322 Reviezo of the Cambridge Course of MatkemaUcs. 



n 



symbol of an indeterminate quantity is-, and an elemen- 

 tary method is given, of ascertaining the true value of ex- 

 pressions which appear to be indeterminate. The general 

 methods of findings the true value of such expressions, be- 

 longs to the higher algebra. Equations of the first degree 

 are concluded by an investigation and application of gene- 

 ral fdrmnlas for their solution, after the manner of Be: 

 zaut. 



The extraction of the square root both of entire and frac- 

 tional numbers, is next introduced, as this operation \s ne- 

 cessary and preparatory to the solution of equations of the 

 second degree. The exposition of the method is founded on 



Hie composition and arralysis of the formuhi a ^ +2 «S + ^^? 

 in V hich a represents the tens and h the units of the num- 

 ber; From this proposition of the theory of numbers, that 

 *' every prime number, which will divide the product oi 

 two numbers, will necessarily divide one of these num- 

 bers," it is shewn to result, that " entire numbers, except 

 such, as are perfect squares, admit of no assignable root, 

 cither amongj whole numbers or fraction^."* Hence, 

 the extraction of the square root, applied to numbers not 

 perfect squares, makes us acquainted with a new kind of 

 numbers, which, having no common measure with UJiity, or 

 no relation to it that can be expressed by whole numbers 

 or fractions, are termed incommensurable ov irrationah A 

 method is here given of approximating ilie square root of 

 aun^bers not perfect squares, and also, the stjuare root of 

 fractions the terms of which are not both perfect squares. 



Proceeding to the solution of equations of the second de- 

 g^e, he shews the reason why t)ie double sign ± is con- 

 siflered as affecting the square root of every quantity, and 

 explains what is to be understood when we say, that the 

 square root of a negative quantity is imaginary. His gene- 

 ral formula for resolving complete equations of this degree, 

 is, x--fi?^=9', in which ^ and q denote known quantities, 

 either positive or negative. After treating of the proper- 

 ties of negative solutions, and examing in what cases prob- 

 lems of the second degree become absurd, he gives an m- 



* The reasonins; here £:iven, exprp&sec! in a summary way is this ; entire 

 numbprs not perfect squares, it is obvious, can liave no entire tool ; there- 

 fore, if they have a root, it mu.--t be among irredocible fi-actions ; but irreda- 

 cible fractions when squared, form still irreducible fractions which cannot 

 become entire numbers. 



