PROPOSITION XV.

96. If a SPHERE BE CIRCUMSCRIBED by a solid bounded by plane surfaces; the capacities of the two solids are as their surfaces.

If planes be supposed to be drawn from the centre of the sphere, to each of the edges of the circumscribing solid, they will divide it into as many pyramids as the solid has faces. The base of each pyramid will be one of the faces; and the height will be the radius of the sphere. The capacity of the pyramid will be equal, therefore, to its base multiplied into of the radius (Art. 48.); and the capacity of the whole circumscribing solid, must be equal to its whole surface multiplied into of the radius. But the capacity of the sphere is also equal to its surface multiplied into of its radius. (Art. 70.)

Cor. The capacities of different solids circumscribing the same sphere, are as their surfaces.

PROPOSITION XVI.

97. A SPHERE has a greater solidity than any regular polyedron of equal surface.

If a sphere and a regular polyedron have the same centre, and equal surfaces; each of the faces of the polyedron must fall partly within the sphere. For the solidity of a circumscribing solid is greater than the solidity of the sphere, as the one includes the other: and therefore, by the preceding article, the surface of the former is greater than that of the latter.

But if the faces of the polyedron fall partly within the sphere, their perpendicular distance from the centre must be less than the radius. And therefore, if the surface of the

polyedron be only equal to that of the sphere, its solidity must be less. For the solidity of the polyedron is equal to its surface multiplied into of the distance from the centre. (Art. 59.) And the solidity of the sphere is equal to its surface multiplied into of the radius.

Cor. A sphere has a less surface than any regular polyedron of the same capacity.

APPENDIX

GAUGING OF CASKS.

ART. 119. GAUGING is a practical art, which does not admit of being treated in a very scientific manner. Casks are not commonly constructed in exact conformity with any regular mathematical figure. By most writers on the subject, however, they are considered as nearly coinciding with one of the following forms:

The second of these varieties agrees more nearly than any of the others, with the forms of casks, as they are commonly made. The first is too much curved, the third too little, and the fourth not at all, from the head to the bung.

120. Rules have already been given, for finding the capacity of each of the four varieties of casks. (Arts. 68, 110, 112, 118.) As the dimensions are taken in inches, these rules will give the contents in cubic inches. To abridge the computation, and adapt it to the particular measures used in gauging, the factor .7854 is divided by 282 or 231; and the quotient is used instead of .7854, for finding the capacity in ale gallons or wine gallons.

If then .0028 and .0034 be substituted for .7854, in the rules referred to above; the contents of the cask will be given in ale gallons and wine gallons. These numbers are to each other nearly as 9 to 11.

PROBLEM I.

To calculate the contents of a cask, in the form of a middle frustum of a SPHEROID.

121. Add together the square of the head diameter, and twice the square of the bung diameter: multiply the sum by of the length, and the product by .0028 for ale gallons, or by .0034 for wine gallons.

If D and d=the two diameters, and 7=the length; The capacity in inches=(2D2+d2)×3×.7854. (Art. 110.) And by substituting .0028 or .0034 for .7854, we have the capacity in ale gallons or wine gallons.

Ex. What is the capacity of a cask of the first form, whose length is 30 inches, its head diameter 18, and its bung diameter 24 ?

Ans. 41.3 ale gallons, or 50.2 wine gallons.

PROBLEM II.

To calculate the contents of a cask, in the form of the middle frustum of a PARABOLIC SPINDLE.

122. Add together the square of the head diameter, and twice the square of the bung diameter, and from the sum

subtract of the square of the difference of the diameters; multiply the remainder by of the length, and the product by .0028 for ale gallons, or .0034 for wine gallons.

The capacity in inches (2D2+d2—§ (D—d)3)×X .7854. (Art. 118.)

Ex. What is the capacity of a cask of the second form, whose length is 30 inches, its head diameter 18, and its bung diameter 24?

Ans. 40.9 ale gallons, or 49.7 wine gallons.

PROBLEM III.

To calculate the contents of a cask, in the form of two equal frustums of a PARABOLOID.

123. Add together the square of the head diameter, and the square of the bung diameter; multiply the sum by half the length, and the product by .0028 for ale gallons, or .0034 for wine gallons.

=

The capacity in inches (D2+d)×XX.7854. (Art. 112 Cor.)

Ex. What is the capacity of a cask of the third form, whose dimensions are, as before, 30, 18, and 24 ?

Ans. 37.8 ale gallons, or 45.9 wine gallons.

PROBLEM IV.

To calculate the contents of a cask, in the form of two equal frustums of a coNE.

124. Add together the square of the head diameter, the square of the bung diameter, and the product of the two diameters; multiply the sum by of the length, and the product by.0028 for ale gallons, or .0034 for wine gallons. The capacity in inches (D'+d+Dd)xlx.7854. (Art. 68.)