the Co-efficients of thefe two Sums, with the Sign of the greater Sum, to the Letters common to each of the said Quantities; and you'll have the Sum required.

1+ 2+ 3+ 415

The Reafon of this Rule is this,

--

All Quantities having Negative Signs, are in Nature directly contrary to fuch as have Affirmative Signs ; and therefore will always deftroy one another: Thus, If a Man have 15001. in Cash, and run in Debt 500l. that is, if to his Cafh he add 500 1. (which is the proper way to exprefs a Debt) there will remain but 1000 l. for the Debt or 5001. will deftroy 500 1. of the Cafe. So alfo if a Man owe 1001. and hath nothing to pay it, then he bath 100l. or is 1001, worse than nothing; and if any one give him 100 1. or add + 100l. to his 100l. the Sum will be nothing; but notwithstanding the Man (tho' worth nothing) will be 100l. better than he was before.

Rule 3. When unlike Quantities are to be added,

Set them down without altering their Signs; and hence will arife Compound Quantities: For unlike Quantities cannot be otherwife added but by their Signs:

Rule 4. When Compound Quantities are to be added,

Find the Sums of the like Quantities, by the first and fecond Rules, and then add these Sums and the unlike Quantities together, by the third Rule, and you'll have the Sum required.

S

CHA P. III.

Subtraction of whole Quantities.

Ubtraction of whole Quantities is perform'd by one general Rule.

Kule.

Change all the Signs of the Subtrahend, viz. of thofe Quantities which are to be Subtracted, or suppose them to be chang'd; then add all the Quantities together, as before in Addition; and their Sum will be the Remainder required.

That to add is the fame thing as to fubtract has been + prov'd in Addition: But this general Rule of Subtraction fuppofes that to fubtract is all one as to add +, which Suppofition may be thus explain'd and prov'd.

-

If a Man owe 10 l. more than he is worth, then his Subftance may, by what has been faid in Addition, be reprefented by 10l. But if any one will pay that 10l. for him, or, which is all one, take away the Debt of 101. or fubtract the 10l. he doth him as much Service, as if he added 10l. to his Substance; for, in either Cafe, he will be worth juft nothing.

6b+ 8bc

+14b

236c

I

2'31a

45+

2597 C

That to Subtract

prov'd:

Ex. 12.

d

a

--

b

Ex. 7.

bc+2

+9

8bc-7

Ex. 10.

2 a

2a

4 b

Ex. 13.

d76—a+b+5d7cb + c

is all one, as to add may be thus

If — c be fubtracted from + b, I say the Remainder r is b + c.

dc=ba, by the Nature of Subtraction. bda, by what hath been faid in

c=

4 r= b+d a +alsirta = = b + d

I, 56r+d C

6-d7r

b + d

7+c8 r= btc. 2. E. D.

(Addition.

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

Note, 1, 5 fignifies from the first and fifth Steps.

CHAP.

CHA P. IV.

Multiplication of whole Duantities. Multiplication of whole Quantities admits of three Cafes. Cafe 1. When two fimple Quantities, whether like or unlike, but having like Signs, are to be Multiplied together.

First, Multiply the Co-efficients one into another, and then to the Product annex the Letters of both Quantities; fo fhall this new Quantity (the Sign+being understood as prefixt before it) be the true Froduct.

Ix231ab, or a b, or ba; for a bisba, by the 16. 7. of the Father of the Mathematicks's Elements.

* Euclid.

Cafe 2. When the Quantities to be Multiplied are Simple and and have unlike Signs.

Join them and the Product of their Co-efficients together as before, but prefix the Sign - before them.

That in Algebraick Multiplication like Signs must give ā Pofitive or an Affirmative Produ&, and unlike Signs a Negative one, may be thus prov'd:

Multiplication being a Compendious Method of adding together (or into one Sum) the Multiplicand fo often repeated as there are Units in the Multiplicator, or Multiplyer; therefore,

When the Multiplicand is Affirmative, according as the Multiplicator is great or small, the Product will be proportionably great or Small; and when the Multiplicator iso (or nothing) the Product will be o too; confequently when the Multiplicator is

C 2

greater

greater or less than o, to wit, Affirmative, or Negative, the Product must be proportionably greater or less than o, viz. Affirmative, or Negative; wherefore,

+Multiplyed by must produce +; viz. +bx+c=+bc, or + cb. Alfo + Multiplyed by must produce; viz. + bx

bc, or

cb.

2, or

4,

When the Multiplicand is Negative, according as the Multiplicator is great, or Small, the Product will be proportionably small, or great (thus 2× 2, or the Sum of 2 and is twice as fmall as 2 X I or -2); And when the Multiplicator is o, the Product will be o too; confequently when the Multiplitator is greater, or less then o; to wit, Affirmative or Negative, the Product must be proportionally lefs, or greater than Nothing; viz. Negative or Affirmative: Wherefore,

Multiplied by + must produce cb. And Multiplyed by bx-cbc, or c b.

--

_bc,

; viz. — bx+c—— must produce +- viz.

;

Cafe 3. If the Multiplyer and Multiplicand, or either of them be compound Quantities.

Then every Term of the Multiplyer must be Multiplyed into all the Terms of the Multiplicand; and the Sum of those particular Products, will be the Product required, by Ax. 6. As in common Arithmetick.

I will add the following Numerical Examples, in order to make it the more plain to Beginners, that like Signs produce +, and unlike Signs produce -.