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

II. For the Radius Vy by help of the obtuse-angled ▵ DVC.

1. Denomination. CA=a as above, DA or DL=b, and putting x again for the sought Vy or VK, CV will be = a −x, DL or DR=b, and consequently DV=b+x, and DC=a−b, for which for brevity's sake we will put c. Now you'l have at least in Denomination in the ▵ CVD the three sides, so that aecording to Problem 9. the segment CW may be determined, for which in the mean while we will put y; then will DW=c+y, which is the same as DL−WL or Vy i. e. b−x.

2. For the Equation. If the □ CW=yy substract it from the □ CV = 〈 math 〉〈 math 〉, and you'l have the □ of the Perpendicular VW, 〈 math 〉〈 math 〉; and if the □ DW = 〈 math 〉〈 math 〉, substract it from the □ DV = 〈 math 〉〈 math 〉, and you'l have the same □ of the Perpendicular VW = 〈 math 〉〈 math 〉 Therefore 〈 math 〉〈 math 〉.

3. Reduction. Therefore taking from both sides xx and yy, 〈 math 〉〈 math 〉; and adding 2cy and 2ax, 〈 math 〉〈 math 〉; and substract∣ing 〈 math 〉〈 math 〉, and dividing by 2c, 〈 math 〉〈 math 〉

But if you add to the same y or CW DC=c, you'l have DW = 〈 math 〉〈 math 〉 i. e. reducing this c to the same Denomination, 〈 math 〉〈 math 〉 = DW. But the same DW =DL−WL=b−x. Therefore 〈 math 〉〈 math 〉, and multiplying 〈 math 〉〈 math 〉, and adding 2cx, and transposing the rest, 〈 math 〉〈 math 〉, and dividing by 〈 math 〉〈 math 〉 just as above in the first Case.

4. The Geometrical Construction therefore will be the same as there. See Fig. 23. n. 4. and 5.

5. The Arithmetical Rule is also the same, but the given quantities in this Example, which the figure of the Problem will shew, thus vary, while a remains 12, b will be 8, and c 4, from which data (or given quantities) there will not∣withstanding come out again, for x or the Radius Vy 2 ⅔.