PROBLEM I.

To describe the conic sections from the cone.

Note, ADN is a section of the plane, passing through its axis at right angles with the sections of the ellipsis, parabola, or hyperbola.

For the ellipsis.

1. Let G H be its transverse-axis in the plane AD N; bisect it at K; through K, draw R Q parallel to A D.

2. Bisect QR, at M, with the radius MR or M Q, describe the semicircle R P Q.

3. From K, draw K H, perpendicular to QR, cutting the circle at H: then K H is the semiconjugate axis, from which the ellipsis may be described as at No. 1.

For the parabola.

1. Let S E be the axis of the parabola, parallel to the other side N D.

2. From E, draw E C at right angles to A D, the base, cutting the semicircle at C: then EC is an ordinate or half the base of the parabola, which may be described as at No. 2.

For the hyperbola.

1. Let IF be the height of the hyperbola, produce it till it cut the opposite side A N, produced at L, then F L is the transverse-axis.

2. From I, draw I B at right angles to A D: then IB is half the base, which may be described as at No. 3. Note, the letters are made to correspond at No. 1, 2, and 3, with those of the cone where they are taken from.

The Sections of Solids.

OF A GLOBE.

DEFINITIONS.

A globe is a solid figure, and may be supposed to be generated by the revolution of a semicircle about its diameter, which becomes the axis of the globe, and the centre of the semicircle is the centre of the globe.

Corollary 1. Hence all right lines drawn from the centre to the circumference of a globe are equal to another, for the semicircle touches the surface of the globe in every point as it revolves round.

Corollary 2. The section of a globe by a plane passing through its centre, is a semicircle, whose diameter is equal to the diameter of the generating semicircle.

Corollary 3. Every section of a globe cut by a plane, is a circle, for all the lines drawn from the centre to its surface, are equal, consequently the generating semicircle may revolve round any line as an axis, therefore every point in the semicircle will generate a circle.

Corollary 4. If a semiglobe is cut at right angles to the plane of its base, the section is a semicircle.

PROBLEM

PROBLEM I.

To find the section of a semiglobe at right angles to the plane A B C, through its centre, and pass through the line A B in that plane.

Bisect A B in D; on D, as a centre with the radius DA, or D B, describe the semicircle A E B, and it will be the section required.

PROBLEM II.

Given two segments of circles A B C, and D E F, equal or unequal, having their two chords. A C, and D F equal to each other, and the segment A B C being placed upon D F, so that AC shall coincide with D F, and the segment A B C, at right angles to DE F, to find the radius of a globe, so that the arc lines ABC, and D E F, shall be in its surface when the two segments are placed in the above position.

1. Make a rectangle A D F C, so that the opposite sides A C, and D F, will be the bases of the segments A B C, and D E F.

2. Find the centres G and H, of these segments.

3. Through H, draw I K parallel to DF and complete the semicircle I D E F K.

4. Through G or H draw HL parallel to G F, cutting A C, and I K, at L and H; make H M equal to LG, join M K or MI, and it will be the radius required.

If upon M, as a centre, with the distance M I, or MK, a segment INK is described, it will be part of the greatest circle that can be drawn in the globe.