doubtedly in the right; but in reality, I make no supposition. I reason with regard to any triangle already constructed, and actually existing ; I study its properties, and find from studying them, that in every triangle the sum of all the angles is equal to two right angles. So soon as this principle is established, I can easily (as I have done in the third edition of my Elements, Prop. 24. I.), construct a triangle having a given base c, and two adjacent angles A and B, whose sum differs from two right angles as little as we please; and by this construction, Euclid's postulate is demonstrated in the most rigorous manner.

I shall extend my observations no farther; I might even have forborne to write on this subject at all, since M. Maurice of the Academy of Sciences has undertaken the task of replying to the objections of Mr Leslie and his learned correspondent, in the Bibliothèque Universelle of Geneva, Oct. 1819; and his dissertation (excepting one or two passages, perhaps liable to. dispute, but having no influence on the question) completely fulfils the object of its author, and establishes in a manner as solid as luminous, the theory which Mr Leslie has attempted to overturn.

May I observe, in conclusion, that Mr Leslie, who has hitherto appeared in the character of an assailant, has not sufficiently secured his own defence, having left without reply the very strong objection alleged against his demonstration of Prop. 22, by Mr Playfair, at p. 88 in the volume of the Edinburgh Review already quoted. In reality, this demonstration supposes, that through a given point, no more than one parallel to a given line can be made to pass, or that there is only one position in which the line meant to be drawn will not meet the given line. Such an assumption is identically the same as Euclid's postulate. There is only this difference between Mr Leslie's method and that of Euclid, that the ancient geometer does not dissemble the difficulty, but presents it, on the contrary, in all its breadth, and requires to have that granted which he cannot prove; while the modern geometer envelopes the difficulty in a shadow of demonstration which, though doubtless it has seduced himself, is certainly any thing but rigorous. It is difficult to conceive how such a mistake could proceed from a mathematician so well versed as Mr Leslie is in the geometry of the ancients; who has shewn himself acquainted with all its most subtle refinements, and has himself invented many demonstrations which the ancients would not have been ashamed to own. One would have expected him to look more narrowly into a subject, which has formed the great difficulty of geometers ancient as well as modern; and not to give out as rigorous a demonstration, which is very far from being so. We subjoin the paper of M. le Baron Maurice, alluded to above, which M. Legendre has taken the trouble to alter in a few points, and adapt for appearing in this Translation.-ED.

On the Application of the Algorithm of Functions to demonstrate the fundamental Propositions of Geometry; in Reply to some Criticisms.-Extracted from the BIBL. UNIV. Geneva, October 1819. In a justly valued work (Elements of Geometry, by John Leslie, 3d edit. 8vo, Edinburgh, 1817), a great number of objections, more

or less specious, are combined together, and directed against the force of the demonstration, which M. Legendre, in Note II. of his Geometry, gives of the fundamental proposition, that, in every rectilineal triangle, the sum of the three angles is equal to two right angles. This demonstration is the more remarkable, as it makes no use of the theory of parallels, but, on the other hand, itself serves rigidly to establish that theory afterwards; and as, being founded solely on the principles of superposition and of homogeneity, the most palpable and evident of all principles, it shares to the full extent in their correctness.

Accordingly, on examining with attention the objections brought forward by Mr Leslie, and by one of his mathematical correspondents, whom he does not name, but speaks of as being at the head of British mathematicians (p. 294 of his work), we shall not fail to discover that their objections, if well-founded, would overthrow the two principles we have mentioned as well as the demonstration which is derived from them. In this paper we mean to establish M. Legendre's demonstration upon an indisputable basis, and to point out, as we proceed, an immediate consequence from it.

Let us first recollect what it is that forms the ground-work of this demonstration: we shall then present the objections and our replies to them.

According to M. Legendre, since the principle of superposition shews that a triangle is determined when one of its sides and the angles adjacent to this side are known, it follows that the third angle must be entirely determined by these three elements which are supposed to be known, or using the terms of Analysis, that it must be a determinate, though as yet an unknown function of them. Thus naming the angles of this triangle A, B, C, and the sides opposite to them a, b, c, we have the symbolical equation C= (A, B, c). Now A, B, C being determinate angles, are also in a determinate ratio with the total angular quantity, or with the right angle, which is a fourth part of it; they are consequently real numbers, if we assume, as the angular unit, the right angle for example. Hence, if the line c could remain in the preceding equation, we might deduce from it c=f (A, B, C), or that a line is a function of the angular unit; an absurd consequence incompatible with the law of homogeneity. By virtue of this law, therefore, we must admit that the proposed equation will reduce itself to this: C=4(A, B), which signifies that the third angle of a triangle is entirely determined by means of the other two; therefore, two angles of one triangle being equal to two of another, the third in the one must be equal to the third in the other; from which the author very easily infers (by a mode which has never been attacked) that the sum of the three angles in a rectilineal triangle is always equal to two right angles. The function (A, B) is therefore an algebraical function equal to 2-A-B; and the proposed function becomes C+B+A=2, a rigorous equation among abstract numbers.

If the triangle were spherical, this reasoning would not be applicable. In spherical triangles there is another element, the radius of the sphere, to which all the sides must be compared. The proposed equation in

that case would be C=(A, B, ); ; and there is no absurdity in sup

posing, an abstract number, to be a function of angles, since this func

tion itself is an abstract number.

Such is very nearly M. Legendre's demonstration: let us now see Mr Leslie's objections.

According to Mr Leslie this demonstration will not bear a rigid examination. 66 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 shews. Wherefore the base c is derived solely from these data, or c=: (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 two 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 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. ***The whole stress of the argument, it may be perceived, lies in the distinction which M. Legendre endeavours to establish between angles and lines,-a distinction which I hold at bottom to be merely arbitrary. Angles and lines are both equally real quantities, though of different kinds; they are capable of being measured, and consequently represented by numbers, by referring each of them to some definite measure or unit of its own denomination. Angles are measured or expressed numerically by angles, and lines by lines. It is true that the mensuration of angles is facilitated by a reference to a subdivision of the circuit or entire revolution; yet even this mode of denoting angular magnitude is evidently only conventional. As standards for measuring straight lines, nature has furnished the limbs of the human body, and the extent of our globe itself. Such units of mensuration are not indeed very definite or readily attainable; but they are not therefore the less real or prominent. Nor is there any essential difference in principle between the expressing of an angle by degrees, of which 360 or 400 are contained in a complete revolution, and the denoting of a straight line on the French system, for instance, by the number of metres it includes, each of which is the forty millionth part of the entire circumference of the earth. Angles and lines hence present to the mind no radical or absolute discrimination, and therefore the argument grounded on such a distinction must lose all its efficacy." (Leslie's Geometry, pp. 293, 294, 297.)

Such are all the criticisms which belong to the ingenious geometer and natural philosopher of Edinburgh. He adds (p. 294), that, with a single exception, he is acquainted with no geometer of any eminence in Britain, who does not admit the fallacy of the argument employed in the demonstration before us. Nevertheless, it is easy, in reading his objections, to perceive that they rest on a palpable error; he confounds, throughout, the natural homogeneity of quantities, with the principle of homogeneity observed in every relation, founded on the laws of the calculus; which two principles are widely different.

Let us begin by resuming this latter principle, and establishing it firmly.

A very little reflection will convince us that no equation can subsist except between homogeneous quantities, that so it may afterwards be reduced to a relation between abstract numbers, when each term of this equation shall have been delivered (if necessary), by division from the unit peculiar to the nature of the quantities considered. Thus no question of arithmetic or algebra ever occurred, in which the unknown quantities were aught but abstract numbers; and it is in knowing how to write the questions, so as to obtain this end, that the art of the analyst consists. So, likewise, in questions of Mechanics, no times or velocities, for example, are ever actually assigned for the value of certain spaces: in such questions, where it seems at first that heterogeneous quantities are combined together, it may still be discovered, that by the skill of the analyst, the true elements of the calculation at length become simply the ratios of these different quantities to their respective units, and the ratios of these ratios; so that, even in questions of this sort, no actual relation is ever definitively established, except between abstract numbers.

Consequently, in the case before us, when we have arrived at the general relation expressed by the symbolical equation C=4 (A, B, c,) it is rigorously essential to its existence that it be capable of being re duced to a relation among abstract numbers. Now, if only the angles A, B, C entered into it, there would be no difficulty: for since each of them expresses a multiple or submultiple of the angular unit, this unit may be made to disappear by means of division. But if the straight line centers also, this relation becomes manifestly absurd, since there occur in it two heterogeneous units, which cannot both be made to disappear from the calculation. If, however, another straight line as a or b, other sides of the triangle, or if a and b occurred at once in the equation, this impossibility of the equation would no longer exist; since we are allowed to suppose that only the ratios of those lines enter into the function; and as these ratios are abstract numbers, the equation would rigorously subsist, inasmuch as these ratios would be factors of the angles or of certain functions of the angles, because in that case, the angular unity disappearing from the calculation, there would only remain a relation between abstract numbers. Hence it is because the line c is the only straight line which occurs in the proposed relation, that we are rigorously authorised a priori to eliminate this line from it, as a quantity which cannot remain without leading to a manifest absurdity.

We can also give an irrefragable mathematical proof, a posteriori,

that 'this elimination must take place, in the case we are now considering. Suppose the triangle in question were a spherical one; we

have seen that the symbolical equation becomes C= (A, B, C), B), and

that we cannot here reason as we did in the case of the rectilineal triangle, being a real number. But if r becomes infinite, that is to say, if the sphere becomes a plane, and so our spherical triangle is changed into a rectilineal one, then it is evident that C, A, B, the determined angles being finite multiples or submultiples of the angular unit, which itself bears a finite proportion to the total angular quantity, those angles can no longer be compared with the quantity, and this, whatever be the magnitude of c; whence it is evident that the magnitude of the angle C cannot be affected by that of the side c, and that hence the proposed relation must of necessity reduce itself to the rigorous equation C=4 (A, B.)

It is objected, that on the same principle, from this other symbolical relation c(a, b, C), it will also be necessary to eliminate C, and therefore to arrive at an absurd conclusion; an objection which proves still more clearly, that Mr Leslie has confounded the homogeneity of quantities with the principle of homogeneity in the calculus. But M. Legendre has shewn sufficiently, that such an elimination is not in this case necessary; and for proof he exhibits the trigonometrical relation b2 C2

cos C =

1+

a2

2

b a

a2

-, an equation between abstract numbers. Now

this example is not, as might in the first glance be supposed, at variance with the principles here laid down: if the angular unit, though not affecting all the terms, has disappeared, the reason is, that the proposed relation turns out to establish, that a certain function of the sides of a triangle is a function of one of its angles; and as this latter function may be transcendental, and consequently, equal to the cosine of the arc which corresponds to that angle, it follows, that just as we have the general relation Angle = so also we have a certain cosine (and every cosine must be an abstract number) equal to a certain algebraical function of the ratios between the sides. By writing the trigonometrical

Arc
Radius'

This supposes r = in a circle whose radius is 1.

r

1, and the angle C to be measured by the arc C taken Note by M. Legendre.