20. On a certain map a square inch represents 4,000 acres. What is the linear scale?

21. The vertical scale of a drawing is 40 feet to an inch, and the horizontal 400 feet to an inch. What is the scale for the area?

22. On a map it is found that 100 acres are represented by 359'48 square inches; but the scale not being attached, it is required to calculate what it is. Give the scale in the form of a ratio, and also in terms of foot per mile.

SECTION XIII.---VOLUME.

ART. 90.-General Unit of Volume. Any unit of volume may be denoted by V. The systematic unit is the volume of a cube whose side is the unit of length, and in such case we have 1 V = L3, where L denotes the same as cubic L. When the unit is not systematic we have some other number instead of 1.

ART. 91.-Imperial Units of Volume. In the imperial system we have cubic units and units of capacity. The base of the cubic unit is the cubic yard, and the relations to it of the cubic foot and the cubic inch follow from the relations of the linear foot and the linear inch to the linear yard.

The primary unit of capacity is the gallon, which involves in its definition the standard of mass. It is the volume of ten imperial pounds of distilled water at the temperature of 62° Fahr. Further, the mass of the water is to be determined by weighing in air against brass weights, the air also being at the temperature of 62° Fahr., and the barometer standing at 30 inches.

In the original definition of the gallon, the volume defined as above was stated to be equivalent to 277 274 cubic inches; but when a more accurate determination of the density of water was made, the alternative part of the definition was repealed. According to the most recent determinations1 the gallon is equivalent to 277.123 cubic inches. The brass gallon, marked “imperial

1 Rankine's Rules and Tables, p. 99.

standard gallon," constructed when the gallon was originally defined, is not the ultimate standard, but pure water taken in conjunction with the standard of mass.

The other units of capacity are defined by means of the gallon.

ART. 92. Metric Units of Volume. In the metric system we have three series of units of volume. The stere and its derivatives are for solid measure, as for example the measuring of wood; the litre and its derivatives are for fluid measure or measure of capacity; while the cubic series is the best adapted for calculations and for science generally. The stere is by definition equivalent to a cubic metre, and the litre to a cubic decimetre. Their derivatives are decimal; while those of the cubic series are millesimal. The authorized abbreviation for cubic is the index3, as in cm.3 for cubic centimetre. That for stere is s., and for litre 7. In the C.G.S. system the primary unit of volume is the cubic centimetre.

For a rectangular

ART. 93. Volume of a Parallelepiped. parallelepiped, V being defined as L3,

1 V L long per L broad per L thick.

==

When V is not defined as L3, we have some number instead of 1.

This rate may be written as an equivalence,

1 V L long by L broad by L thick.

=

For example, in a rectangular slab the number of feet in the thickness is the same as the number of cubic feet per square foot of surface. In the case of a rectangular rod, the number of square inches in the cross section is the same as the number of cubic inches per inch of length.

When the parallelepiped is not rectangular, let the length and breadth be inclined at an angle of 0°, and the thickness be inclined to the base at an angle of 4°, then

sin 0 sin V = L long by L broad by L thick.

EXAMPLES.

Ex. 1. Given the rate of exchange 25.22 francs per £, and that 4.54 litres are equivalent to 1 gallon; deduce the relation of francs per litre to £ per gallon.

4.54 litres = 1 gallon;

.. dividing the one equivalence by the other,

Ex. 2. A schoolroom is 40 ft. long, 26 ft. 6 in. broad, and 19 ft. 3 in. high. If 80 cubic feet of space and 8 square feet of floor must be provided for each scholar, what is the maximum number of scholars which the room can provide for?

Hence the greatest number of scholars provided for is 132.

EXERCISE XIII.

1. Express the litre in terms of the cubic foot.

2. An imperial gallon is 277 274 cubic inches; a Winchester bushel 215042 cubic inches; how many Winchester bushels are equal to 100 imperial bushels? 3. Reduce 85 litres to gallons, and 54 gallons to litres.

4. By how much does a quart exceed a litre?

5. Reduce 234 hectolitres to bushels, and 432 bushels to hectolitres.

6. What is 1.05 dollars per bushel in terms of shilling per quarter?

7. What is one shilling and sixpence a gallon in francs per litre, taking exchange at 25 francs per £?

8. A gas stove burns 7 cubic feet per hour, and the cost of the gas is 4s. per 1,000 cubic feet. Find the hourly cost.

9. A rectangular block of stone is as broad as it is long, and contains a cubic feet. If it were as broad as it is high the bulk would be b cubic feet. Find the length.

10. If a wall, 42 feet long, 10 feet high, and 2 feet thick, contains 12,800 bricks, how many bricks of the same kind will be required for a wall 112 feet long, 6 feet high, and 2 feet thick?

11. If 81 gallons of water will fill a cistern 4 feet 4 inches long, 2 feet 8 inches broad, and 1 foot 1 inch deep, how many cubic inches are contained in a pint? 12. The price of wheat is 36s. 6d. per quarter; express it in terms of francs per hectolitre, the rate of exchange being 25 30 francs = £.

13. The price of brandy is 52 centimes per litre ; express it in pence per gallon, the rate of exchange being 25'80 francs £.

=

14. By taking the decimetre as equal to four inches, what percentage of error is introduced, first, in linear measure; second, in square measure; third, in cubic measure. Take the legal equivalent as the true length of the metre?

15. Find the length of the side of a cubical vessel that shall contain twice as much as one whose side is 8 inches.

SECTION XIV.-VOLUME, Continued.

ART. 94. Volume of a Cylinder, a Cone, a Pyramid. The volume of a cylinder is given by

1 V = S base by L altitude, where V denotes L3 and S denotes L2.

In the case of a cone the altitude is variable, but it can be shown that the average altitude is of the greatest altitude. Hence VS base by L greatest altitude. In a pyramid the altitudes vary in the same manner as in a cone. Hence V = S base by L greatest altitude.

ART. 95.-Volume of a Sphere. A sphere may be considered as made up of a great number of pyramids having their bases on