Page:Cyclopaedia, Chambers - Volume 2.djvu/246

 MUL

(fl>7)

MUL

Thus the Multiplication of 4 by 5 makes 20, /. e. four times five amount to twenty 5 which Algebraills exprefs thus, 4X 5=20. See Char actee.

In Multiplication, the firft Factor, i. e. the Number to be multiplied, or "Multiplicand, \s placed over that whereby it is to be multiplied 5 (Tee Multiplicand) and the Factum or ProduB t under both. An Example or two will make

the FroceJs of Multiplication eafy. Suppofe t would

know the Sum of 169 multiplied by S, or 8 times atfp.

Operation,

Multiplicand

Multiplier

Fatla m, or Product

269

8

2152

Explication.

neath, fas in the Example') I begin with the Multiplicator thus 3 8 times p make 72, fet down 2, and carry 7 tens, as in Addition 5 then 8 times 6 make 48, and 7 I carried, 5 5 ? fet down 5, and carry 5 5 lattly, S times 2 make i6> and with 5 1 carried 21, which I put down : fo as coming to number the feveral Figures placed in order, 2, r, 5,2, I find the Product to be two thoufand one hundred fifty two. See Numeration.
 * The Factors being difpofed, and a Line drawn under-

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 Spe- cies with the Multiplicator.

Thus the Product of 269 Men or Yards multiplied by 8 Pounds or Pence, is 2157 Pom.ds or Peace ; fo many of thefe going to the 269 at the Rate of 8 apiece. Hence the ■vaft Ufe of Multiplication in Commerce, g/e.

If the Multiplicator confilts of more than one Figure, the whole Multiplicand is to be added to itfelf, firft, as often as the right-hand Figure of the Multiplicator mews, then as often as the next Figure of the Multiplicator (hews, and fGon. Thus 421 X23 is equal t0 42iX 9 andalf6 421 X 20. The Product arifing from each Figure of the Multiplicator, multiplied into the whok Multiplicand, is to be placed by itfelf nvfucha manner, that the firit or right-hand Figure thereof may ftan' 4 under that Figure of the Multiplicator from which the faid Product antes. For Inftance 5

Where the Multiplicator is not compofed wholly of Inte- gers 5 as it frequently happens in Bufioefs, where Pounds are accompanied with Shillings and Pence 5 Yards with Feet and Inches ; the Methods of Proeeedure are as fol- low ;

Eirji Method, Suppofe I have bought 27 Ells of Cloth at j;/. 16 s. 6 d. per Ell, and would know the Amount of the whole, — - I fir it multiply 37 Ells by 13 /. in the common Method of 'Multiplication by integers, leaving the two Pro- ducts without adding 'em up; then multiply the fame 37 Eils by 16 s. leaving, in like manner, the two Products without adding 'em. Laftly, I multiply the fame 37 by the 6 d. the Product whereof is 221 d. which divided by 12, (fee Division) gives 16 s. 6 d. and this added to the Products of the 16 s. the Sum will be 610 s. 6 d, the A- mountof 37 Ells at i5 s. 6 d. the Ell. Laftly, the 610 u 6 d. are reduced into Pounds by dividing 'em by 20 : (fee Reduction) upon adding the whole, the Amount of 37 Ells at 1 3/. itS s. 6d. will be found as in the following

37 Ells 37 Ells 37 Ells

At [5 Pounds. At 16 Shillings. At 6 Pence*

1 1 r

57 50 10 6

222

37 18 6

Multiplicand

Multiplicator —

Particular Product of 421 X 3 Particular Product of 42 1 X 20

The Total Product

421

Product 511 10 6 610 6

Second Method. Suppofe the fame Queftinn ; reduce the 13 I. 16 s, into Shillings, the Amount will be 27tf.f> re- duce 276 j. into Pence, adding 6, the Amount will be 3318^. Multiply the 37 Ells by 3318, the Amountwill be 122760V* which divided by 123 and the Quotient 10230 s. 6 d. re- duced into Pounds by cutting oft the lait Figure on the right, and taking half of thofe on the left, yields 511 I. 10 s. 6 d. the Price of the 37 Ells, as before.

Tho by thefe two Methods any Mult plications of this kind may be effected, yet the Operations being long, we fhall add a third much fhorter, by Aliquot and Aliqtttwt Parts: Obferving by the way, that Aliquot Parts of any thing are thofe contained feveral times therein, and which, divide 'em without any remainder; and that Aliquant Parts are other Parts of the fame thing compofed of feveral Ali- quot Parts : Both as in the following Table.

_ __ Aliquot Parts of a Pound of 20 t.

126-3 842

5685

This Difpofition of the right-hand Figure of each Pro- duct, follows from the firlt general Rule ; the right- hand Figure of each Product being always of the fame De- nomination with that Figure of the Multiplicator from which itarifes. ,, , _ c

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 1 X 20 (that is the 2 of 23) — 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 performed without regarding the Cyphers, till the Produft of the other Fi- gures be found : To which they are to be then affix'd on the right. And if the Multiplicator have Cyphers inter- mixed, they need not to be regarded at all. Inftances of each follow.

Aliquant Parts of a Pound of 20 u

an Aliquant Part compos' d of a 10th and a soth. , of a 5th and a loth.

of a 4th and a 10th. , of two jths.

of a 4th and a 5th.

of a half and a £0ch.

of a half and a 10th.

of a half, a 10th and 20th. , of a half and a 5 th. . of a half and a 4th. , of a half, a 5th, and loth.

of a half, a 4th, and rath. . of a half and two 5 tlis. 19/. of a half, a 4th, and 5th>

To Multiply by Aliquot Farts is nothing elfe in effect but to divide a Number by 3, 4, 5, 2>c. which is effected by taking a 3d, 4th, or 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 attff. 8 d. per Ell.

oj. make half of 20 j.

V-

5 j. a fourth.

4 j. a fifth.

6 s.

a j, a tenth.

1 f-

1 j. a twentieth.

Us.

6 s. 8 d. a third.

9 /.

3 s. 4 d. a (ixth.

11 s.

2 j. 6 d. an Eighth.

12 s.

1 j. fid. a twelfth.

13, S,

1 x. 4tf. a fifteenth.

14 J.

I j. 3 d. a fixteenth*

M J-

10 d. a twenty-fourth.

16 J.

5<*. a forty-eighth.

17 X.

Xtt A

3 5 s |

1 |o ilo

100

J/jlOO

3I0

8013

5oo<f

Multiplicand Multiplicator

Procluft

Operation.

- 317 Ells. 6 s.

— —

214.800O

71000

48078 40065

8 it.

115/. 1 3 J.

4 i.

40115070

Thus much for an Idea of Multiplication, where the Mul- tiplicator confifls wholly of Integers ; in the Praxis whereof 'tis fuppofed the Learner is apprized of the ProduB of any of the nine Digits multiplied by one another, eafily learnt from the common Table, (fee Table) or otherwife.

Note, This Multiplication is render'd ftill eafier, and more expeditious by the ufe of certain Rods, whereon are mark'd the feveral Progreffions of Digits in the Table, and which give the feveral Multiples of any Sum by infpection, call d Nef air's Dimes ; the Vefiriftimt and Ufe whereof fee under tie bead N r. r A i b 's B ma.

The Queflion being Stated ; take the Multiplicator, which according to the Table of Aliquot Parts is the third j and fay, the third of 5 is I, fet down 1 ; thethird of 41s I, fet down 1, remains i, that is, one ten, which added 107, makes 175 then the third of 1 7 is 5 5 remains 2 Units, i. e. two thirds, or 13 s. $d. which place after the Pounds. Upon numbering the Figures I, I, and 5, Integers, and j 3 s. 4 d. the Aliquot Part remaining, I rind the Sum 115 /.1 3 1. 4 d.

For hhdtiflicalior. ly Aliquant Tarts: Suppofe I would

multiply by the Aliquant Part 19 i. 1 firft take for 13;. half

the Multiplicand ; then for 5, which is the 4th ; and,

7 N laftly,