particular Product, muft ftand directly underneath the Third Figure of the First Product: And fo on until all is done. Now the Reafon of placing the firft Figure of every particular Product in their Order, will be very obvious to any one that confiders the last Example; wherein the Cyphers are only fet down to fhew the true Diftance of the fift Figure in each particular Product from the Units place. And altho' it is not ufual to fet down Cyphers in this manner; yet they are always fuppofed to be there: That is, their Places are always left void, as in the two following Examples; wherein I have placed Points inftead of Cyphers. EXAMPLE 3. Let it be required to multiply 78094, into or with 7563. 78094} Factors. 234282 The First particular Product with 468564. The Second particular Product with 390470.. The Third particular Product with 546658... The Fourth particular Product with 590624922 The Total, or true Product required. EXAMPLE 4. Suppose it be required to multiply 57498 into 60008. 3 60 500 7000 57498 60008 459984 The Product with 344988... The Product with 8 60000 3450339984-57498x60008, as was required. Here you may obferve, that I pafs over the Cyphers, and only take care of placing the firft Product of the laft Figure, viz. of 60coo acccording to the foregoing Directions. When there is a Cypher or Cyphers, to the Right-hand either of the Multiplicand or Multiplicator, or to both in that cafe multiply the Figures as before; neglecting the Cyphers until the particular Products are added together; Then to their Sum annex fo many Cyphers as are in either or both the Factors. As in thefe Examples: EXAMPLE Take a few Examples without their Work at large. 75649×579=43800771 687000x356=244572000 530674×45007=23884044718 7901375×30000=237041250000 537084000x590700 317255518800000 102030405x504030201=51426405540261405 Note, If it be required to multiply any Number with 10, 100, 1000, 10000, &c. it is only annexing the Cyphers of the Multiplier to the Figures of the Multiplicand, and the Work is done. Thus, $578x10 =5780. 578x1000 578000 Thefe Examples (being well understood) are fufficient to inftruct the Learner all the Varieties that can happen in multiplying of whole Numbers, according to the Method generally practifed: However it may not be amifs to fhew here how Multiplication may be performed (with many Figures) by Addition only. EXAMPLE. Let it be required to multiply 879654 into 79863. In order to perform this (or any other Operation of this kind) by Addition only; you must make a Tariffa or small Table of the given Multiplicand, in this manner: First, Make a fmall Column, and in it place gradually downward the Nine fingle Figures; viz. 1, 2, 3, 4, 5, &c. D 2 Then Then against the Figure 1, fet down the Multiplicand (which in this Example is 879654) and against the Figure 2, fet down the double of the Multiplicand, found by adding it to itfelf; To this double add the Multiplicand, fetting down their Sum against the Figure 3. And the I 879654 2 1759308 3 2638962 43518616 5 4398270 65277924 76157578 87037232 fo proceed on by a continual Addition, until there be Ten times the Multiplicand in the Table; which if the Work is true, will be the Multiplicand itself with a Cypher to Right-hand of it, as in the annexed Table. This being done, it will be eafy to conceive, that the Figures in the fmall Column of the Table, do refpectively reprefent those of the Multiplier: And that the Numbers against any of thofe Figures in the fmall Column, will be the true Product of the Multiplicand agreeing to any Figure of the Multiplier; as plainly appears by the Work of this Example. Then 9 7916886 10 8796540 $79864} The Factors as before. Note, This Method of tabulating the Multiplicand, is both eafy and certain; being neither fubject to Errors, nor burdenfome to the Memory, and therefore in large Calculations it may be found very useful. But for common Practice the ufual Method (as in Page 18, &c.) is beft, and to be preferred before this. Moft Mafters that teach (and feveral Authors that write of) Arithmetick, do teach to prove the Truth of Multiplication, by cafting away all the Nines that are contained in both the Factors, and their Product; but becaufe that Method is very erroneous, as might be easily fhewed; I fhall therefore omit inferting it, and leave the Proof of Multiplication to the next Section, wherein prefume) the Reafon and Proof, both of it, and Division, will plainly appear. Seat. Sect. 5. Of Division. Dibition is a Rule by which one Number may be speedily fubtracted from another, fo many times as it is contained therein. That is, It fpeedily discovers how often one Number is contained (or may be found)'in another: And to perform that there are required Two Numbers to be given. 1. The one of them is that Number which is propofed to be divided, and is called the Dividend. 2. The other is that Number by which the faid Dividend is to be divided, and is called the Divifor. And by comparing thefe Two, viz. the Dividend and the Divifor together, there will arife a Third Number, called the Quotient; which fhews how often the Divifor is contained in the Dividend, or into what Number of Equal Parts the Dividend is then divided. Therefore, Divifion is by Euclid fitly termed the measuring of one Number by another, viz. one Number is faid to measure another by that Number, which when it multiplies, or is multiplied by it, it produceth. Euclid 7. Def. 23. And, if a Number measuring another, multiply that Number by which it meafureth, or be multiplied by it, it produceth the Number which it measureth. Euclid 7. Axiom 9. That is to fay, If that Number which divides another (called the Divifor) be multiplied with the Number which is produced by Divifion (called the Quotient) their Product will be the Number divided or Dividend. Whence it follows, that Division and Multiplication are the Converse and Direct Contrary one to another (as Subtraction is to Addition) and do mutually prove the Truth of each other's Operations. I fhall therefore make choice of the foregoing Examples in Multiplication, in order (as I prefume) to render the Bufinefs of Divifion more plain and easy. Firft, let it be required to find how often 6 is contained in 24. That is, to divide 24 by 6. N. B. Always place down the given Numbers in this Order; First fet down the Divifor, and to the Right-hand of it draw a crooked Line; then fet down the Dividend, and to the Right of it draw another crooked Line, in which must be placed the Quotient Figure, or Figures, as they become found. Thus 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 just 4 times. Therefore 4 is the true Quotient or Answer required. 3 24 18 6 12 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 thofe of Multiplication, viz. at the First Figure to the Left-hand, or that of the higheft 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 muft the Second Figure of the Divifor be taken from the Second Figure of the Dividend, when it is joined with what Remains of the Firft. And as often muft the Third Figure of the Divifor be taken from the Third Figure of the Dividend, &c. But if the First Figure of the Divifor cannot be taken from the Firft Figure of the Dividend. Then; Cafe |