Analogy in the Mathematical Sciences.
Whoever wishes to acquire a deep acquaintance with Nature must observe that there are analogies which connect whole branches of science in a parallel manner, and enable us to infer of one class of phenomena what we know of another. It has thus happened on several occasions that the discovery of an unsuspected analogy between two branches of knowledge has been the starting-point for a rapid course of discovery. The truths readily observed in the one may be of a different character from those which present themselves in the other. The analogy, once pointed out, leads us to discover regions of one science yet undeveloped, to which the key is furnished by the corresponding truths in the other science. An interchange of aid most wonderful in its results may thus take place, and at the same time the mind rises to a higher generalisation, and a more comprehensive view of nature.
No two sciences might seem at first sight more different in their subject matter than geometry and algebra. The first deals with circles, squares, parallelograms, and other forms in space; the latter with mere symbols of number. Prior to the time of Descartes, the sciences were developed slowly and painfully in almost entire independence of each other. The Greek philosophers indeed could not avoid noticing occasional analogies, as when Plato in the Thæetetus describes a square number as equally equal, and a number produced by multiplying two unequal factors as oblong. Euclid, in the 7th and 8th books of his Elements, continually uses expressions displaying a consciousness of the same analogies, as when he calls a number of two factors a plane number, ἐπίπεδος ἀριθμός, and distinguishes a square number of which the two factors are equal as an equal-sided and plane number, ἰσόπλευρος καὶ ἐπίπεδος ἀριθμός. He also calls the root of a cubic number its side, πλευρά. In the Diophantine algebra many problems of a geometrical character were solved by algebraic or numerical processes; but there was no general system, so that the solutions were of an isolated character. In general the ancients were far more advanced in geometric than symbolic methods; thus Euclid in his 4th book gives the means of dividing a circle by purely geometric means into 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30 parts, but he was totally unacquainted with the theory of the roots of unity exactly corresponding to this division of the circle.
During the middle ages, on the contrary, algebra advanced beyond geometry, and modes of solving equations were gradually discovered by those who had no notion that at every step they were implicitly solving geometric problems. It is true that Regiomontanus, Tartaglia, Bombelli, and possibly other early algebraists, solved isolated geometrical problems by the aid of algebra, but particular numbers were always used, and no consciousness of a general method was displayed. Vieta in some degree anticipated the final discovery, and occasionally represented the roots of an equation geometrically, but it was reserved for Descartes to show, in the most general manner, that every equation may be represented by a curve or figure in space, and that every bend, point, cusp, or other peculiarity in the curve indicates some peculiarity in the equation. It is impossible to describe in any adequate manner the importance of this discovery. The advantage was two-fold: algebra aided geometry, and geometry gave reciprocal aid to algebra. Curves such as the well-known sections of the cone were found to correspond to quadratic equations; and it was impossible to manipulate the equations without discovering properties of those all-important curves. The way was thus opened for the algebraic treatment of motions and forces, without which Newton’s Principia could never have been worked out. Newton indeed was possessed by a strong infatuation in favour of the ancient geometrical methods; but it is well known that he employed symbolic methods to discover his theorems, and he now and then, by some accidental use of algebraic expression, confessed its greater power and generality.