5. A pyramid is a folid having any plane figure for a base, and its fides are triangles whofe vertices meet in a point at the top, called the vertex of the pyramid.

The pyramid takes names according to the figure of its bafe, like the prifm; being triangular, or fquare, or hexagonal, &c.

6. A cone is a round pyramid; having a circular bafe.

7. A fphere is a folid bounded by one continued convex furface, every point of which is equally diftant from a point within, called the the center. The fphere may be conceived to be formed by the revolution of a femicircle about its diameter, which remains fixed.

D

8. The axis of a folid, is a line drawn from the middle of one end, to the middle of the oppofite end; as between the oppofite ends of a prifi. Hence the axis of a pyramid, is the line from the vertex to the middle of the bafe, or the end on which it is fuppofed to stand. And the axis of a sphere, is the fame as a diameter, or a line paffing through the center, and terminated by the furface on both fides.

9. When the axis is perpendicular to the bafe, it is a right prifm or pyramid; otherwife it is oblique.

10. The height or altitude of a folid, is a line drawn from its vertex or top, perpendicular to its bafe. This is equal to the axis in a right prifm or pyramid;

pyramid; but in an oblique one, the height is the perpendicular fide of a right-angled triangle, whose hypotenufe is the axis.

11. Alfo a prifm or pyramid is regular or irregular, as its bafe is a regular or an irregular plane figure.

12. The fegment of a pyramid, fphere, or any other folid, is a part cut off the top by a plane parallel to the bafe of that figure.

The fection made by the plane, is a plane fimilar to the base of the figure; and every fection of a fphere is a circle. If the fection be made through the center, it is a great circle of the fphere, having the fame diameter with the fphere; if not, it is a little circle.

13. A fruftum, truncus, or trunk, is the part remaining at the bottom after the fegment is cut off.

If a fruftum be cut by a plane diagonally paffing through the extremity of one fide at the lefs end, and through the extremity of the oppofite fide at the greater end, the two parts into which it is cut, are called ungulas or hoofs; the greater hoof being that including the greater end; and the lefs, that including the lefs.

14. A zone of a sphere, is a part intercepted between two parallel planes; and it is the difference between two fegments. When the ends, or planes, are equally diftant from the center, on both fides, the figure is called the middle zone.

15. The fector of a sphere, is composed of a fegment less than a hemifphere or half sphere, and of a cone having the fame bafe with the fegment, and its vertex in the center of the fphere.

16. A circular fpindle, is a folid generated by the revolution of a fegment of a circle about its chord, which remains fixed.

17. A wedge is a folid having a rectangular base, and two of the oppofite fides ending in an acies or edge.

18. A prifmoid is a folid having for its two ends any diffimilar parallel plane figures of the fame number of fides, and all the upright fides of the folid trapezoids. If the ends of the prifmoid be bounded by diffimilar curves, it is fometimes called a cylindroid.

19. An ungula, or hoof, is a part cut off a folid by a plane oblique to the bafe.

PROBLEM 1.

To find the Surface of a Prifm.

GENERAL RULE.

It is evident, that, if the area of each side and end be calculated feparately, the fum of thofe areas will be the whole furface of any prifin, whether right or oblique; or, indeed, of any other body whatever.* —But for a right prifin obferve the following

PARTICULAR RULE.

Multiply the perimeter of the end by the height, and the product will be the fum of the fides, or upright furface.

If the ends of the prifm be regular plane figures, multiply the perimeter of the end by the fum of the height of the prifm and the radius of the circle infcribed in the end, and the product will be the whole furface.

The furfaces of fimilar prifms, and indeed of any other fimilar bodies, are as the fquares of their like lineal dimenfions. This follows from their being compofed of fimilar plane figures, alike placed.

What is the upright furface of a triangular prifm whofe length is 20 feet, and the ends of its bafe each 18 inches?

Here 18×3 54 inches 4 feet is the perimeter of the bafe.

Therefore 4×20= 90 fquare feet is the upright furface.

Again, by rule 2 problem 4 fection 1, we have · 2×××43301319485585 the area of the

two ends.

Therefore 919485585 is the whole furface.

EXAMPLE II.

What is the furface of a cube, the length of each of whofe fides is 20 feet?

First, 20 x 20

And 400 X6

400 is the area of one fide. 2400 is the whole furface required.

EXAMPLE III.

What must be paid for lining a rectangular ciftern with lead at 2d a lb, the lead being 7 feet to the lb, fuppofing the length within fide to be 3 feet 2 inches, the breadth 2 feet 8 inches, and height 2 feet 6 inches?

First, (3832) × 2 × 30 = 70 X 60 4200 fquare inches, the two fides and two ends together. Then, 38 × 321216 is the area of the bottom. Therefore 4200 + 12165416 fquare inches 37 fquare feet, is the whole area. And 37 X 7263 Therefore 1 lb: 2d ::

coft.

lb, is the whole weight. 263: 21 35 103d, the

What is the convex furface of a round prifm, or cylinder, whofe length is 20 feet, and the diameter of whofe bafe is 2 feet?

First, 31416 x 2 = 6.2832 is the circumference.
Therefore 6.2832 X 20

furface.

125.664 is the convex

EXAMPLE V.

What is the whole furface of a cylinder whose length is 10 feet, and the circumference 3 feet?

Here 3

2 X 3'1416

477463 is the radius of the end.

Theref. 10.477463 × 3 = 31°43239 is the whole furface.

PROBLEM II.

To find the Solidity of a Prifm. Multiply the area of the bafe by the height, and the product will be the folidity.*

*For, if we conceive to be cut off from the prifm, by a plane parallel to the ends, a part whofe height is equal to the lineal measuring unit; and then imagine both its ends to be divided, in a fimilar manner, into as many fquares as are expreffed by the area of each, the fide of each of thofe fquares being equal to the lineal meafuring unit; and, laitly, fuppofe planes to be drawn through the correfponding lines of divifion; it is evident that the part cut off will be divided, by those planes, into as many cubes as there are fquares in each end, and alfo having the fame dimenfion with thofe fquares, viz. the lineal measuring unit; and this number of cubes is the measure of the part.

But the magnitude of the whole prifm, or, indeed, of any other of an equal bafe, is to the magnitude of the part whofe height is the lineal measuring unit, as the length of the whole is to that unit (1).

And