207. The operation may sometimes be facilitated by substituting for the unknown quantities the sum and difference

of two others*.

and x3 + y3 = (2 + v)3 + (2 − v)3

= 8+ 12v +6 v2 + v3 + 8 − 12v + 6v2 – v3

* This artifice may be used, when the unknown quantities in each equation

are similarly involved.

PROBLEMS PRODUCING QUADRATIC

EQUATIONS.

208. PROB. I. A person bought a certain number of oxen for 80 guineas, and if he had bought 4 more for the same sum, they would have cost a guinea a piece less; required the number of oxen and price of each.

Let be the number of oxen,

80

80

and

16

= 5 guineas, the price of each.

In this, and in many other cases, especially in the solution of philosophical questions, we deduce from the algebraical process answers which do not correspond with the conditions. The reason seems to be, that the algebraical expression is more general than the common language; and the equation, which is a proper representation of the conditions, will also express other conditions, and answer other suppositions. In the foregoing instance a may either represent a positive or a negative quantity, and cannot in the operation represent a positive quantity alone (Art. 197); and the equation

when is negative, or represents the diminution of stock, will be a proper expression for the solution of the following problem: A person sells a certain number of oxen for 80 guineas; and had he sold 4 fewer for the same sum, he would have received a guinea a piece more for them; required the number sold.

209. PROB. II. To divide a line of 20 inches into two such parts, that the rectangle under the whole and one part may be equal to the square of the other part.

Let be the greater part, then will 20 - be the less,

and a (20x)×20= 400 20x, by the question,

=

Here

The observation contained in the preceding article may be applied here; and it is to be remarked, that the negative values thus deduced are not insignificant, or useless. the negative value shews, that if the line be produced ✓500 + 10 inches, the square of the part produced is equal to the rectangle under the line given and the line made up of the whole and part produced.

210. PROB. III. To find two numbers, whose sum, product, and the sum of whose squares, are equal to each other. Lety and ay be the numbers,

Since the square of every quantity is positive, a negative quantity has no square root; the conclusion therefore shews that there are no such numbers as the question supposes.

[A collection of Problems may be found in Appendix 11.]

RATIOS.

211. Ratio is the relation which one quantity bears to another in respect of magnitude, the comparison being made by considering what multiple, part, or parts, one is of the other.

Thus, in comparing 6 with 3, we observe that it has a certain magnitude with respect to 3, which it contains twice; again, in comparing it with 2, we see that it has a different relative magnitude, for it contains 2 three times, or it is greater when compared with 2 than it is when compared with 3. The ratio of a to b is usually expressed by two points placed between them, thus, a : b; and the former, a, is called the antecedent of the ratio, the latter, b, the consequent.

212. COR. 1. When one antecedent is the same multiple, part, or parts, of its consequent, that another antecedent is of its consequent, the ratios are equal. Thus, the ratio of 4 : 6 is equal to the ratio of 2: 3, that is, 4 has the same magnitude

when compared with 6, that 2 has when compared with 3, since

416

a

=

2

3

C

; the ratio of a b is equal to the ratio of c: d, if

α

= because and represent the multiple, part, or parts, b ̄ d'

d

that a is of b, and c of d.

213. COR. 2. If the terms of a ratio be multiplied or divided by the same quantity, the ratio is not altered.

For

α ma

b mb

(Art. 101); .. a: b=ma: mb.

214. COR. 3. That ratio is greater than another, whose antecedent is the greater multiple, part, or parts, of its consequent. Thus the ratio 7: 4 is greater than the ratio 8: 5;

7
or is greater than or
4' 20

32

5' 20

These con

clusions follow immediately from our idea of ratio.

[Ex.

Which is greater a+x ax or a2+x2 : a2 – x2 ?

and since the former is the greater by the quantity

2ах

215. DEF. A ratio is called a ratio of greater inequality, of less inequality, or of equality, according as the antecedent is greater than, less than, or equal to, the consequent.

216. A ratio of greater inequality is diminished, and of less inequality increased, by adding any quantity to both its terms.

If 1 be added to the terms of the ratio 7: 4, it becomes the ratio 8 5, which is less than the former, (Art. 214).