eries of squares SA, m a, n e', and p y,)=

2SA2+py- fhp; × Ay S. And circles are to each other as the squares of their radii, hence

3S

2SA+py-hp2X.2618 Ay S, the solidity of onefourth of the middle frustum of the spindle. Hence To find the Solidity of the middle Frustum of a Parabolic Spindle.

To twice the square of the middle diameter, add the square of the diameter of the end; and from the sum subtract four-tenths of the square of the difference between these diameters; the remainder multiplied by the length, and that product by .2618 will give the solidity.

NOTE. This rule is useful in cask-guaging. A cask in the form of the middle frustum of a parabolic spindle, is called by guagers a cask of the second variety; and is the most common of any of the varieties.

EXAMPLES.

1. Required the solidity of the middle frustum of a parabolic spindle EFGH, the length, AB A being 20, the greatest diameter CD 16, and the least diameter H EF or GH 12?

(2CD + GH' —.4× (CD — GH)") × ABX.2618= (512 + 144 —.4× 4× 4)×20× .2618—(656 — 6.4) × 5.236 649.6×5.236—3401.3056, the solid content.

a cask is 32 inches, and length AB, 40

2. The bung diameter CD of head diameter EF, 24 inches, inches required its content in ale gallons? 282 cubic inches being 1 gallon. Ans. 96 4909 gallons.

3. The bung diameter CD, of a cask is 36 inches head diameter EF 20 inches, and length AB 36 inches ; required its content in wine gallons? 231 cubic inches being gallon. Ans. 117.89568 gallons.

To find the Solidity of the middle Frustum of any Spindle, formed by the revolution of a Conic Section about the Diameter of that Section.

RULE.

To the square of the greatest diameter add the square of the least, and four times the square of a diameter taken exactly in the middle between the two; multiply the sum by the length, and that product by 1309 for the solidity.

NOTE. See the latter part of the demonstration of the rule, in section X.

EXAMPLES.

1. Required the solidity of the middle frustum EFGH of any spindle; the length AB, being 40, the greatest or middle diameter CD, 32, the least diameter EF or GH, 24, and the diameter IK in the middle between GH and CD), 30, 157568 ? Ans. 27425.72624.

2. The bung diameter of a cask being 36 inches ; head 20; length 36, and a diameter exactly in the middle, 31.95 inches: what is the content in wine gallons? Ans. 117 gallons 34 quarts.

§ XV. Of the five REGULAR BODIES.

A regular, or platonic body, is a solid contained under a certain number of similar and equal plane figures. Only three sorts of regular plane figures joined together can make a solid angle; for three plane angles, at least, are required to make a solid angle, and all the plane angles which constitute the solid angle, must be less, when added together, than four right angles, (Euclid, XI. and 21.) Now each angle of an equilateral triangle is 60 degrees; each angle of a square 90 degrees; and each angle of a pentagon 180 degrees. Therefore there can be only five regular bodies, for the solid angles of each must consist either of three, four, or five triangles, three squares, or three pentagons.

1. The tetraedron, or equilateral pyramid, which has four triangular faces; hence all the plane angles about one of its solid angles make 180 degrees.

2. The octaedron, which has eight equilateral triangular faces; hence all the plane angles about any one of its solid angles make 240 degrees.

3. The dodecaedron, which has twelve equilateral pentagonal faces; hence all the angles about any one of its solid angles, make 324 degrees.

4. The icosaedron, which has twenty equilateral triangular faces; hence all the angles about any one of its solid angles, make 300 degrees.

5. The hexaedron, or cube, which has six equal square faces; hence all the angles about any one of its solid angles make 270 degrees.

NOTE. If the following figures be exactly drawn on pasteboard, and the lines cut half through, so that the parts may be turned up and glued together, they will represent the five regular bodies above described.

A TABLE, shewing the solidity and superficies of the five regular bodies, the length of a side in each being 1, or unity.

To find the Solidity or Superficies, of any regular Body, by the table.

RULE.

1. Multiply the cube of the length of a side of the body, by the tabular solidity, and the product will give the solidity of the body.

2. Multiply the square of the length of a side of the body, by the tabular superficies, and the product will give the superficies of the body.