248 Further Remarks on the Theory of parallel Lines. 



be impossible, is not the whole of Legendre's process nuga- 

 tory ? What are we to think of the attempt to prove that a 

 function, which is a nonentity, must have a determinate form, 

 the same for all triangles ? 



The truth seems to be, that, in order to render the mode of 

 reasoning imagined by Legendre intelligible, we must strip 

 off the functional dress in which it is clothed. Since the 

 third angle of a triangle has the same magnitude in all cases 

 when the base and the other two angles have the same 

 values, there must be some general relation between the four 

 magnitudes, or between the vertical angle and some of the 

 three things that are given. But, by the principle of homo- 

 geneity, there cannot be an equation between the base and 

 one or more of the angles : wherefore the equation we are 

 seeking must subsist between the three angles. And since 

 this equation has no dependence upon the sides, it must be 

 the same in all cases ; because no reason can be assigned why 

 it should be variable in its form. In this statement of the 

 reasoning, nothing is brought forward except what bears con- 

 clusively upon the point to be proved ; and the full force of 

 the evidence is therefore perceived. It certainly amounts to 

 a great degree of probability. It is such a train of thought 

 as, in a process of invention, would lead, with great certainty, 

 to the desired success. But the requisites of a strict demon- 

 stration are in many respects wanting; and, as the whole pro- 

 cedure is a comparison of triangles that have the angles at 

 their bases common, it depends upon the postulate already 

 mentioned. 



There is one conclusion only that can be reasonably drawn 

 from all that has been said. The same difficulty about pa- 

 rallel lines which has so long baffled the geometer, opposes 

 an equally effectual resistance to the power of the algebraic 

 analysis. The same cause operates in both cases. The de- 

 finition of a straight line is indirect and imperfect, furnishing no 

 property that will enable us by a direct train of reasoning to 

 investigate the relation between the three angles of a triangle. 



3. Legendre estimates all angles by their respective pro- 

 portions to a right angle. But he might, with equal pro- 

 priety, have adopted any other determinate angle as the basis 

 of comparison. Professor Leslie has therefore argued that 

 angles are not to be considered as numbers independent of an 

 arbitrary unit; and that their measures are just as indeter- 

 minate as the measures of lines. In the last Number of this 

 Journal, we proved the justness of Legendre's procedure ; but 

 it may not be improper to add some further elucidation, and 

 for this purpose we shall choose a particular case. 



Let 



