Mathematical recreations and essays . be equal to two right angles. His demonstra-tion assumes that if any number of equal triangles are placedin juxtaposition along a line it is possible to draw a triangleenclosing them all: the same assumption was made by T. But unless we assume that space is infinite thisis not justified: the insufficiency of the argument is clearlybrought out by applying it to spherical triangles. Legendres Analytical Proof of the Proposition. I thinkhowever that the follov^ing is the most ingenious of the proofsgiven by Legendre*. A triangle ABG is completely
Mathematical recreations and essays . be equal to two right angles. His demonstra-tion assumes that if any number of equal triangles are placedin juxtaposition along a line it is possible to draw a triangleenclosing them all: the same assumption was made by T. But unless we assume that space is infinite thisis not justified: the insufficiency of the argument is clearlybrought out by applying it to spherical triangles. Legendres Analytical Proof of the Proposition. I thinkhowever that the follov^ing is the most ingenious of the proofsgiven by Legendre*. A triangle ABG is completely deter-mined by one side and two angles, say, a, B, G. Given these,the triangle can be constructed, and therefore the angle A de-termined. Now if the unit of length be changed the measure ? Elements de Geometrie, Paris, 12th edition, p. 281. CH. XV] THE PARALLEL POSTULATE 319 of a will be changed but the triangle, and therefore A, willnot be altered. Hence A cannot depend on the value of a;accordingly it must depend only on B and 0,. Now take a right-angled triangle DBF, of which D is theright angle. Draw DG perpendicular to EF. The angle EDGin the triangle EDG is calculated from the other two angles ofthat triangle, namely E and a right angle, in the same wayas the angle F in the triangle DEF is calculated from theother two angles of that triangle, namely E and a right F is equal to EDG. Similarly E is equal to the sum of the angles F and E is equal to the sum ofEDG and GDF, and therefore is a right angle. Hence the sumof the angles F, E, and D of the triangle DEF is equal to tworight angles. Thus the result is proved for a right-angledtriangle, and it will follow for any other triangle in the sameway as in Thaless proof J. Leslie criticised this proof on the ground that in thecorresponding theorem of Spherical Trigonometry, we knowthat the expression for the value of the angle A involves a/R,where R is the radius of the sphere, and it is conceivable t
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