[527]. As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited.
But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection.—Lagrange.
Leçons Élémentaires sur les Mathematiques, Leçon cinquiéme. [McCormack].
[528]. The greatest enemy to true arithmetic work is found in so-called practical or illustrative problems, which are freely given to our pupils, of a degree of difficulty and complexity altogether unsuited to their age and mental development.... I am, myself, no bad mathematician, and all the reasoning powers with which nature endowed me have long been as fully developed as they are ever likely to be; but I have, not infrequently, been puzzled, and at times foiled, by the subtle logical difficulty running through one of these problems, given to my own children. The head-master of one of our Boston high schools confessed to me that he had sometimes been unable to unravel one of these tangled skeins, in trying to help his own daughter through her evening’s work. During this summer, Dr. Fairbairn, the distinguished head of one of the colleges of Oxford, England, told me that not only had he himself encountered a similar difficulty, in the case of his own children, but that, on one occasion, having as his guest one of the first mathematicians of England, the two together had been completely puzzled by one of these arithmetical conundrums.—Walker, F. A.
Discussions in Education (New York, 1899), pp. 253-254.
[529]. It is often assumed that because the young child is not competent to study geometry systematically he need be taught nothing geometrical; that because it would be foolish to present to him physics and mechanics as sciences it is useless to present to him any physical or mechanical principles.
An error of like origin, which has wrought incalculable mischief, denies to the scholar the use of the symbols and methods of algebra in connection with his early essays in numbers because, forsooth, he is not as yet capable of mastering quadratics!... The whole infant generation, wrestling with arithmetic, seek for a sign and groan and travail together in pain for the want of it; but no sign is given them save the sign of the prophet Jonah, the withered gourd, fruitless endeavor, wasted strength.—Walker, F. A.
Industrial Education; Discussions in Education (New York, 1899), p. 132.
[530]. Particular and contingent inventions in the solution of problems, which, though many times more concise than a general method would allow, yet, in my judgment, are less proper to instruct a learner, as acrostics, and such kind of artificial poetry, though never so excellent, would be but improper examples to instruct one that aims at Ovidean poetry.—Newton, Isaac.
Letter to Collins, 1670; Macclesfield, Correspondence of Scientific Men (Oxford, 1841), Vol. 2, p. 307.