3. What is the content of the middle frustrum of an hyperbolic spindle, the length being 20, the middle or greatest diameter 16, the diameter at each end 12, and the diameter at of the length 14}? Ans. 3248.938. 4. Required the content of the segment of any spindle, its length being 10, the greatest diameter 8, and the middle diameter 6. Ans. 272.272.

Miscellaneous Questions in Solids.

1. If the diameter of the earth be 7930 miles, and that of the moon 2160 miles, required the ratio of their surfaces, and also of their solidities, supposing both of them to be globular, as they are very nearly.

The surfaces of all similar solids are to each other as the squares of their like dimensions; such as diameters, circumferences, like linear sides, &c. &c. And their solidities, as the cubes of those dimensions.

Hence the surface of the moon: surface of the earth :: 21602 4665600 1 79302 62884900 13.47

21602:79302 and*

or, As 1 :

2. Three persons having bought a sugar-loaf, want to divide it equally among them by sections parallel to the base; it is required to find the altitude of each person's share, supposing the loaf to be a cone whose height is 20 inches.

* The ratio of one quantity to another may be expressed by divid ing the antecedent by the consequent.

By the similar cones we have* 3: 1 :: 203:

203

cube of the height of the top sections; wherefore y 3

3.604 the middle part; wherefore the lower part will be

2.528.

3. Three men bought a tapering piece of timber, which was the frustrum of a square pyramid; one side of the greater end was 3 feet, one side of the less end 1 foot, and the length 18 feet; what is the length of each man's piece, supposing they paid equally, and are to have equal shares?

* This proportion as well as all others of the kind, may be ex

pressed thus: 33:1::20:

20

the height of the top section;

3

and, in some instances, this is the more convenient method.

Let ABCDE be a section of the pyramid (when completed) passing through the vertex, and bisecting the opposite sides of the base, and let IL and MN represent the required sections. Draw EF to the middle of AB, and draw CG parallel to it.

Then by similar triangles BG (1 foot): GC (18) :: BF (1.5): FE(27) and FE—FH=EH=9, the altitude of the pyramid EDC.

Hence Prob. VIII. of solids the solidities of the two pyramids EAB and EDC will be found 81 and 3 cubic feet respectively, and 81-3-78=the solidity of the frustrum ABCD. Also =26, the solidity of each person's share,

78

3

which added to the solidity of EDC, will give the solidity of EIL=29, and the double of it added to EDC will give the solidity of EMN=55.

Now in the similar pyramids, EDC(3): EIL(29) :: EH3 : EO3, the cube root of which will give EO and EO-EH =HO the length of the part adjacent to the less end= 10.172.

Again EIL(29): EMN(55) :: EO3: EP the cube root of which will give EP and EP-EO=OP the length of the middle part=4.559. Lastly, EF-EP=PF the length of the part adjacent to the larger end=3.269.

4. If a round pillar, 7 inches over, have 4 feet of stone in it; of what diameter is the column, of equal length, that contains 10 times as much?

The solidities of cylinders, prisms, parallelopipedons, &c. which have their altitudes equal, are to each other as The same rethe squares of their diameters or like sides. mark is applicable to frustrums of a cone or pyramid when the altitude is the same, and the ends proportional.

Hence, As 4: 40, or As 1: 10 :: 72: 490=the square of the required diameter, and √490=22.1359 the diameter required.

5. There is a mill-hopper, in the form of a square pyramid, whose solid content is 13 feet; but one foot is cut

off its perpendicular altitude to make a passage for the grain, from the frustrum or hopper to the mill-stone: the sides of its greater and less end are in proportion of 4 to 1. Required the content in dry or corn measure.

Ans. 10.7292 bushels.

6. The ditch of a fortification is 1000 feet long, 9 feet deep, 20 feet broad at bottom, and 22 at top; how much water will fill the ditch, allowing 282 cubic inches to make a gallon? Ans. 11581274 gallons.

7. A person having a frustrum of a cone 12 inches in height, and the diameters of the greater and smaller ends 5 and 3 inches respectively, wishes to know the diameter of a frustrum of the same altitude, that would contain 3666 cubic inches, and have its diameters in the same proportion as the smaller one.

Ans. The greater diameter 24.4002, and less 14.6401.

OF THE

REGULAR BODIES.

A REGULAR BODY is a solid contained under a certain number of similar and equal plane figures.

The whole number of regular bodies which can possibly be formed is five.

1. The Tetraedron, or regular pyramid, which has four triangular faces.

2. The Hexaedron, or cube, which has six square faces. 3. The Octaedron, which has eight triangular faces. 4. The Dodecaedron, which has twelve pentagonal faces. 5. The Icosaedron, which has twenty triangular faces.

If the following figures are made of pasteboard, and the lines be cut half through, so that the parts may be turned up and glued together, they will represent the five regular bodies here mentioned.

BADA