ends at the points F and G, and stretching the string by means of a pin H, let the pin be moved round, keeping the string tight, when the tract formed by the pin will be the ellipse required; for the sum of the distances of any point, in the circumference from the two foci, will always be equal to the transverse axis.

METHOD II.

To describe an ellipse by finding a number of points in

the curve,

Let AB and CD (Fig. 1, Plate 6,) be the transverse and conjugate diameters of the ellipse required to be described: by the rule just given, find both or either of the foci, F and G; at each extremity of the transverse axis, A and B, erect the perpendiculars HK and LM, making AH and AK equal to AF, and BL and BM equal to BF, and join HL and KM; then take any point, N in AB, and through it draw the perpendicular OP; and on F, as a centre, with a radius equal to NO or NP, describe an arch of a circle cutting OP in the points Q and R, then will these be two points in the curve of the ellipse.

In the same manner, if another point n be taken in AB, and through it be drawn the perpendicular onp; then, if from F, with a radius equal to no, an arch be described cutting op in the points q and r, these points will also be in the curve of the ellipse: and by continuing such a process, a sufficient number of points may be obtained through which the ellipse may be drawn.

METHOD III. (Fig. 2, Plate 6.)

Let AB be the transverse axis, and CD the conjugate axis of the ellipse intended to be drawn: from the point. H, in the conjugate axis, set off the line HG equal to the difference

3P 2

difference between the two semi-axes OA and OC, and produce HG, making GP equal to OC; then will P be a point in the curve of the ellipse.

In the same way, let the line EF, made equal to the sum of the two semi-axes OA and OC, be placed any where on the conjugate axis, as, for example, at E, and meeting the transverse axis (produced, if necessary, as in this figure) in the point F, and let FK be set off equal to OC, then will the point K be in the curve of the ellipse; and so other points may be found.

On the property of the curve here shown, is founded the construction of the instrument called the Trammels, used for describing elliptic curves. This instrument, which is represented in Fig. 3 of Plate 6, consists of two rulers, ab and cd, firmly secured at right angles in the middle of each, having a groove running lengthwise in their upper surface: a long ruler, or rod ef, having two moveable nuts and pins, g and f, is applied to the rulers, and works in both grooves, while a steel point or pencil at e describes the curve of the ellipse by the motion of the rod. Make the distance between the pins ƒ and g equal to the difference between the semi-axis of the ellipse and the distance between the pin at g, and the pencil at e, equal to half the conjugate axis, when consequently the distance between the pin at f and the pencil at e will be equal to half the transverse axis. When this is done, by placing the pins secured by the nuts at ƒ and g in the grooves of the cross rulers, and moving the rod ef gently round, the pencil at e will describe the required ellipse.

Other methods have been recommended for describing ellipses, by means of a pair of compasses, but these are all erroneous, and can only give a figure approaching to the ellipse; for the curves described by the compasses must, in every part, however small, be portions of circles, whereas the elliptic curve is in every part different from

a

a circle, as must be evident, when it is onsidered that the circle is described with one fixed radius from one centre, but that the ellipse is described with radii continually changing their dimensions, and from two centres. If an ellipse be given, and its axis required, the following method is to be followed:

In Fig. 4, Plate 6, Let there be given an ellipse of which it is required to find the axis; within the ellipse, draw the two chords AB and CD parallel to each other, also the line EF, bisecting these chords, and terminated both ways in the curve: bisect this line in the point G, which will be the centre of the ellipse. Then take any point in the curve, as H, and from G, with the radius GH, describe a circle. If this circle fall wholely without the ellipse, the point H will be the vertex of the greater axis; and if it fall wholely within the ellipse, H will be the vertex of the shorter axis; but if the circle falls neither wholely without nor wholely within the ellipse, it must cut it in some other point, as K; then draw HK, and bisecting it in L, draw GL, and produce it both ways to meet the curve in the points M and N, when MN will be the transverse axis of the ellipse, and OP, drawn perpendicular to it through the centre G, will be the conjugate axis, as required.

MENSURATION

MENSURATION OF SOLIDS.

By a solid body we understand whatever is described under the three dimensions of length, breadth, and depth or thickness. Solids are bounded by plane surfaces, as a cube, a prism, &c. or by curved surfaces, as a globe, a cone, &c.

All solid figures, whose extremities are similar, equal and parallel planes, and all whose other sides are parallelograms, are in general termed prisms, such as are represented in Plate 6, Fig. 5, 6, 7, and 8; and the prism may be produced by the motion of one of the planes at its extremity along a right line, in a position always parallel to that which it had at the beginning of the motion.

The two parallel extremities are called the bases of the prism, and a perpendicular let fall from one base to the other, gives the altitude of the prism.

If a prism be cut across in any place by a plane parallel to the base, this section will be perfectly similar, and equal to the base.

A prism is said to be right when the sides are perpendicular to the base; but it is an oblique prism when the sides are not perpendicular to the base.

Prisms are denominated according to the number of their sides; thus, one of three sides, as in Fig. 5, is a triangular prism; Fig. 6 and 7 are quadrangular prisms, their bases consisting of four sides; and Fig. 8 a hexagonal prism, the base consisting of six sides.

Amongst quadrangular prisms are particularly distinguished the parallelopipedon and the cube; the parallelopipedon is a prism whose bases, as well as sides, are

parallel