PROBLEM V.

To describe a conic section through five given points A, B, C, D, E, provided that all these points are joined by right lines, and that any exterior, or angle, formed by these lines, be less than two right angles.

1. Join any four points A, B, C, E, forming the quadrilateral A B C E.

2. Through the fifth point D, draw Df, and D g, parallel to A E, and B C, meeting A B produced both ways at the points ƒ and g, if necessary.

3. Also through D, draw h i, parallel to E C, meeting BC, and A E, produced at the points h, and i.

4. Divide D h, D i, and Df, D g, into any number of equal parts, as six; likewise divide D F and D G, into the same, viz. six.

5. From the point b, and through the points 1, 2, 3, 4, 5, in Di, draw the lines 1 E, 2 E, 3 E, 4 E, 5 E, cutting the lines Ba, Bb, B c, B d, B e, and Bf, at the points a, b, c, d, e, drawn from B, through 1, 2, 3, 4, 5, in D F, which are all in the curve.

In the same manner, the points between B, and D, will be found, viz. by drawing lines from the points A, and C, through the lines D g, and D h.

And if the lines Di, and D f, are produced, and the equal parts 7, 8, 9, extended upon these lines, you would obtain as many points g, h, i, &c. between A and B.

PROBLEM

PROBLEM VI.

To describe a conic section to touch a right line A B, in a given point C, to pass through three other points D, E, and F.

1. Join D C, E C, and D E; through F, draw F A, and F B, parallel to F E and D C, cutting A B, at A and B.

2. Through F, draw G H parallel to D E, and produce the sides C D, and C E, to cut it at G and H.

3. Divide FG, and F H, F A, and F B, each into any number of equal parts, as four.

4. From C, through 1, 2, 3, in FH, draw Ca, Cb, C c, &c.

5. From E, through 1, 2, 3, in F H, draw 1 E, 2 E, 3 E, &c. cutting the former in the points a, b, c, which are in the curve.

In the same manner may points be found in the other side.

To describe a conic section, to touch two right lines A B, and B C, in the points A and C, and to pass through a given point D.

1. Join the points A, and C; through D, draw D E, and D F, parallel to B A, and B C.

2. Through D, draw G H, parallel to A C, cutting B A, and B C, in G and H, and divide D G, and DH, DE, and D F, each into the same number of equal parts.

3. From A, through the points 1, 2, 3, in D E, draw the lines A a, A b, A c.

4. From C, through the points 1, 2, 3, in D H, draw 1 C, 2 C, 3 C, cutting the former in a, b, c, which are in the

curve.

In the same manner may points be found between A and D.

The Sections of Solids.

OF A CYLINDER.

DEFINITIONS.

1. A cylinder is a solid generated by the revolution of a right angled parallelogram, or rectangle, about one of its sides, and consequently the ends of the cylinder are equal circles.

2. Axis is a right line passing from the centres of the two circles which form the ends of the cylinder.

3. If a cylinder is cut by a plane, parallel to a plane passing through its axis, it will be cut in two parts, which are called segments of the cylinder.

4. A segment of a cylinder is comprehended under three planes, and the curve surface of the cylinder; two of these are segments of circles: the other plane is a parallelogram, which is here for distinction's sake, called the plane of the segment, and the circular segments are called the ends of the cylinder.

5. The two sides of the parallelogram, which is parallel to the axis of the cylinder, is called the sides of the segment of the cylinder, and the other two sides of the parallelogram are chords to the ends of the cylinder.

6. If a cylinder, or segment of a cylinder, stands upon one of its ends, that end on which it stands is called the base.

7. If the segment of a cylinder is cut obliquely by a plane, the intersection of that plane, with the plane of the segment, is called the chord of the section.

s. The section of a cylinder cut by any plane inclined to its axis, is an ellipsis.

This is proved by the writers of conic sections.