>AHALLKI-S 



PARALLELS. 



M 



mtaM* of aToidiac the axiom ot Euclid, without .ubrtitution of 



We dull conclude Uiu article with an account o{ a result 



^oauaper. by L*fendi.m the 12th volume of the 'Memoir* 



iMiitut* ' twine hi* Utot attempt at the solution of the problem. 



H JX2* b. p^parif atyU. a geometrical tranalation of the analytical 



poof TidT^o5oad, and iU chain ol reMoning proceed, through 



three pcopoattiooa. 



1. It b tmpuaiibU that the cum of the angle* of a triangle can in any 

 rn t md two right angle*. 



L If there be any one triangle which has angle* together equal to 

 two right angle*, the aune miuit be true of all triangles. 



g, Nth* aogU* of a triangle be not equal to two right angles, then 

 turte* altme may be made to determine a itraight line. 



I. If it be poeaible, let there be a triangle B A o having iU angle* 

 together greater than two right angle*. In A o produced take c K, K o, 

 o i, Ac. each equal to A c. and make the angle* D o K, T I o, Ac. each equal 

 to B AC, and let en. If, Ac. be each equal to AB. Join BD, DF, *c. 



which are not known to be in the same straight line, though they are 

 o, Then the trianglea DCB, FKO, &c. are severally equal in nil 

 rwpecU to B A c. And because the angle* at A, B, o, are together 

 (by hyp.) greater than two right angles, and two of them are equal to 

 two of the angles at c, it follow* that the angle A B c is greater than 

 BCD, whence A B and B C being severally equal to O c and c B, it follows 

 that A c U greater than B D. Similarly c K is greater than D F, lie., 

 whence A t exceed* B D r H K by as many times the excess of A c over 

 B D a* A c i* contained in A I. Now however small this excess may be, 

 it may be multiplied until the multiple exceeds twice A B, or A B and 

 K I together. That is, A I, one side of a figure, may be made greater 

 than A B, B D, D F . . . . K I, the sum of all the other sides ; which is 

 absurd. Hence, it i* not true that the angles of any triangle are 

 together greater than two right angles. 

 J. Let there be any one triangle AB c in which the sum of all the 



ingle* it two right angle*. Make the angle B c D = A B c, and c B D = 

 A OB, whence the triangle B DC is in all respects equal to ABC. 

 Produce A B and A c each to double of its length, and join E, D, and 

 D r. The angle c B E being equal to B A c and ACS together, and c B D 

 being =* B c A, it follow* that E B D = B A c ; whence the triangle E B D is 

 in all reapeeU equal to B A o, and the same, for similar reasons, is D c F. 

 Hence the three angle* at D are severally equal to those of the triangle 

 A B o, or to two right angle* ; and thus E D F is a straight line, and A E F 

 a triangle equiangular with ABC, and therefore having angles equal to 

 two right angle*. Or, if there be any such triangle, there is another of 

 the aame aort with double aides and the same angles ; and since this 

 prutm* of doubling the aide* may be repeated to any extent, it follows 

 that, if there be any other given triangle, a triangle can be found 

 with longer tide*, and having the sum of iu angles equal to two right 



Heat, any triangle A 11 H which has the angle A, must have the sum 

 of all it* angle* *l*o equal to two right angle*. Continue the preceding 

 BffOB*** until the triangle A H x i* completely inclosed, say in A E F, and 

 join B, >r. Then all the angle* of the three triangles A M N, M E, N K v, 

 mike op the angle* at A, K, r (two right angles), those at M (two right 

 angle*), and thoee at (two right angle*) ; six right angles in all. 

 CofweqaeoUy each nt of angle*, in each triangle, must be equal to two 

 right angle* ; for all three w>U making up *ix right angle*, no one et 

 can fall abort of two right angle*, without another set exceeding it, 

 which ha* been ahown to be impo-jble. 



Laitly, on the preceding awumption (namely, that there is one such 

 triangle), rwry triangle ha* the mini of it* angle* equal to two right 

 angle*. Let ABC be the one triangle, an before, and let pq R be any 

 triangle not equiangular with ABC; one of iu angle* uitut be lea* than 

 one of the angle* of ABC; for if not, the miui of the angle* i-, y, R 

 would be greatw than that of A, B, c. or greater than two right angles. 

 Lei it be that r i* lew than A, and make QPZ equal to BAC, con- 

 structing, by the preceding procee* of doubling the aides, a triangle 

 v r w containing : p <j B. Then since P v x has an angle (at T) common 



with P v w, the angle* of P v x are together equal to two right angle*, 

 and because r v x ha* an angle iu common with r g B, the angle* of 



r Q B are also equal to two right angles. Hence, if any one triangle 

 have it* angle* equal to two right angles, the same- must be tru> 

 triangles whatsoever. 



3. If then we deny the preceding truth in the cose of any one 

 triangle, we must deny it in the cac of all. Let it then be 



universally denied ; and, taking any triangle ABC, take a point r>, at 

 which moke the angle BDE = BCA. Then the angle BED must be 

 greater than BAC; for if not, D E A and E A c are at least equal to two 

 right angles, and, E D c and A c D being together equal to two right 

 angles, the angles of the triangles E D A, A D o, are together at least 

 equal to four right angles, which is denied. For it is denied that 

 either set is equal to two right angles, and it has been shown that 

 neither set can be greater. In like manner it may be shown that if 

 D move from c to B (the angle BDE being always=Bo A), the angle 

 BED must continually increase, and con therefore only have a given 

 value for a thereby determined value of B D. That ia, by assigning the 

 angles B, D, E, the side B D can be absolutely laid down. Now since 

 angles might be given in numbers (taking the right angle, which is 

 absolute, as a unit), it therefore follows that the length of a straight 

 line might be handed down from generation to generation, by mean* 

 of numbers only, without any dependence on a linear unit. This i* 

 the same conclusion as follows from the analytical proof, against those 

 who would deny its conclusions. 



We consider the preceding process as containing the most remarkable 

 addition which has been made to the theory. With regard to the 

 whole question, we do not consider the difficulty as one of a different 

 kind from that of the quadrature of the circle or the trisection of the 

 angle. In the earlier stages of mathematical investigation, all that was 

 not evidently impossible was attempted, and failure was, properly and 

 modestly, attributed to the want of sagacity in the investigators. In 

 the instance before us, the object was to deduce positive properties 

 from a purely negative definition, involving, be it observed, the idea of 

 infinity. For if we say that parallels are lines which never meet, 

 however far produced, we must, in the hypothesis " let A B and c D be 

 two parallel lines," contemplate every point of both, however remote 

 from A and B. The demonstration of M. Bertrand appears to us to 

 assume considerations which are indispensable to the direct treatment 

 of this negative definition; nor can we imagine the positive deduction 

 of properties from the assumption of lines which never meet, without 

 making their intervening space, as compared with other spaces, an 

 object of reasoning. And even if the preceding process of M. Legendre 

 should be allowed, so for as proving that one triangle only need be 

 ahown to have the sum of its angles equal to two right angles, and 

 should the final theorem be ultimately completed by a less objectionable 

 tliinl process (of which we do not entirely dunpairi, it may be doubted 

 whether right reasoning will be promoted by the arbitrary reject 

 notions intimately connected with those which are necessary for the 

 perfect conception of a definition. On this the whole question must 

 at last turn : it will readily be granted that a studied exclusion of a 

 particular figure (for instance, of the equilateral triangle) would be no 

 real gain to the strictness of geometry, even though it should be shown 

 that the whole of Euclid might be established without it. The new 

 considerations brought forward by M. Bertrand have not yet r. 

 the degree of attention which we will venture to prophesy must 

 yet be given to them. When they havr IH.M maturely discussed, 

 tin- following question will arise: In admitting the notion anil 

 definition of parallels, and rejecting the comparison of their intervening 

 spaces with angular spaces, are we, or are we not, in tin- 

 those who admit one notion, while they exclude another which i* as 

 much of kin to the first as that of on equilateral triangle to any other 

 triangle f We should be sorry to see this question settled either way 

 without such an examination of the nature of our ideas of magnitude, 

 and in particular of the connection of finite and infinite, as has not yet 

 been made. But even if our question should be resolved in the 



