No two sciences might seem at first sight more entirely discrete and divergent in their subject matter than geometry and arithmetic, or algebra. The first deals with circles, squares, parallelograms, and various other forms in space; the latter with mere symbols of number, the symbols having form indeed, but bearing a meaning independent of shape or size. Prior to the time of Descartes, too, the sciences actually were developed in a slow and painful manner 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, TiTedos àpioμós, and distinguishes a square number of which the two factors are equal as an equal-sided or plane number, ἰσόπλευρος καὶ ἐπίπεδος ἀριθμός. He also calls the root of a cubic number its side, Tλeupá. 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 us the means of dividing a circle by purely geometric or mechanical means into 2, 3, 4, 5, 6, 8, 10, 12, 15, 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 other hand, algebra advanced beyond geometry, and modes of solving equations were painfully discovered by those who had no notion that at every step they were implicitly solving important

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 some curve or figure in space, and that every bend, point, cusp, or other peculiarity in the curve indicates some peculiarity in the values of the algebraic symbols. It is impossible to describe in any adequate manner the importance of this discovery. The advantage was twofold: algebra aided geometry, and geometry gave reciprocal aid to algebra. Curves such as the long described sections of the cone were found to correspond to quadratic equations of no great difficulty; and it was impossible to manipulate the symbolic 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 strange and, to some extent, unfortunate infatuation in favour of the ancient geometrical methods; but it is well known that he employed symbolic methods to discover his profound truths, and he every now and then, by some accidental use of algebraic expressions, confessed its greater powers and generality.

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Geometry, on the other hand, gave the greatest assistance to algebra, by affording concrete representations of relations which would otherwise be too abstract for comprehension. A curve of no great complexity may give the whole history of the variations of value of a troublesome mathematical expression. As soon as we know, too, that every regular geometrical curve repre

sents some algebraic equation, we are presented by simple observation of many mechanical movements with abundant suggestions towards the discovery of mathematical problems. Every particle of a carriage-wheel when moving on a level road is constantly describing a cycloidal curve, the curious properties of which exercised the ingenuity of all the most skilful mathematicians of the seventeenth century, and led to important advancements in algebraic power. It may well be held even that the discovery of the Differential Calculus is mainly due to geometrical analogy, because mathematicians, in attempting to treat algebraically the tangent of a continuously varying curve, were obliged to entertain the notion of infinitely small quantities d. There can be no doubt that Newton's fluxional, or in fact geometrical mode of stating the differential calculus, however much it subsequently retarded its progress in England, facilitated its apprehension at first, and I should think it almost certain that Newton discovered the calculus geometrically.

We may accordingly look upon this discovery of analogy, this happy alliance, as Bossut calls it, between geometry and algebra, as the chief source of discoveries which have been made for three centuries past in mathematical methods. This is certainly the opinion of no less an authority than Lagrange, who has said, So long as algebra and geometry have been separate, their progress was slow, and their employment limited; but since these two sciences have been united, they have lent each other mutual strength, and have marched together with a rapid step towards perfection.'

The advancement of mechanical science has also been greatly aided by analogy. An abstract and intangible

d Lacroix, Traité Elémentaire de Calcul Différentiel et de Calcul Intégral,' 5 me édit. p. 699.

e Histoire des Mathématiques,' vol. i. p. 298.

existence like force demands much power of conception, but it has a perfect concrete representative in a line, the end of which may denote the point of application, and the direction the line of action of the force, while the length can be made arbitrarily to denote the amount of the force. Nor does the analogy end here; for the moment of the force about any point, or its product into the perpendicular distance of its line of action from the point, is found to be correctly represented by an area, namely twice the area of the triangle contained between the point and the ends of the line representing the force. Of late years a great generalization has been effected; the Double Algebra of De Morgan is true not only of space relations, but of forces, so that the triangle of forces is reduced to a case of pure geometrical addition. Nay, the triangle of lines, the triangle of velocities, the triangle of forces, the triangle of couples, and perhaps other cognate theorems, are reduced by analogy to one simple theorem, which amounts merely to this, that there are two ways of getting from one angular point of a triangle to another, which ways, though different in length, are identical in their final results f. In the wonderful system of quaternions of the late Sir W. R. Hamilton, these analogies are embodied and carried out in the most general manner, so that whatever problem involves the threefold dimensions of space, or relations analogous to those of space, is treated by a symbolic method of the most comprehensive simplicity. Since nearly all physical problems do involve space relations, or those analogous to them, it is difficult to imagine any limits to the work which may be ultimately achieved by this calculus.

It ought to be added that to the discovery of analogy

f See Goodwin, Cambridge Philosophical Transactions' (1845), vol. viii. p. 269. O'Brien, On Symbolical Statics,' Philosophical Magazine, 4th Series, vol. i. pp. 491 &c.

between the forms of mathematical and logical expressions, we undoubtedly owe the greatest recent advance in logical science. Boole based his extension of logical processes entirely upon the notion that logic was an algebra of two quantities, o and 1. His profound genius for the investigation of symbolic methods led him to perceive by analogy that there must exist a general system of logical deduction, of which the old logicians had seized only a few stray fragments. Much mistaken as he was in placing algebra as a higher science than logic, no one can deny that the development of the more complex and dependent science had advanced far beyond that of the simpler science, and that Boole, in drawing attention to the connexion, made one of the most important discoveries in the history of science. As Descartes had wedded algebra and geometry, so did Boole substantially accomplish the marriage of logic and algebra.

Analogy in the Theory of Undulations.

There is no class of phenomena which more thoroughly illustrates alike the power and weakness of analogy than the waves which agitate every kind of medium. All waves, whatsoever be the matter through which they pass, obey certain common principles of rhythmical or harmonic motion, and the subject therefore presents a vast field for mathematical generalization. At the same time each kind of medium may allow of waves peculiar in their conditions, so that it is a beautiful exercise in analogical reasoning to observe how, in making inferences from one kind of medium to another, we must make allowance for difference of circumstances. The waves of the ocean are large and visible, and there are the yet greater tidal waves which extend around the globe. From such palpable cases of rhythmical