EXAMPLE I.

Required the furface and folidity of a cylindric ring, whose curve is 12, and the diameter of the ring 3 inches.

A CONE may be cut various ways; and, according to the

different pofitions of the cutting plane, the five plane figures following will arife, viz. the circle, the ellipfe, the parabola, the hyperbola, and the triangle.

DEFINITIONS.

1. The fection is a circle, when the cone is cut parallel to the bafe.

2. If the fection is obliquely to the bafe, it will form an el lipfe. Fig. 1c2.

3. If the plane cut parallel to one of the fides, the section will be a parabola. Fig. 103.

4. The fectioni s an hyperbola, when the cutting plane meets the oppofite cone, and makes another fection fimilar to the for

mer.

5. The fection forms a triangle, when the plane paffes through the vertex and meets the base.

6. The vertex of any fection is the point in which the plane meets the oppofite fide of the cone.

7. The tranfverfe axis is a line drawn between two vertices. 8. The centre of an ellipfe is the middle point of the tranfverse.

9. The conjugate axis is drawn through the centre perpen

dicular to the tranfverfe.

10. The ordinate is a line perpendicular to the axis.

11. The abfciffa is that part of the axis intercepted between. the ordinate and the vertex.

12. The axis of a parabola is a right line drawn from the vertex, fo as to divide the figure into two equal parts.

13. The tranfverfe diameter of an hyperbola is that part of the axis, intercepted between the vertices of the opposite sections.

PROBLEM I.

To defcribe an ellipfe.

It is a known property of the ellipfe, that any two lines. drawn from the foci, meeting in any point of the curve, are together equal to the tranfverfe diameter. Hence the following method of describing an ellipse.

Find the points x y in the tranfverfe, which you are to confider as your foci; there fix two pins, and take a ftring equal to the transverse, and fasten its ends each to a pin, then stretch the ftring with a pencil, and move it round within the thread, fo fhall its path defcribe an ellipfe. Ee

When

When the tranfverfe and conjugate diameters are given, the foci may be found thus. Draw the transverse AB, and conjugate CD so as they may bisect each other at right angles in the point E, and with the distance AE or EB, and centre C or D, defcribe arches, cutting the tranfverfe in the points x y, fo fhall x and y be the foci.

PROBLEM II.

To find the length of the elliptic curve.

RULE.

Multiply the fum of the tranfverfe and conjugate diameters by 3.1416, and half the product will be the circumference nearly.

EXAMPLE I.

Required the length of an elliptic curve, whofe conjugate is 40 and tranfverfe 60 feet.

Ex. 2. What is the length of the circumference, when the diameters are 30, 40 feet?

Anf. 109.956 feet, Ex. 3. Required the circumference of an ellipfe, whose transverse diameter is 20, and conjugate 10 yards.

Anf. 282.744 feet.

Ex. 4. What is the periphery of an ellipfe, whose axis are

36 feet and 24 feet?

Anf. 94.248.

PROBLEM

PROBLEM III.

To find the area of an ellipfe.

RULE

Multiply the tranfverfe by the conjugate, and this product again by .7854 for the area.

EXAMPLE. I.

Required the area of an ellipfe, whofe two axes are 30 and 40 feet.

30 40

1200

-7854

942.4800

Ex. 2. Required the area of an ellipfe, whofe tranfverfe and conjugate are 20 and 10 feet.

Anf. 157.08. Ex. 3. Required the area of an ellipfe, whofe diameters are 48 and 36 yards. Anf. 1357-1712. Ex. 4. Required the area of an ellipfe, whofe two axes are 14 and 12 feet.

Anf. 131.9472.

PROBLEM IV.

The tranfverfe, conjugate, and ordinate being given, to find

the abfciffa.

E e 2

RULE.

RULE.

As the conjugate

Is to the tranfverse,

So is the fquare root of the difference of the fquares of the ordinate and femi-conjugate

To the distance between the ordinate and centre.

Note. This dift ce is to be added to or fubtracted from the femi-tranfverfe, according as the abfciffa is greater or less than the femi-tranfverfe.

EXAMPLE I.

The tranfverfe AB is 60, the conjugate CD 20, and the ordinate Fx 8. It is required to find the abfciffa.

10

10

8 8

100

64

of the semi-conj. 64 fq. of the ordinate.

Ás 20 606

6

20)360

36(6 root.

36

18 distance between the ordinate and centre.

30 femi-tranfverfe.

48 the abfciffa x B.

12 the abfciffa A x.

Ex. 2. The tranfverfe 90, the conjugate 30, and the ordi

nate 12, required the abfciffas.

Anf..72 and 18.

Ex. 3. The tranfverfe 105, the conjugate 35, and the ordinate 14, required the abfciflas.

Anf. 84 and 21.