EVCLUTION OF ALGEBRA AND MECHANICS. 175 



Just incidentally noticing the cii cumstance that the 

 epoch we are describing witnessed the evolution of algebra, 

 a comparatively abstract division of mathematics, by the 

 union. of its less abstract divisions, geometry and arithme 

 tic a fact proved by the earliest extant samples of alge 

 bra, which are half algebraic, half geometric we go on to 

 observe that during the era in which mathematics and 

 astronomy were thus advancing, rational mechanics made 

 its second step ; and something wap done towards giving a 

 quantitative form to hydrostatics, optics, and harmonics. 

 In each case we shall see as before, how the idea of equal 

 ity underlies all quantitative prevision ; and in what simple 

 forms this idea is first applied. 



As already shown, the first theorem established in me 

 chanics was, that equal weights suspended from a lever with 

 equal arms would remain in equilibrium. Archimedes dis 

 covered that a lever with unequal arms was in equilibrium 

 when one weight &quot;\vas to its arm as the other arm to its 

 weight ; that is when the numerical relation between one 

 weight and its arm was equal to the numerical relation be 

 tween the other arm and its Aveight. 



The first advance made in hydrostatics, which we also 

 owe to Archimedes, was the discovery that fluids press 

 equally in all directions ; and from this followed the solu 

 tion of the problem of floating bodies : namely, that they 

 are in equilibrium when the upward and downward pres 

 sures are equal. 



In optics, again, the Greeks found that the angle of in 

 cidence is equal to the angle of reflection ; and their knowl 

 edge reached no further than to such simple deductions 

 from this as their geometry sufficed for. In harmonics 

 they ascertained the fact that three strings of equal lengths 

 would yield the octave, fifth and fourth, when strained by 

 weights having certain definite ratios ; and they did not 

 progress much beyond this. In the one of which cases Av r e 



