ANALOGY. 291 



sents some algebraic equation, we are presented by simple 

 observation of many mechanical movements with abun- 

 dant suggestions towards the discovery of mathematical 

 problems. Every particle of a carriage- wheel when mov- 

 ing on a level road is constantly describing a cycloidal 

 curve, the curious properties of which exercised the in- 

 genuity 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 attempt- 

 ing 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 sub- 

 sequently 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 e , 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 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 



a Lacroix, < Trait Eldmentaire de Calcul Diff&entiel et de Calcul 

 Integral/ 5 me ddit. p. 699. 



e ' Histoire des Math&natiques/ vol. i. p. 298. 



U 2 



