On the Equation to the Hyperbola between the Asymptotes.

Let CE and CB be the two asymptotes to the hyperbola drD, its vertex being F, and EF, bd, AF, BD ordinates parallel to the asymptotes. Put AF or EF = a, CB = X, 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.

E

d

CAB

If the hyperbola be not rectangular AF. EF. sin. F will be equal to a given square.

3. For the Parabola.

If a 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 ay y p; hence pry is the equation to the parabola. Or, if a= abscissa and b the corres

ponding semiordinate, then y, is the equation.

x =

4. For the Circle.

1

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 t and c equal each to d in the foregoing equa. tions to the ellipse, they become y3 dx - x2, when the absciss begins at the vertex of the diameter: and y2 = d2x2, when the absciss begins at the centre. Or y= (2r x-x), and y=(-), respectively, when r is the radius.

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 any one of these curves may be cut by a right line. Hence also it is 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 having some higher equation, or may be cut in more points by a right line.

We may here add an important observation with regard to all curves expressed by equations: viz. that the origin of the co-ordinates is necessarily on a point of the curve itself

So

when all the terms of its equation are affected by one of the variable quantities zor y; and when, on the contrary, there is in the equation one term entirely known, then the origin of the co-ordinates cannot be on a point of the curve. In proof of this, let the general equation of a curve be ax = 0, +bx? y? +eyn = 0; then, it is evident that if we take x= we shall likewise have cy" =0, or y = 0; and consequently the origin of the co-ordinates is a point in the curve. again, if, in the same equation, we take y = 0, it will result that ax = 0, and x = = 0, which brings us to the same thing as before. But, if the equation of the curve include one known term, as, for example, ax + bx y + cy' —g"=0; then taking x =o, we shall have cy' — g*= 0, or y = '/_8", which proves that the corresponding point r, of the curve, is distant from the origin of the x's by the quantity '/ A similar truth will flow from making y = 0, when the same equation will give x =

C

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 mi

nimum.

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 admir. able 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 Eléments 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 Abridgement, vol. 13, page 653*. The chief propositions deduced by these four geometers, together with a few additional propositions, are reduced into one system in the following theorems.

* Another work on the same general subject, containing many valuable theorems, has been published since the first edition of this volume, by Dr. Creswell of Trinity College, Cambridge.

SECTION I.

SURFACES.

THEOREM I.

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 in position, then is the triangle of least perimeter that whose sides AC, BC, are inclined to the line LM in equal angles.

L

B

D

M

For, let Bм be drawn from в, per. pendicularly 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'B, C'D. Then the triangles DMC, BMC, having the angle DCM = angle ACL (th. 7 Geom.) =мCB (by hyp.), DMC BMC, and DM = BM, and Mc common to both, have also DC = BC (th. 1 Geom.).

So also, we have c'n c'в. Hence Ac+CB = AC + CD AD, is less than ac' + c'r (theor. 10 Geom.), or than its equal Ac'c'B. And consequently, AB + BC + AC is less than AB+BC' + AC'. Q. E. D.

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 LM 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; follows that CAB CBA, and consequently ac = CB (th. 4 Geom.)

it

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 for a 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 triangle 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.

THEOREM II.

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 ARC 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.

A. E

D

B

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. 1 cor. 1), or than ACB (by hyp.). Consequently AC AC; and in the right-angled triangles AEC, AEC, having AE common, we have c'E < CE *. Q. E. D.

* When two mathematical quantities are separated by the character <,