DIVISION. ] 



MATHEMATICS. ARITHMETIC. 



437 



number of noughts, is 2, or 3, or 4 the same as the 

 figure preceding the noughts. It therefore follows, thai 

 whether we divide a number, such as 4326, which is oi 

 course 4000+300+20+G, by 9, or simply divide 

 4+3+2+6, that is 15 the sum of the figures by 9, 

 we must, in each case, get the same remainder. This 

 property, taken in connection with the principle referred 

 to above, and to be proved in Algebra (see the multipli- 

 cation of compound quantities in Algebra), suggests the 

 following rule : 



RULE. Add together the figures of the multiplicand, 

 not counting any 9 that may occur, rejecting also 9, 

 whenever, in adding up, the sum amounts to 9 or more : 

 when all the figures are added, the result will therefore 

 be less than 9 : note this result. Proceed in like man- 

 ner with the figures of the multiplier ; noting the result. 

 Multiply the two results together ; retaining, as before, 

 only what is left after the rejection of all the nines the 

 new result contains. Do the same thing with the figures 

 of the product ; and compare this third result with that 

 just found : if the two be the same, the work may be 

 p.fsuined to be correct; if they differ, it is certainly 

 wrong. 



The usual way of noting the four results is to make 

 a cross, to put the first in the left hand opening; 

 second in the opposite opening; the third above, 

 and the fourth below. If the upper and lower results 

 are the same, the work is most likely correct, but other- 

 wise it is wrong. 



Let us proceed in this way to test the ac- 

 curacy of the work at page 430. Commencing 

 at the right of the multiplicand, we say 7 and 

 4 are 11, therefore rejecting 9, 2 and 6 are 8 and 

 2 are 10 : the lirst result, therefore, rejecting 9 

 from this 10, is ], which we place in the opening of the 

 cross to the left. Taking now the multiplier, we say C and 

 5, 11 ; 2 and 3, 6, the second result, which we place 

 opposite the former. The product of the two is 5, with 

 no 9 to reject : this is the third result, to be placed above. 

 Lastly, taking the product, we say 2 and 3 are 5, and 3 

 are 8, and 2 are 10 : 1 and 4 are 5 ; which is the fourth 

 result, and, as it agrees with the preceding, we conclude 

 the work to be correct. 



It is plain, however, that if any of the figures in the 

 product were made to exchange places, the agreement of 

 the third and fourth results would remain, though the 

 product would bo wrong ; a* would also be the case if 

 one figure of it were increased and another diminished, 

 by the same number : all, therefore, that we can safely 

 infer, is, that the agreement spoken of must have place 

 if the work be correct ; so that if it fail the work is 

 wrong. Suppose, for instance, that we had 

 made 73084163x7684=504270392192: then, 

 applying the test, we get, from the first factor, 

 the result 5 ; from the second, the result 6 ; 

 and from the product of these, the result 3 : 

 but, from the above-stated product of the two numbers, 

 the result is 4 : this product, therefore, is incorrect ; 

 and, upon revising the multiplication, we find that the 

 3, after the nought, should have been a 2. 



Simple Division. The operation by which we find how 

 many times one number or quantity is contained in 

 another number or quantity of the same kind, is called 

 division. It is also the operation by which we find the 

 4th part, the 5th part, etc., of a number or quantity. 

 The number or quantity divided is called the di* 

 that by which we divide it, the divisor ; and the result 

 obtained, the qttntii-nt.. 



Ton must not fall into the common mistake of con- 

 sidering the quotient to express always how maici /:,...., 

 the dividend contains the divisor : the 4th part of a 

 mere number tells us how many times that number con- 

 tains 4 ; but the fourth part of a quantity a sum of 

 money, for instance is just the fourth part, and nothing 

 else : it is itxelf also a sum of money. The division is 

 called xim/ili when the quantities concerned are of but 

 one denomination. On coining to the division of com- 

 pound quantities, you will find some further remarks on 

 the true nature of division in general ; at present both 



dividend and divisor, and therefore the quotient, are to 

 be regarded as mere number*. They are, in fact, only 

 abstract expressions for the idea of quantity. 



I. When the Divisor is not greater than 12. 



RULE. Place the divisor to the left of the divi- 

 dend, with a mark of separation, thus), between the 

 two. 



Draw a line beneath the dividend, and, by the multi- 

 plication table, find how many times the divisor is con- 

 tained in the first figure of the dividend, or in the number 

 expressed by the first two figures, or even in the number 

 expressed by the first three figures, should the number 

 given by the first, and even by the first two, be smaller 

 than the divisor ; and write the quotient under the line, 

 taking care to observe what is oner, as the divisor may 

 be contained a certain number of times in the number 

 expressed by the leading figure or figures, and leave 

 something over. 



Proceed to the next figure of the dividend ; regard 

 what was over, if anything, to be prefixed to it; and 

 find how many times the divisor is contained in the 

 number you thus get ; putting the quotient down, and, 

 as before, carrying what is over to the next figure of the 

 dividend, to which you must regard it as prefixed. And 

 in this way figure after figure of the complete quotient 

 is to be found, till all the figures of the dividend have 

 been used. Should there be anything over at the end, 

 this is called the remainder: it is to be written be.-idt 

 the quotient figures, with the divisor placed under it, 

 and a line of separation between them. 



Suppose, for example, we have to divide 25602 by 3, 

 then placing dividend and divisor (3) as in 

 the margin, we proceed thus : 3 is contained 3)25602 



in 2, no times ; so that nothing is to be placed 



under the 2 : 3 is contained in 25, 8 tin 8531 



and 1 over; 8 and carry 1 : this 1, regarded 



as prefixed to the 6, gives the number 16 : 

 we therefore say : 3 in 16, 5 times and 1 over : 3 in 10, 

 3 times and 1 over : 3 in 12, 4 times. Therefore, the 

 quotient is 8534 ; and this is the complete quotient, as 

 there is no remainder. 



Again, suppose it were proposed to divide 7804623 by 

 5, we should say 5 in 7, 1 ; and 2 over : 5 

 in 28, 5 ; and 3 over : 6 in 30, 6 : 5 in 4, : 5)7804623 

 5 in 46, 9 ; and 1 over : 5 in 12, 2 ; and 2 

 over : 5 in 23, 4 ; and 3 over. As there ia 1560924;; 

 here a remainder 3, we annex it, with the 

 divisor 5 under it, to the figures of the quotient, and 

 call 15C0924?, the complete quvt nt, 



The principle upon which the foregoing operation 

 depends is pretty evident : the leading figure in the 

 dividend above is 7000000: the fifth part of this is 

 1000000 and 2000000 over ; that is, with the local value 

 of the next figure 8, 2800000 ; the fifth part of this is 

 500000, and 300000 besides ; the fifth part of which is 

 60000 : the fifth part of the 4000 the local value of the 

 next figure is thousands, and 4000 over ; this, with 

 the local- value of the 6, is 4600; of which the fifth part 

 is 900, and 100 over; this, with the 20, is 120; the fifth 

 part of which is 20, and 20 over ; and lastly, the fifth 

 part of the remaining 23 is 4, and 3 over ; and, to imply 

 that this 3 still remains to be divided, it is put down 

 with the 5 underneath ; because one number, placed in 

 this way under another, is a form frequently used to 

 denote that the upper number is to be divided by the 

 'awer. Hence the fifth part of the proposed number is 

 15(X}924, and the fifth part of 3 besides : this quotient 

 i>eing made up of the several parts which arise from 

 ;aking a fifth of each of the above-mentioned component 

 portions of the number. 



The sign for division is -=-, which stands for the words 

 divided by: thus, 6 -$- 2=3 is a short way of stating that 

 } divided by 2 is equal to 3. As noticed above, there is 

 another way of indicating division, namely, by putting 

 ;he dividend above and the divisor below, a short line 

 separating the two : thus, | =3 expresses the same thing 

 as the notation above. The learner may exercise himself 

 in the rule just explained by proving by it the truth of 



