The chemist having discovered what he believes to be a new element, will have before him an infinite variety of modes of treating and investigating it. If in any of its qualities the substance displays a resemblance to an aklaline metal, for instance, he will naturally proceed to try whether it possesses other properties of the alkaline metals. Even the simplest phenomenon presents so many points for notice that we have a choice from among many hypotheses.

It would be difficult to find a more instructive instance of the way in which the mind is guided by analogy than in the description by Sir John Herschel of the course of thought by which he was led to anticipate in theory one of Faraday's greatest discoveries. Herschel noticed that a screw-like form, technically called helicoidal dissymmetry, was observed in three cases, namely, in electrical helices, plagihedral quartz crystals, and the rotation of the plane of polarisation of light. As he said, "I reasoned thus: Here are three phenomena agreeing in a very strange peculiarity. Probably, this peculiarity is a connecting link, physically speaking, among them. Now, in the case of the crystals and the light, this probability has been turned into certainty by my own experiments. Therefore, induction led me to conclude that a similar connection exists, and must turn up, somehow or other, between the electric current and polarised light, and that the plane of polarisation would be deflected by magneto-electricity.” By this course of analogical thought Herschel had actually been led to anticipate Faraday's great discovery of the influence of magnetic strain upon polarised light. He had tried in 1822-25 to discover the influence of electricity on light, by sending a ray of polarised light through a helix, or near a long wire conveying an electric current. Such a course of inquiry, followed up with the persistency of Faraday, and with his experimental resources, would doubtless have effected the discovery. Herschel also suggests that the plagihedral form of quartz crystals must be due to a screw-like strain during crystallisation; but the notion remains unverified by experiment.

1 Life of Faraday, by Bence Jones, vol. ii. p. 206.

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 startingpoint 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, ènimedos apieμós, and distinguishes a square number of which the two factors are equal as an equal-sided and plane number, ioóπλeupos καὶ ἐπίπεδος αριθμός. He also calls the root of a cubic number its side, Tλeupá. In the Diophantine algebra inany 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.

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

1

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.1 There can be no doubt that Newton's fluxional, that is, geometrical mode of stating the dif ferential 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,2 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

1 Lacroix, Traité Élémentaire de Calcul Différentiel et de Calcul Intégral, 5me édit. p. 699.

2 Histoire des Mathématiques, vol. i. p. 298.

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.1 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 o and I. 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, made one of the most important discoveries in the history of science. As Descartes had wedded algebra and geo

See Goodwin, Cambridge Philosophical Transactions (1845), vol. viii. p. 269. O'Brien, "On Symbolical Statics," Philosophical Magazine, 4th Series, vol. i. pp. 491, &c. See also Professor Clerk Maxwell's delightful Manual of Elementary Science, called Matter and Motion, published by the Society for Promoting Christian Knowledge. In this admirable little work some of the most advanced results of mechanical and physical science are explained according to the method of quaternions, but with hardly any use of algebraic symbols.