EXAMPLES.

1. How many solid or cubick inches, in a cube of marble whose side is 24 inches? 24X24X24-Ans. 13824 sol. in. 2. What is the solidity of a cube, the side of which is 5 feet? Ans. 125 solid feet.

To find the solidity of a parallelopipedon, that is, a solid contained by six quadrilateral planes, every opposite two of which, are equal and parallel.

RULE.-Multiply the length by the breadth, and that product again by the thickness or height, and it will give the solidity.

EXAMPLES.

1. What is the solidity of a parallelopipedon, whose length is 12 feet, breadth 4 feet, and height 6 feet?

12X4X6

Ans. 288 solid feet. 2. How many solid feet in a load of wood 8 feet long 31 feet wide, and 33 feet in height?

Ans 98 feet. 3. What number of bricks 8 inches long, 4 inches wide, and 2 inches thick, will it require to build a house 46 feet long, 38 feet wide, and 20 feet high, and the walls to be 1 foot thick? Ans. 88560 bricks.

To find the solidity of a cylinder. DEFINITION.-A cylinder is a round body whose bases are circles, like a round column or stick of timber, of equal bigness from end to end.

RULE.-Multiply the area of the base by the perpendicular height, and the product will be the solidity.

EXAMPLES.

1. What is the solidity of a cylinder, the height of which iş 5 feet, and the diameter of the end 2 feet? Ans. 15,708, 2. One evening I chanc'd with a Tinker to sit, Whose tongue ran a great deal to fast for his wit; He talk'd of his art with abundance of mettle; So I ask'd him to make me à flat-bottomed kettle; Let the top and the bottom diameters be, In just such proportion as five are to three; Twelve inches the depth I propos'd, and no more; And to hold in ale gallons, seven less than a score. He promis'd to do it, and straight to work went; But when he had done it he found it to scant. He alter'd it then, but too big he had made it; For though it held right, the diameters fail'd it;

Thus altering it often too big and too little,
The Tinker at last had quite spoil'd his kettle;
But he says that he will bring his said promise to pass,
Or else that he'll spoil every ounce of his brass.
Now to keep him from ruin, I pray find it out
The diameter's length, for he'll ne'er do't I doubt.

Ans. 14,791 inches, bottom diameter. 24,652 inches, top diameter. NOTE.-The kettle is not a cylinder, the top and bottom diameters being unequal, yet there is sufficient given in this and the preceding rules for finding the diameters.

To measure a Sphere or Globe.

DEFINITION. A sphere or globe is a round, solid body, in the middle of which is a point, from which all lines drawn to the surface are equal.

RULE.-Multiply the cube of the given diameter by ,5236, and the product will be the solid contents.

EXAMPLES.

1. The diameter of a globe is 12 inches; how many cubick or solid inches does it contain?

12X12X12: 1728X,5236 equal to Ans. 904,7808 solid in. DEM.-By cubing the diameter, the product, 1728, is the solidity of a cube whose side is 12 inches, we then multiply by the decimal, ,5236, because ,5236 is the solidity of a globe whose diameter is 1.

NOTE. A cube whose side is one inch, contains one cubick or solid inch. A globe whose diameter is one inch, contains ,5236 of an inch.

2. Suppose the diameter of the earth is 7911 miles; how many solid or cubick miles does it contain?

the area.

Ans. 259,235,092,532,6816 solid miles.

QUESTIONS ON MENSURATION.

What is Mensuration of Superficies? A. It teaches how to meas ure surfaces or area. How do you find the surface or area of a square? A. Multiply the side of the square into itself, and the product will be How do you find the area of a parallelogram or long square? A. Multiply the length by the breadth, and the product will be the How do you find the area of a right angled triangle? A. By multiplying the base by one half the perpendicular, the product will be the area. How do you find the area of a circle? A. Multiply the

area.

square of the diameter by ,7854, the product will be the area. Why multiply by ,7854? A. Because the area of a circle is ,7854 when the diameter is one. How do you find the area of a globe or ball? A.— Multiply the whole circumference by the whole diameter, and the product will be the area. What does mensuration of solids teach? A. It teaches how to measure solids. How do you find the solidity of a cube? A. Cube one side, and the product will be the solidity. How would you find the number of cubick feet in a load of wood that is 8 feet long, 3 feet wide, and 4 ft. high? A. Multiply the length, breadth and height together, the product will be the solidity. How do you find the solidity of a cylinder? A. Multiply the area of the base by perpendicular height, and the product will be the solidity. How do you find the solidity of a globe? A. Multiply the cube of the diameter by,5236, the product will be the solidity. Why multiply by,5236? A. Because ,5236 is the solidity of a globe whose diameter is 1.

DUODECIMALS,

Is a rule much used by workmen and artificers, in computing the contents of their work. The rule has derived its name from the Latin word duocecim, which signifies twelve.

A foot, which is called an integer, is divided duodecimally, that is, into twelve parts, called inches or primes; an inch or prime is divided into twelve parts, called seconds; a second is divided into twelve parts, called thirds, and so on. But dimensions are usually taken in feet, inches and quarters; the parts smaller than these are generally neglected, being of little or no consequence.

RULE.-1st. Sat down the two given dimensions, i. e. length and breadth, one under the other, so that feet may stand under feet, inches under inches, &c.

2d. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each directly under its corresponding term, observing to carry 1 for every 12, from the inches to the feet.

3d. In like manner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand; omitting, however, what is below the parts of inches, only carrying to these the proper number of units from the lowest denomination.

Or, instead of multiplying by the inches, take such parts of the multiplicand as there are like parts of a foot in the inches. Then add the products together, as in Compound Addition, carrying 1 to the feet. for every 12 inches; the result will be the answer or area, in square feet and inches.

EXAMPLES.

1. How many square feet in a board, 14 feet 9 inches long, and 2 feet 6 inches wide?

Ans. 36 feet, 10 inches.

2) 14

ft. in.
9

2

6

29

Ans. 36 101

7.41

feet.
1 4,7 5
2,5

6 or thus, 7375
2950
Ans. 36,875
12

DEM.--It is plain, in the first operation, that multiplying the multiplicand, 14 feet 9 inches, by the feet in the multiplier, is repeating the multiplicand by 2 units, the same as in Compound Multiplication; hence this rule is sometimes called Cross Multiplication, becauce we first multiply by the units or left hand denomination of the multipliers. Next, it is evident that we may take half of the multiplicand, for the product of the multiplicand multiplied by 6 inches, or one half of a unit, because that part of the multiplier, is one half of a unit, or foot; and multiplying by one half, is taking one half of the multiplicand; hence the reason of our rule is obvious. 2. Multiply 48 feet 6 inch

Inches, 10,5 0 0

es by 10 feet 9 inches.

in. ft. in.
6=148 6

10 9

485 Ο

3=1)24 3

12 1

Ans. 521ft. 4 in.

3. Multiply 14 feet 9 inches by 4 feet 3 inches.

ft. in.

3=1)14 9

4 3

59 0

381

Ans. 62ft. 8 in.

4. Multiply 4 feet 7 inches by 9ft. 6in.

Ans. 43ft. 6 in.

5. What is a marble slab worth, the length of which is 5 feet six inches, and the breadth 1 foot 9 inches, at 75 cents Ans. $7,21 cents 8 mills. per square foot? 6. What will the painting of a floor come to, at 10 cents per square yard, allowing the floor to be 21 feet, 8 inches and the breadth 14 feet, 10 inches? Ans. $3,57cts. long, NOTE.-Divide square feet by 9, and the quotient will be sq. yards. ft. in. 34)53 6 length. 12 3 height.

7. Required the solid or cubick feet in a wall that is 53 feet 6 inches in length, 12 feet 3in. in height, and its thickness 2 feet.

DEM. It is plain, that we must multiply the 3 dimensions together for the solidity; we then multiply together the two first dimensions as in the preceding examples, and then the product multiplied by the other dimension, must evidently give the solidity, for it is multiblying length, breadth and thickness together.

8. How many cubick feet in a load of wood, 8 feet long, 3 feet wide, and 2 feet 8 inches high?

Ans. 64 feet. 9. How many solid feet of timber in a stick which is 40 feet 8 inches long, 2 feet 6 inches wide, and 2 feet 4 inches thick? Ans. 237 feet 2 inches.

QUESTIONS ON DUODECIMALS.

What is the use of Duodecimals? A. It is a rule by which artificers and workmen find the contents of their work. How do you multiply? A. First by the units of the multiplier and then by the parts of a unit. When lenghth and breadth are given, how do yo find the area? A. By multiplying the length by the breadth. When length, breadth and thickness are given, how do you find the solidity? A. By multiplying the length, breadth and thickness together.

VULGAR FRACTIONS.

Having briefly treated of Vulgar Fractions immediately after the Compound Rules, and given some general definitions, and a few such problems as were necessary to give a limited idea only, of Fractions, the learner is, therefore, requested to read again, those general definitions on page 119.

Vulgar Fractions are either proper, improper, compound or mixed.

1. A proper or simple fraction is when the numerator is less than denominator, as, 4, §, 3 5 &c.

2. An improper fraction, is when the numerator is equal to, or greater than the denominator, as 5, 8, 12, &c.

3. A compound fraction, is a fraction of a fraction, connected by the word of, as of, 2 of 11, of §, &c.

4. A mixed number, consists of a whole uumber and a fraction, as 31, 24, 87, &c.

A whole number may be expressed like a fraction by drawing a line under it, and placing 1 directly below for a denominator, as 9, and 15—15, &c.

CASE I. To reduce a fraction to its lowest terms.

RULE.-Divide the terms of the given fraction by any number that will divide them without a remainder; then divide these quotients again in the same manner; and so on, till it appears there is no number greater than 1 which will divide them; then the fraction will be in its lowest terms.

EXAMPLES.

1. Reduce 18 to its lowest terms.

Ars.