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a

4' + 20

20

The Reason of the Operations in this case may be easily under food by any one that duly considers the comparing of Stock and Debts together, or the Ballancing of Accounts betwixt Debtor and Creditor. That is, the Affirmative Quantities represent the Stock or Creditor : The Negative Quantities represent the Debts; and their Sum represents the Ballance, &c.

Cafe 3. When the Quantities are unlike, set them all down, without altering their Signs; and thence will arise compound Quantities, which can be no otherwise added but by their Signs. Thuslila

s6+ 7 dc 2.6 6 1 +232 +6 a-6156+ 7 di + 4 à

Here follow a few Examples wherein all the 3 Gafes are promiscuously concerned. 11 aa-t-2ab+bbl. 8 ab tbc-37

-7 ab--be + 42-50 i+23 aa- 2 ab tabbl ab-t 5-60

Iaa -2 ab +-bb3bc + ab- 45

+4ab-+6040 6 bi-- ab + da 1+213!2a + 2ab + bb 3 he + 40

salatb-ab -791īca

-d
3+3a4e+f
1+2+3.41 a atbab+70–0+40ff

3a +4abc-bb + 30
2 bb
3 аа

2 abc-25
3

dd + 2 aa 3abc-3 1.7.2 +3141-to dd to 2aa + bb

2

-4ab

2

45 t. da

2

2

• abc + 2

Sect. 2. Subtraction of whole Duantities. SUBTRACTION of whole Quantities is performed by

one general Rule.

RU L E. Change all the signs of the Subtrahend, (viz. of those Quantities which are to be subtracted) or Juppose them in your Mind to be changed; then add all the Quantities together, as before in Addition, and their Sum will be tbe true Remainder or Difference required.

This general Rule is deduced from these evident Truths.

To subtract an Affirmative Quantity, from an Affirmative, is the same as to add a Negative Quantity to an Affirmative: that

is +2a taken from + 3a, is the same with ~ 2a added to +39. · Consequently, to subtract a Negative Quantity from an Affire

mative, will be the same as to add an Affirmative Quantity to an Affirmative; that is - 2a taken from +3a will be the same with +20 added to +3a.

Exam.1. Exam.2. Exam.3. Exam.4-
2a
- 2a 8b

15bc

36 8bc I-2

56

7bc

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Exam. 5. Exam. 6. | Exam. 7. 1150 + 12b15a 12 6 9 ab + 36 (2/2a + 762a 703ab + 24 1313a + 5b)3 a 5616 ab + 12

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Exam.8 Exam.9. Exam. 10. Exam. 11.
1 + 2a

2 a
be

2 abd
2
+3a

6bc

+ 7 abd 1-231+ 5a · 5 a + 7 bc

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9 abd

[blocks in formation]

If these 13 Examples be compared with those in Addition, the Work will appear very evident, these being only the Converse or Proof of those; according to the Nature of Addition and Subtraction in common Aritbmetick.

More Examples in Subtraction.
Ila+b15bc3da8a+5bd +25
12 ab 5bc-4da 20-3bd-

12
I-2131+ 2b +7 dal a +8bd + 37

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20

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C+13

a b

24-46 -2.3130 +13 - 3a + bla 20 +46 1 lat b.- 54

76 Ild3b-bc-75

a-b5dt 70 1113a +46 +bc+210176 a+b+5d-70

That ab taken from a tb leaves + 2b for the Remainder, as in the first of these Examples, may be thus proved :

Let'|a+b= And 24 2763 a%3Dx+6

.

per Axiom 1. 1-3 41 b =Z-*-b

per Axiom 2. 4+615'2b = -x which was to be proved.

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The Truth of all Operations in Subtraction, where any Doubt arises, may be proved, by adding the Subtrahend to the Remainder, as in Common Arithmetick.

E X A M P L E.

From 11+52

o -9bc Take 2 - 2a +36 6 da

Subtrahend. 12 31+7a 361 + 6da9bc Remainder. 2+3+1+52

9bc Prouf.

Sect. 3. Pultiplication of whole Duantities.
M

ULTIPLICATION of whole Quantities admits
of three Cases.

Case 1. When the Quantities have like Signs, and no Coëfficients, set or join them together, and prefix the Sign + before them; and that will be their product. Exam.z. Exam.z. Exam.3. Exam. 4.

a+b

-b 216 -b12

tab ladybdit adtbd

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Thus {

I x2

3 lab

Cafe 2. If there be Coefficients; multiply them, and to their Product adjoin the Quantities set together as before.

Thus

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us {

2

76

Thus {

{:

Exam.5. Exam.6. Exam. 7. Exam. 8.
sa

-6d Thus

3 atab atb 36

6

56
IX 2 3
15 ab 17 42 db 118 a +126

18 a +1261 sab + 5bb Cafe 3. When the Quantities have unlike Signs; join them and the Product of their Coëfficients together (as before) but prefix the Sign -- before them ;

Exam.9. Exam. 10. Exam. u. Exam. 12.
ta

6d
40-76

4 amb
-b
+76
3f

31
1 x 2 3 ab 42 db 112 af -- 21bf - 12 af +216f
-

in But + into —, or

-
That + into + will produce + in the Product, is evident from
Multiplication in Common Arithmetick: viz. + s into + 7 will
give + 35&c. But that + intoor - into + should pro-
duce the Sign, as in the four last Examples: And that . in-
to — should produce the Sign to as in the second, fourth, and
fixth Examples, may perhaps seem somewhat hard to be con-
ceived; and requires a Demonstration

First to prove that -76 into +3f=-21bf. As in Ex. 11.
Suppose 11140-765o
Then will 2 40 = 76

per Axiom 1.
But 31+3f=+3f

2 x 3/412af=zibf per Axiom 3. 4-21 bf|5112 af-216f=0

per Axiom 2.

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Consequently + into —, or - into + produces , which was the Thing to be proved.

Secondly to prove that — 7 b into - 3f gives + 21 bf as in
Example 11.
Let 1/40-7b=0

76
But 31-35=-3f
the 2 x 3 is 4-12af=-21bf by what is proved above.
4-216f|51-12 af + 21 bf=o. per Axiom i.
Consequently --- into
into gives + which was to be proved.

Or

Thus, fuppose {i=19} and {

Or these may be otherwise proved by Numbers.
a = 20 l

{ale

c=12] =

8 bers. Then 4 b6 id=4

per Axiom z. Consequently, a-bxcd=6*4= 24, per Axiom 3. but -bxc-d, according to the precedent Rules, will be, as-cb+bd-da, which if true must be equal to 24. Proof a c= 20 x 12 = 240

cb=12 X 14 = 168

da= 8 x 20 = 160 Hence actbd=352 per Axiom 1. And cb+da= 328 which being subtracted,

Leaves actbd-cb-da=352-328=24, which plainly thews, That + into

in the Product. And into produces +

2: E. D. Note, If the Multiplier consists of several Terms, then every one of those Terms must be multiplied into all the Terms of the Multiplicand; and the Sum of those particular Products, will be the Product required, as in Common Arithmetick.

Proof?bd=14x 8=112

}

E X A M P L E S.

2

atb-d

7b75d -b

13 a5f Ixa 3 a atbada

21 ba+15da 1 x6/4 -ba-bbdb

-35bf-25df 3+41512a-da-66+db21b2+15 de-35bf — 25 df

a amba

ma a tb

25 - 30 」二

32-4b
1*23 aaa-abb6ca-9da-8b0+ 12 db

aa+za+4 aa-ba tbb
a2

a+b
aaa + 2aa +44 aaa-baa tiba
- 2 aa-

2

2

+baa-bbat bbb 1 x 23 aa a-8

laaabbb

40-8

Scat.

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