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THEOREM XXIV (20).
If two Parallels intersect any other two Parallels; the Rectangles of the Segments will be respectively Proportional. That is, CK. KL: DK. KE :: GỊ. IH: NI.IO.
3 theor. 23,
QI:: CK. KL: GI. IH;
QI DK.KE: NI.IO;
For, by cor. and by the same, theref. by equal. CK. KL: DK. KE:: GI. IH: NI. 10. Corol. When one of the pairs of intersecting lines comes into the position of their parallel tangents, meeting and limiting each other, the rectangles of their segments become the squares of their respective tangents. So that the constant ratio of the rectangles, is that of the square of their parallel tangents, namely,
CK.KL: DK.KE:: tang2. parallel to c; tang2, parallel to DE.
THEOREM XXv (21).
If there be Three Tangents intersecting each other; their Segments will be in the same Proportion.
That is, GI IH :: CG GD :: DH HE.
For, through the points G, I, D, H, draw the diameters GK, IL, DM, HN; as also the lines CI, EI, which are double ordinates to the diameters GK, HN, by cor. 1 theor. 16; therefore the diameters GK, DM, HN, bisect the lines CL, CE, LE;
But the 3d terms KL, CK, MN are all equal;
as also the 4th terms LN, KM, ne.
Therefore the first and second terms, in all the lines, are proportional, namely GI IH :: CG : GD :; DH; HE. Q. E.D.
ON THE CONIC SECTIONS AS EXPRESSED BY ALGEBRAIC EQUATIONS, CALLED THE EQUATIONS OF THE CURVE.
Let d denote AB, the transverse, or any diameter;
x = AK, any absciss, from the extremity of the diam.
Then, theor. 2, AB2: HI2:: AK. KÈ : DK2,
that is, d2: c2:: x(d-x): y2, hence d'y2 = c2(dx- x2), or dy = c(dx-2), the equation of the curve.
And from these equations, any one of the four letters or quantities, d, c, x, y, may easily be found, by the reduction of equations, when the other three are given.
Or, if p denote the parameter, = cd by its definition; then, by cor. th. 2, d: p :: x(d-x) : y2, or dy2=p(dx—x2), which is another form of the equation of the curve.
Or, if d AC the semiaxis; c = CH the semiconjugate; pc2d the semiparameter; x = CK the absciss counted from the centre; and y = DK the ordinate as before.
Then is AK=d-−x, and кB=d+x, and AK.KB = (d−x)× (d+x) = d2 — x2
Then, by th. 2, d2: c:: d2-x:y, and d2y2=c2(d2 — x2), or dy = c(d2x2), the equation of the curve.
Or, d:p :: d2x2 : y2, and dy2 = p(d2 — x2), another form of the equation to the curve; from which any one of the quantities may be found, when the rest are given.
2. For the Hyperbola.
Because the general property of the opposite hyperbolas, with respect to their abscisses and ordinates, is the same as
that of the ellipse, therefore the process here is the very same as in the former case for the ellipse; and the equation to the curve must come out the same also, with sometimes only the change of the sign of a letter or term, from + to -, or from to +, because here the abscisses lie beyond or without the transverse diameter, whereas they lie between or upon them in the ellipse. Thus, making the same notation for the whcle diameter, conjugate, absciss, and ordinate, as at first in the ellipse; then, the one absciss AK being a, the other BK will be dx, which in the ellipse was d x; so the sign of a must be changed in the general property and equation, by which it becomes d2: c2::: x(d+ x): y; hence d'y2 = c(dxa) and dy = c(data), the equation of the + x2) x2),
Or, using p the parameter as before, it is, d:p :: x(d+x):y2, or dy = p(dx+), another form of the equation to the
Otherwise, by using the same letters d, c, p, for the halves of the diameters and parameter, and r for the absciss CK counted from the centre; then is AK=rd, and вê=x+d, and the property d2: c2:: (r- d) × (x + d) : y', gives d2y2 = c2(x2 — d2), or dy = c(2 - d2), where the signs of d and r2 are changed from what they were in the ellipse. Or again, using the semiparameter, d:p :: x12 and dy = p(x2d) the equation of the curve.
But for the conjugate hyperbola, as in the figure to theorem 3, the signs of both a2 and d2 will be positive; for the property in that theorem being CA: ca2:: CD2 + CA2 : De2, it is d2: c2x2 + d2 : y2 = De2, or d2y2 c2(x2 + d2), and dy = c√(x2 + d2), the equation to the conjugate hyperbola. Or, as dp x2 + d2: y2, and dy2 = p(x2 + d2), also the equation to the same curve.
-On the Equation to the Hyperbola between the Asymptotes.
Let CE and CB be the two asymptotes to the hyperbola dFD, its vertex being F, and EF, bd, AF, BD ordinates parallel to the asymptotes. Put AF or EF = a, CB = X's and BD = y. Then, by theor. 28, AF. EF CB. BD, or a2 xy, the equation to the hyperbola, when the abscisses and ordinates are taken parallel to the asymptotes:
3. For the Parabola.
If r denote any absciss beginning at the vertex, and y` its ordinate, also p the parameter. Then, by cor. theorem 1,
AK: KD :: KD : p, or xy:: y: p; hence pr = y2 is the equation to the parabola.
4. For the Circle.
Because the circle is only a species of the ellipse, in which the two axes are equal to each other; therefore, making the two diameters d and c equal, in the foregoing equations to the ellipse, they become y2 = dxr, when the absciss begins at the vertex of the diameter; and y2 x2, when the absciss begins at the centre.
In every one of these equations, we perceive that they rise to the 2d or quadratic degree, or to two dimensions; which is also the number of points in which every one of these curves may be cut by a right line. Hence it is also that these four curves are said to be lines of the 2d order. And these four are all the lines that are of that order, every other curve being of some higher, or having some higher equation, or may be cut in more points by a right line.
ELEMENTS OF ISOPERIMETRY.
Def. 1. When a variable quantity has its mutations regulated by a certain law, or confined within certain limits, it is called a maximum when it has reached the greatest magnitude it can possibly attain; and, on the contrary, when it has arrived at the least possible magnitude, it is called a minimum.
Def. 2. Isoperimeters, or Isoperimetrical figures, are those which have equal perimeters.
Def. 3. The Locus of any point, or intersection, &c, is the right line or curve in which these are always situated.
The problem in which it is required to find, among figures of the same or of different kinds, those which, within equal perimeters, shall comprehend the greatest surfaces, has long engaged the attention of mathematicians. Since the admirable invention of the method of Fluxions, this problem has been elegantly treated by some of the writers on that branch
of analysis; especially by Maclaurin and Simpson. A much more extensive problem was investigated at the time of "the war of problems," between the two brothers John and James Bernoulli: namely, "To find, among all the isoperimetrical curves between given limits, such a curve, that, constructing a second curve, the ordinates of which shall be functions of the ordinates or arcs of the former, the area of the second curve shall be a maximum or a minimum." While, however, the attention of mathematicians was drawn to the most abstruse inquiries connected with isoperimetry, the elements of the subject were lost sight of. Simpson was the first who called them back to this interesting branch of research, by giving in his neat little book of Geometry a chapter on the maxima and minima of geometrical quantities, and some of the simplest problems concerning isoperimeters. The next who treated this subject in an elementary manner was Simon Lhuillier, of Geneva, who, in 1782, published his treatise De Relatione mutua Capacitatis et Terminorum Figurarum,
His principal object in the composition of that work was to supply the deficiency in this respect which he found in most of the Elementary Courses, and to determine, with regard to both the most usual surfaces and solids, those which possessed the minimum of contour with the same capacity; and, reciprocally, the maximum of capacity with the same boundary. M. Legendre has also considered the same subject, in a manner somewhat different from either Simpson or Lhuillier, in his Elements de Géométrie. An elegant geometrical tract, on the same subject, was also given, by Dr. Horsley, in the Philos. Trans. vol. 75, for 1775; contained also in the New Abridgment, vol. 13, page 653. The chief propositions deduced by these three geometers, together with a few additional propositions, are reduced into one system in the following theorems.
SECTION I. SURFACES.
Of all Triangles of the same Base, and whose Vertices fall in a right Line given in Position, the one whose Perimeter is a Minimum is that whose sides are equally inclined to that Line.
Let AB be the common base of a series of triangles ABC', ABC, &c, whose vertices c', c, fall in the right line LM, given