A folution, indeed, may be represented

by means of the negative root

5; for

then the other number is (x+15=)—5+

15-25. And x25x-5, is equal to

the cube of 5. Such a folution, though

2

useless, and even absurd, it is plain must correspond to the conditions, if those rules with regard to the figns be used in the application of it, by which it was itself deduced. The fame obfervation may be extended even to impoffible roots, which being affumed as the answer of a question, must, by reversing the steps of the investigation, correfpond to the original equations, by which the conditions of that question were expreffed.

EXAMPLE IV.

To find two numbers whofe fum is 100, and

whofe product is 2059.

Let

Let the given fum 100a, the product 2059-6, and let one of the numbers fought Their probe x, the other will be a-x.

duct is ax-x2

Therefore by queftion |1|ax-x2=b, or x2—ax——

By inferting numbers x=71 or 29, and a-x=29 or 71, fo that the two numbers fought are 71 and 29.

Here it is to be observed, that b must not a2, elfe the roots of the

be greater than

4

equation would be impoffible; that is, the given product must not be greater than the square of half the given fum of the numbers fought. This limitation can easily be fhewn from other principles; for, the greateft poffible product of two parts, into which

any

any number may be divided, is when each of them is a half of it. If b be equal to

a there is only one folution, and x=

4

2'

There are three numbers in continual geometrical proportion: The fum of the first and fecond is 10, and the difference of the fecond and third is 24. What are the numbers?

But, though there are two pofitive roots in this equation, yet only one of them can

here be of use, the other being excluded by a condition in the queftion. For, as the fum of the first and fecond is 10, 25 cannot be one of them: 2 therefore is the firft, and the proportionals will be 2, 8, 32.

This reftriction will alfo appear from the explanation given of the third form, to which this equation belongs. For ≈ may be less than 27, but, from the first condition of the question, it cannot be greater; hence

27

the quantity -—27%+ can have only

27

2

one fquare root, viz. %; and this being

2

just solution of the question.

From the other root, indeed, a folution of the question may be reprefented by means of a negative quantity. If the first then be 25, the three proportionals will be 25,-15, 9. These alfo must answer the conditions, according to the rules given for negative quan

quantities, though such a solution has no proper meaning.

Befides, it is to be obferved, that, if the following queftion be proposed, ‘to find 'three numbers in geometrical proportion, fo ' that the difference of the ft and 2d may be 10, and the fum of the 2d and 3d may be 24?' The equation in step 6th will be produced: For, if the 1ft be %, the 2d is

-10, and the 3d 34-%, and therefore 34%—x2=x2-20%+100, the very fame equation as in ftep 4th. In this question, it is plain that the root 25 only can be useful, and the three proportionals are 25, 15, 9.

But the neceffary limitations of fuch a problem are properly to be derived from a general notation. Let the fum of the two first proportionals be a, and the difference of the two laft b. If a is not greater than b, the first term must be the leaft; but, if a be greater than b, the firft term may be either the greatest or the leaft.

When the first term is the leaft, the proper notation of the three terms is z, a—z,