32 ELEMENTS OF SCIENCE 



If we divide a? + 2ab + b* by a + b (its square 

 root as we just saw*) the quotient must equal the 

 divisor, thus : 



a + b) a? + 2ab + I? (a + b 

 c? + ab 



ab + b* 

 ab + b* 



Sometimes quantities may be continued on indefi- 

 nitely, as when i is divided by i x. 



i - x) i (i + x + x* + x 9 &c. &c. 



+ x 



+ X - 



+ X* 



+ re 2 - x 5 



+ x 3 - x* 



+ a? 4 , &c. &c. 



The foregoing observations must suffice as a first 

 introduction to the principles of algebra, as a branch 

 of science replete with the most beautiful, complex, 

 ingenious, and far-reaching processes, whereby alone 

 many calculations are made possible, or the labours of 

 investigation lessened, while the results arrived at have 

 extraordinary accuracy. Though for these purposes we 

 may employ not only purely imaginary, but even 



* See ante, p. 31. 



