penny), becomes 51d., of which we set down the bd., and“ carry"' the 5d. Multiplying the 8d. by 11, we obtain 88d., to which we add the carried 5d.: we then have 93 pence, and this amount, when divided by 12 (the number £ 8. d. of pence in a shilling), becomes 78. 9d., of which 59 17 8} we set down the 9d., and carry the 78. Multiplying the 178. by 11, and adding the carried 7%. to the product, we obtain 1945. : this amount, 658 14 92 when divided by 20 (the number of shillings in a pound), becomes £9 148., of which we set down the 143., and carry the £9. Lastly, multiplying the 259 by it, and adding the carried £9 to the product, we obtain £658 So that the required product is £658 148. 91d. II Note. In the absence of Reduction, this result (obtained by means of Simple Multiplication) would appear under the form £649 1778. 881d. And £658 148. 9.d. would be obtained—although in a round-about way—by means of Compound Addition if £59 175. 8.jd. were taken as addend 11 times. EXAMPLE II.-- There are 9 silver spoons, each weighing i oz. 6 dwts. II grs.: how much do they all weigh? I II Here we have to multiply i oz. 6 dwts. 11 grs. by 9. Multiplying the 11 grs. by 9, we obtain 99 grs. : this number, when divided by. 24 (the oz. dwts. grs. number of grains in a pennyweight), be 6 comes 4 dwts. 3 grs., of which we set down 9 the 3 grs., and carry the 4 dwts. Multiply 4 ing the 6 dwts. by 9, and adding the carried 18 3 4 dwts. to the product, we obtain 58 dwts. : this number, when divided by 20 (the number of pennyweights in an ounce), becomes 2 oz. 18 dwts., of which we set down the 18 dwts., and carry the 2 oz. Multiplying the i oz. by 9, and adding the carried 2 oz. to the product, we obtain ni oz. So that the required product is 11 oz. 18 dwts. 3 grs. II : 83. Rule for Compound Multiplication : Multiply the several denominations, successively,—beginning with the lowest,—by the multiplier, and reduce as in Compound Addition; taking care to add to each product the number “carried” from the preceding product. 30 19 8 NOTE 1.- When the multiplier exceeds 12, but is resolvable into a pair of factors, neither of which is greater than 12, we proceed as in Simple Multiplication £ s. d. -first, multiplying the multiplicand by one of the 3 17 51 factors, and then multiplying the resulting product by the other factor. Thus, if we wanted to find the rent of a farm containing. 72 acres, at £3 178. 5}d. per acre, we could imagine the 9 farm divided into 9 fields of 8 acres each, in which case the rent of one field would be 278 17 0 £3 178. 53d.x8=£30 198. 8d., and the rent of the whole 9 fields £30 198. 8d. x9=£278 178. od. NOTE 2:-The multiplier, when greater than 12, and not resolvable into factors, is sometimes broken up (as in Simple Multiplication into the parts which its digits individully represent—the rent of 365 acres of land, for instance, at £3 175. 5td. per acre, being calculated as follows: £ s. d. I acre. 1413 12 31 365 In many cases, however, the pupil will find it less troublesome to proceed according to the Rule. Thus, to return to the last exercise :365 $d.=365 halfpence = 182]d. X £ 365 x 5d.= 1825d. 365 X 178.= 2007.d.= 167s. 3}d. £ s. d. 63728. 3}d.= 318 12 33 1095 365 X 43.= оо 365 x £3 178. 54d.= 1413 12 33 * The rent of 60 acres is obtained from the multiplication of £38 148. 7d. (the rent of 10 acres) by 6. COMPOUND DIVISION. 84. Division is called COMPOUND when the dividend, or the divisor, or the resulting quotient is a compound number; or when all three being — simple numbers—the dividend and the divisor (whilst of the same kind) are of different denominations, We perform an operation in Compound Division when we divide £23 by 8, and write the quotient under the form £2 178. 6d. ; also, when we find how many sums of 158. each are contained in £3. 85. Compound Division is performed by means of Simple Division and Descending Reduction. EXAMPLE I.-If £658 145.9}d. were divided equally amongst Il persons, how much would each receive ? Here we have to divide £658 145. 9 d. by 11. The division of £658 by II gives £59 for quotient, and £9 for remainder. This remainder, when reduced, becomes (9X 203) 180 shillings, to which we add £ s. d. the 14 shillings in the dividend. Altogether, 11)658 14 9. we then have (180+143) 194 shillings, the division of which by it gives 178 for 59 17 8.1 quotient, and 78. for remainder. This second remainder, when reduced, becomes (7 X 12 =) 84 pence, to which we add the 9 pence in the dividend. We then have, altogether, (84+9=) 93 pence, the division of which by ii gives 8d. for quotient, and 5d. for remainder. This last remainder, when reduced, becomes (5X4=) 20 farthings, to which we add the 2 farthings (zd.) in the dividend. Altogether, there then are (20+2=) 22 farthings, the division of which by 11 gives 2 farthings or şd. for quotient, and no remainder. We thus find that each person's share would be £59 178. 8}d. This result could be obtained, although in a very roundabout way, by Compound Subtraction—in other words, by Simple Subtraction and Descending Reduction. Thus, taking £11 as often as possible from £658, we should find that £i could be given 59 different times to each of the il persons, and that the number of pounds then remaining would be 9. Taking 11s. as often as possible from 1945. (£9 148.), we should next find that is. could be given 17 different times to each person, and that the number of shillings then remaining would be 7. Taking vid. as often as possible from 93d. (78. 9d.), we should next find that id. could be given 8 different times to each а over. person, and that the number of pence then remaining would be 5. Lastly, taking 11 farthings as often as possible from 22 farthings (53d.), we should find that £d. could be given 2 different times to each person, and that nothing would then remain to be distributed : each person's share being thus found to be, as before, £59 178. 8zd. EXAMPLE II.--How many lengths of 2 ft. 8 in. each could be cut off a ribbon 17 yds. long? In 2 ft. 8 in. there are yds. (2 X 12+8== 32 inches, 17 whilst in 17 yds. there are ft. in. 3 (17 X3X12=)612 inches. 2 8 51 Dividing 612 by 32, there 12 12 fore, we find that in 17 yds. there are 19 lengths of 2 32 ) 612(19 ft. 8 in. each, and 4 inches 4 32 292 288 4 86. Rule for Compound Division, (1.) when the Divisor is an abstract number : Divide the several denominations successively-beginning with the highest—by the divisor, taking care, when a portion of any denomination is left as "remainder," to include this portion, after having reduced it, in the next (lower) denomination. (2.) When the Divisor ( is a concrete number, reduce it and the dividend to the lowest denomination which occurs in either, and then find the quotient by Simple Division. Note I.-- When the divisoris an abstract number greater than 12, but resolvable into a pair of factors of which £ $. d. neither exceeds 12, we 8)365 17 14: 72 can proceed as in Simple Division — first dividing 9)45 14 81+ d. (Ist remainder ) the dividend by one of the factors, and after 5 1 73+ d. (2nd ) wards dividing the result- True quotient=£5 18. 7žd. ". ing quotient by the other True remainder=d.x8+ d. factor. If, for instance, = d.=Ind. it were required to divide £365 178. 11d. by 72, we could resolve 72 into the factors 8 and 9, and proceed as in the margin. THE METRIC SYSTEM OF WEIGHTS AND MEASURES. “ The great diversity of Weights and Measures which has existed in all countries has principally arisen from the lesser communities of which they were originally composed having each adopted its own system. In process of time these lesser communities were amalgamated into separate nations, with whose increase of population and trade the inconvenience of a variety of Weights and Measures soon made itself apparent, and the desire of establishing uniformity arose. France was the first country to relieve itself from its barbarous multiplicity of Weights and Measures, by adopting a uniform system. Louis the XVI., at the recommendation of the Constituent Assembly, invited, by a decree, all the nations of Europe, and particularly the King of Great Britain, to confer respecting the adoption of an international system of Weights and Measures. No response being given to this invitation, France committed the consideration of the subject to some of the most learned men of the age, who devised what is called the Metric system : the most simple, convenient, and scientific system of Weights and Measures in existence.' The Metric system is so called from the circumstance that it is based upon the METRE, which is the standard of length. The metre is the ten-millionth (10.000.000) part of the distance -measured upon the meridian of Paris—between the equator and the pole, and is longer by about 3} inches than the English yard. The standard of superficial measure is a square described * Report of Parliamentary Committee on Weights and Measures of United Kingdom: 1862. н. |