The radius of a sphere is a straight line drawn from the centre to any point of the surface; and a diameter is a straight line passing through the centre, and terminated both ways by the surface.

The circumference of a sphere is the same as that of a circle having for its diameter any diameter of the sphere :-such circle is also called a great circle of the sphere.

A hemisphere is one half of a sphere; that is, one of the two equal parts into which a sphere will be divided by a plane passing through its centre.

Spheroids.

$413. A spheroid is a solid described by revolving an ellipse about one of its axes-that axis remaining unmoved or fixed.

An oblate spheroid is described by revolving an ellipse about its shorter axis, CD; and a prolate spheroid, by revolving the ellipse about its longer axis, AB.

MENSURATION OF SOLIDS.

§ 414. The measure of the surface of any solid is expressed by the area of an equivalent plane surface; solidity, or the volume of a solid, is expressed by the ratio of the space occupied by the solid to some assumed unit of solidity, as a cubic inch, or a cubic foot, &c.

In the following propositions it is to be understood that the solidities will be found in cubic units corresponding in name to the linear units in which the dimensions are taken.

Thus if the dimensions of a solid be taken in inches, the solidity will be found in cubic inches; and so for any other denomination.

I. The Convex Surface of a Right Prism or Cylinder, is equal to the perimeter or circumference of its base × its altitude.

II. The Solidity of any Prism or Cylinder is equal to the area of its base X its altitude.

III. The Convex Surface of a Right Pyramid or Cone, is equal to the perimeter or circumference of its base × its slant height 2.

IV. The Solidity of any Pyramid or Cone is equal to the area of its base X its altitude

3.

V. The Convex Surface of a Frustum of a right pyramid or cone, is equal to the sum of the perimeters or circumferences of its two bases × its slant height ÷ 2.

VI. The Solidity of a Frustum of any pyramid or cone, is equal to (the sum of the areas of its two bases + a mean proportional between those areas) × its altitude ÷ 3.

VII. The Surface of a Regular Polyedron is equal to the area of one of its equal faces the number of its faces; or it is equal to the square of one of the edges of the polyedron the surface of a similar polyedren whose edge is unity; and

VIII. The Solidity of a Regular Polyedron is equal to the cube of one of its edges X the solidity of similar polyedron whose edge is unity.

The edges of a polyedron are evidently the same as the sides of the polygons which form the faces of the polyedron.

IX. The Surface of a Sphere is equal to its diameter × its circumference; or it is equal to the square of its diameter x

3.14159.

X. The Solidity of a Sphere is equal its surface its radius 3; or it is equal to the cube of its diameter X 0.5236.

XI. The Solidity of a Spheroid is of the solidity of a Cylinder the diameter of whose base is equal to the fixed axis of the describing ellipse, and whose altitude is equal to the revolving axis of the ellipse.

The demonstrations of the preceding propositions depend on principles of Geometry, and cannot be presented in this work.

In the Mensuration of Solids.

1. Find the convex surface, and the solidity, of a right triangular prism the sides of whose base are 20, 30, and 40 feet, and whose altitude is 12 feet. Ans. 1080 sq. ft. and 3485.676' cu. ft.

2. Find the solidity of a prism whose altitude is 9 feet, and base a regular pentagon of 20 feet in perimeter, and also of a cylinder of the same altitude, and whose base is 4 feet in diameter. Ans. 247.748' cu. ft.; and 113.097' cu. ft.

3. Find the convex surface of a right pyramid whose slant height is feet, and base a square whose side is 3 feet; and also of a right cone of the same slant altitude, and whose base has a radius of 3 feet. Ans. 30 sq. ft.; and 47.123' sq. ft.

4. Find the solidity of a pyramid whose base is a regular hexagon 30 feet in perimeter, and also of cone whose base is 30 in circumference-allowing each of the two solids to be 10 feet in altitude. Ans. 216.506' cu. ft.; and 238.7' cu. ft.

5. Find the convex surface of a frustum of a right pyramid whose bases are equilateral triangles of 12 and 9 feet, respectively, in perimeters, and its slant height 10 feet-also the solidity of a frustum of a cone of altitude 4 feet, and bases 5 -and 2 feet, respectively, in radius.

Ans. 105 sq. ft.; and 163.362' cu. ft. 6. Find the surface, and also the solidity, of a regular octaedron whose edges are each 1.5 feet.

Ans. 7.794' sq. ft.; and 1.59' cu. ft.

7 Find the surface, and also the solidity, of a sphere whose circumference is 13 feet and 10 inches.

Ans. 60.906' sq. ft.; and 44.684' cu. ft.

8. Find the solidity of the oblate, and also of the prolate, spheroid which would be described by the revolutions of an ellipse whose axes are 3 and 4 feet, respectively.

Ans. 18.849' cu. ft.; and 25.132' cu. ft.

what

9. If the radius of a right cylinder's base be 13 ft., must the altitude be, to make the convex surface 25 square feet? and what will be the solidity?

The convex surface

the altitude.

the circumference of the base, will be Ans. 2.652 ft.; and 18.745' cu. ft.

10. If the radius of the base of a right cone be 2 feet, and the convex surface 20 square feet, what is the slant height of the cone? and also its solidity?

The slant height of the cone will be the hypothenuse of a right angled triangle, the radius of the base one of the perpendicu lar sides, and the altitude of the cone the remaining side.

Ans. 3.182' ft.; and 10.36' cu. ft.

11. How many gallons of wine could be put into a vat, which is in form a frustum of a square pyramid,—each side of the larger end being 5 ft., each one of the smaller 34 ft., and the depth 4 feet? (§ 179) Ans. 546.077' gal.

12. A cistern whose form is a frustum of a right cone, has its diameter at the top 12 ft., at the bottom 9 ft., and its depth is 10 ft. How many barrels of water will the cistern contain? Ans. 112.29' bar.

13. The diameter of a legal Winchester bushel is 18 inches, and its depth is 8 inches. What is the diameter of that bushel Ans. 19.777' in.

whose depth is 7 inches?

30

14. A cubic foot of brass is to be drawn into wire of an inch in diameter. What will be the length of the wire, allowing no waste in the metal? Ans. 31.252' miles.

15. A gentleman has a bowling-green, 500 feet in length, and 300 feet in breadth, which he wishes to raise 1 foot higher, by means of the earth to be dug out of a ditch with which he intends to surround it. What must be the depth of the ditch, if its breadth be everywhere 9 feet? Ans. 10.187' ft.

Solidity of a Wedge and Prismoid.

$415. A wedge is a solid included between a rectangular base, two triangular ends, and two parellelograms or trapezoids which meet in a vertical edge parallel to the base.

ABCD is the base of the wedge; ADE and BCF are the triangular ends; and ABFE and CDEF two parallelograms or trapezoids meeting in the vertical edge EF parallel to the base. When the two faces AF and CE are parallelograms, the wedge becomes a triangular prism, having the two triangular ends for its bases.

The altitude of a Wedge is the perpendicular distance from its vertical edge to the base, or the plane of the base produced.

To find the SOLIDITY of a Wedge,—Add the length of the edge to twice that side of the base which is parallel to the edge; multiply the sum by the product of that side of the base which is not parallel to the edge into of the altitude of the Wedge.

§ 416. A rectangular prismoid is a solid included between two parallel rectangular bases, and four trapezoids which are its lateral faces.

GHIK and LMOP are the two rectangular bases;-the lateral faces KL, GM, HO, and OK are trapezoids.

To find the SOLIDITY of a Rectangular Prismoid,-Multiply the sum of the two parallel sides of one of the trapezoidal faces by the sum of the two parallel sides of one of the contiguous trapezoidal faces; to the product add the areas of the two rectangular faces; and multiply the sum thus found by of the altitude of the Prismoid.

For example, suppose that in the Prismoid above represented, GH=10; HI=6; LM=8; MO=5; and the altitude =9. The SOLIDITY is

((10+8)x(6+4)+10x6+8×5)X(180+100) x 420.