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Let ABCD be any polygon. Having taken naar Mudi, aks any side AB, upon AB as a base, form as many triangles ABC, ABD, &c. as there are angles C, D, E, &c. lying out of it.
Put the baseba AB p; let A and B represent the two d angles of the triangle ABC, which are adjacent t to the side AB; A' and B' the two angles of the triangle ABD, which are adjacent to the same side AB, and so on. The figure ABCDE will be entirely determined, if the side p with G the angles A, B, A', B', A", B", &c. are known, and the number of data will in all amount to 3, n being the number of the polygon's sides. This being granted, any side or line x, anyhow drawn in the polygon, and from the data alone which serve to determine this polygon, will be a function of those given quantities; and
since must be a number, we may suppose
= 4: (A, B, A', B', &c.)
P or x= p↓ (A, B, A', B', &c.), and the function will not contain p. If with the same angles, and another side p', a second polygon be formed, the line a corresponding or homologous to x will have for its value x' = p': (A, B, A', B', &c.); hence x:x::p: p'. Figures thus constructed might be defined as similar figures; hence in similar figures the homologous lines are proportional. Thus, not only the homologous sides and the homologous diagonals, but also any lines terminating the same way in the two figures, are to each other as any other two homologous lines whatever.
Let us name the surface of the first polygon S; that surface is ho
mogeneous with the square p2; hence must be a number, contain
ing nothing but the angles A, B, A' B' &c.; so that we shall have S =p24: (A, B, A', B', &c.); for the same reason, S' being the surface of the second polygon, we shall have S'p'20: (A, B, A' B', &c.) Hence S: S':p2: p'2; hence the surfaces of similar figures are to each other as the squares of their homologous sides.
Let us now proceed to polyedrons. We may take it for granted, that a face is determined by means of a given side p, and of the several given angles A, B, C, &c. Next, the vertices of the solid angles which lie out of this face, will be determined each by means of three given quantities, which may be regarded as so many angles; so that the whole determination of the polyedron depends on one side, p, and several angles A, B, C, &c. the number of which varies according to the nature of the polycdron. This being granted, a line which joins to no vertices, or more generally, any line a drawn in a determinate manner in the polyedron, and from the data alone which serve to construct it, will be a function of the given quantities p, A, B, C, &c.; and since X must be a number, the function equal to will contain nothing but
p' the angles A, B, C, &c., and we may put x = p: (A, B, C, &c.)
The surface of the solid is homogeneous to p2; hence that surface may be represented by p2 ↓ : (A, B, C, &c.): its solidity is homogeneous with p3, and may be represented by p3 n: (A, B, C, &c.), the functions designated by and n being independent of p.
Suppose a second solid to be formed with the same angle A, B, C, &c., and a side p' different from p; and that the solids so formed are called similar solids. The line which in the former solid was po: (A, B, C, &c.), or simply po will in this new solid become p' ; the surface which was p2 in the one, will now become p2 in the other ; and, lastly, the solidity which was p3 п in the one, will now become p'3n in the other. Hence, first, in similar solids, the homologous lines are proportional; secondly, their surfaces are as the squares of the homologous sides; thirdly, their solidities are as the cubes of those same sides.
The same principles are easily applicable to the circle. Let c be the circumference, and s the surface of the circle whose radius is r; since there cannot be two unequal circles with the same radius, the quantities S and must be determinate functions of r: but as these quantities are
numbers, the expression of them cannot contain r; and thus we shall
have = and = 6, a and being constant numbers. Let c' be
the circumference, and the surface of another circle whose radius is
and s:s::r2:2; hence the circumferences of circles are to each other as their radii, and their surfaces as the squares of those radii.
Let us now examine a sector whose radius is r. A being the angle at the centre, let a be the arc which terminates the sector, and y the surface of that sector. Since the sector is entirely determined when r and A are known, x and y must be determinate functions of r and A;
are also similar functions. But is a number, as well
; hence those quantities cannot contain r, and are simply func
y' be the arc, and the surface of another sector, whose angle is A, and radius r'; we shall call those two sectors similar: and since the angle
A is the same in both, we shall have = 4: A, and Y = : A.
Hence x:x::r:r', and y: y'::r2:2; hence similar arcs, or the arcs of similar sectors are to each other as their radii; and the sectors themselves are as the squares of the radii.
By the same method, we could evidently shew, that spheres are as the cubes of their radii.
In all this, we have supposed that surfaces are measured by the pro
duct of two lines, and solids by the product of three; a truth which it is easy to demonstrate by Analysis, în like manner. Let us examine a rectangle, whose sides are P and q; its surface, which must be a function of p and q, we shall represent by ø: (p, q). If we examine another rectangle, whose dimensions are p+p' and q, this rectangle is evidently composed of two others, of one having p and q for its dimensions, of another having p' and q; so that we may put 4: (p + P', q) = : (p,q) + (p', q). Let p' =p; we shall have 4 (2 p, q) = 2 øc (p, q.) Let p'=2p; we shall have 4 (3 p, q)= (p, q) + (2 p, q)= 3 p; we shall have 4 (4 p, q) = 4 ( P, q) + 9 Φ Hence, generally, if k is any whole number, we ( P, ¶ ) _ ↑ ( k p, ¶ ); from
shall have (kp, q)
3(p, q.) Let p'
which it follows that
= k 4 (p, q), or
is such a function of p as not to be
changed by substituting in place of p any multiple of it kp. Hence this function is independent of p, and cannot include any thing except
q. But for the same reason
must be independent of q; hence
℗ (P, q )
and must therefore, be limited to a
(p, q) = « p q; and α = 1, we shall have
constant quantity Hence we shall have as there is nothing to prevent us from taking (P, q) = p q ; thus the surface of a rectangle is equal to the product of its two dimensions.
In the very same manner, we could shew that the solidity of a rightangled parallelepipedon, whose dimensions are p, q, r, is equal to the product p qr of its three dimensions.
We may observe, in conclusion, that the doctrine of functions, which thus affords a very simple demonstration of the fundamental propositions of geometry, has already been employed with success in demonstrating the fundamental principles of Mechanics. See the Memoirs of Turin, vol. II.
ADDITION TO NOTE II.
(Furnished by M. LEGENDRE for this Edition.)
THE celebrity which Professor Leslie of Edinburgh so justly enjoys, forbids me to pass over in silence the objections which this learned geometer has adduced against the foregoing theory, and particularly against my proof of the equation C= (A, B), from which our theorem concerning the three angles of a triangle is derived.
The objections alluded to, first made their appearance in the second edition of Mr Leslie's Elements of Geometry, pp. 403 et seq. and though they were refuted, quite completely, as I think, in the equally severe and judicious criticism of that work, published by Mr Playfair, in the Edinburgh Review, vol. xx; though I replied to them in a private let
ter addressed to the author; yet Mr Leslie in his 3d edition, 1817, (pp. 292 et seq.), has again brought forward his objections, inserting along with them an extract from my letter* (which, as may be gathered from the end of the quotation, was not in any way designed for publication), and subjoining, in favour of his opinion, the testimony of a mathematician whom he does not name, but declares to be at the head of British geometers.
Without entering into any profound discussion of this question, I shall restrict myself to place before the reader the principal point of the difficulty. I am required to shew, in opposition to Mr Leslie's opinion, that a line, which is an absolute length, cannot be determined solely from angles, which are represented in calculation by their ratios to the right angle assumed as unity, that is to say, by numbers always included between 0 and 2. Thus the side c of a triangle cannot be determined solely from the angles A, B, C of this triangle; for these angles being only numbers, they can of themselves serve only for determining numbers. Accordingly, what information would you gain, if the value of the side c, as determined by calculating the function which represented it, were to come out, for example? Is it of an inch, o of a foot, of the earth's radius, or of the sun's distance? You cannot say, unless the question offer some other linear datum as unit, or which may serve for unit. With angles, however, the case is different, because the right angle is their natural unit, and any angle is completely determined, whenever we have discovered the value of its numerical ratio to the right angle.
If it is absurd to suppose, that the line c can be determined by the numerical quantities A, B, C alone; hence there can exist no equation between the quantities e, A, B, C. Hence the equation C= (A, B, c), which is given immediately by the principle of superposition, can exist only on condition that c disappear from the second member of it ; otherwise, we are taught by Analysis that an equation containing at once the four quantities A, B, C, c, would allow us to determine c by means of angles A, B, C. Hence we have simply C (A, B); that is, in every triangle, two given angles determine the third. From this property, it is easy to deduce the theorem concerning the three angles of a triangle, either by the demonstration given above, or by that which Mr Leslie himself proposes, which is very simple, but subject to exception in the case of the equilateral triangle.
Mr Leslie endeavours to draw an argument against this theory, from the case where the angle C, and the two sides a and b which contain it are given. Here, the third side c must be entirely determined by those data, which is expressed thus: C= (a, b, C). Mr Leslie adds, (page 93, 3d edit.)
"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. Such is the extreme absurdity to which this sort of reasoning would lead."
But the only absurdity here is the reasoning employed by the
Two errors of the press have found their way into this extract; they may be corrected by putting c in place of C, page 296, line 31, &c.
objector, of which certainly I did not give him the example. Though an angle C, measured by a number, is undoubtedly heterogeneous to each of the quantities a and b which are lines, it is not on
that account heterogeneous to their ratio; and consequently there is
no reason to expunge C from the function (a, b, C.) In the present case, where the function is to represent the line c, this line c must be homogeneous with the lines a and b, and of one dimension; or what amounts to the same, it must be the product of a
by a function of- and of C; hence we shall have simply ca↓
c); so that the function of three quantities is, in this case, reduced to a function of two quantities only. Such is the common doctrine of analytical writers: See the Introduct. in Anal. of Euler, p. 65. It is
evident, moreover, that the equation c⇒a 4
with the trigonometrical formula, cao 1+
this, therefore, no objection against my theory can be drawn, but rather a full and entire confirmation of it.
To shew in a manner, if possible still clearer, that the law of homogeneity, combined with the known principles of the theory of functions, can lead to no results but what are accurate, let us consider an isosceles triangle formed by two equal sides a, a, and the angle C contained by them. Since this triangle is entirely determined by the given quantities a and C, the angle A opposite the side a must be a determinate function of the quantities a and C; we shall express it thus :A= (a, C.) Now, if the quantity a does not disappear from the function, then from the equation A= (a, C), the value of a might be deduced in terms of A and C. But a line a, not being referred to any unit, cannot be equal to a function of two numbers A and C, which must itself be a number. Hence we have simply A= (C); that is to say, in every isosceles triangle, the angle at the base is determined by the angle at the vertex, and conversely. From the vertex to the middle of the base, draw a straight line; it will divide the isosceles triangle into two equal right-angled triangles, and you will infer that in these right-angled triangles, one acute angle determines the other. Hence we conclude, as in the first demonstration, that the two acute angles of a right-angled triangle are together equal to a right angle; and then that in every triangle the sum of all the angles is equal to two right angles.
Let us now proceed to the chief objection made by the anonymous mathematician whose suffrage Mr Leslie brings to bear against me. He maintains that I have done nothing but elude the difficulty; and that my method, or what he calls the mise en équation of the problem, involves a supposition equivalent to Euclid's postulate, because I have considered as existing, and already constructed a triangle in which the angles amount to a sum as near two right angles as we please. If I had indeed made this supposition, my critic were un