Thus Dividend. Divifor 6) 24 (4 the Quotient. Here I confider how many times 6 there is in 24, and find it 4, viz. 4 times 6 is 24, therefore 4 is the true Quotient or Answer required. This is apparent by Subtraction, as in the Margin; where 24 the Dividend is fet down, and from it 6 the Divifor continually fubtracted fo often as it can be, which is juft 4 times. Therefore 4 is the true Quotient or Anfwer required. 24 6 18 6 3 6 6 COROLLARY From hence it is evident; that Divifion is but a concife or compendious Method of fubtracting one Number from another, fo often as it can be found therein; for if the Divifor be continually fubtracted from the Dividend, accounting an Unit (or 1) for each time it is fubtracted (as above) the Sum of those Units will be the Quotient. All Operations in Divifion do begin contrary to those of Multiplication, viz. at the First Figure to the Left-hand, or that of the highest Value, and decrease the Dividend by a repeated Subtraction of each Product arifing from the Divifor when multiplied into the Quotient Figure. And the only Difficulty in Divifion of whole Numbers (or indeed of any Numbers) lies in making choice of fuch a Quotient Figure, as is neither too big, nor too little; and that may be eafily obtained by obferving the following Rule, which hath two Cafes. RULE. Cafe 1. As often as the First Figure of the Divifor is taken from the First Figure of the Dividend: So often must the Second Figure of the Divifor be taken from the Second Figure of the Dividend, when it is joined with what Remains of the First. And as often must the Third Figure of the Divifor be taken from the Third Figure of the Dividend, &c. But if the First Figure of the Divifar cannot be taken from the Firft Figure of the Dividend. Then; Cafe Cafe 2. So often as the First Figure of the Divifor, is taken from the Two Firft Figures of the Dividend, fo often must the Second Figure of the Divifor be taken from the Third Figure of the Dividend, when it is joined with what remained of the Second: And fo often must the Third Figure of the Divifor be taken from the Fourth Figure of the Dividend, &c. That is, the Quotient Figure must be fuch, as being multiplied into the Divifor, will produce a Product equal to fuch 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 muft be taken, and ordered, as in the following Examples: of which let that in Page 16 be the firft, wherein there was given 8569 the Multiplicand, and 8 the Multiplier. To find the Product 68552. Let us here fuppofe the faid Product 68552, and 8 the Multiplier, both given; thence to find the Multiplicand. That is, Let it be required to divide 68552 by 8. Dividend Divifor 8) 68552 (Quotient when found. According to the Rule, Cafe 1. I compare 8 the Divifor 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 firft Figures of the Dividend, and find it may be taken 8 times; for 8 times 8 is 64, being the greatest Product of 8 (into any Figure) that can be taken from 68. I therefore place 8 in the Quotient, and with it multiply 8 the Divifor, fetting down their Product underneath the faid Two Firft Figures of the Dividend, fubtracting it from them, and then the Work will stand Thus 8) 68552 (8 4 In order to a Second Operations 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 confider 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 8 the Divifor, fetting down and fubtracting their Product, as before. Then the Work will stand Thus Thus 8) 68552 (85 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 refpects, as before; and then the Work will ftand Thus 8) 68552 (856 64.. 45 40 55 48 7 Laftly, I point and bring down the 2, viz. the laft 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 Divifor can be taken juft 9 times from 72, and the Work is finished, and will ftand (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 Reafon of the Operations will be very plain to any one that will a little confider of it, as follows I Divifor Divifor 8) 68552 Subtract (8000. The First Quotient Figure This Product of the Divifor into the Quotient is 64000, viz. 8 times 8000; the 64000 Quotient Figure being always of the fame Value or Degree with that Figure under which the Unit's place of it's Product ftands. Divifor 8) 4552 Subtract 4000 Divifor 8) Subtract Divifor 8) Subtract Remains 15/5/2 (500. The Second Quotient Figure. And here the Product is 4000, viz. 8 times 500, not 8 times 5. (60. The Third Quotient Figure. 480 times 60, for the Reasons abovefaid. (9. The Fourth Quotient Figure. Now here the Product is but 72, viz. 9 times 8, because the 9 ftands in the place of Units. (oo) Now the Sum of all the feveral Quotients, viz. 8000+500+60+9=8569, as before. If the Process of this Example be well confidered and compared with that of Multiplication, Page 17, it will evidently ap pear to be only the Converfe of that; for the particular Products are alike in both, only that which is laft there, is first here; there they are added, here they are fubtracted. So that whoever understands the true Reafon of the one, muft needs understand the Reafon of the other, and then Divifion will become very easy, although the Divifor confifts of feveral places of Figures. EXAMPLE. Let it be required to divide 590624922 by 7563. Dividend. Divifor 7563) 590624922 ( 'Tis plain at the firft fight, that 7563 the Divifor, cannot be taken from 5906, the like Number of Figures in the Dividend. Therefore, by the Second Cafe 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, . Then I proceed (per Cafe 2.) and confider 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 fubftract 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 Divifor, 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 Divifor, fetting down their Product under the Dividend, and fubtract 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 refpects as I did before with the 59062, and find the next Quotient Figure will be 8, with which I multiply the Divifor, &c. and fubtract their Product from the faid 61214. Then the Work will ftand Thus 7563) 590624922 (78 52941. 61214 60504 710 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 obferved, 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 ftand Thus |