Problem XXXII

To describe a parabola, by finding points in the curve; the axis A B, or any diameter being given, and a double ordinate C D.

Through A draw E F. Fig. 33, parallel to C D; through C and D draw D F and C E parallel to A B, cutting E F at E and F. Divide B C and B D, each into any number of equal parts, as four; likewise divide C E and D F into the same number of equal parts. Through the points 1, 2, 3, etc. in C D, draw the lines 1 a, 2 b, 3 c, etc. parallel to A B; also through the points, 1, 2, 3, etc. in C E and D F, draw the lines 1 A, 2 A, 3 A, cutting the parallel lines at the points a, b, c; then the points a, b, c, are in the curve of the parabola.

Problem XXXIII

To describe an hyperbola.

If B and C, Fig. 34, be two fixed points, and a ruler A B be made movable about the point B, a string ADC being tied |to the other end of the ride, and to the point C; and if the point A be moved round the centre B, towards G, the angle D of the string ADC, by keeping it always tight, and close to the edge of the ruler A B, will describe a curve D H G, called an hyperbola.

Fig. 33.

Fig. 34.

Fig. 35.

If the end of the ruler at B were made movable about the point C, the string being tied from the end of the rule A to B, and a curve being described after the same manner, is called an opposite hyperbola.

The foci are the two points B and C, about which the ruler and string revolve.

The transverse axis is the line l H terminated by the two curves passing through the foci, if continued.

The centre is the point M, in the middle of the transverse axis l H.

The conjugate axis is the line N O, passing through the centre M, and terminated by a circle from H, whose radius is M C, at N and O.

A diameter is any line VW, drawn through the centre M, and terminated by the opposite curves.

Conjugate diameter to another, is a line drawn through the centre, parallel to a tangent with either of the curves, at the extremity of the other diameter terminated by the curves.

Abscissa is when any diameter is continued within the curve, terminated by a double ordinate and the curve; then the part within is called the abscissa.

Double ordinate is a line drawn through any diameter parallel to its conjugate, and terminated by the curve.

Parameter, or latus rectum, is a line drawn through the focus, perpendicular to the transverse axis, and terminated by the curve.

Problem XXXIV

To describe an hyperbola by finding points in the curve, having the diameter or axis A B, its abscissa B G, and double ordinate D C.

Through G draw E F, Fig. 35, parallel to C D; from C and D draw C E and D F, parallel to B G, cutting E F in E and F. Divide C B and B D, each into any number of equal parts, as four; through the points of division, 1, 2, 3, draw lines to A. Likewise divide E C and D F into the same number of equal parts, viz. four; from the divisions on C E and D F, draw lines to G; a curve being drawn through the intersections at G, a, b, etc. will be the hyperbola required.