xiv

TABLES FOR REDUCING OTHER MEASURES TO C.G.S. MEASURES.

The abbreviation cm. is used for centimetre or centimetres,

More exactly, according to Captain Clarke's comparisons of standards of length (printed in 1866), the metre is equal to 109362311 yard, or 3.2808693 feet, or 39'370432 inches, the standard metre being taken as correct at o° C., and the standard yard as correct at 16° C. Hence the inch is 2'5399772 centimetres.

More exactly, according to the comparison made by Professor W. H. Miller in 1844 of the "kilogramme des Archives," the standard of French weights, with two English pounds of platinum, and additional weights, also of platinum, the kilogramme is 15432 34874 grains, of which the new standard pound contains 7000. Hence the kilogramme is 2.2046212 pounds, and the pound is 453'59265 grammes.

CHAPTER I.

GENERAL THEORY OF UNITS.

Units and Derived Units.

1. THE numerical value of a concrete quantity is its ratio to a selected magnitude of the same kind, called the unit.

Thus, if L denote a definite length, and 7 the unit

length,

L

ī

is a ratio in the strict Euclidian sense, and

is called the numerical value of L.

The numerical value of a concrete quantity varies directly as the concrete quantity itself, and inversely as the unit in terms of which it is expressed.

2. A unit of one kind of quantity is sometimes defined by reference to a unit of another kind of quantity. For example, the unit of area is commonly defined to be the area of the square described upon the unit of length; and the unit of volume is commonly defined as the volume of the cube constructed on the unit of length. The units of area and volume thus defined are called derived units, and are more convenient for calculation than independent units would be. For example, when the above definition of the unit of area is employed, we can assert J

A

that [the numerical value of] the area of any rectangle is equal to the product of [the numerical values of] its length and breadth; whereas, if any other unit of area were employed, we should have to introduce a third factor which would be constant for all rectangles.

3. Still more frequently, a unit of one kind of quantity is defined by reference to two or more units of other kinds. For example, the unit of velocity is commonly defined to be that velocity with which the unit length would be described in the unit time. When we specify a velocity as so many miles per hour, or so many feet per second, we in effect employ as the unit of velocity a mile per hour in the former case, and a foot per second in the latter. These are derived units of velocity.

Again, the unit acceleration is commonly defined to be that acceleration with which a unit of velocity would be gained in a unit of time. The unit of acceleration is thus derived directly from the units of velocity and time, and therefore indirectly from the units of length and time.

4. In these and all other cases, the practical advantage of employing derived units is, that we thus avoid the introduction of additional factors, which would involve needless labour in calculating and difficulty in remembering." *

* An example of such needless factors may be found in the rules commonly given in English books for finding the mass of a body when its volume and material are given. "Multiply the volume in cubic feet by the specific gravity and by 62'4, and the product will be the mass in pounds; or " multiply the volume in cubic inches by the specific gravity and by 253, and the product will be the mass in grains." The factors 62°4 and 253 here employed would be avoided --that is, would be replaced by unity, if the unit volume of water were made the unit of mass.