MATUEMATICa ARITHMETIC. 



[COMPOUND aU'LTII'LIOATIO.Y. 



exainp!>. if thus worked an<t :11 afford exer- 



ci*o 1 -'ii and addition : 



m 200. 





I ; Ib. 11 ox. 21 ft. 

 ; yd*, from 6S7 mil. 3 fur. 



.11. 1 quart* 1 pint from 570 gallons 1 quart. 

 .1 rood* 7 prrchci 23 jrmrd* from 1\ rr*. 

 \i\ <|. yd*. 7ft. 132 in. from 237 *o,. yd. 3ft. 

 nil in. 

 (8.) Subtm-t 18 cubic yd*. 37 ft. 211 in. from 47 cubic yd*. 13 ft. 



' 'rn/vmnrf Quantitits. From the 



nature of multiplication, it is plain that a concrete quan- 

 ean be multiplied only by an abstract niiwW; indeed 

 whatever IK- the multii>licaud the multiplier, which 

 aim]'lr denotos how many times the former is to be 

 eesxarily be a mere number. Strange to 

 books on arithmetic, of the most recent 

 to bo found, in which the multiplication to- 

 gether of concrete quantities is insisted upon, and pre- 

 > bo taught. 1'eople have disputed over and 

 i again about the product of 19 lib. !!}<<., inulti- 

 .: -elf ! They might as well have disputed about 

 lo by Lombanl Street ; or, 



as Mr. Walker pithily expresses it, abovit multiplying 

 y 3 bars of music. "* This last operation, 

 ' palpably absurd as the thing is, the arithmeticians re- 

 i not for a moment hesitate to undertake, 

 ml music occurred in a rule-of-three 

 I well might ; for they refer 

 in justification of such a process, t 

 Is it not ridiculous to appeal to a rule instead of to rea- 

 son and common sense, in a subject which professes to be 

 founded on rational principles, aud supported by demon- 

 stration I We wish to warn you earnestly against this : 

 receive no rule in any department of mathematics, the 

 truth of which is not evident to your own understanding ; 

 . in strict accordance with common sense. 

 i short way of doing addition, 



and i.i:iy always replace it: you have only to 



writo the multiplicand down the proposed number of 



times, and to add all up. The sum is what in miiltipli- 



'ii is called the proiltttt ; but how could a sum of 



money be written down t'lil 19s. lljd times ? Even in 



imi, similar absurdities are to bo met 



with in the books. If you wish to convert pounds into 



shillings, you are 'told to multiply the pounds by 20, and 



, which is worse, to multiply them by 20s. 



l!ut if you multiply pounds by 20, you get not shillings 



is many juXHids, as is obvious : what 



13 to multiply the number denoting how 



m<my pounds by 20 ; because there must be 20 times 



that number of shillings. 



the rule for multiplying a compound 

 quantity by a number : 



i.E I. When the Multiplier it not greater than 12. 

 Put the multiplier under the quantity of least denomi- 

 n : multiply that quantity by it, and divide the 

 it expresses how many of such 

 : the next denomination : put down 

 v the quotient to the product 

 multiplication of the next term and so 

 on till all the terms have )> en multi; 



".'. and is yet such 



M to adn ultiply by 



each factor in succession, as in short multiplication. 



The tablu of factors at the end will bo found very 

 useful in enabling us to tell at a glance whether any 

 number not exceeding 100,000, can be decomposed into 

 factor*, within the limits of the multiplication table ; and 

 if so, what the factors are. 



mrlie, p. M. 



aer)r 'Tiirr on the Tiolln. Many people 



woold bare iTrti good dl of b,-, f r.ir a few tun of In. imific. Sup- 

 IK^.. In Urn* of need, be had nehanftd 11 ban for i'.h., how man? Ib. 

 iihl ban oem nehanird. at thr ..in,,- rat-, for * barn ! Thl. in a rule- 

 af- HUM qnMUan, and Uwn an pkntjr o( book* Walkuifamc for m.Unce 



Multiply 23 14*. 7}<l by 7. Putting tlie multi] : 



under the farthings, and mult i 

 ,. t. them by the 7, ibe product is -'1 far- 

 23 14 7.' things ; aud dividing L'l by 4, the number 

 7 off.. D a penny, we & ,-t ."i and 1 



over ; so that in 21 farthing) there are 



1 ICO 2 6J down the 



1 uiuler /<u tliiiuji, and carry the 5 pence 



to the 4'.}, the pence product ; which gives 54 pence, or 



: we put down the Crl., and carry the 4 to tho 



shillin-s product, and thus get 102 shillings, or i'.. 



and putting down the L'.<., .ud*' 



-. The complete product is tlius li;ii _'.. lv/. 

 Suppose the multii'lic-r, had been 105, then seeing by 

 the table that 105 - 7 X 5 X 3 : after 

 the multiplication by 7, as above, we .. i. 



should have again multi]'! .ud 23 14 7; 



then by 3, as in the margin ; f r< >n i hie U V 



wo see that 105 times 23 14s. 7jU. is 



1M!>1 17s. '.;./. ICG L' ' , 



RULE II. When Wie Multiplier ex- 



ceeds 12, and is not divisible into factor*, 

 each leas than 13. 



Take that number in the table which 



is nearest to the proposed multiplier, 



r greater or less, and use the fac- 



t-oil 1.' 



17 9J 



tors of this number. To the final product n<l<l, if the 

 niimlxjr be lest, nnd sulitmct from it if the number be 

 greater, the product arising from multiplying the given 

 quantity by the difference between the multiplier and 

 number taken from the table; the result will obviously 

 be the complete product required. 



For example, if the multiplier of the sum above bad 

 been 107 instead of 105, we should still have taken 105, 

 and have used the factors of it, as just 

 shown; but to the product by these $. d. 

 factors, we should have added twice 23 14 7; X3 

 i iplii -and ; we should thus have 8 



got 10.") times the sum and twice the 



sum, that is 107 times the sum as pro- 189 17 2 

 posed. If the multiplier had been 7 



10'.), then from 112 times, that is, from 



8X7X2 times, we should have //>- llili'J 2 

 '' times the original multiplicand 2 



as in the margin, and should thus have 



found 'J3 lit 7' t d. X 109= 2580 Ilia. 2058 4 

 4-1-1. 71 3 11J. 



The following statements are left for 



the learner to verify after tho manner 2080 10 4J 

 now shown : 



(1.) 8 18s. Gd. x6=53 11s. Od. 



(2.) 148 7.0jdx9 1335 ::.-. 2|A 



(3.) 148 7*. (;/. X ''d. 



(4.) ik in >.-/. x!'7 =.r.;i o. 10 



(5.) 15 mil. 3 fur. 2 per. 4 yds. x 75= 1153 mil. 6 



fur. 4 per. 3 j 



Diriniim of Ooatpom* Quantitie*. Division of con- 

 crete quantities may be viewed under two asp' 

 accordingly as the divisor is itself a concrete quantity or 

 merely abstract nunil 



If you have to divide by a concrete quantity, your 

 object is to find hon- .1 the smaller quantity 



the divisor is contained in the larger the dividend. 

 Hut if you have to divide by an abstract number, you 

 then seek to divide tho proposed quantity int. 



as there are units in tho divisor. '!']..-. \o" 

 see, are two dill. -: and prei -isioii and accuracy 



of thought require that you should bear in mind tho 

 distinction. When you divide one < lantity by 



another, your quotient is, of course, an ' ,ilier: 



but when you divide a concrete quantity by an abstract 

 number, your quotient is also I 'y of the 



same kind. You will remember that we are not writing 



that would direct the following xfntiny : 



Ban of mutic. Ib. of beef. ban of music. Ib. of beef. 



II : 5 :: 3 : IrV 



And tn eHthi. 1 4-1 lib. of beef, thry would direct thebrcf and the muiio. 

 in the wonrt nnil third tennn, to be multiplied Uigrllicr ! The author of 

 this would thu* incorporate the beef and muiic. 



