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the Coefficients of these two Sums, with the Sign of the Greater,
է bc +7 abd it2131 +22
obc, or of - 24bd
2 b c d x The Reason of this Rule is this,
All Quantities baving negative Signs are in Nature dire&tly contrary to such as bave Affirmative Signs; and therefore will always destrop one another. Thus, if a Man have 1500 l. in Cash, and run in Debt sool; that is, if to his Cash be add – sool. (which is the proper way to express a Debe) there will remain bur 1000 l. for the Debtor - 500l. will destroy sool of the Cap. So also if a Man owe 100 l. and have nothing to pay it, then he hath — 100l. òr is 100 l. worse than Nothing; and if any one give him 100 l, or add + 100 l to his — 100 l, the Sum will be nothing ; but notwithstanding the the Man (tho worth Nothing) will be soo l. better iban be was before. Rule 3. When unlike Quantities are to be added,
Ser them all down without altering their Signs; and hence
59bd it2 1+2+3+41512068+406-3.00 -5968
When compound Quantities are given to be added, find the Sums of the like Quantities, by the first and Yecond Rules, and then add these Sums and the unlike Quantities together, by the third Rule, and you'll have the Sum required,
244 gbedt 818
CH A P. III.
Bubftration of whole Quantities.
That to add is the same thing as to Subftruet + bas been provid. in Addition ; but this general Rule of Subtraction supposes that to, Substract is all one, as to add t, which supposition may be thus explain'd and provid.
If a Man owes 101. more than he is worth, then bis substance may (by what has been said in Addition) be represented by - 10 12 and if any one will pay ibat 10 l. for bim, or, which is all one, tako away the Debt of 1o. l. or Substract --- 10l, he does him as much Service as if he added 10 l. to bis Calh.
8bcf7bd..6:41 + 8bc 7
sd I - at
21314-25-974-1176 - --+-6+5d - 70
That-+d, taken from b, leaves 6 tod, for the Rea mainder, as in the last Example, may be thus prov'd. Supposed
2 Remainder =r Then by the Nas ture of Substraction.
b. - +4= (by. what has been 3+2 4 said in Addition): b is (by Axion 1.)
arta By Substituting -c+d for a in the 156
5lbermoto Larter part of Step 4.0i sto-01616+o-d=r. l. E. D.
to The Truth of all Operations in Subftra&tion, where any doubt aa rifes; may be prou’d by adding the Substrabend to the Remainder, as in common Arithmetick.
+ 36 6dd Substrahend. I 2131 +7a 36 9bc +6da Remainder. 1+3141 54
CHA P. IV.
uzultiplication of whole Quantities. MUltiplication of whole Quantities admits of Three Cases.
Case 1. If two limple Quantities, whether like or unlike, but having like Signs, are to be Multiplied together.
Firí, Multiply the Coefficients one into another, and then to the Product annex the Letters of both Quantities ; so shall this new Quantity (the Sign +being understood as prefixt before it) be the true Product.
7bc x2.131 Tabs ab abcd fåh
Case 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.
100 da IX2131-ab
42 db 6791600 plcddq That in Algebraick Multiplication like Signs must give a Positive or Affirmative Product, and unlike Signs a Negative one, may be thus
I. Since Multiplication is only adding one Factor (or the Multipli. cand) to it self, as often as there are Onits in the other (or the Multiplicator); therefore + Multiplying + must produce f. Since the Sum arising from the Addition of positive Quantities must be positive,
II. A Quantity with an affirmative Sign, Multiplying one sbat hath a negative one, muft produce a negative Product; for 'tis only C 2.
adding adding the negative Fa&tor to it self, as often as there are Units in the other. Nom nover fo many Negativas added togeoler, will fill. be Negative ; and so the Produ& must have a negative Sign.
III. Negative Quantities, Multiplying positive ones, must give a. Negative Produkt; because in this case, the Multiplicator, having a negative Sign, works on the Multiplicand by Subftra&ion; which therefore must be Substra&ed, or made Negative (by changing its Sign) as often as there are négative Units in tbe Multiplyer.
IV. Negatives Multiplying Negatives, must produce an Affirma. tive, or positive Produét; because Multiplication by a Negative Quantity, being only a Substra&tion or changing the sign of the Mula tiplicand as often as there are negative Units in the Multiplicator ; and since Substrading is the Jame as adding to (as was fhewed in Substraction) the defeet of the Multiplicand is by this means taken away, and consequently the Produ& will be Affirmative.
Case 3. If the Multiplyer and Multiplicand, or either of them, be Compound Quantities; then every Terms 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.
Examples. (1) 114 + bed
b ixa 3 na tab - ad
ab bb + bd 34 3+ 415lanto-ad - bb + bd