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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 x, the other BK will be d + X', which in the ellipse was d X'; so the sign of x must be changed in the general property and equation, by which it becomes d:c::: x(d + x): yé; hence d'y= oʻ(dx + 2.2) and dy = CV (lx + x2), the equation of the
Or, using p the parameter as before, it is, ú:p :: X(1 + x):y?, or dy = pdx + x), 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 x for the absciss CK counted from the centre; then is AK=X - il, and BK=x+d, and the property dla : 0 :: 6.7 – d) x (x + d) : y’, gives d’y= c+(x" – d?), or dy = CV (.232 – d), where the signs of d and x are changed from what they were in the ellipse.
Or again, using the semiparameter, d:p :: 22 and dy' = P(x2 – d) the equation of the curve.
But for the conjugate hyperbola, as in the figure to theorem 3, the signs of both .22 and d will be positive; for the property in that theorem being CA?: ca’ :: CD? + CA: De', it is d: 0 : : x2 + d2 :
:yo De, or d’y2 = co(.272 + d’), and dy = CN(+ d), the equation to the conjugate hyperbola.
Or, as à :p :: 22+ dry, and dy = P(x + d), also the equation to the same curve. On the Equation to the Hyperbola between the Asymptotes.
Let CE and co be the two asymptotes to the hyperbola dfd, its vertex being F, and
ch EF, bd, AF, BD ordinates parallel to the asymptotes. Put AF or EF =
a, CB X, EL F and BD = Y.
Then, by theor. 28, AF. EF = CB.BD, or a? = xy, the equation to the
съAB hyperbola, when the abscisses and ordinates are taken parallel to the asymptotes:
3. For the Parabola. If x 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 X: Y::y: p; hence p.x = y’ 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 y? = dx – r?, when the absciss x begins
= at the vertex of the diameter : and y: = – 22, when the absciss begins at the centre.
Scholium, 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 magni. tude it can possibly attain; and, on the contrary, when it has arrived at the least possible inagnitude, 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, &c. 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
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
Q. E. D.
in position, then is the triangle of least perimeter that whose sides AC, BC, are inclined to the line LM in equal angles.
For, let BM be drawn from B, perpendicularly to LM, and produced till DM = BM: join AD, and from the point c where ad cuts LM draw BC: also, from any other point c', assumed in LM, draw c'a, c'é, c'd. Then the triangles DMC, BMC, having the angle DCM = angle Act (th. 7 Geom.) = MCB (by hyp.), DMC = BMC, and DM = BM, and mc common to both, have also DC = BC (th. 1 Geom).
So also, we have c'd = C'B. Hence AC + CB = AC+CD – AD, is less than ac' + c'd (theor. 10 Geom.), or than its equal ac' + c'B. And consequently, AB + BC + Ac is less than AB + BC' + AC'.
Cor. 1. Of all triangles of the same base and the same altitude, or of all equal triangles of the same base, the isosceles triangle has the smallest perimeter.
For, the locus of the vertices of all triangles of the same altitude will be a right line im parallel to the base; and when LM in the above figure becomes parallel to AB, since MCB = ACL, MCB = CBA"(th. 12 Geom.), ACL = CAB; it follows that caB = CBA, and consequently AC = CB (th. 4 Geom.).
Cor. 2. Of all triangles of the same surface, that which has the minimum perimeter is equilateral.
For the triangle of the smallest perimeter, with the same surface, must be isosceles, whichever of the sides be considered as base: therefore, the triangle of smallest perimeter has each two or each pair of its sides equal, and consequently it is equilateral.
Cor. 3. Of all rectilinear figures, with a given magnitude and a given number of sides, that which has the smallest perimeter is equilateral.
For so long as any two adjacent sides are not equal, we may draw a diagonal to become a base to those two sides, and then draw an isosceles triangle equal to the triangle so cut off, but of less perimeter: whence the corollary is manifest.
Scholium. To illustrate the second corollary above, the student may proceed thus : assuming an isosceles triangle whose base is not equal to either of the two sides, and then, taking fora new base one of those sides of that triangle, he may construct another isosceles triangle equal to it, but of a smaller perimeter. Afterwards, if the base and sides of this second isosceles triVOL. III.
angle are not respectively equal, he may construct a third isosceles triangle equal to it, but of a still smaller perimeter: and so on. In performing these successive operations, he will find that the new triangles will approach nearer and nearer to an equilateral triangle.
Of all Triangles of the Same Base, and of Equal Perimeters,
the Isosceles Triangle has the Greatest Surface. Let ABC, ABD, be two triangles of the same base AB and with equal perimeters, of which the one ABC is isosceles, the other is not : then the triangle ABC has a surface (or an altitude) greater than the surface (or than the altitude) of the triangle ABD.
Draw c'd through D, parallel to AB, to cut ce (drawn perpendicular to AB) in c': then it is to be demonstrated that CE is greater than c'e.
The triangles AC'B, AdB, are equal both in base and altitude; but the triangle AC'B is isosceles, while ADB is scalene: therefore the triangle ACB has a smaller perimeter than the triangle ADB (th. I cor. 1), or than ACB (by hyp.). Consequently Ac'<AC; and in the right-angled triangles A EC', AEC, having Ae common, we have c'e < CE*. Q. E. D.
Cor. Of all isoperimetrical figures, of which the number of sides is given, that which is the greatest has all its sides equal. And in particular, of all isoperimetrical triangles, that whose surface is a maximum, is equilateral.
For, so long as any two adjacent sides are not equal, the surface may be augmented without increasing the perimeter.
Remark. Nearly as in this theorem may it be proved that, of all triangles of equal beights, and of which the sum of the two sides is equa., that which is isosceles has the greatest base. And, of all triangles standing on the same base and having equal vertical angles, the isosceles one is the greatest.
When two mathematical quantities are separated by the character <, it denotes that the preceding quantity is less than the succeeding one: when, on the contrary, the separating character is >, it denotes that the preceding quantity is greater than the succeeding one.