Ch. VII. I. In Decimal CHAP. VII. Of the Addition, Substraction, Decimal Fractions. N Integers, the Order of the Places (as has been obferv'd, Chap. 2. §. 24.) the Order is reckon'd from the Right hand to the of Places Left; but in Decimal Fractions it is recis reckon'd from the kon'd quite contrary, namely, from the Left-hand Left-hand to the Right, Thus, fuppo to the Right. 2. why fo called. fing 20 to be an Integer, the Cypher ftands in the firft Place, and 2 in the fecond; but fuppofing 20 to be a Decimal, the 2 ftands in the first Place, and the Cypher in the fecond. In Integers, a Figure fignifies ten Times Decimal as much in any Place, as it does in the Fractions, Place next before it in Order, (according to the Table, Chap.. 2. S. 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-hand 2 denotes 2 hundredth Parts, or 2 Tenths of a Tenth. And hence it ང་བ is, that Fractions of this Sort are call'd Ch. VII. decimal Fractions, namely, because the denominative Value, belonging to any Place in this Sort of Fractions, is a decimal or tenth Part of the denominative Value, belonging to the Place next before it in Order. Namely, an hundredth Part, is the tenth Part of a tenth Part; and a thousandth Part, is the tenth Part of an hundredth Part, &c. 3. Use of Cy The denominative Value of decimal Fractions, (as well as of Integers,) being The Ufe denoted by their respective Places, hence phers in it is requifite, that thofe Places in deci- Decimals, the fame mal (as well as whole) Numbers, which as in whole are void of any fignificant Figure, fhould Numbers. nevertheless be filled up with Cyphers, that thereby may be known the true Places of the fignificant Figures. Thus, if I would exprefs two hundredth Parts without any Tenths, I write it thus, 02 where the Cypher fills up the first decimal Place, and thereby fhews, that the 2 ftands in the fecond decimal Place, i. e. the Place of Hundredths. Thus, again, 002 denotes two thousandth Parts, &c. 4. the Diffe But now, although the Ufe of Cyphers in Decimals is the fame (confider'd in it Excepting felf) as in Integers, namely, to fill vacant Places, and fo to fhew the Places of the fignificant Figures; E 4 up rence aritrue fing from the diffe yet rent Ways there of reckon Ch. VII. there is this Difference arifing from the ing the Or- two different Ways of reckoning the Order of der of Places in Decimals, and in InteDecimals gers. Namely, a Cypher set on the and inte- Right-hand of an integral Figure, pro Places in gers. motes or increases (as has been obferv'd, Chap. 1. S. 30.) its denominative Value, because it promotes the faid Figure into an higher Place; for Inftance, 2 Two, 20 Twenty, 200 two Hundred, &c. But a Cypher fet on the Right-hand of a decimal Figure, does not promote or alter its denominative Value, because it does not alter the Place of the faid Figure; for Inftance, 2 (fuppofing it a Decimal, when alone) denotes two Tenths, and the decimal 20 denotes likewife the fame, as also the decimal 200 denotes the fame; because the faid 2 in each of the three Instances stands in the fame, viz. the first decimal Place. Whereas, on the contrary, a Cypher fet on the left-hand of a decimal Figure, does alter and leffen its denominative Value, because it alters the Place of the faid Figure, by thrufting it, or fhewing it ftands in a Place of a lefs denominative Value; for Instance, the Decimal 2 denotes two Tenths, o2 two Hundredth, 002 (*) two Thousandth Parts, (*) It is to be observ'd, that tho' in Integers the denominative Value Hundred is greater than that of Ten, and that of Parts, c. But a Cypher on the Left- Ch. VIL hand of an integral Figure, does not alter its denominative Value, because it does not alter the Place of the faid Figure: for Inftance, the Integer 2, or 02, or 002, does each denote no more than two Units. 5. Numbers from Inte gers, &c It is next to be observ'd, that (whereas the Places of Integers, are reckon'd Decimal from the Place of Units inclufively) the how diPlaces of decimal Numbers, are reckon'd ftinguith'd from the Place of Units exclufively. Hence Tens is the Denomination of the fecond Place in Integers, but Tenths of the first Place in Decimals, &c. And it is ufual and neceffary to denote, whether the Number given be an Integer or a Decimal, by putting fome Mark before it, viz. either a full Point, or a Comma, or this Mark (L). And by the fame Marks the decimal Figures are distinguish'd from the integral in Numbers given, and confifting partly of Integral, and partly of decimal Figures. Thus .2 or 2 or 12 denote two Tenths; L03 three Hundredths; L004 of a Thousand, greater than that of an Hundred, &c. Yet in Decimals the denominative Value Hundredth is leffer then that of Tenth, and that of Thousandth, leffer than that of Hundredth, &c. Forafmuch as the Hundredth Part of a Thing is Ten times less than the Tenth Part of its and a Thousandth Part of it is Ten times less than a Hundredth. The like of which is obferv'd alfo in the next Section of this Chapter, on another Occafion Ch. VII. 6. tween 1004 four Thousandths; 2143, or 2,43, or 2.43, denote two Units (or Wholes) four Tenths, and three Hundredth Parts (or forty-three Hundredth Parts) of one fuch Whole. Laftly, It is to be obferv'd, that alThe Agree- though the Places of Integers are reckon'd ment be in a different Manner (as has been obworking ferv'd) from the Places of Decimals as to with inte- their Order, namely, the Places of Inteworking gers from the Right-hand, the Places of with De Decimals from the left; yet there is an gers, and cimals, whence it arifes. exact Agreement between them in this, viz. that the denominative Value of every left-hand Place, is ten Times as much as the denominative Value of the next fore-going right-hand Place. Namely, in decimal Numbers, ten Thousandth Parts make one Hundredth Part, and ten Hundredth Parts make one Tenth Part, and ten Tenth Parts make one Unit or Whole; and fo on likewise in Integers, ten Units make one Ten, ten Tens make one Hundred, ten Hundreds make one Thousand, &c. And on this Agreement as to the Increase of the denominative Value of Integers and Decimals, depends and is founded the Agreement between Integers and Decimals as to the Manner of Working, i. e. as to Addition, Subftraction, Multiplication, Divifion, &c. For thefe Operations are perform'd in Decimals, |