Further Remarks on the Theory of parallel Lines. 249 



Let a, b denote the two remaining sides of the triangle ; 

 then the angle C opposite to the base c, must be a determi- 

 nate function of a, b, c ; for the angle has always the same 

 magnitude when the sides have the same values. But, ac- 

 cording to the principle of homogeneity, the expression of C 

 can contain the ratios only of the sides of the triangle ; it will 

 therefore be of this form, viz. 



Again, by trigonometry, we have 



CosC= "* + **-*' . 



Now the value of C deduced from this equation does not de- 

 pend upon any preconceived mode of measuring. It is not 

 the ratio of the angle to a right angle ; it is the length of the 

 arc subtending the angle in a circle of which the radius is 

 unit. In another circle of which the radius is r, and C the 

 length of the arc subtending the angle, the same equation will 

 become 



cos C _ at + bi — c 1 

 r Hab 



We can find — in terms of — — ; and by substituting the 

 value of , we shall obtain an equation of this form, viz. 



r V b c • 



Again, let Q denote the length of the quadrant, or any other 

 determinate arc, in the same circle : thus 



C r i a a \ / a a » 



4= ft x nT» 7j=*W» Tj* 



c 

 And, since — - is the proportion of the angle of the triangle 



to a right angle, this last equation coincides with that found 

 by Legendre's process. 



Professor Leslie is therefore not borne out in his philoso- 

 phical argument respecting the similarity of the measures of 

 lines and angles. But it is no more than justice to observe 

 that his reasoning is very naturally suggested by Legendre's 

 definition. 



An angle is a magnitude sui generis. It cannot be directly 

 compared with a magnitude of a different kind. If it enter 

 into the same equation with the sides of a triangle, this can 

 be effected in no other way than by the intervention of some 



Vol. 63. No. 312. April 1824. I i magnitude 



