If it be required, from To fubftra&t 9800403459 4743865263 The Remainder will be found 5056538196 For, beginning with the Right-hand Figure, and taking 3 from 9, there remains 6 Units, to be wrote underneath the Line: going then to the next Place, 6 I find, cannot be taken from 5; wherefore, from the Place of hundreds 4, I borrow 1, which is equivalent to 10, in the Place of tens; and from the Sum of this 10 and 5, viz. 15, fubftracting 6, I find 9 tens remaining, to be put down under the Line. Proceeding to the Place of hundreds, 2 with the 1 borrowed at the laft, make 3, which fubftracted from 4, leave 1. Again, 5 in the Place of thousands, cannot be fubftracted from 3; for which Reason, taking 1 from 4, in the Place of hundreds of thousands, into the empty Place of tens of thoufands, the Cypher is converted into 10 tens of thousands, whence one 10 being borrowed, and added to the 3, and from the Sum 13 thoufand, 5 thousand being fubftracted, we shall have 8 thousand to enter under the Line: Then fubftracting 6 tens of thousands from 9, there remain 3. Coming now to take 8 from 4; from the 8 further on the Left, I borrow 1, by means whereof, the two Cyphers will be turned each into 9. And after the like manner is the rest of the Subtraction eafily performed. If heterogenous Numbers be to be fubftracted from each other; the Units borrowed are not to be equal to ten; but to so many as there go of Units of the lefs kind, to conftitute an Unit of the greater: For example; 1. 5. d. 27 19 9 17 16 9 For fince 9 Pence cannot be fubftracted from 6 Pence ; of the 16 Shillings, one is converted into 12 Pence; by which means, for 6 we have 18 Pence; whence 9 being fubftracted, there remain 9. In like manner, as 19 Shillings cannot be fubftracted from the remaining 15; one of the 45 Pounds is converted into 20 Shillings, from which, added to the 15, 19 being fubftracted, the Remainder is 16 Shillings. Laftly, 27 Pounds fubtracted from 44 Pounds, there remains 17. If a greater Number be required to be fubftracted from a lefs, it is evident the thing is impoffible.-The lefs Number, therefore, in that Cafe, is to be fubftracted from the VOL. I. K greater; greater; and the Defect to be noted by the negative Character. E. gr. If I am required to pay 8 Pounds, and am only Mafter of 3; when the 3 are paid, there will still remain 5 behind; which are to be noted,-5. Subtraction is proved, by adding the Remainder to the Subtrahend, or Number to be fubftracted: for if the Sum be equal to the Number whence the other is to be fubtracted, the Subtraction is juftly performed.-For example; I MULTIPLICATION S the Act, or Art of multiplying one Number by another, to find the Product. Multiplication, which is the third Rule in Arithmetic, confifts in finding fome third Number, out of two others given; wherein, one of the given Numbers is contained as often as Unity is contained in the other. Or, Multiplication is the finding what will be the Sum of any Number added to itself, or repeated, as often as there are Units in another.-So Multiplication of Numbers is a compendious Kind of Addition. Thus Thus the Multiplication of 4 by 5 makes 20, i. e. four times five amount to twenty. In Multiplication, the firft 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 ul iplication eafy. Suppose I would know the Sum 269 multiplied by 8, or 8 times 269. Multiplicand Factum, or Product 269 8 2152 The Factors being thus difpofed, and a Line drawn underneath, (as in the Example) I begin with the Multiplicator thus: 8 times 9 makes 72, fet down 2, and carry 7 tens, as in Addition; then 8 times 6 make 48, and 7 I carried, 55; fet 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 fo on.Thus 421 and 23 is equal to 421 and 3 and also 421 and 20. The Product arifing from each Figure of the Multiplicator, multiplied into the whole Multiplicand, is to be placed by itself 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 arises. 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. 110 24000 8013 5006 48078 40065 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, eafily 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 Bufinets, where Pounds are accompanied with Shillings and Pence; Yards with Feet and Inches: the Method of Procedure, if you multiply by a single Digit, is the fame as in fimple Numbers, only carrying from one Denomination to another, as the Nature of each Species requires. E. gr. to multiply 123. 14 s. 9 d. 39. by five: Say 5 times 3 Farthings is 15 Farthings, that is, 3 d. 39. write down the 3 q. 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 4 s. is 245. fet down 4 s. and fay, 5 times 10 s. is 50 s. and 10 s. 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 1 or one 10, saying, 5 times 2 is 10 and I is 11; fet down I and carry one, as before, faying, 5 times I is 5 and 1 is 6. Thus it will appear that: 1234, 145. 9 d. 39. multiplied by produces 618 4 5 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 137. 16 s. 6d, per Ell, and would know the Amount of the whole.-I first multiply 37 Ells by the 13. 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 18 s. 6 d. and this added to the Products of the 16s, the Sum will be 610 s. 6d. the Amount of 37 Ells at 16 s. the Ell. Laftly, the 610 s. 6 d. are reduced into Pounds by dividing them by 20: upon adding the whole, the Amount of 37 Ells at 131. 16 s. 6d. will be found as in the following. 37 Ells 37 Ells 37 Ells. At 16 Shillings. At 6 Pence, At 13 Pounds. III 222 222 Product 511 10 6 610 6 Or thus: Suppose the fame Question; reduce the 137. 165, into Shillings, the Amount will be 276 s. reduce 276 s. into Pence, adding 6, the Amount will be 3318 d. Multiply the 37 Ells by 3318, the Amount will be 122766 d. which divided by 12; and the Quotient 10230 s. 6 d. reduced into Pounds by cutting off the laft Figure on the right, and taking half of thofe on the left, yields 511. 10 s. 6d. the Price of the 37 Ells, as before. Though by thefe two Methods any Multiplications of this Kind may be effected, yet the Operations being long, we shall add a third much fhorter -Suppose the fame Queftion: Multiply the Price by the Factors of the Multiplicator, if refolvable into Factors: if not, by thofe that come neareft it; adding the Price for the odd one, or multiplying it by what the Fac K 3 tors |