322 Revierv of the Cambridge Course of Mathematics. 



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 finding 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 formulas for iheir 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 is ne- 

 cessary and preparatory to the solution of equations of the 

 second degree. The exposition of the method is founded on 

 the composition and analysis of the formula a ^ -|- 2 a6 -f 6*, 

 in which 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 of 

 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, 

 either among whole numbers or fractions."* 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 unity, or 

 no relation to it that can be expressed by whole numbers 

 or fractions, are termed incommensnrable ov irrational. A 

 method is here given of approximating the square root of 

 numbers not perfect squares, and also, the square root of 

 fractions the terms of which are not both perfect squares. 



Proceeding to the solution of equations of tlie second de- 

 gree, he shews the reason why the double sign 1: is con- 

 sidered 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" -\-px=q, in which jo 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 in- 



* The reasoning; here siven, expressed in a summar)' way is this ; entire 

 numbers not perfect squares, it is obvious, can liave no entire root ; there- 

 fore, if they have a root, it must be among irreducible fractions ; but irredu- 

 cible fractions when squared, form still irreducible fractions which cannot 

 become entire numbers 



