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30 half the spindle.
30

900

30

3)27000

gooc one-third cube spindle. 4320.3125

4679.6875

4

18718.7500

3.1416

1123125000

187187500

748750000

187187500

561572500

58806.82500000 folidity.

Ex. 2. Required the folidity of a circular fpindle, whofe length is 30, and thickness 224 inches. Anf. 7350.853125. Ex. 3. Required the folidity of a circular spindle, whose middle diameter is 36, and length 40 inches.

Anf. 29919 cubic inches.

PROBLEM XVIII.

To find the folidity of the middle zone of a circular spindle.

RULE.

From the fourth part of the fquare of the length of the whole fpindle, fubtract the fquare of half the length of the middle fruftum, and multiply the remainder by the length

of

of the fruftum: Multiply the central distance by the revolving area which generates the fruftum; then fubtract this latter product from the former, and multiply the remainder by 3.1416, and twice the product will be the folidity.

EXAMPLE I.

Required the folidity of the fruftum of a circular spindle, whofe length is 40, greatest diameter 36, and least 16 inches.

Draw EG parallel to mn, then EF shall be equal mn, =20 and EF2+FB2=EB2=500 chord.

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Ex. 2. Required the folidity of a 'circular fpindle, whofe length is 40, its greatest diameter 32, and least 24 inches.

Anf. 27287 cubic inches.

PROBLEM

PROBLEM XIX.

To find the fuperficies and folidity of the five regular or Platonic

bodies.

RULE.

Multiply the fquare of the given fide into the correfponding tabular area for the fuperficies. And

Multiply the cube of the given side by the proper tabular solidity, for the folidity of the given body.

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This table exhibits the area and folidity of any of the above bodies, the fide being unity.

The areas of the above figures are fo related to those of regular polygons, and their folidities to problems already treated of, that we shall leave the conftruction of the table for the exercife of the learner.

30.

EXAMPLE I. Fig. 97.

Required the area and folidity of a tetraedron, whofe fide is

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Ex. 2. Required the superficial and folid content of a hexae

dron, whofe fide is 6. Fig. 98.

S Superficies 216 Anf. Solidity

216

Ex. 3. Required the area and folidity of an octraedron,

whofe fide is 3. Fig. 99.

Anf.

Superficies 31.176918
Solidity 12.7279215

Ex. 4. Required the fuperficies and folidity of the icofaedron,

whose fide is 2. Fig. 100.

Superficies 34.641 Anf.Solidity 17-4535

Ex. 5. Required the fuperficies and folidity of a dodecae

dron, the fide being 4. Fig. 101.

(Surface 33.03312

Anf.

Solidity 139.62848

PROBLEM XX. Fig. 102.

To find the furface and folidity of a cylindric ring.

RULE

Multiply the circumference of the ring by its length for the fuperficies.

Multiply the area of a section of the ring by the curve, for the folidity.

EXAMPLE

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