Mr Meikle on the Theory of Parallel Lines. 



317 



180°, or those of a quadrilateral to 360", which is just the whole affair to 

 be proved. For let E F H (Fig. 7) be a right angle, and H F K L a 

 Fig. 7. biaogle every way equal to CABD (Fig. 6) and 



which will also be equal to area C M N D, if, as 

 Legendre assumes, the angle M N D =: 90*^. In 

 that case, too, the area E K L would be just as great 

 as the original area E F H, and the rest of the pro- 

 cess would seem equally satisfactory. 



But if angle B N M be acute, and of course 

 M N D obtuse, it is easy to shew how an area equal 

 to C M N D, or one much less, may be taken from E F H, so as to 

 diminish that infinite area, and, by repetition, exhaust it altogether. 

 Thus, whether F K be taken as great as M N, or in any ratio less, if we 

 only make angle E K L equal to the acute angle G N D, the area H F KL 

 may be equal to or much less than C M N D, and yet the area E K L 

 will evidently be less than area E F H in the same ratio that the acute 

 angle E K L is less than E F H. Draw F V, making angle HFV =: 

 L K E ; then the sum of the angles HFV, F V L will evidently exceed 

 the sura of H F K, F K L by whatever the three angles of the triangle 

 F V K want of 180" ; for if in one case the angles are deficient, so 

 (Prop. II.) they must in every other. If, therefore, K O be taken of any 

 magnitude not exceeding F V, and angle K R be made equal F V L, 

 the difference between angle E O R and E K L will exceed the 

 difference between E K L and E F H by whatever the three angles 

 of F V K want of 180°. Hence, by continuing the like construction, the 

 residuary angles which the upper sides of the successive lines make with 

 E F, will continually decrease with increasing differences, till they vanish 

 altogether, and thereby exhaust the whole area E F H. Thus, Ber- 

 trand's famous demonstration fails entirely, unless we first assume that 

 the angles of every triangle amount to at least 180°. 



Another demonstration by Bertrand, and which, like the two just 

 noticed, is highly commended by Legendre, amounts to this: Since 

 the entire indefinite area around a point A, only differs from the sum 

 of the infinite areas of the exterior angles DAE, E B F, F C D of any 

 Fig. 8. triangle A B C, by the finite area of that 



triangle itself; which is supposed to bear 

 no proportion to the indefinite area of the 

 plane ; it is, therefore, concluded that the 

 sum of the angles around the point A is 

 just equal to the sum of these exterior 

 angles. But such reasoning, in this case, 

 is as faulty as in the two preceding, where 

 the doctrine that a finite quantity has no 

 effect on an infinite, was found to fail, 

 unless we first assumed the very thing to be proved, that the angles of a 

 triangle cannot be less than 180*. For if they may be less, produce A C 



