6. Specious Notation, is that wherein Species, or Letters are made ufe of to reprefent Quantities, as the Letters of the Alphabet, a, b, c, or A, B, C, &c. as in Algebra. Or the fame Letters with Points over them, thus, x, y, z, in that Part of the Science called Fluxions. 7. Linear Notation, is the Reprefentation of Quantities by Lines, and Figures compofed of Lines, as is done in all the Parts of common Geometry. 8. The Quantities confidered in Arithmetic are called Numbers, of which there are two Sorts, whole and broken, which are otherwise called Integers and Fractions. The leaft whole Number is Unity, or 1 One; that is, any one Thing is called an Unite; and Nothing, or Nullity is represented by the Cypher o. 9. A Number of Units under Ten, is reprefented by a fingle Digit, as 2, 5, 7, &c, but ten Units are defigned by the first Digit 1, with a Cypher annexed to the right Hand, thus, 10. And as an Unit is made Ten by one Cypher annexed, so it is made Ten times Ten, or an Hundred, by two annexed Cyphers, viz. 100; fo another Cypher makes Ten Hundred, or One Thousand, viz. 1000, and fo on as in the following Table, 10. If the Number confifts not of even Tens, then fuch a Digit is annexed to the Unit as will exprefs the faid Number; thus Seventeen is expreffed by 17. Alfo twice Ten, or Twenty, is expressed by the Figure 2 and a Cypher, thus, 20; and Thirty by 30; Forty by 40; and fo on to an Hundred. The intermediate Numbers are alfo expreffed by annexing proper Digits in the Place of the Cypher; thus 25 is Twenty-five; 37 Thirty-feven, Br. II. From 11. From hence we obtain the Method of Enumeration, or expreffing the Number of Quantities contained in any given Sum, as fhewn below. NUMERATION TABLE. Thousands Hund. of Thouf. o Tens of Thouf. in Millions Hundred of Millions Tens of Millions 77 9 8 Tens-2 Units - 32 I 4 3 2 I 5432 I 6 5 4 3 2 1 321 7 6 5 4 3 2 1 One I Twenty-one Three Hundred, Twenty-one Here it is plain, that in order to numerate the Figures in any Sum, you have only need to mention firft each Figure, and then the Place in which it ftands, according to its Name of Valuation in the Table, in the fame Manner as you see done for each Sum, or Line of Figures in the Table on the Right-hand Side. Thus for Inftance, the Sum 850943 you read or value thus, 8 Hundred, 50 Thoufand, 9 Hundred, 43; or thus at twice, 850 Thoufand, 943; and fo the Sum 406528035 is thus read, 406 Million, 528 Thousand, 35; and fo of others. 12. As an Unit may have its Value encreased ten Times, by annexing a Cypher to the Right-hand, fo its Value is diminished in a ten-fold Proportion by prefixing Cyphers thereto; thus 0,7 is Seven Tenths Thus alfo 53 is Fifty-three Hundredths 0,375 is Three Hundred Seventy-five of an Unit. Thoufandths 13. In this Cafe, the Cypher on the Left-hand, cut off with a Comma (,)ftands in the Unit's Place, and fhews the Number does not amount to Unity, but is a certain Number of fuch Parts as the Unit contains 10, or 100, or 1000, &c. and these Parts are expreffed by the Figures on the Right-hand of the Comma. This kind of Notation of the Parts of a broken or divided Unit is called Decimal, (from Decem, Ten) and those Parts of Unity are called Decimal Numbers or Decimal Fractions. 14. Sometimes a Number confifts of Integers and Decimals together, and is then called a mix'd Number; thus 7,3 is Seven and three Tenths; 84,53 is Eighty-four and Fifty-three hundredth Parts of another; and fo of others. That Part of the Science which treats of these Numbers is called Decimal Arithmetic. 15. If Unity be divided in any other than a ten-fold Proportion, then another Species of Computation will enfue; thus in Aftronomy we divide a Degree into 60 equal Parts or Minutes; these Minutes are each divided into 60 Seconds; each Second into 60 Thirds; and fo on to Fourths, Fifths, &c. And they are thus denoted, viz. 35° : 47′: 31′′ : 23′′''; Thirty-five Degrees, Forty-seven Minutes, Thirty-one Seconds, Twenty-three Thirds. The Rules for managing these Numbers is called Sexagenary or Sexagefimal Arithmetic. 16. It frequently happens that we are obliged to divide an Unit indefinitely, or into any Number of Parts as Occafion requires for comparing a Part with the whole Unit, in Parts of fuch a Divifion In this Cafe, the Way to exprefs fuch a Fraction, is to place the Unit divided into its whole Number of Parts, below a Line, and the Parts of the Unit which are given, above it; thus is three Parts of fuch as the Unit contains Four of; and is five Thirteenths of the whole Unit. Thefe are called Vulgar. Fractions. 234 17. A Vulgar Fraction is faid to be pure, when it confifts only of fractional Parts, as 1, 4, 14, &c. and mixed, when joined with Integers, as 5, 23, 1, &c. 18. The Number placed below the Line, is called the Denominator of the Fraction, because it denominates the Fraction, or Number of Parts into which Unity is broken or divided; and the Number above the Line is called the Numerator, because it enumerates or fhews how many of those Parts make the Fraction propofed. 19. The Fraction is faid to be proper, when the Numerator is lefs than the Denominator, as; but improper, when the Contrary happens; as †, 13, &c. 20. When 20. When any two Quantities are compared together, to obferve the Relation of their Magnitude, fuch Comparison is called a Ratio; and is thus expreffed, a: b; of this Ratio, the first Term (a) is called the Antecedent, and the latter (b), the Confequent. 21 When any two Quantities have the fame Ratio with any other two, it is denoted by this Character::, thus a: b::c:d; the Quantity (a) is to (b) as (c) is to (d); which are therefore said to be analogous or proportionate; and fuch a Comparison, or Expreffion, is call'd Analogy or Proportion. 22. When any Calculation is to be made, it is done either by Addition, Subftraction, Multiplication or Divifion of Quantities; which four fundamental Rules are called the Algorithm of Quantities, and which we now proceed to explain. viz. fignifies Characters for Abbreviation explained, More; as 3+4, is 3 added to 4 Lefs; as 4-3, is 3 taken from 4 Multiplied by; as 3X4, is 3 multiplied by 4 CHAPTER I. ADDITION OF INTEGERS, or WHOLE NUMBERs. 23. A DDITION of Numbers confifts in adding together all the Units contained in feveral particular Numbers, pro perly difpofed, into one Sum, Aggregate, or Total, expreffing the Value of all together. And this is performed in the following Manner, viz. 24. Let the feveral particular Sums, or Numbers, be difpofed one under another, in fuch a Manner, that the Place of Units, Tens, Hundreds, &c. in each, may confitute a perpendicular Column of Figures; thus, let it be required to add together the Numbers 57, 762, 5389, 97615; in order to do this they must first be difpofed thus, 25. The 25 The Numbers placed, as above, you proceed to add together by the following RULE, Reckon up all the Digits in the firft, or Right-hand, Column, and obferve, for every Ten to carry One to the Place of Tens in the second Column, fetting down the remaining Digits under the first Column of Units: Thus, 5+9+2+7=23, which is thus expreffed, five more nine is fourteen, and two is fixteen, and feven is twenty-three br23, in which Sum there are two Tens and 3 over; you must then fet down the 3 and carry Two to the next Place of Tens, and proceed as before; thus, 2+1+8+6+5=22; here again are two Tens, and Two over to be fet down under the fecond Column; then carrying the Two to the third Column of Hundreds, you fay again, 2+6+3+7=18; here is but one Ten, and 8 to be fet down; then carrying One to the next Column, fay 1+7+5=13; here again is one Ten, and 3 to be fet down; laftly, carry One to the laft Place, and fay 1+9=10, which Number, be it what it will, is always fet down, and the Sum total is compleat in one Number, as in the Examples. 26. The Reason why you carry Ten from every Column to the next, is because the Value of the Figures in each Column encreases in a ten-fold Proportion, as is evident from Article 9; and the Digits fet down under each Column are Units, Tens, Hundreds, Thoufands, &c. according to Inft. 11. which will exprefs the Value of all the Columns feverally collected and added together. Thus in the first Example, the Sums of each Column will ftand thus, viz. |