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

W

ADDITION,

HICH is the firft of the four fundamental Rules, 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 Numbers, 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

Suppose, 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 I 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 Thoufands a fourth, and fo on-Then fay, 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 make 19, 19 and 4 make 23, 23 and 8 make 31; then referving 3, write down I as before, and fay again, 3 and make 4, 4 and 3 make 7, 7 and 7 make 14; wherefore write 4 underneath: And lastly, fay I and I make 2, 2 and 8 make 10, which in the laft Place write down, and you will have the Sum of them all.

9

87899 13403 1920

885

104107

ADDITION of Numbers of different Denominations, for inftance, 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 muft 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.

L S. d.

120
65 12 5

15 9

9

8 o

For an inftance, 5 Pence and 9 Pence make 14 Pence; now in 14 there is one 12, or a Shilling, and two remaining Pence; the Pence, fet down; and referve I 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 I to the Column of Tens; I and I and 1 make three Tens of Shillings, or 30 Shillings; in 30 Shillings there is once twenty Shil- 195 16 2 lings, or a Pound, and 10 over: Write one in the Column of Tens of Shillings, and carry I to the Column of Pounds; and continue the Addition of Pounds, according to the former Rules.

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

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

3

Pence.

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, Subftraction 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 Subftraction 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° Subftra&t, 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 thefe Rules, any Number may be fubftra&ted out of another greater. For example;

If

If it be required, from

To fubftract

9800403459

47438652ft3

The Remainder will be found 5056538196 For, beginning with the Right-hand Figure, and taking 3 from 9, there remains 6 Units, to be wrote underneath the Line: going then to the next Place, 6 I find, cannot be taken from 5; wherefore, from the Place of hundreds 4, I borrow 1, which is equivalent to 10, in the Place of tens; and from the Sum of this 10 and 5, viz. 15, fubftracting 6, I find nine tens remaining, to be put down under the Line. Proceeding to the Place of hundreds, 2 with the I borrowed at the laft, make 3, which fubftracted from 4, leave 1. Again, 5 in the Place of thoufands, cannot be fubftracted from 3; for which Reafon, taking I from 4, in the Place of hundreds of thousands, into the empty Place of tens of thousands, the Cypher is converted into 10 tens of thousands, whence one 10 being berrowed, and added to the 3, and from the Sum 13 thoufand, 5 thousand being fubftracted, we fhall have 8 thoufand to enter under the Line: Then fubftracting 6 tens of thousands from 9, there remain 3. Coming now to take 8 from 4; from the 8 further on the Left, I borrow 1, by means whereof, the two Cyphers will be turned each into 9. And after the like manner is the reft of the Subfraction eafily performed.

If heterogeneous Numbers be to be fubftracted from each other; the Units borrowed are not to be equal to ten; but to fo many as there go of Units of the lefs kind, to conftitute an Unit of the greater: For example;

d.

1. 5. 45 16

6

27 19 9

17 16 9

For fince 9 Pence cannot be fubftracted from 6 Pence; of the 16 Shillings, one is converted into 12 Pence; by which means, for 6 we have 18 Pence; whence 9 being fubftracted, there remain 9. In like manner, as 19 Shillings cannot be fubftracted from the remaining 15; one of the 45 Pounds is converted into 20 Shillings, from which, added to the 15, 19 being fubftracted, the Remainder is 16 Shillings. Laftly, 27 Pounds fubftracted from 44 Pounds, there remain 17.

If a greater Number be required to be fubftracted from a less, it is evident that the thing is impoffible.-The lefs Number, therefore, in that Cafe, is to be fubftracted from the greater; VOL. I.

K

and

3;

and the Defect to be noted by the negative Character, E. gr. If I am required to pay 8 Pounds, and am only Master of when the 3 are paid, there will ftill remain 5 behind; which are to be noted,-5.

Subftraction is proved, by adding the Remainder to the Subtrahend, or Number to be substracted: for if the Sum be equal to the Number whence the other is to be substracted, the Subftraction is justly performed.-For example;

9800403459 fubtrahend-
4743865263

1. 5. d.

MULTIPLICATION

S the Act, or Art of multiplying one Number by another, to find the Product.

Multiplication, which is the third Rule in Arithmetic, confifts in finding fome third Number, out of two others given; wherein, one of the given Numbers is contained as often as Unity is contained in the other.

Or, Multiplication is the finding what will be the Sum of any Number added to itself, or repeated, as often as there are Units in another.-So Multiplication of Numbers is a compendious Kind of Addition.

3

Thus