That is, if B is in debt, C is c times as much in debt. Hence if a negative quantity be multiplied by a positive, the product is negative.

A gentleman owned a number a of farms, and each farm was worth a number c of dollars, which was his whole property. He hired money and fitted out a number b of vessels, and each vessel was worth as much as one of his farms. All the vessels were lost at sea. How much was he then worth.

He was worth a lars.

-b times c dollars. That is, a c-b c dol

с

Now if the number of farms exceeded the number of vessels, he still had some property, but if the number of vessels exceeded the number of farms, (that is, if b is larger than a,) the quantity a cbc is negative, and he owed more than he could

pay.

Hence if a positive quantity be multiplied by a negative the product will be negative.

Let it be remembered that a b has the same numerical value as ba, they differ only in the sign.

Hence (a—b) c + (b − a) d — — cm + d m = m (d—c).

8

Now if d is greater than c, (which is the case when c→d is negative,) the quantity m (dc) is positive.

Hence if a negative quantity be multiplied by a negative, the product will be positive.

Another demonstration. Suppose both ab and cd to be negative, as before; then b -a and d c will both be po

sitive, and their product will be positive.

This product is precisely the same as that produced by multiplying a-b by c-d. Therefore if two negative quantities be multiplied together, the product will be the same as that of two positive quantities of the same numerical value, and will have the positive sign.

b

It is required to find the second power of a-b, and also of

α.

The second power of each is a2 + b2 — 2 a b.

Now if a-b is positive, then ba is negative; or if a-b is negative, then ba is positive.

That is, the second power of any quantity, whether positive or negative, is necessarily positive.

The rules for division will necessarily follow from those of multiplication.

Hence the rules which apply to terms affected with the sign - in compound quantities, extend to isolated negative quantities.

We might also derive the same rules in the following manner. It has been shown that a negative quantity is derived from some contradiction in the conditions of question, by which that quantity entered into the equation with the wrong sign. Now, in order to make it right, the sign of that quantity must be changed in all places where it is used. That is, if it was before added, it must now be subtracted; and if it was subtracted before, it must now be added, and that whether multiplied by another quantity or not.

This shows that x was used in all cases with the wrong sign, therefore to insert in each term where x is found.

m in place of x we must change the sign

Ifm be inserted by the rules found above, the same result will be produced.

When a negative value has been found for the unknown quantity, we have observed it shows that there was some inconsistency in the question. If then the unknown quantity be put again into the same equation, with the contrary sign, as we introduced-m above, that is, if the unknown quantity be taken with the negative sign, and introduced by the above rules into all the terms where it was found before, a new equation will be produced, differing from the former only in some of the signs. Then if the conditions of the question be altered so as to correspond with the new equation, it will be consistent, and a positive value will be obtained for the unknown quantity. The new value of the unknown quantity however will be the same as the former, with the exception of the sign. Therefore, when once we are accustomed to interpret this kind of results, it will be unnecessary to go through the calculation a second time.

The following examples are intended to exercise the learner in interpreting these results.

1. A father is 55 years old, and his son is 16. In how many years will the son be one fourth as old as the father?

Here r has a negative value, consequently it entered into the equation with the wrong sign. Putting now - x instead of x into the equation, it becomes

This shows that something must be subtracted from the present age; that is, the son was a fourth part as old as the father some years before.

This equation gives

x = 3.

Therefore he was one fourth part as old 3 years before, when the father was 52, and the son 13.

2. A man when he was married was 45 years old, and his wife 20. How many years before, was he twice as old as she?

There is a wrong supposition in this question. Putting JC into the equation it becomes

This shows that she was not half as old as he when they were married, but that it was to happen 5 years afterward, when the man was 50, and the wife 25.

3. A labourer wrought for a man 15 days, and had his wife and son with him the first 9 days, and received $14.25. He afterwards wrought 12 days, having his wife and son with him 5 days, and received $13.50. How much did he receive per day himself, and how much for his wife and son?

4. A labourer wrought for a man 11 days, and had his wife with him 4 days, and received $17.82. He afterwards wrought 23 days, having his wife with him 13 days, and received $39.78. How much did he receive per day for himself, and how much for his wife?

5. A labourer wrought for a gentleman 7 days, having his wife with him 4 days, and his son 3 days, and received $7.89. At another time he wrought 10 days, having his wife with him 7 days, and his son 5 days, and received $11.65. At a third time he wrought 8 days, having his wife with him 5 days, and his son 8 days, and received $7.54. How much did he receive per day himself, and how much for his wife and son severally?

6. What number is that, whose fourth part exceeds its third part by 16?

The question as it was proposed involves some contradiction. Putting in x it becomes

This shows that the question should have been as follows ; What number is that, whose third part exceeds its fourth part by 16?

7. What number is that, of which exceeds of it by 18?

8. What fraction is that, to the numerator of which if 1 be added, its value will be 3, but if 1 be added to its denominator, its value will be ?

9. What fraction is that, from the denominator of which, if 2 be subtracted, its value will be 14, but if 2 be subtracted from its numerator, its value will be ?

10. It is required to divide the number 20 into two such parts, that if the larger be multiplied by 3, and the smaller by 5, the sum of the products will be 125.