and divide the sum by the number of them for the mean breadth, which multiply by the length for the area.*

Note 1. Take half the sum of the extreme breadths for one of the said breadths.

Note 2. If the perpendiculars or breadths be not at equal distances, compute all the parts separately as so many trapezoids, and add them all together for the whole area.

Or else add all the perpendicular breadths together, and divide their sum by the number of them for the mean breadth, to multiply by the length; which will give the whole area not far from the truth.

Ex. 1. The breadths of an irregular figure, at five equidistant places, being 8.2, 7·4, 9.2, 10·2, 8.6; and the whole length 39: required the area.

First, (828.6) 28.4, the mean of the two

Ex. 2. The length of an irregular figure being 84, and the breadths at six equidistant places, 17·4, 20.6, 14.2, 16.5, 20·1, 24·4; what is the area?

Ans. 1550.64.

F

H

K

C

e

B

*This rule is made out as follows: Let D ABCD be the irregular piece, having the several breadths AD, EF, GH, IK, BC at the a equal distances AE, EG, GI, IB. Let the several breadths in order be denoted by the corresponding letters a, b, c, d, e, and the whole length AB by 1; then compute the areas of the parts into which the figure is divided by the perpendiculars, as so many trapezoids by Problem 3, and add them all together. Thus the sum of the parts is,

a+b x AE+b+c

2

× EG +c+d

A E G I

d+e

2

2

× GI+ X IB ;

= :({ a+b+c+d+}e) × 41 = (m+b+c+d) il,

which is the whole area, agreeing with the rule; m being the arithmetic mean between the extremes and 4 the number of the parts, And the same for any other number of parts.

MENSURATION OF SOLIDS.

By the Mensuration of Solids are determined the spaces included by contiguous surfaces; and the sum of the measures of these including surfaces is the whole Surface or Superficies of the body.

The measure of a solid is called its solidity, capacity, or content. A better term is volume.

Solids are measured by cubes, whose sides are inches, or feet, or yards, &c. And hence the volume of a body is said to be so many cubic inches, feet, yards, &c., as will fill its capacity or space, or another of equal magnitude.

The least ordinary solid measure, or measure of volume, is the cubic inch, other cubes being taken from it according to the proportion in the following table:

Table of Cubic or Solid Measures.

1728 cubic inches make 27 cubic feet make 1662 cubic yards make. 64000 cubic poles make 512 cubic furlongs make

1 cubic foot.

1 cubic yard.

1 cubic pole.

1 cubic furlong.

PROBLEM I.

1 cubic mile.

To find the superficies of a prism.

Multiply the perimeter of one end of the prism by the altitude of one of the parallelograms, and the product will be the lateral surface. To which add, also, the area of the two ends of the prism, when required.*

* And the rule is evidently the same for the surface of a cylinder, which may be regarded as a prism of an infinite number of lateral faces.

Or, compute the areas of all the sides and ends separately, and add them all together.

Ex. 1. To find the surface of a cube, the length of each side being 20 feet. Ans. 2400 feet. Ex. 2. To find the whole surface of a triangular prism whose length is 20 feet and each side of its end or base 18 inches. Ans. 91.948 feet. Ex. 3. To find the convex surface of a round prism, or cylinder, whose length is 20 feet and diameter of its base is 2 feet. Ans. 125.664.

Ex. 4. What must be paid for lining a rectangular cistern with lead at 2d. a pound weight, the thickness of the lead being such as to weigh 7 lbs. for each square foot of surface; the inside dimensions of the cistern being as follows, viz., the length 3 feet 2 inches, the breadth 2 feet 8 inches, and depth 2 feet 6 inches? Ans. £2 3s. 10d.

To find the superficies of an irregular polyhedron. Find the superficies of each of its bounding polygonal faces, and add the results.

To find the superficies of a regular polyhedron. Find the area of one of its faces by Prob. VI., and multiply this by the number of faces.

PROBLEM II.

To find the surface of a regular pyramid or cone. Multiply the perimeter of the base by the slant height, or length of the side, and half the product will evidently be the convex surface or the sum of the areas of all the triangles which form it. To which add the area of the end or base, if requisite. Note. The slant height of a regular pyramid is the perpendicular from the vertex to the middle of one of the sides of the base.

Ex. 1. What is the upright surface of a triangular pyramid, the slant height being 20 feet, and each side of the base 3 feet? Ans: 90 feet.

Ex. 2. Required the convex surface of a cone, or circular pyramid, the slant height being 50 feet, and the diameter of its base 8 feet. Ans. 667.59.

PROBLEM III.

To find the surface of the frustum of a regular pyramid or cone, being the lower part, when the top is cut off by a plane parallel to the base.

RULE I. Add together the perimeters of the two ends, and multiply their sum by the slant height, taking half the product for the answer. Because the

lateral faces of the frustum of a pyramid are trapezoids, having their opposite sides parallel, and the frustum of a cone is the frustum of a pyramid of an infinite number of lateral faces.

RULE II. Multiply the perimeter of the section midway between the two bases by the slant height. This depends upon the fact that the perimeter of the middle section is half the sum of the perimeters of the bases, as may be easily shown.

Ex. 1. How many square feet are in the surface of the frustum of a square pyramid whose slant height is 10 feet; also, each side of the base or greater end being 3 feet 4 inches, and each side of the less end 2 feet 2 inches? Ans. 110 feet.

Ex. 2. To find the convex surface of the frustum of a cone, the slant height of the frustum being 12 feet, and the circumferences of the two ends 6 and 8.4. Ans. 90 feet.

PROBLEM IV.

To find the volume of any prism or cylinder. Find the area of the base, or end, whatever the figure of it may be; and multiply it by the altitude of the prism, or cylinder, for the volume.

*The altitude of a prism is the perpendicular distance between its parallel bases. The cylinder, as well as the prism, may be oblique. Prop. 3 of Solid Geom., upon which, with the note to Prop. 6 of the same, the demonstration of this depends, may evidently be extended to an oblique cylinder.