BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 355 



If w— 1 be odd, then the several pairs give as before the product unity ; but 

 there remains the factor — AB, which has for its coeflBcient -1. 

 Therefore, in either case, (APj) x (APJ (AP„_i) = // R»~i. 



XIII. The symbolism employed in the foregoing propositions appears to be 

 applicable to Plane Trigonometry in all its parts. To the elementary proposi- 

 tions of Geometry it is either inapplicable, or applicable by processes and con- 

 siderations unsuitable to the demonstration of elementary truths. Thus, if by 

 this method we undertake to prove that the angles at the base of an isosceles tri- 

 angle are equal to one another, we have (AC) = (BC). (Fig. 5.) 



But 



But 



.-. (AC). [a + o' + \/" 



■6J 



AC=(AC) . 1^' = (AC) . [a + ' 



CB = AD = (AC) . 1~2^=(AC) . [n' + A/-T'] 

 AC + CB=AB. 



~b + V^'] = AB a positive quantity ; consequently the im- 

 possible or sinal parts of the coefiicient of du'ection must destroy one another, or 

 \/^~b = -\/^' or 6= -b'- Therefore the angles A and B have their sines equal 

 in length, but of different affections. The angles themselves, therefore, being to- 

 gether less than -tt, are geometrically equal to one another. 



CoR. Much in the same way we might prove that in every triangle the greater 

 angle has the greater side opposite to it ; and, conversely, that the greater side 

 has the greater angle opposite to it. 



