the Coefficients of these two Sums, with the Sign of the Greater, to the Letters common to each of the laid Quantities; and you'll have the Sum required.

The Reason of this Rule is this,

2 b c d x

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 1500 1. in Cafe, and run in Debt 5001; that is, if to his Cafh he add sool. (which is the proper way to exprefs a Debt) there will remain but 1000 l. for the Debt or- 500l. will deftroy sool of the Cash. So alfo if a Man owe 100 l. and have nothing to pay it, then he hath — 100 l. or is 100 li worfe than Nothing; and if any one give him 100 1. or add + 100 1. to bis 100 1. the Sum will be nothing; but notwithstanding the the Man (tho worth Nothing) will be 100 l. better than he was before.

--

-

Rule 3. When unlike Quantities are to be added,

Set them all 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 given to be added, find the Sums of the like Quantities, by the firft and Tecond Rules, and then add these Sums and the unlike Quantities together, by the third Rule, and you'll have the Sum required.

1+2+3+45 1740 : +176+f=&

CHA P. III.

Substration of whole Quantities.

Subtraction of whole Quantities is perform'd by one general

Rule.

Hule.

Change all the Signs of the Subftrahend; viz. of thofe Quantities which are to be Subftracted, or fuppofe them to be changed; then add all the Quantities together, as before in Addition, and their Sum will be the Remainder or Difference required.

That to add is the fame thing as to Subftract, has been prov'd. in Addition; but this general Rule of Subftraction fuppofes that to Subftract is all one, as to add +, which Suppefition may be thus explain'd and prov'd.

If a Man owes 101. more than he is worth, then bis Substance may (by what has been faid în Addition) be represented by and if any one will pay that 10 l. for him, or, which is all one, take away the Debt of 101. or Substract 101. he does him as much Service as if he added 10 1. to his Cash.

7

d\

+2bla+46 +21 + bc — dl — 2a +46

cd; for the Re

That +d, taken from b, leaves b mainder, as in the last Example, may be thus prov'd.

By Subftituting S

cd for a in the

latter part of Step 4.C

a=r

-bata = (by what has been faid in Addition) b is (by Axiem 1.) =rta

of Step 4.6lb

5+c-d6b+c_d = r. Q. E. D.

The Truth of all Operations in Subftraction, where any doubt a rifes; may be prov'd by adding the Subftrahend to the Remainder, a's in common Arithmetick.

CHA P. IV.

Multiplication of whole Quantities.

Multiplication of whole Quantities admits of Three Cafes.

Cafe 1. If two fimple Quantities, whether like or unlike, but having like Signs, are to be Multiplied together.

Firft, Multiply the Coefficients 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 Product.....

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

Join them and the Product of their Coefficients together, as before. But prefix the Sign - before them.

That in Algebraick Multiplication like Signs must give a Pofitive or Affirmative Product, and unlike Signs a Negative one, may be thus prov'd.

I. Since Multiplication is only adding one Factor (or the Multiplicand) to it felf, as often as there are Units in the other (or the Multiplicator); therefore + Multiplying must produce +. Since the Sum arifing from the Addition of pofitive Quantities must be pofitive.

II. A Quantity with an affirmative Sign, Multiplying one that bath a negative one, must produce a negative Product; for 'tis only

C2

adding

adding the negative Factor to it felf, as often as there are Units in the other. Now never fo many Negatives added together, will ftillbe Negative; and fo the Product must have a negative Sign.

III. Negative Quantities, Multiplying pofitive ones, must give a. negative Product; because in this Cafe, the Multiplicator, having a negative Sign, works on the Multiplicand by Subftraction; which therefore must be Subftra&ted, or made Negative (by changing its Sign) as often as there are negative Units in the Multiplyer.

IV. Negatives Multiplying Negatives, must produce an Affirmative, or pofitive Product; because Multiplication by a Negative Quantity, being only a Subftraction or changing the Sign of the Multiplicand as often as there are negative Units in the Multiplicator; and fince Subftracting is the fame as adding †, (as was fhewed in Subftraction) the defect of the Multiplicand is by this means taken away, and confequently the Product will be Affirmative.

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. As in common Arithmetick.