ted geometrically; but the geometrical construction of problems may be effected also, without refolving the equation, and even without deducing a final equation, by the methods afterwards to be explained.

If the final equation is fimple or quadratic, the roots being obtained by the common rules, may be geometrically exhibited by the finding of proportionals, and the addition or fubtraction of squares.

By inferting numbers for the known quantities, a numeral expreffion of the quantities fought will be obtained by refolving the equation. But, in order to determine fome particulars of the problem, be fides finding the unknown quantities of the equation, it may be farther neceffary to make a fimple conftruction; or, if it is required that every thing be expreffed in numbers, to fubftitute a new calculation in place of that conftruction.

PROP.

PROP. I.

To divide a given ftraight line AB into two parts, fo that the rectangle contained by the whole line, and one of the parts, may be equal to the fquare of the other part.

This is Prop. 11th II. B. of Euclid.

Let C be the point of divifion, and let AB-a, AC=x, and then CB-a-x. From the problem a2-ax=x2; and this equation being refolved (Chap. V. P. II.)

nuse of a right angled triangle of which the

two fides are a, and, and is therefore

gives x=AC, which is the proper folution. But if a line Ac be taken on the opposite

fide of A, and equal to the above mention

ed hypothenuse, together with, it will

and will give another folution; for in this cafe alfo ABX Bc Ac2. But c is without the line AB, and therefore, if it is not confidered as making a division of AB, this negative root is rejected.

This folution coincides with what is gi

ven by Euclid. For

2

a2+ is equal (fee

4

the fig. of Prop. 11th II. B. Eucl. Simfon's edit.) to EB or EF, and therefore x=

a2+

2 a

4 2

EF-EA=AF=AH; and

the point H corresponds to C in the preceding figure.

Befides, if on EF+EA=CF (instead of EF-EA=FA) a fquare be described on the oppofite fide of CF from AG, BA produced will meet a fide of it in a point, which if it be called K, will give KBXBA =KA2. K corresponds to c; and this folution will correfpond with the algebraical folution, by means of the negative root.

If

If CB had been called x, and AC=a—x,

the equation would be axa2-2ax+x2, . 3a±√5a, in which both

which gives x= 2

roots are positive, and the solutions derived from them coincide with the preceding. If the folution be confined to a point within the line, then one of the pofitive roots must be rejected, for one of the roots of the compound fquare from which it is derived

is x

за

2

a negative quantity, which in this

ftrict hypothefis is not admitted. In fuch a problem, however, both conftructions are generally received, and confidered even as neceffary to a complete folution of it.

If a folution in numbers be required, let AB=10, then x=√125-5. It is plain, whatever be the value of AB, the roots of this equation are incommensurate, though they may be found, by approximation, to any degree of exactness required. In this case, x=±11.1803-5, nearly, that is 6.1803 nearly; and Ac=16.1803

AC nearly.

PROB.

In a given Triangle ABC to infcribe a Square.

Suppofe it to be done, and let it be EFHG; from A let AD be perpendicular on the base BC, meeting EF in K.

it be called x: Then (KD=EG=) EF=p

-x.

On account of the parallels EF, BC, AD: BC:: AK: EF; that is, pax: p-x, and p2-px=ax, which equation being rep2 folved, gives x=;

pta

Therefore x or AK is a third proportional to p+a, and p, and may be found by 11. VI. El. The point K being found, the construction of the square is fufficiently obvious.