are The existing weights which are multiples of 1 lb. avoirdupois, The quarter: =28 lb.; the hundredweight=112 lb.; and the ton=2240 lb. The proposed new measures are There is no denomination higher than pounds in trey or apothecaries weight. With the help of the preceding tables, a compound number may be reduced to a decimal fraction expressed in terms of the principal unit of that number. Taking 17s. 8d. as an example. Also, a decimal fraction of any principal unit may be reduced to an equivalent compound number. The reductions from troy and apothecaries weights (expressed as compound numbers) are to avoirdupois weight. Taking 8 oz. 15 dwts. 17 gr. troy. 8 oz. troy=548571426 lb. av. 15 dwt. 17 gr. adding. This, if reduced to ounces and drams avoirdupois (as in the last example)=9 oz. 10.222714 dr. The equivalents of the decimal parts of 1 lb. av. are given as compound numbers in avoirdupois weight, troy weight, and apothecaries weight. b. The French system of names of weights and measures has been explained in Art. 142., and illustrated by the Table of Measures of Length. The other French measures and weights are Measures of Surface. The principal unit is the are, which is a square decametre, that is, a square having the decametre for side; and which consequently contains 100 square metres. The names of the multiples and parts of the are, formed according to the nomenclature already explained, are the Myriare, kilare, hectare, decare, (are), deciare, centiare, milliare. Of these only the myriare, hectare, are, and centiare (or square metre) are in use. Measures of Volume.—The principal unit is the cubic metre. The measures are, the cubic metre, the cube of the decimetre (=the Tooth part of the cubic metre), and the cube of the centimetre (=the 1000000th part of the cubic metre). No other multiples or parts of the unit of volume are in use. 00 1 cubic metre=1·308022 cubic yd. Measures of Capacity for Liquids, Corn, &c.—The principal unit is the litre, which is a cube having the decimetre for edge. The multiples and parts in use are the Hectolitre, decalitre, (litre), decilitre, centilitre. The litre=22009687 gallons imperial British measure. Weights. The principal unit of weight is the gramme, or the weight of the centimetre cube of distilled water at its greatest density. The weights in use are the Myriagramme, kilogramme, hectogramme, decagramme, (gramme), decigramme, centigramme, milligramme. The gramme='002205 lb. avoirdupois. Money. The principal unit is the franc, which contains 4.5 grammes of pure silver and 5 gramme of alloy (the alloy used being copper). No names are given to the multiples of the franc. Its tenth part is called decime, and its hundredth part, centime. All these weights and measures are derived from the metre. REDUCTION OF COMPOUND NUMBERS. 144. From the reciprocal relations of the different denominations of a compound number, as multiples and parts of each other, it is easy to express one of any denomination in terms of every other denomination of the same measure or weight. For instance, 1 lb. troy=12 oz. = 12 × 20 dwt. or 240 dwt.=240 x 24 gr. or 5760 gr. The process is the same for any multiple of one and for any other weight or measure: whence, if units of any proposed denomination are to be expressed in terms of a lower denomination, the change or reduction to a lower denomination is made by multiplying the number in the proposed denomination by a factor expressing the number of units of the lower denomination which make one of the higher denomination an operation which may be repeated as often as the table of the weight or measure allows. And if units of any proposed denomination are to be expressed in terms of a higher denomination, the reduction is made by dividing the number in the proposed denomination by a divisor expressing the number of units of the proposed denomination which make one of the next higher denomination - an operation which may be also repeated as often as the table allows. Hence, likewise, a compound number may be expressed in terms of any denomination contained in the weight or measure to which the compound number belongs. For if each denomination of the compound number is expressed in terms of the required denomination, and the reduced expressions are added together, their sum is the equivalent of the compound number, expressed in the required denomination. As an illustration, let it be required to express £8 16s. 44d., as farthings, as pence, as shillings, and as pounds. When a compound number is to be reduced to the denomination of some unit inferior to the highest contained in it; a convenient practice is to reduce the number in the highest denomination to the next lower denomination; to add to the result the number in this lower denomination; to reduce the sum to the next lower denomination, and so on. Taking the last example as an instance of this form of reduction, = 8£= 8 x 20s. 160s... £8 16s. = 160s. +16s. 1768. 176s. 176 x 12d. =2112d. .. 176s. 4d.=2112d. + 4d. =2116d. 2116d.=2116 × 4qr.=8464gr. .'. 21161d. 8464qr.+ 1gr.=8465qr. or £8 16 41-8465qr. This calculation is usually represented in the following manner : The process is the same as that of Art. 138., for the reduction of a number expressed in any system to an equivalent number expressed in the decimal system. The reduction of a number from a lower to a higher denomination, through one or more intermediate denominations, may be made by dividing at once by a divisor expressing the number of the lowest denomination, which make 1 of the highest denomination, if the result is to be expressed in terms of the highest denomination only; but by a divisor expressing the number of the lowest denomination which make 1 of the next higher denomination, then by a divisor expressing the number of this denomination which make one of the next higher denomination, and so on, if the result is to be expressed as a compound number. For instance, if it be required to reduce 16573 farthings to pounds, the reduction may be effected at once by dividing 16573 by 960, the number of farthings which make 1 pound. The quotient 16573-17253£, or 17-2635416£, 960 960 |