of the allowance ofthe Fall, for euery hundred foote: or of the Ventills (if the waterslabour be farre, and great) I neede not: Seing, at hand (about vs) manyexpert men can sufficiently testifie, in effecte, the order: though theDemonstration of the Necessitie thereof, they know not: Nor yet, if theyshould be led, vp and downe, and about Mountaines, from the head of theSpring: and then, a place being assigned: and of them, to be demaunded, how low or high, that last place is, in respecte of the head, from which(so crokedly, and vp and downe) they be come: Perhaps, they would not, or could not, very redily, or nerely assoyle that question. _Geometrie_therefore, is necessary to _Hydragogie_. Of the sundry wayes to forcewater to ascend, eyther by _Tympane_, _Kettell mills_, _Skrue_, _Ctesibike_, or such like: in _Vitruuius_, _Agricola_, (and other, )fully, the maner may appeare. And so, thereby, also be most euident, howthe Artes, of _Pneumatithmie_, _Helicosophie_, _Statike_, _Trochilike_, and _Menadrie_, come to the furniture of this, in Speculation, and tothe Commoditie of the Common Wealth, in practise. 

+‡Horometrie‡, is an Arte Mathematicall, which demõstrateth, how, at alltimes appointed, the precise vsuall denominatiõ of time, may be knowen, for any place assigned. + These wordes, are smoth and plaine easieEnglishe, but the reach of their meaning, is farther, then you wouldelightly imagine. Some part of this Arte, was called in olde time, _Gnomonice_: and of late, _Horologiographia_: and in Englishe, may betermed, _Dialling_. Auncient is the vse, and more auncient, is theInuention. The vse, doth well appeare to haue bene (at the least) abouetwo thousand and three hundred yeare agoe:

 [4. Reg. 20. ]

in * King _Achaz_ Diall, then, by the Sunne, shewing the distinction oftime. By Sunne, Mone, and Sterres, this Dialling may be performed, andthe precise Time of day or night knowen. But the demonstratiuedelineation of these Dialls, of all sortes, requireth good skill, bothof _Astronomie_, and _Geometrie_ Elementall, Sphæricall, Phænomenall, and Conikall. Then, to vse the groundes of the Arte, for any regularSuperficies, in any place offred: and (in any possible apt positiontherof) theron, to describe (all maner of wayes) how, vsuall howers, maybe (by the _Sunnes_ shadow) truely determined: will be found no sleightPainters worke. So to Paint, and prescribe the Sunnes Motion, to thebreadth of a heare. In this Feate (in my youth) I Inuented a way, +Howin any Horizontall, Murall, or Æquinoctiall Diall, &c. At all howers(the Sunne shining) the Signe and Degree ascendent, may be knowen. +Which is a thing very necessary for the Rising of those fixed Sterres:whose Operation in the Ayre, is of great might, euidently. I speake nofurther, of the vse hereof. Bur forasmuch as, Mans affaires requireknowledge of Times & Momentes, when, neither Sunne, Mone, or Sterre, canbe sene: Therefore, by Industrie Mechanicall, was inuented, first, how, by Water, running orderly, the Time and howers might be knowen: whereof, the famous _Ctesibius_, was Inuentor: a man, of _Vitruuius_, to the Skie(iustly) extolled. Then, after that, by Sand running, were howersmeasured: Then, by _Trochilike_ with waight: And of late time, by_Trochilike_ with Spring: without waight. All these, by Sunne or Sterresdirection (in certaine time) require ouersight and reformation, according to the heauenly Æquinoctiall Motion: besides the inæqualitieof their owne Operation. There remayneth (without parabolicall meaningherein) among the Philosophers, [A perpetuall Motion. ]

a more excellent, more commodious, and more marueilous way, then allthese: of hauing the motion of the Primouant (or first æquinoctiallmotion, ) by Nature and Arte, Imitated: which you shall (by furder searchin waightier studyes) hereafter, vnderstand more of. And so, it is tymeto finish this Annotation, of Tymes distinction, vsed in our common, andpriuate affaires: The commoditie wherof, no man would want, that cantell, how to bestow his tyme. 

+‡Zographie‡, is an Arte Mathematicall, which teacheth anddemonstrateth, how, the Intersection of all visuall Pyramides, made byany playne assigned, (the Centre, distance, and lightes, beyngdetermined) may be, by lynes, and due propre colours, represented. + Anotable Arte, is this: and would require a whole Volume, to declare theproperty thereof: and the Commodities ensuyng. Great skill of_Geometrie_, _Arithmetike_, _Perspectiue_, and _Anthropographie_, withmany other particular Artes, hath the _Zographer_, nede of, for hisperfection. For, the most excellent Painter, (who is but the propreMechanicien, & Imitator sensible, of the Zographer) hath atteined tosuch perfection, that Sense of Man and beast, haue iudged thingespainted, to be things naturall, and not artificiall: aliue, and notdead. This Mechanicall Zographer (commonly called the Painter) ismeruailous in his skill: and seemeth to haue a certaine diuine power:As, of frendes absent, to make a frendly, present comfort: yea, and offrendes dead, to giue a continuall, silent presence: not onely with vs, but with our posteritie, for many Ages. And so procedyng, Consider, How, in Winter, he can shew you, the liuely vew of Sommers Ioy, and riches:and in Sommer, exhibite the countenance of Winters dolefull State, andnakednes. Cities, Townes, Fortes, Woodes, Armyes, yea whole Kingdomes(be they neuer so farre, or greate) can he, with ease, bring with him, home (to any mans Iudgement) as Paternes liuely, of the thingesrehearsed. In one little house, can he, enclose (with great pleasure ofthe beholders, ) the portrayture liuely, of all visible Creatures, eitheron earth, or in the earth, liuing: or in the waters lying, Creping, slyding, or swimming: or of any foule, or fly, in the ayre flying. Nay, in respect of the Starres, the Skie, the Cloudes: yea, in the shew ofthe very light it selfe (that Diuine Creature) can he match our eyesIudgement, most nerely. What a thing is this? thinges not yet being, hecan represent so, as, at their being, the Picture shall seame (in maner)to haue Created them. To what Artificer, is not Picture, a greatpleasure and Commoditie? Which of them all, will refuse the Directionand ayde of Picture? The Architect, the Goldsmith, and the Arras Weauer:of Picture, make great account. Our liuely Herbals, our portraitures ofbirdes, beastes, and fishes: and our curious Anatomies, which way, arethey most perfectly made, or with most pleasure, of vs beholden? Is itnot, by Picture onely? And if Picture, by the Industry of the Painter, be thus commodious and meruailous: what shall be thought of _Zographie_, the Scholemaster of Picture, and chief gouernor? Though I mencion not_Sculpture_, in my Table of Artes Mathematicall: yet may all menperceiue, How, that _Picture_ and _Sculpture_, are Sisters germaine: andboth, right profitable, in a Commõ wealth. And of _Sculpture_, aswell asof Picture, excellent Artificers haue written great bokes incommendation. Witnesse I take, of _Georgio Vasari_, _Pittore Aretino_:of _Pomponius Gauricus_: and other. To these two Artes, (with other, ) isa certaine od Arte, called _Althalmasat_, much beholdyng: more, then thecommon _Sculptor_, _Entayler_, _Keruer_, _Cutter_, _Grauer_, _Founder_, or _Paynter (&c)_ know their Arte, to be commodious. 

 [An objection. ]

+‡Architecture‡+, to many may seme not worthy, or not mete, to bereckned among the _Artes Mathematicall_. To whom, I thinke good, to giuesome account of my so doyng. Not worthy, (will they say, ) bycause it isbut for building, of a house, Pallace, Church, Forte, or such like, grosse workes. And you, also, defined the _Artes Mathematicall_, to besuch, as dealed with no Materiall or corruptible thing: and also diddemonstratiuely procede in their faculty, by Number or Magnitude. First, [The Answer. ]

you see, that I count, here, _Architecture_, among those _ArtesMathematicall_, which are Deriued from the Principals: and you know, that such, may deale with Naturall thinges, and sensible matter. Ofwhich, “some draw nerer, to the Simple and absolute MathematicallSpeculation, then other do. 

 [☞]

And though, the _Architect_ procureth, enformeth, & directeth, the_Mechanicien_, to handworke, & the building actuall, of house, Castell, or Pallace, and is chief Iudge of the same: yet, with him selfe (aschief _Master_ and _Architect_, ) remaineth the Demonstratiue reason andcause, of the Mechaniciens worke: in Lyne, plaine, and Solid: by_Geometricall_, _Arithmeticall_, _Opticall_, _Musicall_, _Astronomicall_, _Cosmographicall_” (& to be brief) by all the formerDeriued _Artes Mathematicall_, and other Naturall Artes, hable to beconfirmed and stablished. If this be so: then, may you thinke, that_Architecture_, hath good and due allowance, in this honest Company of_Artes Mathematicall_ Deriuatiue. I will, herein, craue Iudgement of twomost perfect _Architectes_: the one, being _Vitruuius_, the Romaine: whodid write ten bookes thereof, to the Emperour _Augustus_ (in whose daiesour Heauenly Archemaster, was borne): and the other, _Leo BaptistaAlbertus_, a Florentine: who also published ten bookes therof. _Architectura_ (sayth _Vitruuius_) _est Scientia pluribus disciplinis &varijs eruditionibus ornata: cuius Iudicio probantur omnia, quæ abcæteris Artificibus perficiuntur opera. _ That is. +Architecture, is aScience garnished with many doctrines & diuerse instructions: by whoseIudgement, all workes, by other workmen finished, are Iudged. + Itfolloweth. _Ea nascitur ex Fabrica, & Ratiocinatione. &c. Ratiocinatioautem est, quæ, res fabricatas, Solertia ac ratione proportionis, demonstrare at[que] explicare potest. +Architecture, groweth of Framing, and Reasoning. &c. Reasoning, is that, which of thinges framed, withforecast, and proportion: can make demonstration, and manifestdeclaration. +_ Againe. _Cùm, in omnibus enim rebus, tùm maximè etiam inArchitectura, hæc duo insunt: quod significatur, & quod significat. Significatur proposita res, de qua dicitur: hanc autem SignificatDemonstratio, rationibus doctrinarum explicata. +Forasmuch as, in allthinges: therefore chiefly in Architecture, these two thinges are: thething signified: and that which signifieth. The thing propounded, whereof we speake, is the thing Signified. But Demonstration, expressedwith the reasons of diuerse doctrines, doth signifie the same thing. +_After that. _Vt literatus sit, peritus Graphidos, eruditus Geometriæ, &Optices non ignarus: instructus Arithmetica: historias compluresnouerit, Philosophos diligenter audiuerit: Musicam sciuerit: Medicinænon sit ignarus, responsa Iurisperitorũ nouerit: Astrologiam, Cæli[que]rationes cognitas habeat. +An Architect+_ (sayth he) +_ought tovnderstand Languages, to be skilfull of Painting, well instructed inGeometrie, not ignorant of Perspectiue, furnished with Arithmetike, haueknowledge of many histories, and diligently haue heard Philosophers, haue skill of Musike, not ignorant of Physike, know the aunsweres ofLawyers, and haue Astronomie, and the courses Cælestiall, in goodknowledge. _+ He geueth reason, orderly, wherefore all these Artes, Doctrines, and Instructions, are requisite in an excellent _Architect_. And (for breuitie) omitting the Latin text, thus he hath. +_Secondly, itis behofefull for an Architect to haue the knowledge of Painting: thathe may the more easilie fashion out, in patternes painted, the forme ofwhat worke he liketh. And Geometrie, geueth to Architecture many helpes:and first teacheth the Vse of the Rule, and the Cumpasse: wherby(chiefly and easilie) the descriptions of Buildinges, are despatched inGroundplats: and the directions of Squires, Leuells, and Lines. Likewise, by Perspectiue, the Lightes of the heauen, are well led, inthe buildinges: from certaine quarters of the world. By Arithmetike, thecharges of Buildinges are summed together: the measures are expressed, and the hard questions of Symmetries, are by Geometricall Meanes andMethods discoursed on. &c. Besides this, of the Nature of thinges (whichin Greke is called φυσιολογία) Philosophie doth make declaration. Which, it is necessary, for an Architect, with diligence to haue learned:because it hath many and diuers naturall questions: as specially, inAqueductes. For in their courses, leadinges about, in the leuell ground, and in the mountinges, the naturall Spirites or breathes are ingendreddiuers wayes: The hindrances, which they cause, no man can helpe, buthe, which out of Philosophie, hath learned the originall causes ofthinges. Likewise, who soeuer shall read Ctesibius, or Archimedesbookes, (and of others, who haue written such Rules) can not thinke, asthey do: vnlesse he shall haue receaued of Philosophers, instructions inthese thinges. And Musike he must nedes know: that he may hauevnderstanding, both of Regular and Mathematicall Musike: that he maytemper well his Balistes, Catapultes, and Scorpions. &c. Moreouer, theBrasen Vessels, which in Theatres, are placed by Mathematicall order, inambries, vnder the steppes: and the diuersities of the soundes (whichy^e Grecians call ηχεῖα) are ordred according to Musicall Symphonies &Harmonies: being distributed in y^e Circuites, by Diatessaron, Diapente, and Diapason. That the conuenient voyce, of the players sound, whẽ itcame to these preparations, made in order, there being increased: withy^t increasing, might come more cleare & pleasant, to y^e eares of thelokers on. &c. And of Astronomie, is knowẽ y^e East, West, South, andNorth. The fashion of the heauen, the Æquinox, the Solsticie, and thecourse of the sterres. Which thinges, vnleast one know: he can notperceiue, any thyng at all, the reason of Horologies. Seyng therforethis ample Science, is garnished, beautified and stored, with so manyand sundry skils and knowledges: I thinke, that none can iustly accountthem selues Architectes, of the suddeyne. But they onely, who from theirchildes yeares, ascendyng by these degrees of knowledges, beyng fosteredvp with the atteynyng of many Languages and Artes, haue wonne to thehigh Tabernacle of Architecture. &c. And to whom Nature hath giuen suchquicke Circumspection, sharpnes of witt, and Memorie, that they may bevery absolutely skillfull in Geometrie, Astronomie, Musike, and the restof the Artes Mathematicall: Such, surmount and passe the callyng, andstate, of Architectes:

 [A Mathematicien. ]

and are become Mathematiciens. &c. And they are found, seldome. As, intymes past, was Aristarchus Samius: Philolaus, and Archytas, Tarentynes:Apollonius Pergęus: Eratosthenes Cyreneus: Archimedes, and Scopas, Syracusians. Who also, left to theyr posteritie, many Engines andGnomonicall workes: by numbers and naturall meanes, inuented anddeclared. _+

Thus much, and the same wordes (in sense) in one onely Chapter of thisIncõparable _Architect Vitruuius_, shall you finde. And if you should, but take his boke in your hand, and slightly loke thorough it, you wouldsay straight way:

 [Vitruuius. ]

This is _Geometrie_, _Arithmetike_, _Astronomie_, _Musike_, _Anthropographie_, _Hydragogie_, _Horometrie_. _&c_. And (to cõclude)the Storehouse of all workmãship. Now, let vs listen to our other Iudge, our Florentine, _Leo Baptista_: and narrowly consider, how he dothdetermine of _Architecture_. _Sed ante[que] vltra progrediar. &c. +Butbefore I procede any further +_(sayth he) +_I thinke, that I ought toexpresse, what man I would haue to bee allowed an Architect. For, I willnot bryng in place a Carpenter: as though you might Compare him to theChief Masters of other Artes. For the hand of the Carpenter, is theArchitectes Instrument. _+

 [VVho is an Architect. ]

+_But I will appoint the Architect to be “that man, who hath the skill, (by a certaine and meruailous meanes and way, ) both in minde andImagination to determine and also in worke to finish: what workes soeuer, by motion of waight, and cuppling and framyng together of bodyes, may most aptly be Commodious for the worthiest Vses of Man. ” And that hemay be able to performe these thinges, he hath nede of atteynyng andknowledge of the best, and most worthy thynges. &c. The whole Feate ofArchitecture in buildyng, consisteth in Lineamentes, and in Framyng. Andthe whole power and skill of Lineamentes, tendeth to this: that theright and absolute way may be had, of Coaptyng and ioyning Lines andangles: by which, the face of the buildyng or frame, may be comprehendedand concluded. And it is the property of Lineamentes, to prescribe vntobuildynges, and euery part of them, an apt place, & certaine nũber:a worthy maner, and a semely order: that, so, y^e whole forme and figureof the buildyng, may rest in the very Lineamentes. &c. And we mayprescribe in mynde and imagination the whole formes, *

 [* The Immaterialitie of perfect Architecture. ]

all material stuffe beyng secluded. Which point we shall atteyne, byNotyng and forepointyng the angles, and lines, by a sure and certainedirection and connexion. Seyng then, these thinges, are thus:_+

 [What, Lineament is. ]

+_Lineamente, shalbe the certaine and constant prescribyng, conceiued inmynde: made in lines and angles: and finished with a learned minde andwyt. _+ “We thanke you Master _Baptist_, that you haue so aptly broughtyour Arte, and phrase therof, to haue some Mathematicall perfection:

 [Note. ]

by certaine order, nũber, forme, figure, and _Symmetrie_ mentall:” allnaturall & sensible stuffe set a part. Now, then, it is euident, (Gentlereader) how aptely and worthely, I haue preferred _Architecture_, to bebred and fostered vp in the Dominion of the pereles _Princesse_, _Mathematica_: and to be a naturall Subiect of hers. And the name of_Architecture_, is of the principalitie, which this Science hath, aboueall other Artes. And _Plato_ affirmeth, the _Architect_ to be _Master_ouer all, that make any worke. Wherupon, he is neither Smith, norBuilder: nor, separately, any Artificer: but the Hed, the Prouost, theDirecter, and Iudge of all Artificiall workes, and all Artificers. For, the true _Architect_, is hable to teach, Demonstrate, distribute, describe, and Iudge all workes wrought. And he, onely, searcheth out thecauses and reasons of all Artificiall thynges. Thus excellent, is_Architecture_: though few (in our dayes) atteyne thereto: yet may notthe Arte, be otherwise thought on, then in very dede it is worthy. Norwe may not, of auncient Artes, make new and imperfect Definitions in ourdayes: for scarsitie of Artificers: No more, than we may pynche in, theDefinitions of _Wisedome_, or _Honestie_, or of _Frendeshyp_ or of_Iustice_. No more will I consent, to Diminish any whit, of theperfection and dignitie, (by iust cause) allowed to absolute_Architecture_. Vnder the Direction of this Arte, are thre principall, necessary _Mechanicall Artes_. Namely, _Howsing_, _Fortification_, and_Naupegie_. _Howsing_, I vnderstand, both for Diuine Seruice, and Manscommon vsage: publike, and priuate. Of _Fortification_ and _Naupegie_, straunge matter might be told you: But perchaunce, some will be tyred, with this Bederoll, all ready rehearsed: and other some, will nycely nipmy grosse and homely discoursing with you: made in post hast: for feareyou should wante this true and frendly warnyng, and tast giuyng, of the_Power Mathematicall_. Lyfe is short, and vncertaine: Tymes areperilouse: &c. And still the Printer awayting, for my pen staying: Allthese thinges, with farder matter of Ingratefulnes, giue me occasion topasse away, to the other Artes remainyng, with all spede possible. 

+The Arte of ‡Nauigation‡, demonstrateth how, by the shortest good way, by the aptest Directiõ, & in the shortest time, a sufficient Ship, betwene any two places (in passage Nauigable, ) assigned: may becõducted: and in all stormes, & naturall disturbances chauncyng, how, tovse the best possible meanes, whereby to recouer the place firstassigned. + What nede, the _Master Pilote_, hath of other Artes, herebefore recited, it is easie to know: as, of _Hydrographie_, _Astronomie_, _Astrologie_, and _Horometrie_. Presupposing continually, the common Base, and foundacion of all: namely _Arithmetike_ and_Geometrie_. So that, he be hable to vnderstand, and Iudge his ownnecessary Instrumentes, and furniture Necessary: Whether they beperfectly made or no: and also can, (if nede be) make them, hym selfe. As Quadrantes, The Astronomers Ryng, The Astronomers staffe, TheAstrolabe vniuersall. An Hydrographicall Globe. Charts Hydrographicall, true, (not with parallell Meridians). The Common Sea Compas: The Compasof variacion: The Proportionall, and Paradoxall Compasses

 [Anno. 1559. ]

(of me Inuented, for our two Moscouy Master Pilotes, at the request ofthe Company) Clockes with spryng: houre, halfe houre, and three houreSandglasses: & sundry other Instrumẽtes: And also, be hable, on Globe, or Playne to describe the Paradoxall Compasse: and duely to vse thesame, to all maner of purposes, whereto it was inuented. And also, behable to Calculate the Planetes places for all tymes. 

Moreouer, with Sonne Mone or Sterre (or without) be hable to define theLongitude & Latitude of the place, which he is in: So that, theLongitude & Latitude of the place, from which he sayled, be giuen: or byhim, be knowne. Whereto, appertayneth expert meanes, to be certifiedeuer, of the Ships way. &c. And by foreseing the Rising, Settyng, Nonestedyng, or Midnightyng of certaine tempestuous fixed Sterres: ortheir Coniunctions, and Anglynges with the Planetes, &c. He ought tohaue expert coniecture of Stormes, Tempestes, and Spoutes: and such lykeMeteorologicall effectes, daungerous on Sea. For (as _Plato_ sayth, )_Mutationes, opportunitates[que] temporum presentire, non minus reimilitari, quàm Agriculturæ, Nauigationi[que] conuenit. +To foresee thealterations and opportunities of tymes, is conuenient, no lesse to theArt of Warre, then to Husbandry and Nauigation. +_ And besides suchcunnyng meanes, more euident tokens in Sonne and Mone, ought of hym tobe knowen: such as (the Philosophicall Poëte) _Virgilius_ teacheth, inhys _Georgikes_. Where he sayth, [Sidenote: Georgic. 1. ]

 _Sol quo[que] & exoriens & quum se condet in vndas, Signa dabit, Solem certissima signa sequuntur. &c. -------- Nam sæpe videmus, Ipsius in vultu varios errare colores. Cæruleus, pluuiam denunciat, igneus Euros. Sin maculæ incipient rutilo immiscerier igni, Omnia tum pariter vento, nimbis[que] videbis Feruere: non illa quisquam me nocte per altum Ire, ne[que] a terra moueat conuellere funem. &c. Sol tibi signa dabit. Solem quis dicere falsum Audeat? -------- &c. _

And so of Mone, Sterres, Water, Ayre, Fire, Wood, Stones, Birdes, andBeastes, and of many thynges els, a certaine Sympathicall forewarnyngmay be had: sometymes to great pleasure and proffit, both on Sea andLand. Sufficiently, for my present purpose, it doth appeare, by thepremisses, how _Mathematicall_, the _Arte_ of _Nauigation_, is: and howit nedeth and also vseth other _Mathematicall Artes_: And now, if Iwould go about to speake of the manifold Commodities, commyng to thisLand, and others, by Shypps and _Nauigation_, you might thinke, that Icatch at occasions, to vse many wordes, where no nede is. 

Yet, this one thyng may I, (iustly) say. In _Nauigation_, none ought tohaue greater care, to be skillfull, then our English Pylotes. Andperchaunce, Some, would more attempt: And other Some, more willinglywould be aydyng, it they wist certainely, What Priuiledge, God hadendued this Iland with, by reason of Situation, most commodious for_Nauigation_, to Places most Famous & Riche. And though, [* Anno. 1567 S. H. G. ]

(of * Late) a young Gentleman, a Courragious Capitaine, was in a greatreadynes, with good hope, and great causes of persuasion, to haueventured, for a Discouerye, (either _Westerly_, by _Cape de Paramantia_:or _Esterly_, aboue _Noua Zemla_, and the _Cyremisses_) and was, at thevery nere tyme of Attemptyng, called and employed otherwise (both then, and since, ) in great good seruice to his Countrey, as the Irish Rebelshaue * tasted:

 [* Anno. 1569]

Yet, I say, (though the same Gentleman, doo not hereafter, dealetherewith) Some one, or other, should listen to the Matter: and by goodaduise, and discrete Circumspection, by little, and little, wynne to thesufficient knowledge of that +Trade+ and +Voyage+: Which, now, I wouldbe sory, (through Carelesnesse, want of Skill, and Courrage, ) shouldremayne Vnknowne and vnheard of. Seyng, also, we are herein, halfeChallenged, by the learned, by halfe request, published. Therof, verely, might grow Commoditye, to this Land chiefly, and to the rest of theChristen Common wealth, farre passing all riches and worldly Threasure. 

+‡Thaumaturgike‡, is that Art Mathematicall, which giueth certaine orderto make straunge workes, of the sense to be perceiued, and of mengreatly to be wondred at. + By sundry meanes, this _Wonder-worke_ iswrought. Some, by _Pneumatithmie_. As the workes of _Ctesibius_ and_Hero_, Some by waight. Wherof _Timæus_ speaketh. Some, by Stringesstrayned, or Springs, therwith Imitating liuely Motions. Some, by othermeanes, as the Images of Mercurie: and the brasen hed, made by _AlbertusMagnus_, which dyd seme to speake. _Boethius_ was excellent in thesefeates. To whom, _Cassiodorus_ writyng, sayth. +_Your purpose is to knowprofound thynges: and to shew meruayles. By the disposition of yourArte, Metals do low: Diomedes of brasse, doth blow a Trumpet loude:a brasen Serpent hisseth: byrdes made, sing swetely. Small thynges werehearse of you, who can Imitate the heauen. &c. _+ Of the straungeSelfmouyng, which, at Saint Denys, by Paris, [* Anno. 1551]

* I saw, ones or twise (_Orontius_ beyng then with me, in Company) itwere to straunge to tell. But some haue written it. And yet, (I hope) itis there, of other to be sene. And by _Perspectiue_ also straungethinges, are done. As partly (before) I gaue you to vnderstand in_Perspectiue_. As, to see in the Ayre, a loft, the lyuely Image of another man, either walkyng to and fro: or standyng still. Likewise, tocome into an house, and there to see the liuely shew of Gold, Siluer orprecious stones: and commyng to take them in your hand, to finde noughtbut Ayre. Hereby, haue some men (in all other matters counted wise)fouly ouershot thẽ selues: misdeaming of the meanes. Therfore sayd_Claudius Cælestinus_. 

 [De his quæ Mundo mirabiliter eueniunt. Cap. 8. ]

_Hodie magnæ literaturæ viros & magna reputationis videmus, opera quedamquasi miranda, supra Naturã putare: de quibus in Perspectiua doctuscausam faciliter reddidisset. _ That is. +_Now a dayes, we see some men, yea of great learnyng and reputation, to Iudge certain workes asmeruaylous, aboue the power of Nature: Of which workes, one that wereskillfull in Perspectiue might easely haue giuen the Cause. _+ Of_Archimedes Sphære_, _Cicero_ witnesseth. 

 [Tusc. 1. ]

Which is very straunge to thinke on. +_For when Archimedes_+ (sayth he)+_did fasten in a Sphære, the mouynges of the Sonne, Mone, and of thefiue other Planets, he did, as the God, which (in Timæus of Plato) didmake the world. That, one turnyng, should rule motions most vnlike inslownes, and swiftnes. _+ But a greater cause of meruayling we haue by_Claudianus_ report hereof. Who affirmeth this _Archimedes worke_, tohaue ben of Glasse. And discourseth of it more at large: which I omit. The Doue of wood, which the _Mathematicien Archytas_ did make to flye, is by _Agellius_ spoken of. Of _Dædalus_ straunge Images, _Plato_reporteth. _Homere_ of _Vulcans Selfmouers_, (by secret wheles) leauethin writyng. _Aristotle_, in hys _Politikes_, of both, maketh mention. Meruaylous was the workemanshyp, of late dayes, performed by good skillof _Trochilike. &c. _ For in Noremberge, A flye of Iern, beyng let out ofthe Artificers hand, did (as it were) fly about by the gestes, at thetable, and at length, as though it were weary, retourne to his mastershand agayne. Moreouer, an Artificiall Egle, was ordred, to fly out ofthe same Towne, a mighty way, and that a loft in the Ayre, toward theEmperour comming thether: and followed hym, beyng come to the gate ofthe towne. *

 [* ☞]

Thus, you see, what, Arte Mathematicall can performe, when Skill, will, Industry, and Hability, are duely applyed to profe. 

 [A Digression. ]

And for these, and such like marueilous Actes and Feates, Naturally, Mathematically, and Mechanically, wrought and contriued:

 [Apologeticall. ]

ought any honest Student, and Modest Christian Philosopher, be counted, & called a +Coniurer+? Shall the folly of Idiotes, and the Mallice ofthe Scornfull, so much preuaile, that He, who seeketh no worldly gaineor glory at their handes: But onely, of God, the threasor of heauenlywisedome, & knowledge of pure veritie: Shall he (I say) in the meanespace, be robbed and spoiled of his honest name and fame? He that seketh(by S. Paules aduertisement) in the Creatures Properties, and wonderfullvertues, to finde iuste cause, to glorifie the Æternall, and AlmightieCreator by: Shall that man, be (in hugger mugger) condemned, as aCompanion of the Helhoundes, and a Caller, and Coniurer of wicked anddamned Spirites? He that bewaileth his great want of time, sufficient(to his contentation) for learning of Godly wisdome, and Godly Veritiesin: and onely therin setteth all his delight: Will that mã leese andabuse his time, in dealing with the Chiefe enemie of Christ our Redemer:the deadly foe of all mankinde: the subtile and impudent peruerter ofGodly Veritie: the Hypocriticall Crocodile: the Enuious Basiliske, continually desirous, in the twinke of an eye, to destroy all Mankinde, both in Body and Soule, æternally? Surely (for my part, somewhat to sayherein) I haue not learned to make so brutish, and so wicked a Bargaine. Should I, for my xx. Or xxv. Yeares Studie: for two or three thousandMarkes spending: seuen or eight thousand Miles going and trauailing, onely for good learninges sake: And that, in all maner of wethers: inall maner of waies and passages: both early and late: in daunger ofviolence by man: in daunger of destruction by wilde beastes: in hunger:in thirst: in perilous heates by day, with toyle on foote: in daungerousdampes of colde, by night, almost bereuing life: (as God knoweth): withlodginges, oft times, to small ease: and somtime to lesse securitie. Andfor much more (then all this) done & suffred, for Learning and attainingof Wisedome: Should I (I pray you) for all this, no otherwise, nor morewarily: or (by Gods mercifulnes) no more luckily, haue fished, with solarge, and costly, a Nette, so long time in drawing (and that with thehelpe and aduise of Lady Philosophie, & Queene Theologie): but atlength, to haue catched, and drawen vp, * a Frog?

 [* A prouerb. Fayre fisht, and caught a Frog. ]

Nay, a Deuill? For, so, doth the Common peuish Pratler Imagine andIangle: And, so, doth the Malicious skorner, secretly wishe, & brauelyand boldly face down, behinde my backe. Ah, what a miserable thing, isthis kinde of Men? How great is the blindnes & boldnes, of theMultitude, in thinges aboue their Capacitie? What a Land: what a People:what Maners: what Times are these? Are they become Deuils, them selues:and, by false witnesse bearing against their Neighbour, would they also, become Murderers? Doth God, so long geue them respite, to reclaime themselues in, from this horrible slaundering of the giltlesse: contrary totheir owne Consciences: and yet will they not cease? Doth the Innocent, forbeare the calling of them, Iuridically to aunswere him, according tothe rigour of the Lawes: and will they despise his Charitable pacience?As they, against him, by name, do forge, fable, rage, and raiseslaunder, by Worde & Print: Will they prouoke him, by worde and Print, likewise, to Note their Names to the World: with their particulardeuises, fables, beastly Imaginations, and vnchristen-like slaunders?Well: Well. O (you such) my vnkinde Countrey men. O vnnaturall Countreymen. O vnthankfull Countrey men. O Brainsicke, Rashe, Spitefull, andDisdainfull Countrey men. Why oppresse you me, thus violently, with yourslaundering of me: Contrary to Veritie: and contrary to your owneConsciences? And I, to this hower, neither by worde, deede, or thought, haue bene, any way, hurtfull, damageable, or iniurious to you, or yours?Haue I, so long, so dearly, so farre, so carefully, so painfully, sodaungerously sought & trauailed for the learning of Wisedome, &atteyning of Vertue: And in the end (in your iudgemẽt) am I become, worse, then when I begã? Worse, thẽ a Mad man? A dangerous Member in theCommon Wealth: and no Member of the Church of Christ? Call you this, tobe Learned? Call you this, to be a Philosopher? and a louer of Wisedome?To forsake the straight heauenly way: and to wallow in the broad way ofdamnation? To forsake the light of heauenly Wisedome: and to lurke inthe dungeon of the Prince of darkenesse? To forsake the Veritie of God, & his Creatures: and to fawne vpon the Impudent, Craftie, ObstinateLier, and continuall disgracer of Gods Veritie, to the vttermost of hispower? To forsake the Life & Blisse Æternall: and to cleaue vnto theAuthor of Death euerlasting? that Murderous Tyrant, most gredilyawaiting the Pray of Mans Soule? Well: I thanke God and our Lorde IesusChrist, for the Comfort which I haue by the Examples of other men, before my time: To whom, neither in godlines of life, nor in perfectionof learning, I am worthy to be compared: and yet, they sustained thevery like Iniuries, that I do: or rather, greater. Pacient _Socrates_, his _Apologie_ will testifie: _Apuleius_ his _Apologies_, will declarethe Brutishnesse of the Multitude. _Ioannes Picus_, Earle of Mirandula, his _Apologie_ will teach you, of the Raging slaunder of the MaliciousIgnorant against him. _Ioannes Trithemius_, his _Apologie_ willspecifie, how he had occasion to make publike Protestation: as well byreason of the Rude Simple: as also, in respect of such, as were countedto be of the wisest sort of men. “Many could I recite: But I deferre theprecise and determined handling of this matter: being loth to detect theFolly & Mallice of my Natiue Countrey men. *

 [* ☞]

Who, so hardly, can disgest or like any extraordinary course ofPhilosophicall Studies: not falling within the Cumpasse of theirCapacitie: or where they are not made priuie of the true and secretecause, of such wonderfull Philosophicall Feates. ” These men, are offower sortes, chiefly. The first, I may name, _Vaine pratling busiebodies_: The second, _Fond Frendes_: The third, _Imperfectly zelous_:and the fourth, _Malicious Ignorant_. To eche of these (briefly, and incharitie) I will say a word or two, and so returne to my Præface. 

 [1. ]

_Vaine pratling busie bodies_, vse your idle assemblies, andconferences, otherwise, then in talke of matter, either aboue yourCapacities, for hardnesse: or contrary to your Consciences, in Veritie. 

 [2. ]

_Fonde Frendes_, leaue of, so to commend your vnacquainted frend, vponblinde affection: As, because he knoweth more, then the common Student:that, therfore, he must needes be skilfull, and a doer, in such matterand maner, as you terme _Coniuring_. Weening, thereby, you aduaunce hisfame: and that you make other men, great marueilers of your hap, to hauesuch a learned frend. Cease to ascribe Impietie, where you pretendAmitie. For, if your tounges were true, then were that your frend, _Vntrue_, both to God, and his Soueraigne. Such _Frendes_ and_Fondlinges_, I shake of, and renounce you: Shake you of, your Folly. 

 [3. ]

_Imperfectly zelous_, to you, do I say: that (perhaps) well, do youMeane: But farre you misse the Marke: If a Lambe you will kill, to feedethe flocke with his bloud. Sheepe, with Lambes bloud, haue no naturallsustenaunce: No more, is Christes flocke, with horrible slaunders, duelyædified. Nor your faire pretense, by such rashe ragged Rhetorike, anywhit, well graced. But such, as so vse me, will finde a fowle Cracke intheir Credite. Speake that you know: And know, as you ought: Know not, by Heare say, when life lieth in daunger. Search to the quicke, & letCharitie be your guide. 

 [4. ]

_Malicious Ignorant_, what shall I say to thee? _Prohibe linguam tuam amalo. A detractione parcite linguæ. +Cause thy toung to refraine frõeuill. Refraine your toung from slaunder. +_ Though your tounges besharpned, Serpent like, & Adders poyson lye in your lippes:

 [Psal. 140. ]

yet take heede, and thinke, betimes, with your selfe, _Vir linguosus nonstabilietur in terra. Virum violentum venabitur malum, donecpræcipitetur. _ For, sure I am, _Quia faciet Dominus Iudicium afflicti:& vindictam pauperum. _

Thus, I require you, my assured frendes, and Countrey men (youMathematiciens, Mechaniciens, and Philosophers, Charitable and discrete)to deale in my behalf, with the light & vntrue tounged, my enuiousAduersaries, or Fond frends. And farther, I would wishe, that at leysor, you would consider, how _Basilius Magnus_, layeth _Moses_ and _Daniel_, before the eyes of those, which count all such Studies Philosophicall(as mine hath bene) to be vngodly, or vnprofitable. Waye well_S. Stephen_ his witnesse of _Moses_. 

 [Act. 7. C. ]

_Eruditus est Moses omni Sapientia Ægyptiorũ: & erat potens in verbis &operibus suis. +Moses was instructed in all maner of wisedome of theÆgyptians: and he was of power both in his wordes, and workes. +_ You seethis Philosophicall Power & Wisedome, which _Moses_ had, to be nothingmisliked of the Holy Ghost. Yet _Plinius_ hath recorded, _Moses_ to be awicked _Magicien_. And that (of force) must be, either for thisPhilosophicall wisedome, learned, before his calling to the leading ofthe Children of _Israel_: or for those his wonders, wrought before King_Pharao_, after he had the conducting of the _Israelites_. As concerningthe first, you perceaue, how _S. Stephen_, at his Martyrdome (being fullof the Holy Ghost) in his Recapitulation of the olde Testament, hathmade mention of _Moses_ Philosophie: with good liking of it: And_Basilius Magnus_ also, auoucheth it, to haue bene to _Moses_ profitable(and therefore, I say, to the Church of God, necessary). But ascõcerning _Moses_ wonders, done before King _Pharao_: God, him selfe, sayd: _Vide vt omnia ostenta, quæ posui in manu tua, facias coramPharaone. +See that thou do all those wonders before Pharao, which Ihaue put in thy hand. +_ Thus, you euidently perceaue, how rashly, _Plinius_ hath slaundered _Moses_, [Lib. 30. Cap. 1. ]

of vayne fraudulent _Magike_, saying: _Est & alia Magices Factio, a Mose, Iamne, & Iotape, Iudæis pendens: sed multis millibus annorumpost Zoroastrem. &c. _

 [1. ]

Let all such, therefore, who, in Iudgement and Skill of Philosophie, arefarre Inferior to _Plinie_, “take good heede, least they ouershoote themselues rashly, ” in

 [☞]

Iudging of _Philosophers straunge Actes_: and the Meanes, how they aredone. 

 [2. ]

But, much more, ought they to beware of forging, deuising, and imaginingmonstrous feates, and wonderfull workes, when and where, no such weredone: no, not any sparke or likelihode, of such, as they, without allshame, do report. 

 [3. ]

And (to conclude) most of all, let them be ashamed of Man, and afraideof the dreadfull and Iuste Iudge: both Folishly or Maliciously todeuise: and then, deuilishly to father their new fond Monsters on me:Innocent, in hand and hart: for trespacing either against the lawe ofGod, or Man, in any my Studies or Exercises, Philosophicall, orMathematicall: As in due time, I hope, will be more manifest. 

Now end I, with +‡Archemastrie‡+. Which name, is not so new, as thisArte is rare. For an other Arte, vnder this, a degree (for skill andpower) hath bene indued with this English name before. And yet, this, may serue for our purpose, sufficiently, at this present. +This Arte, teacheth to bryng to actuall experience sensible, all worthy conclusionsby all the Artes Mathematicall purposed, & by true Naturall Philosophieconcluded: & both addeth to them a farder scope, in the termes of thesame Artes, & also by hys propre Method, and in peculier termes, procedeth, with helpe of the foresayd Artes, to the performance ofcomplet Experiẽces, which of no particular Art, are hable (Formally) tobe challenged. + If you remember, how we considered _Architecture_, inrespect of all common handworkes: some light may you haue, therby, tovnderstand the Souerainty and propertie of this Science. _Science_ I maycall it, rather, then an Arte: for the excellency and Mastershyp ithath, ouer so many, and so mighty Artes and Sciences. And bycause itprocedeth by _Experiences_, and searcheth forth the causes ofConclusions, by _Experiences_: and also putteth the Conclusions themselues, in _Experience_, it is named of some, _Scientia Experimentalis_. The +_Experimentall Science_+. _Nicolaus Cusanus_ termeth it so, in hys_Experimentes Statikall_, And an other _Philosopher_, [R. B. ]

of this land Natiue (the floure of whose worthy fame, can neuer dye norwither) did write therof largely, at the request of _Clement the sixt_. The Arte carrieth with it, a wonderfull Credit: By reason, itcertefieth, sensibly, fully, and completely to the vtmost power ofNature, and Arte. This Arte, certifieth by _Experience_ complete andabsolute: and other Artes, with their Argumentes, and Demonstrations, persuade: and in wordes, proue very well their Conclusions. *

 [☞]

But wordes, and Argumentes, are no sensible certifying: nor the full andfinall frute of Sciences practisable. And though some Artes, haue inthem, _Experiences_, yet they are not complete, and brought to thevttermost, they may be stretched vnto, and applyed sensibly. As forexample: the Naturall Philosopher disputeth and maketh goodly shew ofreason: And the Astronomer, and the Opticall Mechanicien, put somethynges in _Experience_: but neither, all, that they may: nor yetsufficiently, and to the vtmost, those, which they do, There, then, the_Archemaster_ steppeth in, and leadeth forth on, the _Experiences_, byorder of his doctrine _Experimentall_, to the chief and finall power ofNaturall and Mathematicall Artes. Of two or three men, in whom, thisDescription of _Archemastry_ was _Experimentally_, verified, I haue readand hard: and good record, is of their such perfection. So that, thisArt, is no fantasticall Imagination: as some Sophister, might, _Cum suisInsolubilibus_, make a florish: and dassell your Imagination: and dashyour honest desire and Courage, from beleuing these thinges, so vnheardof, so meruaylous, & of such Importance. Well: as you will. I haueforewarned you. I haue done the part of a frende: I haue discharged myDuety toward God: for my small Talent, at hys most mercyfull handesreceiued. To this Science, doth the _Science Alnirangiat_, greatSeruice. Muse nothyng of this name. I chaunge not the name, so vsed, andin Print published by other: beyng a name, propre to the Science. Vnderthis, commeth _Ars Sintrillia_, by _Artephius_, briefly written. But thechief Science, of the Archemaster, (in this world) as yet knowen, is another (as it were) OPTICAL Science: wherof, the name shall be told (Godwillyng) when I shall haue some, (more iust) occasion, therof, toDiscourse. 

Here, I must end, thus abruptly (Gentle frende, and vnfayned louer ofhonest and necessary verities. ) For, they, who haue (for your sake, andvertues cause) requested me, (an old forworne Mathematicien) to take penin hand: (through the confidence they reposed in my long experience: andtryed sincerity) for the declaryng and reportyng somewhat, of the fruteand commodity, by the +Artes Mathematicall, to be atteyned vnto+: euenthey, Sore agaynst their willes, are forced, for sundry causes, tosatisfie the workemans request, in endyng forthwith: He, so feareththis, so new an attempt, & so costly: And in matter so slenderly(hetherto) among the common Sorte of Studentes, considered or estemed. 

And where I was willed, somewhat to alledge, why, in our vulgare Speche, this part of the Principall Science of _Geometrie_, called _EuclidesGeometricall Elementes_, is published, to your handlyng: being vnlatinedpeople, and not Vniuersitie Scholers: Verily, I thinke it nedelesse. 

 [1. ]

For, the Honour, and Estimation of the +Vniuersities, and Graduates+, is, hereby, nothing diminished. Seing, from, and by their NurseChildren, you receaue all this Benefite: how great soeuer it be. 

 [2. ]

Neither are their Studies, hereby, any whit hindred. No more, then theItalian _Vniuersities_, as _Academia Bononiensis_, _Ferrariensis_, _Florentina_, _Mediolanensis_, _Patauina_, _Papiensis_, _Perusina_, _Pisana_, _Romana_, _Senensis_, or any one of them, finde them selues, any deale, disgraced, or their Studies any thing hindred, by _FraterLucas de Burgo_, or by _Nicolaus Tartalea_, who in vulgar Italianlanguage, haue published, not onely _Euclides Geometrie_, but of_Archimedes_ somewhat: and in Arithmetike and Practicall Geometrie, verylarge volumes, all in their vulgar speche. Nor in Germany haue thefamous _Vniuersities_, any thing bene discontent with _AlbertusDurerus_, his Geometricall Institutions in Dutch: or with _GulielmusXylander_, his learned translation of the first sixe bookes of_Euclide_, out of the Greke into the high Dutch. Nor with _Gualterus H. Riffius_, his Geometricall Volume: very diligently translated into thehigh Dutch tounge, and published. Nor yet the _Vniuersities_ of Spaine, or Portugall, thinke their reputation to be decayed: or suppose anytheir Studies to be hindred by the Excellent _P. Nonnius_, hisMathematicall workes, in vulgare speche by him put forth. Haue you not, likewise, in the French tounge, the whole Mathematicall Quadriuie? andyet neither Paris, Orleance, or any of the other Vniuersities ofFraunce, at any time, with the Translaters, or Publishers offended: orany mans Studie thereby hindred?

 [3. ]

And surely, the Common and Vulgar Scholer (much more, the Gramarian)before his comming to the _Vniuersitie_, shall (or may) be, now(according to _Plato_ his Counsell) sufficiently instructed in_Arithmetike_ and _Geometrie_, for the better and easier learning of allmaner of _Philosophie_, _Academicall_, or _Peripateticall_. And by thatmeanes, goe more cherefully, more skilfully, and spedily forwarde, inhis Studies, there to be learned. And, so, in lesse time, profite more, then (otherwise) he should, or could do. 

 [4. ]

Also many good and pregnant Englishe wittes, of young Gentlemen, and ofother, who neuer intend to meddle with the profound search and Studie ofPhilosophie (in the _Vniuersities_ to be learned) may neuerthelesse, now, with more ease and libertie, haue good occasion, vertuously tooccupie the sharpnesse of their wittes: where, els (perchance)otherwise, they would in fond exercises, spend (or rather leese) theirtime: neither seruing God: nor furdering the Weale, common or priuate. 

 [5. ]

And great Comfort, with good hope, may the _Vniuersities_ haue, byreason of this _Englishe_ +Geometrie, and Mathematicall Præface+, thatthey (hereafter) shall be the more regarded, esteemed, and resortedvnto. For, when it shall be knowen and reported, that of the_Mathematicall Sciences_ onely, such great Commodities are ensuing (as Ihaue specified): and that in dede, some of you vnlatined Studentes, canbe good witnesse, of such rare fruite by you enioyed (thereby): aseither, before this, was not heard of: or els, not so fully credited:“Well, may all men coniecture, that farre greater ayde, and betterfurniture, to winne to the Perfection of all Philosophie, [Vniuersities. ]

may in the Vniuersities be had: being the Storehouses & Threasory of allSciences, [☞]

and all Artes, necessary for the best, and most noble State of CommonWealthes. ”

 [6. ]

Besides this, how many a Common Artificer, is there, in these Realmes ofEngland and Ireland, that dealeth with Numbers, Rule, & Cumpasse: Who, with their owne Skill and experience, already had, will be hable (bythese good helpes and informations) to finde out, and deuise, newworkes, straunge Engines, and Instrumentes: for sundry purposes in theCommon Wealth? or for priuate pleasure? and for the better maintayningof their owne estate? I will not (therefore) fight against myne owneshadowe. For, no man (I am sure) will open his mouth against thisEnterprise. No mã (I say) who either hath Charitie toward his brother(and would be glad of his furtherance in vertuous knowledge): or thathath any care & zeale for the bettering of the Cõmon state of thisRealme. Neither any, that make accompt, what the wiser sort of men (Sageand Stayed) do thinke of them. To none (therefore) will I make any_Apologie, _ for a vertuous acte doing: and for cõmending, or settingforth, Profitable Artes to English men, in the English toung. “But, vntoGod our Creator, let vs all be thankefull: for that, +_As he, of hisGoodnes, by his Powre, and in his wisedome, [☞]

hath Created all thynges, in Number, Waight, and Measure_+: So, to vs, of hys great Mercy, he hath reuealed Meanes, whereby, to atteyne thesufficient and necessary knowledge of the foresayd hys three principallInstrumentes: Which Meanes, I haue abundantly proued vnto you, to be the_Sciences_ and _Artes Mathematicall_. ”

And though I haue ben pinched with straightnes of tyme: that, no way, I could so pen downe the matter (in my Mynde) as I determined: hopyng ofconuenient laysure: Yet. If vertuous zeale, and honest Intent prouokeand bryng you to the readyng and examinyng of this Compendious treatise, I do not doute, but, as the veritie therof (accordyng to our purpose)will be euident vnto you: So the pith and force therof, will persuadeyou: and the wonderfull frute therof, highly pleasure you. And that youmay the easier perceiue, and better remember, the principall pointes, whereof my Preface treateth, [The Ground platt of this Præface in a Table. ]

I will giue you the +Groundplatt+ of my whole discourse, in a Tableannexed: from the first to the last, somewhat Methodically contriued. 

If Hast, hath caused my poore pen, any where, to stumble: You will, (I am sure) in part of recompence, (for my earnest and sincere good will to pleasure you), Consider the rockish huge mountaines, and the perilous vnbeaten wayes, which (both night and day, for the while) it hath toyled and labored through, to bryng you this good Newes, and Comfortable profe, of Vertues frute. 

 So, I Commit you vnto Gods Mercyfull direction, for the rest: hartely besechyng hym, to prosper your Studyes, and honest Intentes: to his Glory, & the Commodity of our Countrey. _Amen_. 

 _Written at my poore House At Mortlake. _

 _Anno. 1570. February. 9. _

 [Decoration]

 [Transcriber’s Note:

 The “Groundplat” was printed in the form of a stemma, or tree, on an oversized fold-out page. The layout was impossible to reproduce for this e-text, so the information has been rearranged in nested-list form. Size markings (see note at beginning of e-text) are relative within each paragraph. ]

 _J. DEE_

 +‡Here haue you (according to my promisse) the Groundplat of‡+ +my MATHEMATICALL Præface: annexed to _Euclide_ (now first)+ published in our Englishe tounge. An. 1570. Febr. 3. 

+‡Sciences, and Artes Mathematicall, ‡ are, either+

 +‡Principall, ‡ which are two, onely, +

 +Arithmetike. +

 +‡Simple‡+, Which dealeth with Numbers onely: and demonstrateth all their properties and appertenances: where, an Vnit, is Indiuisible. +‡Mixt‡+, Which with aide of Geometrie principall, demonstrateth some Arithmeticall Conclusion, or Purpose. 

 +Geometrie. +

 +‡Simple‡+, Which dealeth with Magnitudes, onely: and demonstrateth all their properties, passions, and appertenances: whose Point, is Indiuisible. +‡Mixt‡+, Which with aide of Arithmetike principall, demonstrateth some Geometricall purpose, as +EVCLIDES ELEMENTES+. 

 +‡The vse‡ whereof, is either, +

 In thinges Supernaturall, æternall, & Diuine: By Application, _Ascending_. In thinges Mathematicall: without farther Application. In thinges Naturall: both Substãtiall, & Accidentall, Visible, & Inuisible. &c. By Application: _Descending_. 

 The like Vses and Applications are, (though in a degree lower) in the +Artes Mathematicall Deriuatiue+. 

 +‡Deriuatiue‡ frõ the Principalls: of which, some haue+

 +‡The names of‡ the Principalls: as, +

 +_Arithmetike_, vulgar: which considereth+

 --Arithmetike of most vsuall whole numbers: And of Fractions to them appertaining. --Arithmetike of Proportions. --Arithmetike Circular. --Arithmetike of Radicall Nũbers: Simple, Compound, Mixt: And of their Fractions. --Arithmetike of Cossike Nũbers: with their Fractions: And the great Arte of Algiebar. 

 +_Geometrie_, vulgar: which teacheth Measuring+

 +‡At hand‡+

 All Lengthes. --+Mecometrie. + All Plaines: As, Land, Borde, Glasse, &c. --+Embadometrie. + All Solids: As, Timber, Stone, Vessels, &c. --+Stereometrie. +

 +‡With distãce‡+ from the thing Measured, as, +‡How farre‡+, from the Measurer, any thing is: of him sene, on Land or Water: called +Apomecometrie+. +‡How high or deepe‡+, from the leuell of the Measurers standing, any thing is: Seene of hym, on Land or Water: called +Hypsometrie+. +‡How broad‡+, a thing is, which is in the Measurers view: so it be situated on Land or Water: called +Platometrie+. 

 +‡Of which‡ are growen the Feates & Artes of+

 +Geodesie+: more cunningly to Measure and Suruey Landes, Woods, Waters. &c. +Geographie. + +Chorographie. + +Hydrographie. + +Stratarithmetrie. +

 +‡Propre names‡ as+, +Perspectiue, +--Which demonstrateth the maners and properties of all Radiations: Directe, Broken, and Reflected. 

 +Astronomie, +--Which demonstrateth the Distances, Magnitudes, and all Naturall motions, Apparences, and Passions, proper to the Planets and fixed Starres: for any time, past, present, and to come: in respecte of a certaine Horizon, or without respecte of any Horizon. 

 +Musike, +--Which demonstrateth by reason, and teacheth by sense, perfectly to iudge and order the diuersitie of Soundes, hie or low. 

 +Cosmographie, +--Which, wholy and perfectly maketh description of the Heauenlym and also Elementall part of the World: and of these partes, maketh homologall application, and mutuall collation necessary. 

 +Astrologie, +--Which reasonably demonstrateth the operations and effectes of the naturall beames of light, and secrete Influence of the Planets, and fixed Starres, in euery Element and Elementall body: at all times, in any Horizon assigned. 

 +Statike, +--Which demonstrateth the causes of heauines and lightnes of all thinges: and of the motions and properties to heauines and lightnes belonging. 

 +Anthropographie, + Which describeth the Nũber, Measure, Waight, Figure, Situation, and colour of euery diuers thing contained in the perfecte body of MAN: and geueth certaine knowledge of the Figure, Symmetrie, Waight, Characterization, & due Locall motion of any percell of the said body assigned: and of numbers to the said percell appertaining. 

 +Trochilike, +--Which demonstrateth the properties of all Circular motions: Simple and Compound. 

 +Helicosophie, +--Which demonstrateth the designing of all Spirall lines: in Plaine, on Cylinder, Cone, Sphære, Conoïd, and Sphæroid: and their properties. 

 +Pneumatithmie, +--Which demonstrateth by close hollow Geometricall figures (Regular and Irregular) the straunge properties (in motion or stay) of the Water, Ayre, Smoke, and Fire, in their Continuitie, and as they are ioyned to the Elementes next them. 

 +Menadrie, +--Which demonstrateth, how, aboue Natures Vertue, and power simple: Vertue and force, may be multiplied: and so to directe, to lift, to pull to, and to put or cast fro, any multiplied, or simple determined Vertue, Waight, or Force: naturally, not, so, directible, or moueable. 

 +Hypogeiodie, +--Which demonstrateth, how, vnder the Sphæricall Superficies of the Earth, at any depth, to any perpendicular line assigned (whose distance from the perpendicular of the entrance: and the Azimuth likewise, in respecte of the sayd entrance, is knowen) certaine way, may be prescribed and gone, &c. 

 +Hydragogie, +--Which demonstrateth the possible leading of water by Natures law, and by artificiall helpe, from any head (being Spring, standing, or running water) to any other place assigned. 

 +Horometrie, +--Which demonstrateth, how, at all times appointed, the precise, vsuall denomination of time, may be knowen, for any place assigned. 

 +Zographie, +--Which demonstrateth and teacheth, how, the Intersection of all visuall Pyramids, made by any plaine assigned (the Center, distance, and lightes being determined) may be, by lines, and proper colours represented. 

 +Architecture, +--Which is a Science garnished with many doctrines, and diuers Instructions: by whose iudgement, all workes by other workmen finished, are iudged. 

 +Nauigation, +--Which demonstrateth, how, by the Shortest good way, by the aptest direction, and in the shortest time: a sufficient Shippe, betwene any two places (in passage nauigable) assigned, may be conducted: and in all stormes and naturall disturbances chauncing, how to vse the best possible meanes, to recouer the place first assigned. 

 +Thaumaturgike, +--Which geueth certaine order to make straunge workes, of the sense to be perceiued: and of men greatly to be wondred at. 

 +Archemastrie, +--Which teacheth to bring to actuall experience sensible, all worthy conclusions, by all the Artes Mathematicall purposed: and by true Naturall philosophie, concluded: And both addeth to them a farder Scope, in the termes of the same Artes: and also, by his proper Method, and in peculiar termes, procedeth, with helpe of the forsayd Artes, to the performance of complete Experiences: which, of no particular Arte, are hable (Formally) to be challenged. 

+¶ Imprinted by _Iohn Day_. +

An. 1570. Feb. 25. 

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Errors and Anomalies:

Unless otherwise noted, spelling and punctuation are unchanged. Errors are listed below, with the original form, if changed, shown in[brackets]. Unusual words include “fatch” (probably used as a variantof “fetch”) and the mathematical terms “sexagene” and “sexagesme”. 

 How, worldly goods: how, worldly dignitie [_“o” in second “worldly” invisible_] his most diligent hearers (so infinitely mought [hearers) so] the boundes, and duety of an Hydrographer [Hydographer] of the Grekes it is called _Eteromekes_ [_text unchanged: correct form is “Heteromekes”_] τὸ ὁτὶ [_accent unchanged_] in our worldly affaires [wordly] fall to worke. ❉. [_Some text readers may not display the oversized-asterisk symbol. _] _Emptying the first. _ [Emptyting] Απὸ τάυτης τῆς ἡμέρας, περὶ παντὸς, Αρχιμήδει λέγοντι πιϛευτεόν [ἡμήρας ... πιϛευτέομ] of the suddeyne [snddeyne] that the right and absolute way may be had [he had] Georgic I: [_The quoted segments, each ending in “&c. ”, are 438-439; 451-457; 463-464. _]

Additional Notes:

 The Greek letter η (eta) was consistently printed as if it were the ou-ligature ȣ. The Latin “-que” was written as an abbreviation resembling “-q´;”. It is shown here as [que]. 

 Mathematical symbols seen in the section accompanying the diagrams could not be reproduced. The following substitutions were made: --The curly “P” used for “Pounds” is shown as {P}. --The “potestas” symbol, used to represent “x” (the unknown), is shown as {x}. --All roots were expressed as the “root” sign √ combined with symbols for the power of 2 (doubled for power of 4, or fourth root) and 3. They are shown as ²√ ³√ ⁴√. 

Euclid:

The following Propositions were identified by number. 

6. 12: (How) to find a fourth (line) proportional to three given straightlines. 

11. 34: In equal parallelepipedal solids the bases are reciprocallyproportional to the heights; and those parallelepipedal solids in whichthe bases are reciprocally proportional to the heights are equal. 

11. 36: If three straight lines are proportional, then theparallelepipedal solid formed out of the three equals theparallelepipedal solid on the mean which is equilateral, but equiangularwith the aforesaid solid. 

12. 1: Similar polygons inscribed in circles are to one another as thesquares on their diameters. 

12. 2: Circles are to one another as the squares on their diameters. 

12. 18 (“last”): Spheres are to one another in triplicate ratio of theirrespective diameters.