addresses delivered before the california teachers association

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Purchase of this book includes free trial access to where you can read more than a million books for free. This is an OCR edition with typos. Excerpt from book: The values of x are represented by lines drawn horizontally from the origin O, to the left or right, from the extremities of which lines are drawn upwards to represent the values of x2. Each pair of lines thus determined by a pair of values x, x2, fixes a point in the plane, and the aggregate of all such points range themselves along the curve known as the parabola. The second quantity x2, it is now customary to represent by a second letter, as y, and to speak of x and y as functions of each other. In a similar manner any algebraic equation between two quantities, x and;)/, has its graph. Now mark the radical departure here taken in the interpretation of algebraic quantities. A straight line stands as the representative of any such quantity, an interpretation wholly repugnant to Greek geometry. So long as his practice conformed to the canon which, from time immemorial, had been his guide in the geometrical interpretation of quantity, x2 could mean for the Greek only a square, an area, never a straight line. But his orthodoxy was the barrier to his further progress, and from the abandonment of that orthodoxy the modern mathematician dates the possibility of achievement beyond the limits of investigation which the Greeks had set for themselves. Henceforth, moreover, the interests of algebra and geometry were one and the same; progress in the one was to mean a simultaneous progress in the other. Twenty-eight years pass, and we stand at the threshold of the greatest of the discoveries in mathematics of modern times—that of the differential calculus. The year is 1665, when Sir Isaac Newton communicated to some of his friends the fundamental ideas of the new method. Taken in connection with the new interpretation of the algebraic equation by Descartes, the scope of this method ...
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