apply to it a square LEG; fasten at G one end of a cord, equal in length to E G; fix the other end to the focus F; slide the square steadily along

the straight edge, holding the cord taut against the edge of the square by a pencil D, and it will describe the curve.

PROB. LXIII.-To construct a parabola when the vertex A, the axis A B, and a point M of the curve are given (fig. 167).

Construct the rectangle A B M C; divide M C into any number of equal parts, four for instance; di

vide A C in like manner; con

nect A 1, A 2, A 3; through 1' 2' 3', draw parallels to the axis. The intersections I, II, III, of these lines are points in the required curve.

C

d e

Fig. 167.

M

PROB. LXIV. To draw a tangent to a given point II of the parabola (fig. 167).

From the given point II let fall a perpendicular on the axis at b; extend the axis to the left of A; make A a equal to A b; draw a II, and it is the tangent required.

The lines perpendicular to the tangent are called normals. To find the normal to any point I, having the tangent to any other point II.-Draw the normal II c; from I let fall a perpendicular I d on the axis A B; lay off de equal to be; connect I e, and we have the normal required. The tangent may be drawn at I by a perpendicular to the normal I e.

The Hyperbola.

An hyperbola is a curve from any point P in which, if two straight lines be drawn to two fixed points, F F' the foci, their difference shall always be the same.

PROB. LXV. To describe an hyperbola (fig. 168).

From one of the foci F, with an assumed radius, describe an arc, and from the other focus F', with another radius exceeding the former by the given difference, describe two small arcs, cutting the first as at P and p. Let this operation be repeated with two new radii, taking care that the second shall exceed the first by the same difference as before, and two new points will be determined; and this determination of points in the curve may thus be continued till its track is obvious. By making use of the same radii, but transposing, that is, describing with the greater about F, and the less about F', we have another series of points equally belonging to the hyperbola, and answering the definition; so that the hyperbola consists of two separate branches.

The curve may be described mechanically (fig. 169).—By fixing a ruler to one focus F', so that it may be turned round on this point, connect the extremity of the ruler R to the other focus F by a cord shorter than the whole length F R of the ruler by the given difference; then a pencil P keeping this cord always stretched, and at the same time pressing against the edge of the ruler, will, as the ruler revolves around F', describe an hyperbola, of which F F' are the foci, and the differences of distances from these points to every point in the curve will be the same.

PROB. LXVI.-To draw a tangent to any point P of an hyperbola (fig. 170).

On F' P lay off P p equal to F P; connect Fp, and from P let fall a perpendicular on this line F p, and it will be the tangent required.

The three curves, the ellipse, the parabola, and the hyperbola, are called conic sections, as they are formed by the intersections of a plane with the surface of a cone (plate III).

If the cone be cut through both its sides by a plane not parallel to the base, the section is an ellipse;

F'

F

Fig. 170.

if the intersecting plane be parallel to the side of the cone, the section is a parabola; if the plane have such a position, that when produced it meets the opposite cone, the section is a hyperbola. The opposite cone is a reversed cone formed on the apex of the other by the continuation of its sides.

The Cycloid.

The cycloid is the curve described by a point in the circumference of a circle rolling on a straight line.

PROB. LXVII.-To describe a cycloid (fig. 171).

Draw the straight line A B as the base; describe the generating circle tangent to the centre of this line, and through the centre C draw the line E E parallel to the base; let fall a perpendicular from C upon the base;

E6

32

D
Fig. 171.

E

B

divide the semicircumference into any number of equal parts, for instance six; lay off on A B and C E distances C 1', 1' 2'..., equal to the divisions of the circumference; draw the chords D1, D2..., from the points 1', ‚2',3'... on the line C E, with radii equal to the generating circle, describe arcs;

from the points 1', 2', 3', 4', 5' on the line B A, and with radii equal successively to the chords D 1, D 2, D 3, D 4, D 5, describe arcs cutting the preceding, and the intersections will be points of the curve required.

2d Method (fig. 172).—Let 0 9' be the base line, 0 4 9 the half of the generating circle; divide the half circle into any number of equal parts, say 9, and draw the chord 0 1, 0 2, 0 3, &c.; lay off on the base 01', 1'2', 2' 3'...., equal respectively to the length of one of the divisions of the half circle 01; draw through the points 1', 2', 3'..... lines parallel to the chords 0 1, 0 2, 0 3....; the intersections I,

II, III.... of these lines are centres of the arcs 0 a, ab, bc...., of which the cycloid is composed.

The Epicycloid.

The epicycloid is formed by a point in the circumference of a circle revolving either externally or internally on the circumference of another circle as a base.

PROB. LXVIII.-To describe an epicycloid.

Let us in the first place take the exterior curve. Divide the circumference A B D (fig. 173) into a series of equal parts 1, 2, 3 ...., beginning from the point A; set off in the same manner, upon the circle A M, A N, the divisions 1', 2', 3'.... equal to the divisions of the circumference ABD. Then, as the circle A B D rolls upon the circle A MAN, the points 1, 2, 3 will coincide successively with the points 1', 2′, 3'; and, drawing radii from the point O through the points 1', 2', 3', and also describing arcs of circles from the centre O, through the points 1, 2, 3, ...., they will intersect each other successively at the points c, d, e.... Take now the distance 1 to c, and set it off on the same arc from the point of intersection, c, of the radius AC; in like manner, set off the distance 2 to d, from 6 to A', and the distance 3 to e to A', and so on. Then the points A', A3, A', will be so many points in the epicycloid; and their frequency may be in

creased at pleasure by shortening the divisions of the circular arcs. Thus the form of the curve may be determined to any amount of accuracy, and completed by tracing a line through the points found.

As the distances 1 to c,.... which are near the commencement of the curve, must be very short, it may, in some instances, be more convenient. to set off the whole distance i to 1 from c, and in the same way the distance b to 2 from d to A', and so on. In this manner the form of the curve is the more likely to be accurately defined.

2d Method. To find the points in the curve, find the positions of the centre of the rolling circle corresponding to the points of contact 1', 2′, 3′, &c., which may be readily done by producing the radii from the centre O, through the points 1', 2', 3',.... to cut the circle B C. From these centres describe arcs of a circle with the radius of C A, cutting the corresponding arcs described from the centre O, and passing through the points A', A2, A,.... as before.

When the moving circle A B D is made to roll on the interior of the circumference A M, A N, as shown (fig. 174), the curve described by the point A is called an interior epicycloid. It may be constructed in the