Therefore, 3HX (r2+h2)=3h×(R2+H'.)

Or, зHr2+3Hh2-3hR3—3hH2=0. (Alg. 178.)

To reduce the expression for the solidity of the zone to the required form, without altering its value, let these terms be added to it: and it will become

(зHR'+3Hr-—3hR3—3hr2+H3—3H3h+зHh3—h3)

Which is equal to

1π × з(Н—h) × (R2+r2+1(H—h)3)

Or, as equals .5236 (Art. 71.) and H-h equals d,
The zone .5236 × 3d × (R2+r2+‡ď2.)

Ex. 1. If the diameter of one end of a spherical zone is 24 feet, the diameter of the other end 20 feet, and the distance of the two ends, or the height of the zone 4 feet; what is the solidity? Ans. 1566.6 feet.

2. If the earth be a sphere 7930 miles in diameter, and the obliquity of the ecliptic 23° 28'; what is the solidity of one of the temperate zones?

Ans. 55,390,500,000 miles.

3. What is the solidity of the torrid zone?

4. What is the convex surface of a spherical zone, whose

breadth is 4 feet, on a sphere of 25 feet diameter ?

5. What is the solidity of a spherical segment, whose height is 18 feet, and the diameter of its base 40 feet?

PROMISCUOUS EXAMPLES OF SOLIDS.

Ex. 1. How much water can be put into a cubical vessel three feet deep, which has been previously filled with cannon balls of the same size, 2, 4, 6, or 9 inches in diameter, regularly arranged in tiers, one directly above another?

Ans. 96 wine gallons.

2. If a cone or pyramid, whose height is three feet, be divided into three equal portions, by sections parallel to the base; what will be the heights of the several parts ?

Ans. 24.961, 6.488, and 4.551 inches.

3. What is the solidity of the greatest square prism which can be cut from a cylindrical stick of timber, 2 feet 6 inches in diameter and 56 feet long ?*

Ans. 175 cubic feet.

4. How many such globes as the earth are equal in bulk to the sun; if the former is 7930 miles in diameter, and the latter 890,000? Ans. 1,413,678.

* The common rule for measuring round timber is to multiply the square of the quarter-girt by the length. The quarter-girt is one-fourth of the circumference. This method does not give the whole solidity. It makes an allowance of about one-fifth, for waste in hewing, bark, &c. The solidity of a cylinder is equal to the product of the height into the area of the base.

If C the circumference, and #=3.14159, then (Art. 31.)

If then the circumference were divided by 3.545, instead of 4, and the quotient squared, the area of the base would be correctly found. See note B.

5. How many cubic feet of wall are there in a conical tower 66 feet high, if the diameter of the base be 20 feet from outside to outside, and the diameter of the top 8 feet; the thickness of the wall being 4 feet at the bottom, and decreasing regularly, so as to be only two feet at the top? Ans. 7188.

6. If a metallic globe filled with wine, which cost as much at 5 dollars a gallon, as the globe itself at 20 cents for every square inch of its surface; what is the diameter of the globe? Ans. 55.44 inches.

7. If the circumference of the earth be 25,000 miles, what must be the diameter of a metallic globe, which, when drawn into a wire of an inch in diameter, would reach round the earth? Ans. 15 feet and 1 inch.

8. If a conical cistern be 3 feet deep, 7 feet in diameter at the bottom, and 5 feet at the top; what will be the depth of a fluid occupying half its capacity?

Ans. 14.535 inches.

9. If a globe 20 inches in diameter, be perforated by a cylinder 16 inches in diameter, the axis of the latter passing through the centre of the former; what part of the solidity, and the surface of the globe, will be cut away by the cylinder?

Ans. 3284 inches of the solidity, and 502,655 of the surface.

10. What is the solidity of the greatest cube which can be cut from a sphere three feet in diameter ? *

Ans. 5 feet.

11. What is the solidity of a conic frustum, the altitude of which is 36 feet, the greater diameter 16, and the lesser diameter 8?

12. What is the solidity of a spherical segment 4 feet high, cut from a sphere 16 feet in diameter ?

SECTION V.

ISOPERIMETRY.

ART. 77. It is often necessary to compare a number of different figures or solids, for the purpose of ascertaining which has the greatest area, within a given perimeter, or the greatest capacity under a given surface. We may have occasion to determine, for instance, what must be the form of a fort, to contain a given number of troops, with the least extent of wall; or what the shape of a metallic pipe to convey a given portion of water, or of a cistern to hold a given quantity of liquor, with the least expense of materials.

78. Figures which have equal perimeters are called Isoperimeters. When a quantity is greater than any other of the same class, it is called a maximum. A multitude of straight lines, of different lengths, may be drawn within a circle. But among them all, the diameter is a maximum. Of all sines of angles, which can be drawn in a circle, the sine of 90° is a maximum.

When a quantity is less than any other of the same class, it is called a minimum. Thus, of all straight lines drawn from a given point to a given straight line, that which is perpendicular to the given line is a minimum. Of all straight lines drawn from a given point in a circle, to the circumference, the maximum and the minimum are the two parts of the diameter which pass through that point. (Euc. 7, 3.)

In isoperimetry, the object is to determine, on the one hand, in what cases the arca is a maximum, within a given perimeter; or the capacity a maximum, within a given surface and on the other hand, in what cases the perimeter is

a minimum for a given area, or the surface a minimum, for a

given capacity.

PROPOSITION I.

79. An ISOSCELES TRIANGLE has a greater area than any scalene triangle, of equal base and perimeter.

If ABC be an isosceles triangle whose equal sides are AC and BC; and if ABC' be a scalene triangle on the same base AB, and having AC' + BC' AC+BC; then the area of ABC is greater than that of ABC'.

=

Let perpendiculars be raised from each end of the base, extend

H'

D'

D

B

AC to D, make C'D' equal to AC', join BD, and draw CH and C'H' parallel to AB.

As the angle CAB=ABC, (Euc. 5, 1.) and ABD is a right angle, ABC+CBD=CAB+CDB=ABC+CDB. Therefore CBD CDB, so that CD=CB; and by construction, C'D'= AC'. The perpendiculars of the equal right angled triangles CHD and CHB are equal; therefore, BH=1BD. In the same manner, AH'=AD'. The line AD=AC+BC=AC' +BC' D'C'+BC'. But D'C'+BC'>BD'. (Euc. 20, 1.) Therefore, AD>BD'; BD>AD', (Euc. 47, 1.) and 1⁄2 BD>

AD'. But BD, or BH, is the height of the isosceles triangle; (Art. 1.) and AD' or AH', the height of the scalene triangle; and the areas of two triangles which have the same base are as their heights. (Art. 8.) Therefore the area of ABC is greater than that of ABC'. Among all triangles, then, of a given perimeter, and upon a given base, the isosceles triangle is a maximum.

Cor. The isosceles triangle has a less perimeter than any scalene triangle of the same base and area.

The triangle