Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

SOLƲTION.

Make the area of the given rectangle, or the square of LM 〈◊〉〈◊〉 to bb, and the side sought of the rectangle BI=x; the o∣••er side BC will be = 〈 math 〉〈 math 〉, and having substracted out of it 〈◊〉〈◊〉 and GK (i. e. 2x) the side of the lesser square FG will ••〈 math 〉〈 math 〉−2x, i. e. 〈 math 〉〈 math 〉; whose square since it is the ••orth part of the greater square by the Hypothesis, you'l have 〈 math 〉〈 math 〉; 〈◊〉〈◊〉 multiplying both sides by 〈 math 〉〈 math 〉; 〈◊〉〈◊〉 taking away 4b{powerof4}, and adding 〈 math 〉〈 math 〉;

and dividing by 4, 〈 math 〉〈 math 〉.

Therefore according to the third case of affected quadratick ••¦quations, 〈 math 〉〈 math 〉 i. e. 〈 math 〉〈 math 〉.

Therefore 〈 math 〉〈 math 〉.

Geometrical Construction. If the given line b be taken 〈◊〉〈◊〉 unity, b{powerof4} and bb will be equal to it. Therefore if betwe•• LM as unity, and MN=3 ⅓b, you find a mean proporti••¦nal MO (n. 2. Fig. 42.) 'twill be 〈 math 〉〈 math 〉; which subst••¦cted from MQ=2b, or added to it, will give two values the quantity xx, viz. PQ and IQ; the first whereof will 〈◊〉〈◊〉 only a true one, and of use here. Now therefore a mea•• proportional QR found between PQ and unity will expre•• the quantity sought x.

Therefore for forming the square it self, since its side AB=〈 math 〉〈 math 〉, you may proceed as in the former Constructon, (vi•• n. 3.)