AND OIHER SURD EQUATIONS. 233 



(Art. 8.) If we make s/a: n .r, and s/x + \ zzi x^ we 



shall have 



X ■\- X n: 0, 



x^—l? —— 1 ; 



1 2 



whence we get x zz. — » and x'zz — : the only value of 



I 2.0 2 2,0 



X being rejective, the equation is impossible; x "=1 x^ (or 



x^ — 1) ^^ — > is the solution of the equation 



K^x -j- 1 — V^ ^^ 0. 



Examples of this kind might be multiplied indefinitely ; 

 these, however, are sufficient to show, that the method of 

 the ancient algebraists (adapted) is admirably suited for 

 the ready exclusion of foreign roots from the solution, and 

 likewise for enabling us at once to determine, without trial, 

 to what equations those rejective roots severally belong. 



10. Garnier seems to have been the first mathematician 

 who distinctly affirmed the existence of impossible equa- 

 tions. In his Ajialyse AlgSrique, p. 335, art. 92, he says, 

 in speaking of the equation 



— \/x — 1 rz 1 — \/x — 4, 

 it " cannot be satisfied when the radicals are taken with 

 the sign plus;"* and in, the same place he further remarks, 

 that " the operations by means of which the radicals are 

 made to disappear, introduce roots foreign to the proposed" 

 equation.* 



Subsequently the subject received some attention from 

 the late Mr. Homer of Bath, a gentleman whose valuable 



* The above translations are taken from Mr. Cockle's ITorm, X., before 

 referred to, and are followed by the subjoined remark : — " So that Gaknier 

 must be understood as having distinctly asserted the existence of surd 

 equations vi^ithout roots, and also that the appearance of roots which 

 such equations present, are the roots introduced by the processes through 

 which we seek to rationalize the equations." — Mechanics' Magazine, 

 vol. xlix. p. 557. 



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