Thus the Multiplication of 4 by 5 makes 20, i. e. four times five amount to twenty. In Multiplication, the first Factor, i. e. the Number to be multiplied, or the Multiplicand, is placed over that whereby it is to be multiplied; and the Factum or Product under both." An Example or two will make the Procefs of Multiplication eafy. Suppofe I would know the Sum 269 multipled by 8, or 8 times 269. Multiplicand Factum, or Product 269 2152 The Factors being thus difpofed, and a Line drawn underneath, (as in the Example) I begin with the Multiplicator thus: 8 times 9 make 72, fet down 2, and carry 7 tens, as in Addition; then 8 times 6 make 48, and 7 I carried, 55; set down 5, and carry 5; laftly, 8 times 2 make 16, and with 5 I carried 21, which I put down: fo as coming to number the feveral Figures placed in order, 2, 1, 5, 2, I find the Product to be 2152. Now fuppofing the Factors to exprefs Things of different Species, viz, the Multiplicand Men, or Yards, and the Multiplier Pounds; the Product will be of the fame Species with the Multiplicator. Thus the Product of 269 Men or Yards multiplied by 8 Pounds or Pence, is 2152 Pounds or Pence; fo many of thefe going to the 269 at the Rate of 8 a-piece. Hence the vaft Ufe of Multiplication in Commerce, &c. If the Multiplicator confift of more than one Figure, the whole Multiplicand is to be added to itself, firft, as often as the Right-hand Figure of the Multiplicator fhews, then as often as the next Figure of the Multiplicator fhews, and so on.Thus 421 and 23 is equal to 421 and 3 and alfo 421 and 20. The Product arifing from each Figure of the Multiplicator, multiplied into the whole Multiplicand, is to be placed by itfelf in fuch a Manner, that the firft or Right-hand Figure thereof may stand under that Figure of the Multiplicator from which the faid Product arifes. For inftance; This Difpofition of the Right-hand Figure of each Product, follows from the first general Rule; the Right-hand Figure of each Product being always of the fame Denomination with that Figure of the Multiplicator from which it arifes. Thus in the Example, the Figure 2 in the Product 842, is of the Denomination of tens, as well as the Figure 2 in the Multiplicator. For I and 20 (that is the 2 of 23) is equal to 20, or 2 put in the place of tens, or fecond place. Hence if either of the Factors have one or more Cyphers on the Right-hand, the Multiplication may be formed without regarding the Cyphers, till the Product of the other Figures be found: To which they are to be then affixed on the right. And if the Multiplicator have Cyphers intermixed, they need not to be regarded at all.-Inftances of each follow. 24/00 8013 120 2148000 100 72.000 40113078 Thus much for an Idea of Multiplication, where the Multiplicator confifts wholly of Integers; in the Praxis whereof, it is fuppofed, the Learner is apprized of the Product of any of the nine Digits multiplied by one another, cafily learnt from the Table annexed. There are alfo fome Abbreviations of this Art.-Thus to multiply a Number by 5, you need only add a Cypher to it, and then halve it. To multiply by 15, do the fame, then add both together. The Sum is the Product. Where the Multiplicator is not compofed wholly of Integers; as it frequently happens in Bufinefs, where Pounds are accompanied with Shillings and Pence; Yards with Feet and Inches: the Method of Procedure, if you multiply by a fingle Digit, is the fame in fimple Numbers, only carrying from one Denomination to another, as the Nature of each Species requires. E. gr. to multiply 1237. 14s. 9d. 39. by five : Say 5 times 3 Farthings is 15 Farthings, that is, 3d. 39. write down the 39. and proceed, faying, 5 times 9 Pence is 45 Pence, and 3 Pence added from the Farthings is 48 Pence, which is 45. fet down a Cypher, as there are no Pence remaining, and proceed, faying, 5 times 4 s. is 20s. and 4s. is 24 s. fet down 4 s. and fay, 5 times 10s. is 50 s. and 10s. is 60s. which make 3 Pounds, to be carried to the Place of Pounds. Therefore continue thus; 5 times 3 is 15 and and 3 is 18; fet down 8 and carry I or one 10, faying, 5 times 2 is 10 and 1 is 11; fet down i and carry one, as before, faying, 5 times is 5 and I is 6. Thus it will appear that : 1237. 145. 9d. 39. multiplied by produces 5 618 4 0 3 If you multiply by two or more Digits, the Methods of Procedure are as follow. Suppose I have bought 37 Ells of Cloth at 131. 165. 6d. per Ell, and would know the Amount of the whole.-I first multiply 37 Ells by the 137. in the common Method of Multiplication by Integers, leaving the two Products without adding them up; then multiply the fame 37 Ells by 16s. leaving, in like manner, the two Products without adding them. Laftly, I multiply the fame 37 by the 6d. the Product whereof is 222 d. which divided by 12, (fee DIVISION) gives 18s. 6d. and this added to the Products of the 16s. the Sum will be 610s. 6d. the Amount of 37 Eils at 165. the Ell. Laftly, the 610s. Ed. are reduced into Pounds by dividing them by 20: upon adding the whole, the Amount of 37 Ells at 131. 16s. 6d. will be found as in the following. 37 Ells 37 Ells 37 Ells. At 13 Pounds. At 16 Shillings. At 6 Pence. Product 511 10 6 186 610 6 222 Or thus: Suppofe the fame Question: reduce the 137. 16s. into Shillings, the Amount will be 276s. reduce 276s into Pence, adding 6, the Amount will be 3318d. Multiply the 37 Ells by 3318, the Amount will be 122766 d. which divided by 12; and the Quotient 10230s. 6d. reduced into Pounds by cutting off the laft Figure on the right, and taking half of thofe on the left, yields 5117. 10s 6d. the Price of the 37 Ells, as before. Though by these two Methods any Multiplications of this Kind may be effected, yet the Operations being long, we fhall add a third much fhorter-Suppofe the fame Queftion: Multiply the Price by the Factors of the Multiplicator, if refolvable into Factors: if not, by thofe that come nearest it; adding the Price for the odd one, or multiplying it by what the Fac tors want of the Multiplier. So, the Work will ftand thus: 37 Ells at 16s. 6d. : 6 times 6 is 36 and 1 is 37: Therefore 6 82 19 O 497 14. 13 16 6 511 10 6 The Price of the 37 Ells. But the most commodious 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 those 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 6s. 8d. Suppofe I have 347 Ells of Ribbon at 6s. 8d. per Ell. Multiplicand 347 Ells. 6s. 8 d. 115. 135. 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. 4d. which place after the Pounds. Upon numbering the Figures I, I, and 5 Integers, and 13s. 4d. the aliquot Part remaining, I find the Sum 115%. 135. 4d. For MULTIPLICATION by aliquant Parts: Suppose I would multiply by the aliquant Part 19 s. I firft take for ios. half the Multiplicand; then for 5, which is the fourth, and laftly, 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 serve as a Model for Multiplication by any aliquant Part that may occur. Multiplicand Product 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. Thefe 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 duodecimal 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 used in meafuring, &c.-The Method is thus. 5 2 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 10 12 Parts, or one Inch; the whole Sum makes 12 Feet 3 Inches.-In the fame Manner you may manage Shillings and Pence, &c. I 12 F. I. 3468 I 3 { { 8 64 9 72 24 9 times 9 81 27 36 12 times< 25 8 96 9 108 16 20 24 28 32 30 35 8 40 45 36 42 48 10 120 |