ARITHMETIC.

RITHMETIC is the Art of Numbering; or, that Part of the Mathematics which confiders the Powers and Properties of Numbers, and teaches how to compute or calculate truly, and with Expedition and Eafe. Arithmetic confifts chiefly in the four great Rules, or Operations, of Addition, Subtraction, Multiplication and Divifion. It is true, for the facilitating and expediting of Calculations, Mercantile, Aftronomical, &c. divers other ufeful Rules have been contrived; as, the Rules of Proportion, of Alligation, of false Position, Extraction of Square and Cube Roots, Progreffion, Fellowship, Intereft, Barter, Rebate, Reduction, Tare and Tret, &c. But thefe are only various Applications of the first four Rules; and, as they are the Foundation of all Computation, an Introduction to them feems not to be improper in this Place; which we shall therefore give in the most short, plain, and familiar manner.

NUMERATION

S the Art of eftimating or pronouncing any Number, or Se

Iries of Numbers.

The Characters whereby Numbers are ordinarily expressed, are the ten following ones, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. It being the Law of the common Numeration, that when you are arrived at ten, you begin again, and repeat as before; only expreffing the Number of Tens.

That the ten numerical Notes may exprefs not only Units, but alfo Tens or Decads, Hundreds or Centuries, Thousands, &c. they have a local Value given them; fo, as that when either alone, or when placed in the Right-hand Place, they denote Units; in the fecond Place, Tens; in the third, Hundreds; in the fourth, Thousands.

Now, to express any written Number, or affign the proper Value to each Character, divide the propofed Number by Comma's into Claffes, allowing three Characters in each Class;

beginning

beginning at the Right-hand. Over the Right-hand Figure of the third Clafs, add a fmall Mark, or tranfverfe Line; over the Right-hand Figure of the fifth Clafs, add two Marks, or trans verfe Lines; over that of the feventh, three, &c. The Number to the Left of the firft Comma, exprefs by thousands; that which has over it the first tranfverfe Line, exprefs by millions; that with two by billions; that with three, by trillions, &c. Laftly, the Left-hand Character of each Clafs, exprefs by hundreds; the middle one by tens; and the Right-hand one, by units. Thus will the Numeration be effected.

E. gr. The following Numbers, 2", 125, 473′′, 613, 578', 432, 597, is thus expreffed or read: Two trillions, one hundred twenty five millions of billions, four hundred seventy three billions, fix hundred thirteen thousands of millions, and five hundred feventy eight millions, four hundred and thirty two thousand, five hundred and ninety seven.

And thus it appears, that by Numeration we learn the different Value of Figures, by their different Places; and, of confequence, to read or write any Sum, or Number.

From this Table may be observed:

1. The Names of the feveral Places, viz. Units, Tens, Hundreds, &c. which proceed (increafing by a tenfold Proportion) from the Right-hand to the Left.

2. That every Figure hath two Values; one in itself; the other from the Place it stands in. Thus, on the Left-fide of the Table, the Figure 9 in the upper Line, ftanding in the Unit's place, is only nine; but in the fecond Line, being removed into the place of Tens, becomes ninety; and in the third Lineis nine hundred, &c.

3. That

3. That tho' a Cypher is nothing in itself, yet it gives Value to other Figures, by removing them into higher Places:

All which being very obvious, I proceed to

WHI

ADDITION,

WHICH is the firft of the four fundamental Rules, [CH is or Operations in Arithmetic. Addition confifts in finding the Amount of feveral Numbers, or Quantities; feverally added one to another.-Or, Addition is the Invention of a Number, from two or more homogeneous ones given, which is equal to the given Numbers taken jointly together. The Number, thus found, is called the Sum, or Aggregate of the Numbers given.

The Addition of fimple Numbers is eafy. Thus it is readily perceived that 7 and 9 make 16; and 11 and 15

make 26.

In longer, or compounded Numbers, the Bufinefs is performed by writing the given Numbers in a Row downwards; homogeneous under homogeneous, i. e. Units under Units, Tens under Tens, &c. and fingly collecting the Sums of the refpective Columns.

To do this we begin at the Bottom of the outmoft Row or Column to the Right; and if the Amount of this Column do not exceed 9, we write it down at the Foot of the fame Column: If it do exceed 9, the Excefs is only to be wrote down, and the reft referved to be carried to the next Row, and added thereto; as being of the fame Kind or Denomination.

Suppofe, e. gr. the Numbers 1357 and 172, were given to be added; write either of them, v. gr. 172, under the other, 1357; fo, as the Units of the one, viz. 2, ftand under the Units of the other, viz. 7; and the other 1357 Numbers of the one, under the correfpondent ones of 172 the other, viz. the place of Tens under Tens, as 7 under 5; and that of Hundreds, viz. i, under the place 1529 of Hundreds of the other, 3.-Then, beginning, fay,

2 and 7 make 9; which write underneath; alfo 7 and 5 make 12; the laft of which two Numbers, viz. 2, is to be written, and the other 1 referved in your Mind to be added to the next Row, 1 and 3: Then fay, 1 and 1 make 2, which added to 3 make 5; this write underneath, and there will remain only 1, the firft Figure of the upper Row of Num

bers,

bers, which alfo must be writ underneath; and thus you have the whole Sum, viz. 1529.

So, to add the Numbers 87899-13403-885-1920 into one Sum, write them one under another, fo as all the Units make one Column, the Tens another, the Hundreds a third, and the place of Thousands a fourth, and so on-Then say, 5 and 3 make 8; 8 and 9 make 17; write 7 underneath, and the I add to the next Rank; faying, I and 8 make 9, 9 and 2 make 11, 11 and 9 make 20; and having writ the o underneath, fay again, 2 and 8 make 10, 10 and 9 make 19, 19 and 4 make 23, 23 and 8 make 31; then referving 3, write down as before, and fay again, 3 and I make 4, 4 and 3 make 7, 7 and 7 make '4; wherefore write 4 underneath: And lastly, say and I make 2, 2 and 8 make 10, which in the laft Place write down, and you will have the Sum of 104107.

them all.

87899 13403

1920 885

ADDITION of Numbers of different Denominations, for instance, of Pounds, Shillings and Pence, is performed by adding or fumming up each Denomination by itself, always beginning with the loweft; and if after the Addition there be enough to make one of the next higher Denomination, for inftance, Pence enough to make one or more Shillings; they must be added to the Figures of that Denomination, that is, to the Shillings; only referving the odd remaining Pence to be put down in the Place of Pence.-And the fame Rule is to be obferved in Shillings with regard to Pounds.

1. S. d.

120

15 9

65 12

8 0

9

For an inftance, 5 Pence and 9 Pence make 14 Pence; now in 14 there is once 12, or a Shilling, and two remaining Pence; the Pence, fet down; and reserve 1 Shilling to be added to the next Column, which confifts of Shillings. Then I and 8 and 2 and 5 make 16: the 6 put down, and carry the 1 to the Column of Tens; 1 and 1 and make three Tens of Shillings, or 30 Shillings; în 30 Shillings there is once 20 Shillings, or a Pound, and 10 over: Write one in the Column of Tens of Shillings, and carry 1 to the Column of Pounds and continue the Addition of Pounds, according to the former Rules.

195 16 2

;

So, half of an even Sum will be carried to the Pounds; and the odd one (where it so happens) fet under the Tens of the Shillings.

To facilitate the cafting up of Money, it will be neceffary to learn the following Table.

2

Pence.

O

SUBSTRACTION,

R SUBTRACTION, in Arithmetic, the fecond Rule, or rather Operation, in Arithmetic; whereby we deduct a lefs Number from a greater, to learn the precife Difference: Or, more juftly, Substraction is the finding a certain Number from two homogenous ones given; which, with one of the given Numbers, is equal to the other.

The Doctrine of Subtraction is reducible to what follows: To SUBSTRACT a lefs Number from a greater.—1° Write the lefs Number under the greater, in fuch manner, as that homogenous Figures anfwer, to homogenous, i. e. Units to Units, Tens to Tens, &c. as directed under ADDITION, 2° Under the two Numbers draw a Line. 3° Subftract, feverally, Units from Units, Tens from Tens, Hundreds from Hundreds; beginning at the Right-hand, and proceeding to the Left: and write the feveral Remainders in their correfpondent Places, under the Line. 4° If a greater Figure come to be fubftracted from a lefs; borrow an unit from the next Left-hand Place; this is equivalent to 10, and added to the lefs Number, the Subftraction is to be made from the Sum: or if a Cypher chance to be in the next Left-hand Place, borrow the Unit from the next further Place.

By these Rules, any Number may be fubfstracted out of another greater. For example;