tors want of the Multiplier. So, the Work will stand thus : 37 Ells at 165. 6 d. 6 times 6 is 36 and 1 is 37: Therefore 6 37 But the most commodicus is the fourth Method, which is performed by aliquot and aliquant Parts-where you are to obferve by the way, that aliquot Parts of any thing are thofe contained feveral times therein, and which divide without any Remainder; and that aliquant Parts are other Parts of the fame thing compofed of feveral aliquot Parts. To MULTIPLY by aliquot Parts, is in Effect only to divide a Number by 3, 4, 5, &c. which is done by taking a 3d, 4th, 5th, &c. from the Number to be multiplied. Example. To multiply, v. g. by 6 s. 8 d. Suppofe I have 347 Ells of Ribbon at 6 s. 8 d. per Ell. Multiplicand 347 Ells. 6 s. 8 d. 115. 13s. 4d. The Queftion being ftated, take the Multiplicator, which according to the Table of aliquot Parts is the third; and fay, the third of three is 1, fet down 1; the third of 4 is 1, fet down 1, remains 1, that is, ten, which added to 7, makes 17; then the third of 17 is 5; remain 2 Units, i. e. two thirds, or 135 4 d. which place after the Pounds. Upon numbering the Figures 1, 1, and 5 Integers, and 13 s. 4 d. the aliquot Part remaining, I find the Sum 115l. 13 s. 4 d. For MULTIPLICATION by aliquant Parts: Suppose I would multiply by the aliquant Part 19s. I first take for 10 s. half the Multiplicand; then for 5, which is the fourth, and lastly, for 4, which is the 5th. The Products of the three aliquot Parts that compofe the aliquant Part, being added together, the Sum will be the total Product of the Multiplication, as in the following Example; which may ferve as a Model for Multiplication by any aliquant Part that may occur. Multiplicand Produc 356 Ells. For the Proof of MULTIPLICATION.-The Operation is right when the Product divided by the Multiplier quotes the Multiplicand; or divided by the Multiplicand quotes the Multiplier.-A readier Way, though not abfolutely to be depended on (fee ADDITION) is thus: Add up the Figures of the Factors, cafting out the nines; and fetting down the Remainders of each. These multiplied together, out of the Factum, caft away the nines, and fet down the Remainder. If this Remainder agree with the Remainder of the Factum of the Sum, after the nines are caft out; the Work is right, Crofs MULTIPLICATION, or otherwife called du decimal Arithmetic, in an expeditious Method of multiplying Things of feveral Species, or Denominations, by others likewife of different Species, &c. E. gr. Shillings and Pence by Shillings and Pence; Feet and Inches by Feet and Inches; much ufed in meafuring, &c.-The Method is thus. Suppofe 5 Feet 3 Inches to be multiplied by 2 Feet 4 Inches; fay, 2 times 5 Feet is 10 Feet, and 2 times 3 in 6 Inches: Again, 4 times 5 is 20 Inches, or 1 Foot 8 Inches; and 4 times 3 is 12 Parts, or one Inch; the whole Sum makes 12 Feet 3 Inches. In the fame Manner you may manage Shillings and Pence, &c. F. I. 3 2 4 10 6 8 8 56 24 { 8 64 9 72 78 88 12 3 4 times 456789 8 96 10 120 11 132 12 144 DIVISION S the laft of the four great Rules, being that whereby we find how often a lefs Quantity is contained in a greater; and the Overplus. Divifion, in Reality, is only a compendious Method of Subftraction; its Effect being to take a lefs Number from another greater, as often as poffible; that is, as oft as it is contained therein. There are, therefore, three Numbers concerned in Divifion: 1. That given to be divided, called the Dividend. 2. That whereby the Dividend is to be divided, called the Divifor. 3. That which expreffes how often the Divifor is contained in the Dividend; or the Number refulting from the Divifion of the Dividend by the Divifor, called the Quotient. There are diverse ways of performing Divifion, one called the English, another the Flemish, another the Italian, another the Spanish, another the German, and another the Indian way, all equally juft, as finding the Quotient with the fame Certainty, and only differing in the manner of arranging, and difpofing the Numbers. The Italian way is moft generally used. Divifion is performed by feeking how often the Divifor is contained in the Dividend; and when the latter confifts of a greater Number of Figures than the former, the Dividend must be taken into parts, beginning from the left, and proceeding to the right, and feeking how often the Divifor is found in each of those Parts. For Example, it is required to divide 6759 by 3: I first feek how oft 3 is contained in 6, viz. twice; then how oft in 7, which is likewife twice, with one remaining. This 1, therefore, is joined to the next Figure 5, which makes 15, and I feek how oft 3 in 15; and laftly, how oft 3 in 9. All the Numbers expreffing how oft 3 is contained in each of thofe Parts, I write down according to the Order of the Parts of the Dividend, that is, from left to right, and feparate them from the Dividend itself, by a Line, thus: Divifor. Dividend. Quotient. 3) 6759 (2253 It appears, therefore, that 3 is contained 2253 times in 6759; or that 6759 being divided into 3, each Part will be 2253. If there be any Remainder, that is, if the Divifor repeated a certain Number of times is not equal to the Dividend, what remains is wrote over the Divifor Fraction-wife. Thus, if inftead of 6759 the Dividend were only 6758, the Quotient will be the fame as in the former Cafe, except for the the laft Figure 8; for 3 being only contained twice in 8, the Number in the Quotient will be 2; and as twice three is only 6, there remains 2 of the Dividend; which I write after the Quotient, with the Divifor underneath it, and a Line to feparate the two; thus, ABBER. VIATIONS. ift. If there are any Cyphers on the Right-hand of your Divifor, you may cut off fo many Cyphers, or Figures, on the Right-hand of your Dividend; but remember to bring them down (if Figures) to the Remainder. EXAMPLE. 24 21 35 1429 2dly. By the foregoing Rule, you may obferve, that to divide by 10, 100, 1000, &c. is only to cut fo many Figures from the Right-hand of the Dividend, as there are Cyphers in the Divifor. EXAMPLE. So the Quotient is 43682, the Remainder 735. 3dly, 3dly. When your Divifor is 12, or confifts only of one fingle Figure, or can be reduced to one, by cutting off Cyphers, from its Right-hand, the Work may be easily performed in one Line, thus: RULE. Drawing a Line under the Dividend, fet down under its first Figure, how often the Divifor is contained in it; what remains imagine placed before the next Figure; and, confidering how often your Divifor is contained in the Sum it makes, fet down the Number underneath, as before; and fo proceeding through all the Figures, fet down what remains at last, in the Place where your Quotient used to stand. 4)93645(I 23411 If you are to divide several Numbers by one common Divifor (as in the calculating of Tables, &c.) that you may know exactly at once how often your Divifor will go, in fome convenient Corner make a Table of your Divifor, by multiplying it severally by all the nine Digits: Thus, fuppofe 562 your Divifor : Divifion is proved by multiplying the Quotient by the Divifor, or the Divifor by the Quotient; and adding what remains of the Divifion, if there be any thing. If the Sum be found equal to the Dividend, the Operation is juft, otherwise there is a Mistake. PART |