INTRODUCTION. 



order that h x 65. may make b^zna^. So that simple fractional num- 

 bers serve as indices of the number of times that the quantity must 

 be multiplied together, in order that the product may be the com- 

 mon multiplier of the scries, or the simple number b. 



Scholium. Fractional powers are sometimes denoted by the 

 mark *y, meaning root : thus ^ azz.ai, s/ azzai. The second power 

 of a number a being called its square, and the third its cube, the 

 fractional powers are called square and cube roots. 



The sums of geometrical progressions may be thus computed, if 

 a-^ab+ab^ . . . -f aft»— ^zza-, ab-^ab^-}-a¥ . . . -^ab^zzbx, and sub- 

 tracting the former equation from the latter ab» — azzbx — x, therefore 

 ab^- 



b—l 



which, when i<l and wzz co, or infinite, becomes 



1—6 



The binomial theorem, for involution, is (rt-|-6)«=:a"-fn.a«— 1 b-j-n. 



In simple cases, its truth 



may be shown by induction. See 244. 



POWERS OF NUMBERS. 



