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

PROBLEM II.

THE square of the Hypothenusa in a right-angled ▵ being given, as also the difference of the other two squares to find the sides. E. g. If the Hypothenusa be BC (Fig. 25.) and the difference of the squares of both the legs, and conse∣quently its Leg also BE given (for the squares being given the sides are also given geometrically) to find the sides of the right-angled ▵ which shall have these conditions; or more plainly, to find one side e. g. the lesser which being found, the other, or the greater, will be found also.

SOLƲTION.

Let the □ of the given Hypothenusa = aa, and the square by which the two other differ = bb. Let the less side = x, and its □=xx. Wherefore the greater will be xx+bb. And since the sum of these is = to the □ of the Hypothenusa, you'l have 〈 math 〉〈 math 〉; and substracting 〈 math 〉〈 math 〉; and dividing by 〈 math 〉〈 math 〉. Therefore 〈 math 〉〈 math 〉.

Geometrical Construction. Having described a semi-circle on BC, and applyed therein BE, the □ EC will = aa−bb; and having described another semi circle upon EC divided into two Quadrants the □ DC will be 〈 math 〉〈 math 〉, and so DC = 〈 math 〉〈 math 〉 or the side sought; which being also transferr'd upon the other semi-circle describ'd on BC, viz. from C to A gives the other side AB and the whole ▵ sought.

The Arithmetical Rule. From the square of the Hypothe∣nusa substract the given difference, and the square root extra∣cted out of half the remainder gives the lesser side of the ▵ sought.