Mr Meikle on the Theory of Parallel Lines, 315 



KL M. Now angle A E H being obviously half the angle of an equi- 

 lateral triangle whose base would be E F, it is at least as great as angle 

 L K M ; and, therefore also, because E H exceeds E A, while the angle 

 E A G being obtuse, and E H G a right angle, every straight line drawn 

 from E to meet A G or G H, will exceed E A, since it will be the greatest 

 side of a triangle having either E A or E H for another side ; so that 

 the quadrilateral A E H G will be more than capable of containing the 

 triangle K L M, and so will the equal quadrilateral C F H G. It is also 

 evident that by continually increasing D E, D F by parts each equal 

 A E, and drawing other lines to join their extremities, the triangle E D F 

 may be increased to exceed any given area, for it would thereby acquire 

 an unlimited number of increments each greater than area K L M. But 

 if one triangle had the sum of its angles less than 180° by the nth part of 

 that quantity, another triangle whose area is n times as great, would 

 (Prop. II.) have no angles at all, which is absurd, and more especially 

 in the present case, where angle A D C is considerable. 



Such being a deduction from the premises in Prop. II., shews that the 

 angles of a triangle can never be less than 180°, which is equivalent to 

 proving Euclid's 12th Axiom, the one being so easily deduced from the 

 other. But if one triangle had its angles greater than 180° by an nth 

 part, another whose area is half of n times as great, would (Prop. II.) 

 have its angles equal to 270° ; and, consequently, some two of them 

 would amount to at least 180°, which (Euclid I. 17) is impossible. 

 Hence, there is no triangle whose angles exceed 180** ; and, therefore, 

 the three angles of every triangle are just equal to 180°. 



The equilateral triangle in Fig. 5 might have been omitted, and a 

 right angle used for ADC, but that would scarcely have conduced to 

 greater brevity or simplicity. If anything could be reckoned to have 

 been assumed above, it would be that the sum of the angles of a triangle 

 can never be inappreciable or so small that it could not be multiplied to 

 exceed any given angle. But if this cannot be regarded as already self- 

 evident, yet the assumption absolutely necessary here to complete the 

 proof would be smaller than any that could be assigned ; and, there- 

 fore, Euclid's assumption of the angles of a triangle amounting to 180° 

 would exceed the amount required here, in a ratio greater than any that 

 could be assigned. Thus, whatever part of the 180° were supposed to 

 be necessary here as the least sum of the angles ; for example, though it 

 were only the millionth, billionth, or any other assigned part, a still 

 smaller would obviously suffice. Under the anticipated objection, there- 

 fore, the question would still be reduced within limits incomparably nar- 

 rower than, so far as I am aware, it had ever been before ; especially 

 since most of the authors who have attempted it, assume more than 

 even Euclid assumes. 



I shall now endeavour to examine the professedly direct demonstra- 

 tions depending on infinite quantities, as employed by Bertrand, in his 

 Devehppement de la Partie Elementaire des Math^matiques, and by Le- 



