Geometry, on the other hand, gave great assistance to algebra, by affording concrete representations of relations which would otherwise be too abstract for easy 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 represents some algebraic equation, we are presented by observation of 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 be held that the discovery of the Differential Calculus was mainly due to geometrical analogy, because mathematicians, in attempting to treat algebraically the tangent of a curve, were obliged to entertain the notion of infinitely small quantities.3 There can be no doubt that Newton’s fluxional, that is, 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 principles of the calculus geometrically.
We may accordingly look upon this discovery of analogy, this happy alliance, as Bossut calls it,4 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 Lagrange, who says, “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 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 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 generalisation 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 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.5 In the 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.
It ought to be added that to the discovery of analogy between the forms of mathematical and logical expressions, we owe the greatest advance in logical science. Boole based his extension of logical processes upon the notion that logic is an algebra of two quantities 0 and 1. His profound genius for symbolic investigation 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 fragments. 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 connection,