of the 16 parts into which the square is divided; or the area is of a square foot-i.e., & × of a square foot.

Obs. It must be observed that in multiplying together the numbers, fractional or otherwise, which express the number of units in the sides of a rectangle, only one denomination must be used. The fact is that we cannot talk of multiplying two geometrical magnitudes together. We cannot, for example, talk of multiplying 3 feet by an inch, or by 2 feet; but we can multiply two numbers together which indicate the lengths of the two lines, with reference to some one standard unit, and then deduce the geometrical result which corresponds to the numerical result thus obtained.

8. The following table of Square Measure is by the above principle deduced from that of the Measures of Length. The learner is recommended to do this for himself.

SQUARE MEASURE.

144 square inches (sq. in.) = I square foot......written I sq. ft.

The acre contains, as will be found by calculation, 10 square chains, or 100000 square links, or 4840 square yards.

Flooring, roofing, plastering, &c., are often calculated by a "square" of 100 square feet.

A hide of land is 100 acres.

MEASURES OF SOLIDITY OR VOLUME.

-CUBIC MEASURE.

9. Definitions. A solid figure is that which has length, breadth, and thickness. A cube is a solid con

tained by six squares, of which every opposite two are parallel. The sides of the squares are called the edges of the cube.

All solids, or spaces which could be filled by solids, are measured by means of the number of cubic inches, cubic feet, &c., which they contain, i.e., by cubes, the edges of which are respectively 1 inch, I foot, &c., in length.

The magnitude of any solid figure is sometimes called its volume.

10. To find the magnitude of a Cube, the length of an edge being given.

Raise the number expressing the number of linear units in the edge to the third power. This will give

the number of cubic units of the same kind in the given cube.

For instance, a cube of which the edge is 4 inches long contains 64 cubic inches; a cube of which the edge is 5 feet long contains 125 cubic feet.

The truth of this will appear from the following diagram:

Take a cube, as in the diagram, of which the edge is supposed to be 4 inches long, and divide each edge into lengths of one inch. Then, by drawing parallel planes, as indicated in the figure, we can divide the cube into a number of

Now, any

cubes, each of which is a cubic inch. one slice such as that which is shaded clearly contains 4 x 4, or 16 cubic inches, and there are 4 such slices. Hence the cube contains 4 X 4 X 4, or 64 cubic inches.

11. Definitions.-A rectangular parallelepiped is a solid figure contained by six rectangular figures, of which every opposite two are parallel.

This differs from a cube in the fact that the length, breadth, and thickness are not equal.

The volume of (ie., the number of cubic units in) a parallelepiped is obtained by multiplying the numbers together which express the number of linear units in the length, breadth, and thickness respectively.

This will perhaps be sufficiently apparent from the accompanying diagram of a rectangular parallelepiped, of which the length, breadth, and height are supposed to be 6, 5, 4 inches respectively.

There will evidently be six such slices as that we have shaded, each containing 5 × 4, or 20 cubic inches. The volume of the solid will therefore be 6 × 5 × 4, or 120 cubic inches.

1728 cubic inches

CUBIC MEASURE.

I cubic foot. = I cubic yard

27 cubic feet

.written I c. ft. I c. yd. This measure is used in estimating the magnitude of timber, stone, boxes of goods, the capacity of rooms, ships, &c.

For example, 42 cubic feet are defined to be one ton of shipping.

For liquids and dry commodities other systems are adopted, which we will give after we have explained

THE MEASURES OF WEIGHT.

12. The smallest weight in use is called a grain, and by Act of Parliament is defined in the following manner:-A vessel, of which the capacity is a cubic inch, when filled with distilled water at a temperature of 62° (Fahrenheit's thermometer), has its weight increased by 252.458 grains. Of the grains thus determined, 7000 are a pound Avoirdupois, and 5760 a pound Troy.

*

TROY WEIGHT.

13. The derivation of the word Troy is doubtful. One theory is that it comes from the town Troyes, in France, because of the pound Troy having first been used there. Another derivation is "Troynovant," the mythological name of London; a third derives it from trois (three), because it is the money weight, and that money and money weight have each three denominations-penny, shilling, pound; pennyweight, ounce, pound. Troy weight is used in weighing gold, silver, precious stones, &c., and also in scientific investigations. The fineness of gold-that is, the ratio of the weight of pure gold in any given mass to the weight of the whole-is generally estimated by the number of carats (about 3 grains) of pure gold contained in 24 carats of the given substance. Standard gold—that is, the gold of our coinage-is "22 carats fine." This means that out of 24 carats of sovereign gold 22 are pure gold. Sometimes this is also expressed by saying that standard gold is 1 fine, this being the ratio of the pure to the alloyed metal. Diamonds and other precious stones are weighed by carats.

The following are the different denominations in Troy weight:

24 grains (24 grs.) = I pennyweight ...written i dwt.

14. The weights used by apothecaries are aliquot parts of the pound Troy, and are as follows:

* In scientific calculations and measurements, a decimal system is most generally now used, as being much more convenient.

15. The pound avoirdupois contains 7000 grains, and a cubic foot of distilled water, at 62° Fahrenheit, weighs 62.321 pounds avoirdupois very nearly.

The following are the subdivisions :—

A stone is the name given to the weight of 14 lbs.

A sack of coals is 2 cwt.

A ton of shipping is 42 cubic feet.

A load of rough timber is 40 cubic feet.

A load of squared timber is 50 cubic feet.

IMPERIAL LIQUID AND DRY MEASURES.

16. The gallon contains 277 274 cubic inches, and contains 10 pounds avoirdupois of distilled water at a temperature of 62° Fahrenheit.

* The weight used for weighing heavy goods, goods of weight

(avoir du pois.)