that order does not impair the practical utility of the relation of the metre to the quadrant, but it shows that it is the material metre which is now to be considered as the ultimate standard.

According to the same authority the length of the polar axis is 500,482,296 inches; and Sir John Herschel pointed out that to define the inch as the five-hundred-millionth part of the polar axis would involve a correction which would be inappreciable in ordinary measurements, and sensible only with the nicest scientific instruments.

ART. 60.-Derived Metric Units. The great excellence of the metric system lies in the mode in which the subsidiary units are defined. Each is a decimal multiple or sub-multiple of the standard unit, and the scale of units is thus in harmony with the decimal nature of the notation of Arithmetic. The prefixes for multiples are derived from the Greek, and those for sub-multiples from the Latin.

The abbreviations given are those authorized by the Comité International, which has for its object the development of the metric system. The abbreviation for metre is m.; hence km. for kilometre, mm. for millimetre.

10

4

ART. 61.-Comparison of Yard and Metre. The metric system has now been adopted over the greater part of Fig.1. Europe and has begun to spread over America. Its use in contracts was legalized in Britain in 1864, and it has also been legalized in the United States. A careful comparison of the yard and metre was 9 made by Captain Kater, who gave as his result

This equivalence was adopted in the Act of 1864. A more recent determination by Colonel Clarke 1 metre 39.37043 inches.

gives

=

Recently, Professor Stoney has deduced a convenient equivalent

which agrees with the above, so far as they do agree, and cuts off the uncertain difference.

These are determinations for the platinum metre at 32° Fahr., and the bronze yard at 62° Fahr. With 4 a brass metre correct at 32° Fahr., but used at an ordinary temperature such as 62° Fahr., the equivalence is

1 metre 39-382 inches.

=

A sufficient approximation for ordinary purposes is

1 metre = 393 inches;

while a rougher approximation, easily remembered, 1 decimetre = 4 inches.

is

In Fig. 1 the decimetre is compared with 4 inches.

3

2

Centimetres Inches

ART. 62.-The C.G.S. System. The British Association, through their Committee on Electrical Standards, have extended the metric system and developed it so as to suit the measurement of electrical quantities. This development of the metric system is commonly called the Centimetre-Gramme-Second system, or by abbreviation * Nature, vol. xxix., p. 278.

E

the C.G.S. system. The standard of length is the metre, but the primary unit of length is not the metre but the centimetre. This system was adopted as the basis for electrical measurements by the International Congress of Electricians, which met at Paris in 1881.

ART. 63.-Change of Length. When a straight rod is changed in length uniformly, the rate of expansion is expressed by

a L increment per L original length;

(1)

and the ratio of the expansion is expressed by

1 + a L expanded length per L original length.

(2)

The reciprocal of (1) is

1

1+ a

Loriginal length per L expanded length;

and the rate of diminution is

i.e.,

(3)

(1-1) decrement per L expanded length; (4)

When a is a small quantity, 1-a is a sufficient approximation for

1

1+ a

(Art. 29), and then (3) becomes

(1a) L contracted length per L original length ;

and (4) becomes

a L decrement per L original length.

Here we have the same ideas as in the change in value of a sum of money discussed under simple interest.

Ex. 2. Reduce 5

76.43 miles.

francs per metre to pence per yard, when ex

1. Reduce 80 metres to yards, and 40 yards to metres.

2. Reduce 321 miles to kilometres.

3. Find the reciprocal of 66 feet = link.

4. One metre contains 3.2809 English, and 3.1862 Prussian feet. fraction of an English inch does the Prussian exceed the English foot?

By what

5. Determine the height of Mont Blanc in old Parisian feet, its height being given as 15,784 in English feet, and the ratio of the English to the old Parisian foot being as 1'0297 to 9662.

6. Taking a centimetre at two fifths of an inch, find the number of centimetres in 30 yds. 3 inches.

7. Pliny says that the side of the base of the Great Pyramid was 883 pedes. If the pes is equivalent to the semi-cubit, what was the side of the Pyramid in English feet? Assume 1 cubit 20.7 inch.

=

8. Three per cent. of the length of a wire is 4 inches too long for a certain purpose, and 3 feet 9 inches is ten per cent. too short. What is the length of the wire ?

9. Reduce 48. 6d. per yard to francs per metre when exchange is at 25 francs 22 c. per £.

10. Find the successive convergents to the ratio of the metre to the yard, taking 1 yard 9144 m.

=

11. The scale of a map is six inches to the mile; express the scale in terms of nch to inch.

ART. 64.

SECTION X.-ANGLE.

Sexagesimal Units. The circumference of the circle is divided into four equal parts called quadrants, and also into six equal parts called sextants. The sixtieth part of a sextant is denominated a degree; the sixtieth part of a degree is denomi