Arithmetic, Part 1 · Section 15 – Section 16
Diviſion.
DIviſion is the juſt contrary to Multiplication, for that turnes ſmall denominations to greater, as Multiplication turns greater to ſmaller. Or (in whole Numbers, of which only we yet ſpeak) Diviſion is the asking how many times one Sum is contained in another? and the number which anſwereth to that queſtion is called the Quotient.
And the number containing, which is to be divided, is called the Dividend.
And the number contained, or by which the Dividend is to be divided, is called the Diviſor.
And as often as the Dividend contains the Diviſor, ſo often doth the quotient containe Unity.
The wayes of performing Diviſion are divers I will begin with that which is moſt uſed and taught, which is as followeth.
THE RƲLE.
Place the Diviſor under the Dividend, ſo that the figures next to the left hand ſtand directly one under the other, if the reſt of the Diviſor be not greater: or if all the Diviſor be greater then that above it, then the ſaid Diviſor muſt be devolved one place further toward the right hand; having ſo placed them, try how many times the lower figures are contained in the upper figures, and write that figure which anſwereth that queſtion within a crooked line in the margine of the work, which is called the Quotient, and by it multiply the firſt figure of the Diviſor, and take the product out of the figures directly over it, beginning the Subtraction toward the left hand; then cancel that figure of the Diviſor, and alſo, that of the Dividend which hath been already uſed, with a light daſh of a pen, and write the remaine (when the product of the firſt figure multiplied by the quotient is ſubtracted as before) juſt over the figure uſed and cancelled; the… proceed to doe the like with the ſecond, third, and fourth figure of the Diviſor if there be ſo many; till having cancelled it all, and ſet the remaine orderly above the Dividend, you have finiſhed one work.
Now if the Dividend have ſtill ſome figures untouched towards the right hand, then remove the Diviſor ſtill toward the right hand, but one place at a time, and then againe aske or try how many times the lower may be had in the upper, and write the anſwer in the quotient whether it be 1 or more, (onelyonly it cannot be above 9) or nothing, then put 0 in the quotient, and multiply the Diviſor by this new figure, and ſubtract the product, ſetting the remainer orderly above, as before, this work muſt be repeated by removing the Diviſor ſtill one place towards the right hand, until Ʋnity in the Diviſor ſtand under Unity in the Dividend, and then the work is done.
Example. 1.
Let it be required to divide 4096 by 3 Place them thus …
Aske how many times 3 in 4 the quotient is 1, which is put within a crooked line by it ſelf.
Then in your mind multiply the Diviſor 3 by the quotient 1. And having ſaid theſe words Once 3 is 3, … preſently cancel the 3. And having added theſe words, Out of 4, cancel the 4. And after theſe words, and there remains 1, write 1 juſt over the 4, as you ſee here done.
Then remove the diviſor one place towards the right hand, … ſaying, how many times 3 in 10, the anſwer is 3, which write in the quotient; and in your mind multiplie the Diviſor 3 by the quotient 3, the product is o, wherefore ſay three times three is nine out of ten, and there remains one, then (having cancelled the 10 and the 3, write over them 1.
Again, remove the Diviſor 3 one place more, … asking how many times 3 in 19? the anſwer being 6, write 6 in the quotient, and ſay 6 times 3 is 18 out of 19, and there remains 1, wherefore having cancelled the 3, and the 19, write 1 over 3, and remove the Diviſor once more, and aske how many times 3 in 16? anſwer is 5, which write in the quotient, … then in mind ſay 5 times 3 is 15 out of 16, and there remains 1, and cancelling the 16 and the 3, write over 3, 1: Now becauſe the Diviſor 3 is advanced ſo far till it is come to ſtand under 6 in the Dividend, which 6 is the place of Unity there, the ſaid Diviſor cannot be removed any more, and therefore the diviſion is ended, and the quotient being 1365, ſhewes that the Diviſor 3 is contained in the Dividend 4096, 1365 times; and 1 remaining, which 1 being leſſe then the Diviſor 3, doth not contain it once, but one third part of once, which after the Reader hath skill in broken numbers muſt be joined to the quotient thus 1365⅓. The beſt proof of this Diviſion is by multiplying the quotient into the Diviſor, and to the product adde the remaine; then if the work be well done, the ſum ſhall be equal to the Dividend.
So 1365 multiplied by 3 produceth 4095, to which adding the remain 1, the ſum 4096 is equal to the Dividend.
Here we divided only by one figure 3, becauſe the firſt Example being eaſie and clear, ſhould be a faire Introduction to the ſecond,
Note that if the Diviſor had been greater then 4, as 5, … the work muſt have begun thus So the quotient would have been 819½, which is one place leſſe.
Example 2.
Let it be required to divide 1310720 by 4096, Place them thus …
Now the queſtion to be asked is, how many times 4096 is there in 13107? to find an anſwer to this queſtion, the Reader which hath but an indifferent faculty of judging; may do it, (for the moſt part) by conſidering the firſt figure in the diviſor, as here 4, for preſently he knowes that 4 times 4 is 16, which cannot be had in 13, therefore the firſt figure in the quotient muſt be leſſe then 4.
Again, it cannot be much leſſe, becauſe the ſecond place in the Diviſor is 0.
He may therefore venture on 3, And having put 3 in the quotient, ſay 3 times 4 is 12, … which ſubtract out of 13 over it thus, ſay 2 out of 3, and there remains 1, and 1 out of 1 and there remaines 0, cancelling the 0 and the 13, and ſet the remaine 1 over the 4, as you ſee.
Then go on ſaying 3 times 0 is 0, out of 11, and remains ſtill 11, againe 3 times 9 is 27, take 7 out of 10 there remains 3, to be ſet over the 0, which is over 9, and 2 with 1 borrowed (to make the 0 10, from the which the 7 was taken) is 3, and 3 out of 11, remains 8, which write over the place of 0 in the Diviſor, cancelling the 9 in it, and alſo thoſe figures of the Dividend 110, out of which you have taken any thing; then laſtly, ſay 3 times 6 is 18, take 8 out of 7 I cannot, therefore (borrowing 10,) ſay 8 out of 17 remains 9, which write over the 7, then in the next place, the 10 borrowed is 1, and the 10 in the 18 is 1. Say therefore 1 and 1 is 2, out of 3, and the remain is 1, which write over the 3, having ſtill cancelled the figures which you have uſed orderly as you go.
Now the whole Diviſor being cancelled, it muſt be removed on one place further, and placed as here.
Now aske again how many times 4096 in 8192? … it will be found (becauſe of the 0 after the 4) as often as 4 in 8, that is two times, put therefore in the quotient 2, and work as before, ſaying 2 times 4 is 8, out of 8 remains 0, then 2 times 0 is 0 out of 1 remains 1, then 2 times 9 is 18, take 8 out of 9 remains 1, (which put over the 9,) and 1 out of 1 in the next place remains 0, then laſtly, 2 times 6 is 12, that is, 2 out of 2 there remaines 0, and 1 out of 1, there alſo remaines 0, cancel and put the remainer over as formerly.
Now againe, the Diviſor being all cancelled, ſhould be removed; and aske how many times 4096 in 0000, the anſwer is 0, which being put in the quotient, the work is all done.
And the quotient 320, ſhewes that the Diviſor 4096 is contained in the Dividend 1310720, three hundred and twenty times.
And whether the Diviſor have 2, 3, 5, 7, or more places, the working is ſtill like this, not differing from it at all.
Except for brevity in ſome caſes, where the Diviſor toward the right hand hath one Cypher or more; for thoſe Cyphers may be placed orderly under the Dividend at firſt, and remain there till the work be done, with the reſt of the Diviſor, which muſt needs ſhorten the Diviſion.
As if 2587645 were to be divided by 15000, place them thus …
And ſay how many times 15 in 25? … anſwer is 1, which write in the quotient, and multiply in minde the diviſor by the quotient, ſaying once one is one, out of 2 there remains 1, then cancel the 2, and the 1 under it, and write the remaining 1 over it, as here is done, then ſay once 5 is 5 out of 5, and there remains 0, therefore cancel the two 5, and over the uppermoſt write 0.
Now remove the Diviſor one place, and aske how many times 15 in 108? anſwer is 7, which write in the quotient, and multiply, ſaying 7 times 1 is 7, out of 10 remains 3, which write over the 0, then ſay 7 times 5 is 35, take 5 out of 8 remains 3, which write over 8, and 3 out of 3 remains 0.
Now againe, remove the Diviſor, and aske how many times 15 in 37? anſwer is 2, which write in the quotient, and ſay two times 15 is 30, out of 37 remaines 7, and the Diviſion is ended, the remaine being 7645.
Proof of this.
| Multiply the quotient | 172 | ||
| by the Diviſor | 15 | 860 | 172 |
| The Product is | 2580 |
| Before which put the three Cyphers, And then it is | 2580000 |
| To which adde the remaine | 7645 |
| The Total is | 2587645 |
Which is equal to the Dividend, and therefore the work is right.
So if one would divide any ſum by 10, 100, 1000, 10000, &c. he need but cut off the firſt, two firſt, three firſt, or four firſt figures towards the right hand, the other figures ſhall be the quotient, and thoſe cut off, the remaine.
As if 1587645 be divided by 1000, the quotient is 1587, and the remaine 645.
If one would know how many pounds Sterling are in 95670 ſhillings; having placed them thus, that is, 2 under 9, … and 0 under 0, then divide by 2, which is very eaſie. So the quotient 4783, ſhewes that in the number of ſhillings given, is contained 4783 pounds, and 10 ſhillings remaining.
The reaſon why the Diviſor was 2, is becauſe there are 20 ſhillings in one pound, and therefore any number of ſhillings is turned into pounds, by dividing by 20, that is by 2, putting the cypher under unity, onelyonly to fill a place at laſt. And this way of turning any number of ſhillings into pounds may be eaſily effected by memory, if you ſuppoſe the laſt figure of your given number to be cut off with a line, or comma, and taking the half of the other figures. Thus, let the given number of ſhillings be 5739; Imagine the laſt figure 9 to be ſeperated from the reſt by a line thus 573 9, now by memory take the half of 573, by ſaying in your mind the half of 5 is 2, (and 1 remaining, which makes the 7 following 17,) the half of 17 is 8, (and 1 remaing) the half of 13 is 6, (and 1 remaing, which 1 is 10 s. to be added to the 9, and the whole is 286 li. 19 s.
In this way of Diviſion, as in all others, if the remaine at laſt be greater then the Diviſor, the quotient is not juſt, but too little, which may be remedied (without beginning the work againe) by dividing the remainer onelyonly by the ſame diviſor; for thereof will ariſe a new quotient, which added to the former quotient, the ſumme will be the juſt quotient.
So if 7290 be divided by 27, the numbers being placed thus,
Becauſe 27 can be had in 72 but twice, … put 2 in the quotient, ſaying 2 times 27 is 54, out of 72, and there remains 18 which write over 72, cancelling as before is ſhewed; then removing the Diviſor, ſay how many times 27 in 189? anſwer is 7, but if one ſhould miſtake, and write 5 in the quotient, and ſay 5 times 27 is 135, out of 189, remains 54 and write it over as before; and remove the Diviſor, and ſay, how many times 27 in 540? anſwer is 20, but ſhould not be above 9, ſay therefore, 9 times 27 is 243, out of 540, and the remain is 297, which being the laſt remain, and greater then the Diviſor, ſhewes the quotient 259 is too little.
Wherefore divide the laſt remain 297, … By the Diviſor 27 ſaying, twenty ſeven in 29 once, and write 1 in the new quotient; and ſay once 27 is 27, out of 29, remains 2 which write over 9, and remove the Diviſor, and ſay, 27 in 27, juſtly once, ſo write 1 in the quotient, ſo the quotient is 11, which added to the former quotient 259, gives 270, which is the true and whole quotient.
A Second way of Diviſion.
Although (as I have ſaid) the former way is more uſed, yet this may ſeem plainer and more natural to ſome, I will therefore give one example of it.
Example.
Let it be required to divide 6477734 by 334, … where the firſt figure in the quotient is eaſily ſeen 1, ſubtract once 334 out of 647, and write the remain under a line, then ſee how many times 334 in 3137, anſwer 9, by which multiply the diviſor 334 in another paper, the product is 3006, which ſubtract out of the remaine, then the new remaine is 131734.
Again, how many times 334 in 1317, anſwer is 4, for the third figure in the quotient; by which multiply the Diviſor, the product is 1002, which take out of the later remaine, as in the margine, then will remain 31534.
Now try how many times 334 in 3153, anſwer is 9, for the fourth figure in the quotient, by which multiply the diviſor (in a by paper) the product is 3006, (as might have been ſeen above in the ſecond ſubtraction) which ſubtracted out of the later remaine, there remains now 1474.
Laſtly, aske how many times 334 in 1474, anſwer is 4, for the laſt figure in the quotint, by which multiplying the diviſor 334, the product is 1336, which ſubtracted from 1474, there finally remaines 138, which being leſſe then the diviſor, ſhewes the diviſion is done.
Proof of this.
The ſeveral ſubtractions and the final remaine added together. …
The Total equal to the dividend 6477734
If the former (as I have ſaid more uſual) way ſeem difficult to beginners, becauſe the products of the diviſor into the ſeveral figures in the the quotient are not ſet down but mentally made; and alſo becauſe the ſubtraction of them begins towards the left hand: and laſtly, becauſe the remaine is ſtill ſet above: yet this later way which agrees altogether with plain Subtraction before taught, I hope is ſo plain, that any diligent reader may acquire it, without a Tutor. And yet for the better ſatisfaction and help of the young Learner, I will adde another way or two more of Diviſion.