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with Proposition 3.—(3.) While the rod AC retains its place, let two rods AB and CB of unlimited length, and applied at the ends A and C, be opened gradually till the one forms with AC a given angle CAB, and the other a given angle ACB ; it is evident that AB and BC will then rest crossing each other in those positions, and containing a determinate triangle, of which the vertex B is their point of mutual intersection. This property corresponds with Proposition 20–(4.) Let the rod AB of a given length make a given angle with the unlimited rod AC, and applying at the end B another given rod, turn this gradually round till it meets AC. If BC exceeds the distance of B from AC, it will evidently, after stretching beyond AC, again come to meet that boundary. With such conditions, therefore, the rods might contain two determinate triangles, the one acute and the other obtuse, and which are hence distinguished from each other by those obvious characters. This qualified property, omitted in most elementary works, is yet of extensive application, and was requisite to complete the conditions of the equality of triangles. It corresponds with Proposition 21. The four preceding theorems are reducible, however, to a single property, which includes all the different requisites to the equality of triangles. The sides of a triangle are obviously independent of each other, being subject to this condition only, that any one of them shall be less than the remaining two sides. But since all the angles of a triangle are together equal to two right angles, the third angle must, in every case, be the necessary result of the other two angles. A triangle has, therefore, only five original and variable parts—the three sides and two of its angles. Any three of these parts being ascertained, the triangle is absolutely determined. Thus—when (1.) all the three sides are given,-when (2.) two sides and their contained angle are given,-when (4.) two sides and an opposite angle are given, with the affection of the triangle, or when (3.) one side and two angles, and thence the third angle are given,-the triangle is completely marked out.
M. Legendre, in a very elaborate note to his Elemens de
Geometrie, has sought, with much ingenuity, to deduce à priori the radical properties of triangles from the theory of functions. But, like other similar attempts, his investigation actually involves in it a latent assumption. This subtle logician sets out with the principle which would seem almost intuitive, that a triangle is determined when the base and its adjacent angles are given. The vertical angle, therefore, depends wholly on these data, the base and its adjacent angles. Call the base c, its adjacent angles A, B, and the vertical or opposite angle C. This third angle, being derived from the quantities A, B and c, must be a determinate function of them, or formed from their combination. Whence, adopting his notation, C-4 : (A, B, c). But the line c is of a nature heterogeneous to the angles A and B, and therefore cannot be compounded with these quantities. Consequently C-4 : (A, B), or the vertical angle C is a function merely of the angles A and B at the base; and hence the third angle of a triangle must depend wholly on the other two. To a speculative mathematician this argument is very seductive, though it will not bear a rigid examination. Many quantities in fact appear to result from the combined relation of other quantities that are altogether heterogeneous. Thus, the space which a moving body describes, depends on the joint elements of time and velocity, things entirely distinct in their nature; and thus, the length of an arc of a circle is compounded of the radius, and of the angle it subtends at the centre, which are obviously heterogeneous magnitudes. For aught we previously knew to the contrary, the base c might, by its combination with the angles A and B, modify their relation, and thence affect the value of the vertical angle C. In another parallel case, the force of this remark is easily perceived. Thus, when the sides a, b and their contained angle C are given, the triangle is determined, as the simplest observation shows. Wherefore the base c is derived solely from these data, or c=p: (a, b, C). But the angle C, being heterogeneous to the sides a and b, cannot coalesce with them into an equation, and consequently the base c is simply a function of a and b, or it is the necessary result merely of the other twe sides. In other words, as the third angle of a triangle depends on the other two angles, so the base of a triangle must have its magnitude determined by the lengths of the two incumbent sides. Such is the extreme absurdity to which this sort of reasoning would lead In both of these instances, indeed, the conclusion is admitted by implication, only in the one it is consistent with truth, while in the other it is palpably false. —That such an acute philosopher could overlook the fallacy of his argument, can only be ascribed to the influence which peculiar trains of thought acquire over the mind, and to the extreme facility with which elementary principles insinuate and blend themselves with almost every process of reasoning. The objections here directed against the celebrated abstruse attempt to demonstrate, à priori, the equality of triangles from the nature of equations, and the properties of homogeneous quantities, have, generally, I believe, been deemed conclusive. I have scarcely heard, indeed, of a geometer of any emimence in the island, (except the learned writer of a critique which appeared in the Edinburgh Review), who is not perfectly convinced of the fallacy lurking in the argument advanced by its very ingenious inventor. On this occasion, I shall take the liberty of introducing an extract from a letter to me, dated October 20, 1816, from an old friend and fellow-student, who now stands decidedly at the head of our mathematicians. “With regard to Legendre's demonstration, I am of opinion, that there is involved in the mise en equation, (reduced to an equation,) a principle which is equivalent to Euclid's 12th axiom, (If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, will at length meet on that side on which are the angles which are less than two right angles.) Using , the notation of your book, his assumption is, that C = q : (A, B, c): Now, this means that we shall get the angle C, by combining the angles A and B with the line c, in a certain way; and it is implied, that this is true, whatever value the line c may have; or, in other words, it is true for all values of c. Suppose then an individual triangle, of which c is the base,
and A, B, the angles at its extremities; conceive an indefinite number of lines, of any lengths, c', c”, c”, &c. and at the ends of each of these lines, angles to be made equal to A and B, will a triangle be thus formed upon each of the lines c', c”, c'", &c. or not If you say that you cannot allow the existence of such triangles without proof, you agree with the Greek geometer, but then you must deny the legitimacy of Legendre’s equation, C = 0: (A, B, c); for it supposes the possibility of such triangles, since it is a determination of the third angle of each of them from knowing the base and the other two angles. If you grant the possibility of the triangles, then Legendre’s equation will be established; but you also admit Euclid's 12th axiom: For you assume, that two lines drawn at the extremities of any third line, so as to make with it two angles equal to any two angles of a triangle, do meet one another when produced. On examination you will find, that the only relation generally true of two angles of a triangle is this, that they are together less than two right angles. I cannot, therefore, admit, that Legendre's demonstration contributes in any degree to remove the difficulty in geometry. The intrinsic evidence of a principle, or proposition, is the same whether it be expressed in common language, or translated into the language of functions. Grant to the geometer the same assumption which is implied in the functional equation of the analyst, and he will be no longer embarrassed with the theory of parallel lines. Legendre endeavours to justify his equation, by saying that two triangles are identical when they have their bases equal, and likewise the angles adjacent to their bases equal, each to each. But this does not prove, that of all the infinite number of triangles which can be formed upon a line greater or less than the base of a given triangle, there is always one that has the angles at its base equal to the angles at the base of the given triangle. If this be thought a more self-evident principle than those that geometers have employed, let it be transferred to geometry, and that science will no longer have need to borrow aid from the theory of functions.” To these acute and judicious remarks, I think it unnecessary to subjoin any farther observations; but, in justice to the
illustrious author of the argument drawn from the higher analysis, I must state, that he still remains persuaded of its legitimacy. In a very flattering letter, which he did me the honour to write, bearing date, Paris, 5th February 1816, he thus adverts to the subject in dispute. * Ayant un très grande idée de la superiorité de vos lumières, Monsieur, j'eprouve un regret d'autant plus vif de voir que vous n'approuviez pas, ou meme que vous regardiez comme illusoire la demonstration que j'ai donnée dans mes notes du principe sur les trois angles . du triangle. J'ai cependant la conviction intime que cette demonstration est parfaitement rigoureuse, et j'ose vous prier d'y donner encore quelqu'attention, persuadé que vous reconnoitrez son exactitude. La loi de l'homogenité est une loi generale, qui n'est jamais en defaut, et qui doit être rangeé parmi les principes elementaires les plus generaux et les plus simples. L'angle est un quantité que je mesure toujours, par son rapport avec l'angle droit, car l'angle droit est l'unité naturelle des angles : Dans cette notion très simple, une angle est toujours un nombre. Il n'en est pas de même des lignes : une ligne ne peut entrer dans le calcul, dans une equation quelconque, qu'avec une autre ligne que sera prise pour unité, ou qui aura un rapport connu avec la ligne unité. * Ainsi l'equation C = q : (A, B, c) rapportée, pag.403, ou A, B, C, sont des angles, et par consequent des nombres, ne sauroit subsister, à moins que c ne disparoisse. Car si c ne disparoit pas, il faudra qu'une longueur absolue c soit determinée par des nombres, sans que l'unité de longueur soit connue, ce qui est une absurdité. L'objection faite, pag. suiv. sur l'equation c = p : (a, b, C) se résout très facilement. Rien n'empêche que C, qui est un nombre, (par rapport à l'angle droit pris pour unité), ne soit une fonction de a, b, C, pourvu que cette fonction soit de nulle dimension, c'est-à-dire, pourvu que le fonction de a, b, C se reduise à une fonction de deux rapports,
tels que #. -#. Et en effet, c'est ce qui a lieu d'après l'equa(Z
tion trigonometrique, cos C = 2 ab T 2 b