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4. 1 qr. 7 lbs. to the decimal of 1 ton; 3 cwt. 3 qrs. 20 lbs. to the decimal of 3 tons.

5. 3} lbs. to the decimal of lu cwt.; 14 oz. to the decimal of 3 cwt. 2 qrs.

6. 20 grs. to the decimal of 1 lb. Troy; 3 dwts. 16 grs. to the decimal of 4 oz. 11 dwts.

7. 1 rood, 10 poles to the decimal of 1 acre; 3 roods, 15. square yards to the decimal of 5 acres.

8. hours to the decimal of 10 weeks; 7 h. 18' to the decimal of 1 year (365 days).

9. Bring the sum of 14 of 9 hours, of 121 days, of 7 minutes, to the decimal of a week.

10. Express a pound troy as the decimal of a pound avoirdupois.

11. Reduce the sum of 6 lbs. 6.4 oz. avoirdupois, and 8 oz. 6 dwts. 16 grs. to the decimal of 1 ton. 12. Express 2.36 of -5i 8 of Os. 20. + 1.4583 of 6d. as the decimal of £5,

CHAPTER IV.

THE METRIC SYSTEM.

32.-The fundamental unit of the metric system is the

1 metre. -,

A metre is the ten-millionth, or 107 part of 90° of the earth's meridian, and measures 39.3708 English inches. In order to express multiples and sub-multiples of this unit, and, indeed, of any unit in the metric system, we make use of one or more of the following prefixes :Deka, : 10 times. Deci,

10th, Hecto, 100 Centi,

100th. Kilo,

1,000
Milli,

1,000th. Myria, 10,000

We will arrange these prefixes and the word unit in order according to their signification, thus

Myria, Kilo, Hecto, Deka, Unit, Deci, Centi, Milli. Now, as we read this line from left to right, it is evident

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Kilom.

Dekam.

Metres.

Millim.

Hectom. Dekam.

Metre.

Decim.

Centim,

Millim.

that the words have a signification decreasing tenfold in value; and as we read it from right to left, they have a signification increasing tenfold in value.

It therefore follows that figures placed under the above words have a local as well as an intrinsic value; and further, if when a figure is wanting to complete the series, its place be filled up by a cipher, it will be seen that the local value corresponds exactly with the ordinary decimal notation.

Moreover, we have only to place a mark in fact, a decimal point) at the right of the figure standing under any of the words of the above memorial line, and the given quantity is at once expressed in the denomination corresponding to that figure. Thus, taking the metre as our unit: Myriam.

Decim.
3 2 5 4 7 8
Myriam, Kilom.

3 2 0 5 4 7 0 8
3.2054708 myriametres.
32.054708 kilometres.
320-54708 hectometres.
3205.4708 dekametres.
32054.708 metres.
320547.08 decimetres.
3205470.8 centimetres.

32054708. or 32054708 millimetres. The following rule for expressing any quantity in terms of any one multiple or sub-multiple of the unit, or of the unit itself, is therefore evident:

Rule.-Put ciphers in the place of any multiple, unit, or sub-multiple absent in the series, and write the figures in close order, as in the ordinary decimal notation. Then place a decimal point at the right of the figure corresponding to the denomination in which we wish to express the given quantity.

Ex. 1.-Express 5 myriam. 3 hectom. 6 decim. as metres.
Filling up with ciphers the vacant spaces, we have-
Myriam. Kilom. Hectom. Dekam. Metres.

0 3 0 0 6

= 50300.6 metres.

Decim,

Decim.

Decim.

Ex. 2.-Express 3 dekam. 4 decim, as myriametres.
Filling up with ciphers, we have-
Myriam. Kilom. Hectom. Dekam, Metres.

Centim. Millim
0 0 0 3 0 4 0 0

0.0030400 myriametres 0.00304 myriametres. (The student will see that it was unnecessary here to extend the series beyond decimetres.)

Ex, 3.-Express 13 metres 502 millimetres as kilometres.
We may write the given quantity thus-
Kilom. Hectom. Deckam. Metres.

Centim. Millim.
0 0 1 3 5 0 2

= 0.013502 kilometres. We have hitherto spoken only of the fundamental unit, and its multiples and sub-multiples. We shall hereafter (Art. 35) explain the principal derived units, viz., the Gram, the Are, the Stere, the Litre, and the Franc; but as the multiples and sub-multiples of these derived units bear the same relation respectively to the corresponding derived unit, as in the case of the fundamental unit, all the preceding remarks relative to the multiples and sub-multiples of the fundamental unit apply equally to those of the Gram, the Are, the Stere, the Litre, and the Franc.

With regard to the units, multiples, and sub-multiples of square and cubic measure, properly so called, it is necessary to make a few remarks.

33. Square Measure.—The unit of square measure is the square metre; and since the series myriam., kilom., hectom., dekam., metre., &c., decrease in value tenfold when read from left to right, and increases similarly when read from right to left, it follows that the series square myriam., square kilom., square hectom., square dekam., square metre, &c., will decrease or increasė 102 or 100-fold. Hence we see that in square measure the multiples and sub-multiples increase or decrease successively 100-fold, and, therefore, when quantities in square measure are expressed by the ordinary decimal notation, each multiple or sub-multiple must occupy the place of two figures, a cipher being supplied when we have less than ten of any multiple or sub-multiple, and two ciphers when there is any blank in the series,

Sq. kilom. Sq. hectom. Sq, dekam. Sq. metre. Sq. centim.
Ex. 10 3 15

3

5 Sq. kilom. Sq. hectom. Sq. dekam. Sq. metre.

Sq. decim.

Sq. centim. 10 03

15
03

00 05
10·0315030005 square kilometres.
1003.15030005 hectometres.
100315.030005 dekametres.
10031503.0005 metres.
1003150300.05 decimetres.
100315030005 centimetres.

34. Cubic Measure. The unit of cubic measure is the subic metre, and hence after the remarks in the last article, since 103 1000, when quantities in cubic measure are expressed by the ordinary decimal notation, the units, multiples, and sub-multiples must respectively occupy the place of three figures, ciphers being supplied to fill up blank spaces when necessary. Ex, 1. 325 cubic metres 51 cubic decimetres.

051 325.051 cubic metres. 325051 cubic decimetres.

= 325

Ex. 2.-25 cubic metres 3 cubic decim. 40 cubic centim.

25 cubic metres 003 cubic decim. 040 cubic centim,
25.003040 cubic metres.
25003.040 decimetres.
25003040 centimetres.

35. Derived Units. The principal derived units of the metric system are

1. The Gram, for measures of weight,

The gram is the weight of a cubic centimetre of distilled water at the temperature of 4° C.

1 gram = 18.4323 grains, 1 grain = 10648 gram. 2. The Are, for land mogsure.

The are is a square whgge side measures 10 metres ; it is therefore equal to a square dekametre, er 100 square metres, 1 are = 119.6033 square yards, 1 hectare = 2:471 acres, 1 acre = •405 hectare.

3. The Stere, for fire-wood.

The stere is equivalent to a cubic metre. It is therefore the solidity of a cube whose edge measures 1 metre.

4. The Litre, for measures of capacity.

The litre is a capacity equal to the volume of a cube whose edge measures a decimetre or 10 centimetres. It is therefore equal to a cubic decimetre or 1,000 cubic centimetres, and 1,000 litres are equivalent to a cubic metre.

1 litre = .2201 gallon, 1 gallon = 1.543 litres, 11 gallons 50 litres nearly 5. The Franc, for money.

The franc is a coin weighing 5 grams, and composed of an alloy, nine-tenths of which are silver and one-tenth copper.

The following table exhibits at a glance the fundamental unit and the above derived units, together with the multiples and sub-multiples at present in use :TABLE OF THE METRIC SYSTEM OF WEIGHTS AND MEASURES.

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A quintal = 100 kilog. = 2 cwt. nearly; a millier, or tonneau de mer, = 10 quintals

= 20 cwt. nearly.

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