2. Let there be any one triangle en c in which the sum of all the angles is two right angles. Make the angle B 0 D= A n c, and c B D= • e n, whence the triangle n D 0 is in all respects equal to A n c. Produce A B and A C each to double of its length, and join r., D, and n r. The angle CBE being equal to u A c and A c B together, and 0 B D bOitig.BCA, it follows that E B D= n A C; whence the triangle EBD is in all respect. equal to B • o, and the same, for similar reasons, is D 0 r. Hence the three angles at D are severally equal to those of the triangle A BC, or to two right angles ; end thus E D F is a straight line, and A E a triangle equiangular with • 12 c, and therefore having angles equal to two right angles. Or, if there be any such triangle, there is another of the same sort with double aides and the same angles ; and since this process of doubling the sides may be repeated to any extent, it follows that, If there be any other given triangle, a triangle can be found with longer sides, and having the sum of its angles equal to two right angles.
Next, any triangle A as E which has the angle A, must have the sum of all it.. angles also equal to two right angles. Continue the preceding promos until the triangle a st x is completely inclosed, say in A E r, and join a, E. Then all the angles of the three triangles A II N, M N x F. make np the angles at a, a, r (two right angles), those at al (two right angles), and those at E (two right angles); six right angles in all. Consequently each sat of angles, in each triangle, must be equal to two right angles ; for all three sets making up six right angles, no one set can tell short of two right angles, without another act exceeding it, which has been shown to be impassible.
Lastly, on the preceding assumption (namely, that there Is one such triangle), every triangle has the rum of its angles equal to two right angles. Let a B c be the one triangle, as before, and let r Q a be any triangle nut equiangular with A n c ; one of its angles must be less than one of the angles of A a C ; for if not, the sum of the angles r, Q, would be greater than that of A, 11, c, or greater than two right angles. Let it be that r is loss than A, and make Q P Z equal to n A c, con. structing, by the preceding proems of doubling the aides, a triangle r sr containing q a. Then since r v x has an angle (at v) common universally denied ; and, taking any triangle A n c, take a point n, at which make the angle nDE=BO A. Then the angle n E n must be greater than B A C ; for if not, D E A and it 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 ED A, A DC, 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. ln like manner it may bo shown that if D move from C to B (the angle B D II being always= B o a), the angle B E D must continually increase, and can therefore only have a given value for a thereby determined value of 13 D. That is, 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 means of numbers only, without any dependence on a linear unit. This is
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. ln 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 n and o D be two parallel lines," contemplate every point of both, however remote from a and n. The demonstration of 11I. 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 far as proving that one triangle only need be shown to have the sum of its angles equal to two right angles, and should the final theorem be ultimately completed by a less objectionable third process (of which wo do not entirely despair), it may bo doubted whether right reasoning will be promoted by the arbitrary rejection of 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 received the degree of attention which we will venture to prophesy must yet be given to them. When they have been maturely discussed, the following question will arise:—ln :Omitting the notion and definition of parallels, and rejecting the comparison of their intervening spaces with angular spaces, are we, or are we not, in the position of those who admit one notion, while they exclude another which is Al much of kin to the first as that of an equilateral triangle to any other triangle t 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 affirmative, it does not follow that comparisons of iofinites can be successfully introduced into elementary teaching.