Note. The five Regular Bodies are sometimes called Platonic Bodies.

It appears from an ancient Greek Epigram, quoted by Scarburgh in his "English Euclid," (Oxford, 1705,) that the five Platonic Bodies, which the wise Pythagoras found out, were indeed discovered by him; but Plato elucidated and taught them in the clearest manner; and Euclid took them as the foundation of his own imperishable renown."

Pythagoras was a native of the island of Samos, and was born about 590 years before Christ; Plato was born at Athens B. c. 429; and Euclid flourished at Alexandria, about 300 years before the Christian æra.

PROBLEM XVIII.

To find the solidity of an irregular solid.

RULES.

1. Divide the irregular solid into different figures; and the sum of their solidities, found by the preceding Problems, will be the solidity required.

2. If the figure be a compound solid, whose two ends are equal plane figures; the solidity may be found by multiplying the area of one end by the length.

3. To find the solidity of a piece of wood or stone, that is craggy or uneven, put it into a tub or cistern, and pour in as much water as will just cover it; then take it out, and find the content of that part of the vessel through which the water has descended, and it will be the solidity required.

4. If a solid be large and very irregular, so that it cannot be measured by any of the above Rules, the general method is to take lengths, in two or three different places; and their sum divided by their number, is considered as a mean length.

A mean breadth and a mean depth are found by similar processes.

Sometimes the length, breadth, and depth, taken in the middle, are considered as mean dimensions.

Note. The unhewn blocks, in the freestone quarries, in the vicinity of Leeds, are generally measured by the method described in the last Rule. The dimensions, however, are not, in general, taken to the extremities of the stones, in order to make an allowance for the waste in hewing.

EXAMPLES.

1. The lower part of a stone is a parallelopipedon, the breadth of whose end is 7, and its depth 5 feet. The upper part is a triangular prism, the perpendicular of whose end is 4 feet; required the solidity of the stone, its length being 18 feet.

Here 7 x 5 x 18

By Rule I.

35 × 18:

=

630, the solidity of the lower part; and 7 × 2 × 18 = 14 × 18=252, the solidity of the upper part; then 630 +252882 feet, the solidity of the whole stone.

By Rule II.

Here 7 x 5 + 7 x 2 = 35+ 14 49, the area of the end; and 49 x 18 882 feet, the solidity as before.

2. Being desirous of finding the solidity of an irregular piece of wood, I immersed it in a cubical vessel of water; and when it was taken out, the water descended 6 inches; required the solidity of the wood, the side of the vessel being 30 inches.

By Rule III.

Here 30 x 30 x 6=900 × 6=5400 inches, the solidity required.

3. The lengths of an irregular block of marble, taken in different places, are 8 feet 6 inches, 8 feet 10 inches, and 9 feet 2 inches; the breadths 4 feet 7 inches, 4 feet 2 inches, and 4 feet 9 inches; the depths 3 feet 2 inches, 3 feet 5 inches, and 3 feet 11 inches; required its solidity.

4. Wanting to know the solidity of an irregular block of marble, I immersed it in a cylindrical tub of water, whose diameter was 34.8 inches; and on taking it out, I found the fall of the water to be 12.6 inches; what was the solidity of the marble? Ans. 11984.50028 inches.

5. Required the solidity of an irregular block of Yorkshire-stone, whose dimensions are as follow: viz. lengths taken in different places, 11 feet 3 inches, and 11 feet 9 inches; breadths, 5 feet 5 inches, 5 feet 9 inches, and 6 feet 4 inches; depths, 4 feet 5 inches, 4 feet 8 inches, and 5 feet 2 inches. Ans. 318 ft. 7 in. 9 pa.

PROBLEM XIX.

To find the magnitude or solidity of a body from its weight.

RULE.

As the tabular specific gravity of the body,
Is to its weight in avoirdupois ounces,
So is one cubic foot, or 1728 cubic inches,
To its content in feet, or inches, respectively.

Note 1. The specific gravities of bodies are their relative weights contained under the same given magnitude, as a cubic foot, a cubic inch, &c.

2. As a cubic foot of water weighs just 1000 ounces avoirdupois, the numbers in the foregoing Table express not only the specific gravities of the several bodies, but also the weight of a cubic foot of each, in avoirdupois ounces.

3. The several sorts of wood, mentioned in the preceding Table, are supposed to be dry.

4. In some tables, sea-water is 1026, and in others 1030.

EXAMPLES.

1. What is the solidity of a marble chimney-piece, whose weight is 210 lb. 15 oz. avoirdupois?

2. What is the solidity of an irregular Yorkshire-stone, whose weight is 228 lb. 15 oz. avoirdupois? Ans. 11⁄2 foot. 3. A piece of carved mahogany weighs 49 lb. 14 oz. avoirdupois; what is its solidity?

Ans. 1297.21919 inches.

PROBLEM XX.

To find the weight of a body from its magnitude or

solidity.

RULE.

As one cubic foot, or 1728 cubic inches,

Is to the content of the body,

So is its tabular specific gravity,

To the weight of the body.

EXAMPLES.

1. The solidity of a grindstone is 3 feet; what is its weight?

2. The solidity of a beam of dry oak is 25 feet; what is its weight? Ans. 1445 lb. 5 oz.

3. Required the weight of a block of marble, whose length is 63 feet, and its breadth and thickness each 12 feet ; these being the dimensions of one of the stones in the walls of Balbec.

Ans. 683.4375 tons, which is nearly equal to the burthen of an East India ship.

SECTION II.

DESCRIPTION AND USE OF THE
CARPENTER'S RULE.

THIS instrument is commonly called Cogeshall's Sliding Rule, and is much used in measuring timber and artificers' work; not only in taking the dimensions, but also in casting up the contents.

It consists of two pieces of box, each one foot in length; and connected together by a folding joint.

One side or face of the rule is divided into inches and half-quarters, or eighths; and on the same face there are also several plane scales, divided into twelfth parts, which are designed for planning such dimensions as are taken in feet and inches.

On one part of the other face are four lines marked A, B, C, D; the two middle ones, B and C, being upon a slider.

Three of these lines, viz. A, B, C, are exactly alike, and are called double lines; because they proceed from 1 to 10, twice over. The fourth line, D, is a single one, proceeding from 4 to 40, and is called the girt line.