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

PROBLEM V.

HAving given for a right-angled Triangle the difference of both the Legs from the Hypothenusa, to find the sides and so the whole Triangle. E. g. Suppose the difference of the less side to be BD and of the greater DE (n. 1. Fig. 35.) to find the sides themselves, and so make the Triangle.

SOLƲTION.

For BD put a, for DE, b. Let the greater side be x; the Hypothenusa will be x+b; therefore the lesser side will be x+b−a. Now the □□ of the sides are = to the □ of the Hypothenusa, i. e. 2xx−2ax+2bx+bb−2ab+aa

=xx+2bx+bb; and taking away aa, 〈 math 〉〈 math 〉; and adding 2ax and 2ab, and taking away 〈 math 〉〈 math 〉. Therefore 〈 math 〉〈 math 〉 i. e. 〈 math 〉〈 math 〉.

The Geometrical Construction. Between the given diffe∣rences BD and DE▪ find (n 2.) a mean proportional DF▪ and join to it at right angles the equal line FG, and cut off DH equal to DG; and so you'l have BH the greater side of the triangle sought. This being prolong'd to C, so that HC shall be = b, having described a semi-circle upon the whole line BC apply therein BA=BH; and having drawn AC, the Triangle sought ABC, will be formed.

The Arithmetical Rule. If the square root extracted from the double rectangle of the differences be added to the greater difference, you'l have the greater side sought, &c.