8. the Proof of Multiplication by Divifion, and of Divifion by Multiplication. As for the Method of dividing larger of Divifi- Numbers by Logarithms, it fhall be exon, by Lo- plain'd in the laft Chapter of this Treagarithms. tife; together with other Particulars relating to Logarithms. The fitch particular Rule rela ting to Divifion. 9. There are fome other compendious Ways of working Division. Namely, when the Divifor confifts of the Figure I, with one or more Cyphers to the Right-hand of it, then the Divifion is perform'd by cutting off fo many Figures on the Right-hand of the Dividend, as there are Cyphers in the Divifor, and taking the Reft for the Quotient, the Figures cut off, if fignificant Ones, being to be plac'd by the faid Quotient, with the Divifor written under them in Manner of a Fraction, as has been afore fhewn. For Inftance. 10. particular Rule. 10) 100 (10. 10) 122 (12. 10) 12/4 (125 If the Divifor confifts of any other fig. The fixth nificant Figure than 1, and of Cyphers to the Right-hand, then cutting off fo many right-hand Figures of the Dividend, as there be Cyphers in the Divifor, the Divifion may be perform'd only by the fignificant Figure (or Figures) of the Divifor placing (as afore) the Fi 1 gures gures cut off from the Dividend, if fignificant Ones, by the Quotient found. Thus, 6000) 21481000 (358. And 2400) 72000 (30. Where it is to be noted, that the Examples of Divifion, contain'd in this and the fore-going Section of this Chapter, being compard with the Examples of Multiplication contain'd in §. 6, of the fore-going Chapter, will further fhew, how Multiplication and Divifion mutually prove one the other. Numbers Hitherto I have spoken of the Divifion of Numbers of one external Denominati- of dividing on: As for the Divifion of Numbers of of several feveral external Denominations, if the external Divifor be a fingle Figure, it may easily ins be perform'd, obferving the general Rules. Thus, S. d. 1. S. d. 7) 1717. 7 • 10. (245 .6 . 10 Denomina 12. Shillings, and the 7 Shillings added thereto, make a new Dividend to be divided by the fame Divisor given, 7, viz. 7) 47 (6 42 5 And this remaining 5 Shillings, being turn'd into Pence, and added to the 10 Pence given in the Dividend, makes another Dividend to be divided by 7, the given Divifor, viz. Whence it appears, that 1717 l. 7 s. 10d. being divided among feven Perfons, each Man must have for his Quotient or Share 2451. 6s. 10d. And this Example compar'd with the Example of Multiplication, Chap. 5. S. 12. further fhews, how Multiplication and Divifion are mutually of Divifi- prov'd one by the other. on by Reduction. If the Divifor confifts of two or more Figures, then the Divifion may best be perform'd by Reduction, as fhall be shewn, Chap. 10. S. 2.—§. 6. It remains only to fhew, how Divifion 13. of Numbers of one external Denominati- of proving Divifion on may be prov'd (well enough for com- by cafting mon Ufe) by cafting away 9: Namely, away 9. if it be confider'd, that the Dividend in Divifion always anfwers to the Product in Multiplication, and the Divifor and Quotient here, to the two Factors there. Hence, if, 9 being caft away as often as may be, out of the Divifor and Quotient diftinctly; and after that, out of the Product of the two Remainders first found; the third Remainder last found, be the fame with the Remainder of the Dividend, after that 9 is caft away from it likewife, as often as may be, then the Divifion is rightly perform'd. Thus, In like manner, 5006) 40113078 (8013. And thus much for the Divifion of In tegers or whole Numbers. I. In Decimal the Order from the CHAP. VII Of the Addition, Substraction N Integers, the Order of the Places (as has been obferv'd, Chap. 2. §. 24) is reckon'd from the Right-hand to the of Places Left; but in decimal Fractions it is recis reckon'd kon'd quite contrary, namely, from the Left-hand Left-hand to the Right. Thus, fuppoto the fing 20 to be an Integer, the Cypher ftands in the first Place, and 2 in the second; but fuppofing 20 to be a Decimal, the 2 ftands in the firft Place, and the Cypher in the fecond. Right. Decimal why fo called. In Integers, a Figure fignifies ten Times as much in any Place, as it does in the Fractions Place next before it in Order, (according to the Table, Chap. 2. §. 23.) but in Decimals, a Figure fignifies ten Times as little in any Place, as it does in the Place next before it in Order. Thus, fuppofing 220 to be an Integer, the Righthand 2 fignifies 2 Tens, the Left-hand 2 fignifies 2 Hundreds, or Tens of Tens but fuppofing 220 to be a Decimal, the Left-hand 2 denotes 2 tenth Parts, the Right |