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Case 2. So often as the First Figure of the Divisor, is taken from the Two First Figures of the Dividend, so often muft the Second Figure of the Divisor be taken from the Third Figure of the Dividend, when it is joined with what remained of the Second:

And so often must the Third Figure of the Divifor be taken from tbe Fourth Figure of the Dividend, &c.

That is, the Quotient Figure must be fuch, as being multiplied into the Divisor, will produce a Product equal to such a part of the Dividend as is then taken for that Operation : But if fuch a Product cannot be exactly found, then the next lefs must be taken, and ordered, as in the following Examples: of which let that in Page 16 be the first, wherein there was given 8569 the Multiplicand, and 8 the Multiplier. To find the Product 68552. Let us here suppose the said Product 68552, and 8 the Mulo tiplier, both given; thence to find the Multiplicand. That is, Let it be required to divide 68552 by 8.

Dividend Divisor 8) 68552 ( Quotient when found. According to the Rule, Cafe 1. I compare 8 the Divisor with 6 the First Figure of the Dividend, and finding I cannot take it from that; I then confider (by Cafe 2.) how often 8 can be taken from 68, the two first Figures of the Dividend, and find it may be taken 8 times; for 8 times 8 is 64, being the greatest Produt of 8 (into any Figure). that can be taken from 68. I therefore place 8 in the Quotient, and with it multiply 8 the Divisor, fetting down their Product underneath the said Two Firft Figures of the Dividend, subtracting it from them, and then the Work will ftand

Thus 8) 68552 (8

64

4

In order to a Second Operation, I make a Point under the next Figure of the Dividend, viz. -under the 5, and bring it down underneath in it's own place to the Remainder 4, which will by that means become 45. Then I consider how many times 8 can be taken from 45, and find it may be 5 times; for 5 times 8 is 40, I therefore place 5 in the Quotient, and with it multiply & the Divifor, ' letting down and subtre&ting theis Produå, as before. Then the Work will stand

Thus

Thus 8) 68552 (85

64.

45 40

5 For a Third Operation, I make a Point under the next Figure of the Dividend, viz. under the 5, and bring it down, as before, proceeding in all respects, as before ; and then the Work will stand

Thus 8) 68552 (856

64..

45 40

55
48

7 Lastly, I point and bring down the 2, viz. the last Figure of the Dividend to the Remainder 7, which will then become 72, and proceeding as in the other Operations, I find that 8 the Divisor can be taken just 9 times from 72, and the Work is finished, and will stand

Thus 8) 68552 (8569

64...

45
40

55 48

72 72

(0) The true Quotient is found to be 8569, being exactly the Eighth part of 68552, or the Multiplicand of the proposed Example of Multiplication. As was required.

The Reason of the Operations will be very plain to any one that will a little consider of it, as follows 1

Divisor Divifor 8) 68 5 5 2

(8000. The Firft Quotient Figure

Subtraa

This Product of the Divifor into the

Quotient is 64000, viz. 8 times 8000; the 640003.Quotient Figure being always of the same

Value or Degree with that Figure under

wbich the Unit's place of it's Product stands. 4552 (500. The Second Quotient Figure.

And here the Product is 4000, viz. 8

Divifor 8)
Subtract

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oo{times 500, not 8 times 5.

151512

Divifor 8)
Subtract

(60. The Third Quotient l'igure.

Allo here the Product is 480, viz. 8

80{times 60, for the Reasons abovelaid.

Divisor 8) (712 (9.

The Fourth Quotient Figures

Now here the Product is but 72, viz. Subtra Et

9 times 8, because the g stands in the place

of Units. Remains (00) Now the Sum of all the several Quotients, viz. 8000+500+60+9=8569, as before.

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If the Process of this Example be well considered and compaa. red with that of Multiplication, Page 17, it will evidently apa pear to be only the Converse of that; for the particular Produels are alike in both, only that which is last there, is for here; there they are added, here they are subtracted. So that whoever understands the true Reafon of the one, must needs understand the Reason of the other, and then Division will become very easy, although the Divifor consists of several places of Figures.

EXAMPLE
Let it be required to divide 59062.4922 by 7563.

Dividend. Divifor 7563) 590624922 ( 'Tis plain at the firft fight, that 7563 the Divisor, cannot be taken from 5906, tbe like Number of Figures in the Dividend.

'Therefore, by the Second Case of the Rule (Page 23.) there must be allowed Five Figures of the Dividend, viz. 59062 for the First Operation or Quotient; that fo the First Figure 7 of the Divifor may be taken out of the two First Figures, viz. 59 of the Dividend, at

Theo

Then I proceed (per Case 2.) and consider how often 7 may be taken from 59, and find it may be taken 8 times, for 8 times 7 is but 56, which I mentally subftract from 59, and there remains 3 ; to this 3 I mentally adjoin the Third Figure of the Dividend, viz. o, which makes it 30, out of which I must take the Second Figure of the Divisor, viz. 5, so often as I took the 7 from 59, which was 8 times : But that cannot be, for 8 times 5 is 40, which is more than 30, therefore 8 is too big a Figure to be placed in the Quotient; yet, hence I conclude, that the next lefs, viz. 7 may be taken without any further Trial. I therefore place 7 in the Quotient, and with it multiply the Divisor, setting down their Product under the Dividend, and fubtraét it from thence, as in the other Example, and then the Work will stand

Thus 7563) 590624922 (7

52941

6121 In order to a Second Operation, I make a Point under the next Figure of the Dividend, viz. under the 4, and bring it down to the Remainder 6121, which will then become 61214, with which I proceed in all respects as I did before with the 59062, and find the next Quotient Figure will be 8, with which I multiply the Divisor, &c. and subtract their Product from the faid 61214. Then the Work will stand

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To this Remainder 710, I point and bring down the next Figure of the Dividend, viz. 9, which makes it 7109; now because the Divifor 7563 cannot be taken from 7109, I therefore place a Cypher in the Quotient.

And this must always be carefully observed, viz. That for every Figure or Cypher, which is brought down from the Dividend, in order to a new Operation, there must always be either a Figure or Cypher, fet down in the Quotient. Then the Work will stand

Thus

Thus 7563) 590624922 (780

52941

61214
60504

7109 To this 7109, I bring down another Figure of the Dividend, viz. 2, and then it will become 71092; then I consider how often 7 can be taken from 71, &c. (just as at the first Operation) and find it may be taken 9 times, therefore I set down 9 in the Quotient, and with it multiply the Divisor, setting down and fubtracting their Product, as before; Then the Work will stand

Thus 7563) 590624922 (7809

52941

61214 60504

71092
68067

3025 To this Remainder 3025, I point and bring down the last Figure 2 of the Dividend, which makes it 30252; then proceeding in all respects as before, I find the Quotient Figure to be 4, with it I multiply the Divisor, setting down and subtracting their Product as before, and then the Work will stand

Thus 7563) 590624922 (78094

52941

61214
60504

71092
68067

30252
30252

(00000) Here the Work is ended, and I find the Quotient to be 78094, being the true Multiplicand of the proposed Example of Multia plication, Page 18. That is, 7563 is contained in 590624922 just 78094 times, &c. E 2

If

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