I A TICS. ARITHMETIC. 





I. ICfon thr Multiplier i$ tut grtattr than 12. 

 Rri.r Put the multiplier under the multiplicand, 

 uniU under unite ; and, by aid of the table, multiply 

 each figure of the multiplicand, commencing at the 

 unit*' figure, by the multiplier. Sot down the right- 

 hand figure only of the product hen it in a number of 

 more than one figure, and carry a* in addition. 

 For example : multiply in :'.'_' l.\- 4. The multiplier 4 

 ^ placed under the mnltipli- 

 cand 0432, as in the margin, we 

 proceed thus : 4 times 2 are 8 : 

 4 times 3 are 12 ; 2 and carry 1 : 

 4 times 4 are 16, and 1 are 17 ; 

 7 .ind . .nn 1 : 4 times 6 are 24, 

 and 1 an 



A beginner, with the table before him, can easily per- 

 form operations of this kind ; but he must learn to work 

 tin-in without looking at the table. It is as 

 wull to show him the time and trouble saved, 

 by actually exhibiting the work of such exam- 

 ples by addition, as here annexed. The f<>l- 

 lowiii},', worked like the example above, require 

 no fuithcr explanation : 



0073214 07387 264135 

 8 9 10 12 



Multiplicand 6432 



Mlllt: 4 



Product . . 25728 





1000 when 



673870 3160620 



The multiplication by 10, as in tin- third of these exam- 

 ples, requires, in fact, no actual work, or reference to the 

 table. You know that a nuinler becomes ten times as 

 great by simply putting a after the figures ; this c 

 each figure to advance a place to the lift, so that its local 

 value is increased tenfold. In like manner, a number be- 

 comes multiplied by 100 when two O's are added to it ; by 

 three are added, and so on, as is evident from 

 i. The ci/'li-r, though in itself of no value, 

 thus plays an important part in our notation ; by filling 

 up what would otherwise be gaps between figures, it 

 keeps them in their proper places, and preserves their 

 local values; and by being put after a number, it has 

 the effect of multiplying that number by 10, 100, <tc., 

 according as it is written once, twice, Arc. 



The .-i'ii' for multiplication is X placed between the 

 / , thus : ( 1 . ) 340 x 7 = 2422. (-'. ) 0(47 X o = :tt)235. 

 :$X6=1470:U8. (4.) 53274X800=42019200. 

 In working this fourth example, the plan 

 is to consider 8 only as the multiplier, and 53274 

 to put tlie ciphers to the right of it, as in 800 



the margin, annexing them afterwards to -- 

 tli.' product by 8. (5.) 470329X11 = 42G19-'Otl 

 :r)b42xl-'=4510104. 



II. When the Multiplier it greater than 12. 



RULE. Place the multiplier under the multiplicand, 



under units, tens under tens, A-c. 

 Commencing with the units' figure, multiply by each 

 in succcsxion. ami arrange the several rows of results, so 

 that the first figure on the right in each row may be 



IT the multiplying tigiirc that pro- 

 duced it. Add up all these products, and the 

 urn will be tin product. 



example, if we ha vr to multiply 426 by 34, 



we place-tin' i'. I n ndi'i tie 'JO. and proceed thus : 



.1* 6 are 24 ; 4 and carry 2 : 4 times 2 are 



8, and 2 are 10 ; and carry 1 : 4 times 4 are 



16, and 1 are 17. The first row is now . 14484, 



I. and we begin anew, with tin- IP 

 It, an multiplier, taking care to put the tint figure we 



ii the new row directly und.-r this 3. 3 

 time* 6 are 18 ; 8 and carry 1 : 3 times 2 are 1704 

 0, and 1 are 7 : 3 times 4 are 12. The rows 12780 

 are now completed, so that, drawing a line - 

 and adding up, we find the product to be 14484 

 lit-l 



Hee, from the local position of our 

 multiplier, 3, that it is in reality 30, and 420x30 

 127HO: adding this product to the former product, that 

 given by the 4, M in the margin, the whole product by 34 



must necessarily be the result ; and you see that it agrees 

 with that in the example. 



If our multiplier had been a number of three figures, 

 as 634, then, to the partial products obtained we must 

 have added the product duo to the 5 ; which, 

 having regard to its local value, is 500 ; anil 426 



if we retain the noughts, the wl, 'ion 534 



would be as here annexed. And it is plain 

 that we may always omit the noughts, pro- 1704 



vided wo take care, as the rule directs, to put 1 .'7*0 



rst figure of each partial product directly 213000 

 under the multiplying figure, which supplies 

 that product It is worthy of notice, too, 227484 

 that the product will always be the ;> 

 whichever of the two numbers ' the inulti 



plier: you may easily satisfy yourself that l-ii multiplied 

 by 634, is the same as 534 multiplied by -I 'JO. To bo con- 

 vinced that this principle is perfectly general, you have 

 only to assure yourself of the fact within the limits of 

 the multiplication table, which you may do by replacing 

 multiplication by addition, as shown in a previous ex- 

 amide; that is, proving to yourself that 3 times 7 is the 

 same as 7 times 3 ; that f> times 8 is the same as 8 times 

 5, and so on, as the table declares : 

 because, whatever be the two fac- 

 tors, the multiplication of one by '.'< < LV, |~ 

 the other is made up only of mul- 

 tiplications within the limits of 1 ^-' 

 the table. It is in general most ]:;:':;."> II .4 

 convenient to take that for the 7941 21)10 



multiplier which gives the fewer 71- 



partial products, or rows of figures. 942332 



(Seethe operations in the margin.) !(lL':;:;j 



The learner may now exere 



himself in the process, by showing that the following 

 statements are true : 



(1. 4214x24 = 101136. 



(2. 658X243 = 15'.X!H. 



(3. 8364x3300-7607200, 



(4. 16607x3094-48288068. 



(.").) Show that 24:;xoio = !txOxllx8x7x3. 



(0.) Show that 9048x198664x121x33x16. 



When the multiplier consists of two figures, forming a 

 number greater than 12, there are two partial prodi 

 or rows of figures, to add up; but, with a little ad 

 the product may be written down at once, whenever the 

 multiplier does not exceed 20. Supi 

 fur instance, it were 16, then, if wo mid- 2378 



tiply by the 6, and, as we go on, add in 10 



not only what we carry from any figure of 

 the multiplicand, but also the immediately 

 preceding figure of the multiplicand, the This is the 

 complete product will be obtained in same as 



line, as in the margin ; the operation being 



1 on thus: times 8 are 48, 8 and 16 



carry 4 : 6 times 7 are 42 and 4 a 



8 are 54; 4 and curv ."> : times 3 are 

 18 and 6 are 23 and 7 are 30 ; and carry 2378 



3 : C times 2 are 12 and 3 are 15 and 3 are 



18 ; 8 and carry 1 : 1 and 2 are 3. It will :NO IS 



lie advisable for the learner to practise 

 this short way with the multipliers, 13, 14, 15, 10, 17, 

 18, 19. Multiplication whicli is thus performed in one 

 line, is called Am-t multiplication : when there are more 

 lines, it is luinj multiplication. 



Method of proving multiplication '>;/ to /inei. 



\Ve shall here mention a useful method of trying 

 whether the product of two numbers is correct ; but 

 must the explanation of the /' t" the 



method till yon arrive at \Ve can only men- 



tion here, that if any number be divided by 9, the re- 

 mainder will be the same as would arise from dividing 

 the rum of the figures in that number by 9 : for In is 

 ec|ii:d to once 9-f-l ; 100 is equal to 11 times 9+1 ; KKK) 

 to 111 times 9+1; and so on: that is, the remainder 

 arising from dividing 1, followed by any number of 

 \tt, by !(. is always 1. Consequently the remainder 

 arising from dividing 2, or 3, or 4, ifcc. , followed by any 



