xxviti.] ANALOGY. 633 



equation, we are presented by observation of mechanical 

 movements with abundant suggestions towards the dis- 

 covery 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 mathe- 

 matical methods. This is certainly the opinion of La- 

 grange, who says, " So long as algebra and geometry have 

 been separate, their progress was slow, and their employ- 

 ment 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 perfec- 

 tion." 



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 perpen- 

 dicular distance of its line of action from the point, is 



1 Lacroix, Traite Element aire de Calcul Di/erentiel et de Calcul 

 Integral, 5 me edit. p. 6^9. 



2 Histoire des Mathetnatiques, vol. L p. 298- 



