3. Required the solidity of the octaedron ABCDE, whose linear side is 22 inches. 4. The linear side of the dodecaedron ABCDE is 5.68 feet; what is its solidity? E D C Ans. 1404.26984 feet. 5. What is the solidity and superficies of the icosae dron ABCDEF, whose linear edge is 15? Solidity 7363.22028. 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 Era. PROBLEM XVIII. To find the solidity of an irregular solid. 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 di mensions, 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. By Rule I. Here 7 x 5 x 18 35 × 18 = 630, the solidity of the lower part; and 7 × 2 × 18 = 14 x 18 = 252, the solidity of the upper part; then 630 +252882 feet, the solidity of the whole stone. By Rule II. Here 7 x 57 x 2 = 35 + 14 49, the area of the end; and 49 × 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 × 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, A TABLE Of the specific gravities of bodies. Fine Gold...........19640 Chalk................................. Standard Gold...... Steel....... Cast Iron.... ..... .1793 Sand....... .1520 Lignum-vitæ...... .1333 Ebony... ..1331 1250 .....1150 ........ ..1063 ....1030 8000 Sea-water..... ..1026 Pitch..... Box-wood.. 7425 Gunpowder, shaken.....922 Logwood............. Tin........ .7320 ..913 ..4930 ....3000 Beech... .....852 .........800 ..2700 ....2642 Portland-stone.......2496 Cherry-tree... ....715 Mill-stone.............2484 Yorkshire-stone......2442 Pear-tree... Cedar of Lebanon.......613 .... 661 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. Q ૩ |