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IF a Positive Quantity be multiplied by a Negative one, or contrariwise, the Product will be a Negative Quantity.
If a−b is to be multiplied by c; it is certain, that a multiplied by c, makes ac a Positive Quantity, by Consect. 1. Definit. 28. Moreover b by the same c (a Negative by a Positive) will make −bc; and so the whole Product of a−b by +c, will be ac−bc.
Suppose a−b=e; therefore ec will be = to the Product of a−b by c: and since a−b is = e by the Hypoth. adding on both sides h, you'l have a=e+b. by Schol. Definit. 26. and multiplying both sides by c, ac=ec+bc, by Consect. 2. Definit. 28. and by further subtracting from each side bc, you'l have ac−bc=ec, that is, to the Product of a−b by c. Q. E D.
SInce ac−bc is the Product of a−b by c, it is manifest also, that if ac−bc be divided by c, you'l have a−b for the Quotient; and so always a Positive Quantity (as ac) di∣vided by a Positive one, c, will give a Positive Quotient; but a Negative Quantity −bc divided by a Positive one, will give a Negative Quotient.