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Ex. 2. Required the solidity of a circular spindle, whose length is 40, its greatest diameter 32, and least 24 inches.
Anf. 272874 cubic inches.
PROBLEM PROBLEM XIX.
To find the superficies and folidity of the five regular or Platonic
Multiply the square of the given fide into the corresponding tabular arca for the superficies. And
Multiply the cube of the given fide by the proper tabular solidity, for the solidity of the given body.
This table exhibits the area and folidity of any of the above bodies, the side being unity.
The areas of the above figures are so related to those of regular polygons, and their solidities to problems already treated of, that we shall leave the construction of the table for the exercise of the learner.
EXAMPLE I. Fig. 97.
Required the area and solidity of a tetraedron, whose fide is 10.
Ex. 2. Required the superficial and solid content of a hexaedron, whose fide is 6. Fig. 98.
S Superficies 216
3. Required the area and solidity of an octraedron, whose fide is 3. Fig. 99.
Anf. (Superficies 31.176918
Solidity 12.7279215 Ex. 4. Required the superficies and solidity of the icofaedron, whose fide is 2. Fig. 100.
S Superficies 34.641 Anf.
Solidity 17.4535 Ex. 5. Required the superficies and folidity of a dodecae. dron, the fide being 4. Fig. 101.
Surface 33.03312 An
PROBLEM XX. Fig. 102.
To find the surface and folidity of a cylindric ring.
Multiply the circumference of the ring by its length for the fuperficies.
Multiply the area of a section of the ring by the curve, for the folidity.
Required the surface and solidity 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 positions of the cutting plane, the five plane figures following will arisc, viz. the circle, the ellipse, the parabola, the kyperbola, and the triangle.
1. The fection is a circle, when the cone is cut parallel to the base.
2. If the section is obliquely to the base, it will form an el lipse. l'ig. 102.
3. If the plane cut parallel to one of the fides, the section will be a parabola. Fig. 103.
4. The sectioni s an hyperbola, when the cutting plane mcets the opposite cane, and makes another section similar to the former.
5. The section forms a triangle, when the plane países through the vertex and meets the base.
6. The vertex of any section is the point in which the plane meets the opposite side of the cone.
7. The transverse axis is a line drawn between two vertices.
7. The centre of an ellipse is the middle point of the tranfverse.
9. The conjugate axis is drawn through the centre perpendicular to the transverse.
10. The ordinate is a line perpendicular to the axis.
1. The abscissa 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, so as to divide the figure into two equal parts.
13. The transverfe diameter of an hyperbola is that part of the axis, intercepted between the vertices of the opposite fections.
To describe an ellipfis
It is a known property of the ellipse, that any two lines drawn from the foci, meeting in any point of the curve, are together equal to the transverse diameter. Hence the following method of describing an elliple.
Find the points x y in the transverse, which you are to conSider as your foci; there fix two pins, and take a string equal to the transverse, and fasten its ends each to a pin, then streich the string with a pencil, and move it round within the thread, so fhall its path describe an ellipse. Еe