EYCLTJTION OF ALGEBRA AND MECHANICS. 175 



Just incidentally noticing the circumstance that the 

 epoch we are describing witnessed the evolution of algebra, 

 a comi^aratively abstract division of mathematics, by the 

 union of its less absti^act 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 was 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 was to its arm as the other arm to its 

 weight ; that is— rwhen the numerical relation between one 

 weight and its arm was equal to the numerical relation be- 

 tween the other arm and its weight. 



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 doAvnward 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 we 



