To find the Solidity of a Prismoid, fig. 32. RULE.—Add into one sum the areas of the two ends and four times the middle section, parallel to them; then, this sum multiplied by one-sixth of the height, will give the content.

Note.—The length of the middle section is equal to half the sum of the lengths of the two ends; and its breadth is equal to half the sum of the breadths of the two ends.

To find the Convex Surface of a Sphere, or Globe,

fig. 33.

RULE.-Multiply the diameter of the sphere by its circumference.

Or, multiply 3.1416 by the square of the diameter; the product will be the convex surface required.

Note.-The convex surface of any zone or segment may be found, in like manner, by multiplying its height by the whole circumference of the sphere.

To find the Solidity of a Sphere, or Globe, fig. 33. RULE.-Multiply the cube of the axis by 5236; the product will be the solidity.

To find the Solidity of a Spherical Segment, fig. 34.

RULE.-To three times the square of the radius of its base add the square of its height; then, multiply the sum by the height, and the product by •5236.

To find the Solidity of a Spherical Zone or Frustum, fig. 35.

RULE.-To the sum of the squares of the radius of each end add one-third of the square of the height of the zone; this sum, multiplied by the said height, and the product by 1.5708, will give the solidity.

OF SPHEROIDS.

To find the Solidity of a Spheroid, fig. 36. RULE.-Multiply the square of the revolving axis, e d, by the fixed axis, a b; the product, multiplied by 5236, will give the content.

To find the Solidity of the Segment of a Spheroid, figs. 37 and 38.

RULE.- When the base, e f, is circular or parallel to the revolving axis, c d, fig. 37, multiply the

fixed axis, a b, by 3, the height of the segment, a g, by 2, and subtract the one product from the other; then multiply the remainder by the square of the height of the segment, and the product by 5236.

Then, as the square of the fixed axis is to the square of the revolving axis, so is the last product to the content of the segment.

RULE.- When the base, e f, is perpendicular to the revolving axis, e d, fig. 38, multiply the revolving axis by 3, and the height of the segment, c g, by 2, and subtract the one from the other; then, multiply the remainder by the square of the height of the segment, and the product by 5236.

Then, as the revolving axis is to the fixed axis. so is the last product to the content.

To find the Solidity of the Middle Frustum of a Spheroid, figs. 39 and 40.

RULE.— When the ends, e f and g h, are circu

lar, or parallel to the revolving axis, c d, fig. 39, to twice the square of the revolving axis, e d, add the square of the diameter of either end, e f, or gh; then multiply this sum by the length, a b, of the frustum, and the product again by 2618; this will give the solidity.

RULE.- When the ends, e f and g h, are elliptical, or perpendicular to the revolving axis, 1 k, fig. 40, to twice the product of the transverse and conjugate diameters of the middle section, a b, add the product of the transverse and conjugate of either end; multiply this sum by the length, k, of the frustum, and the product by 2618; this will give the solidity.

To find the Surface of a Circular Spindle, fig. 41. RULE.-Multiply the length, a b, of the spindle by the radius, o c, of the revolving arc. Multiply also the said arc, a c b, by the central distance, o e, or distance between the centre of the spindle and

centre of the revolving arc. Subtract this last product from the former; double the remainder; multiply it by 3.1416, and the product will give the surface of the spindle.

Note.-The same rule will serve for any segment, or zone, cut off perpendicularly to the chord of the revolving arc; but, in this case, the particular length of the part, and the part of the arc which describes it, must be used, instead of the whole length and whole arc.

To find the Solidity of a Circular Spindle, fig. 41.

RULE.-Multiply the central distance, o e, by half the area of the revolving segment, a c be a. Subtract the product from one-third of the cube, a e, of half the length of the spindle. Then, multiply the remainder by 12.5664, or 4 times 3·1416, and the product will be the solidity required.

To find the Solidity of the Frustum, or Zone, of a Circular Spindle, fig. 42.

RULE. From the square of half the length, h i, of the whole spindle, take one-third of the square of half the length, n i, of the frustum, and multiply the remainder by the said half-length of the frustum. Multiply the central distance, o i, by the revolving area, which generates the frustum. Subtract the last product from the former; and the re