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

Demonstration.

Suppose the side of the cone to be AB here also = a and BM parallel to the section = b and the intercepted AM=c, and EI=eb; and NI will be again = ec, all as in the hyperbola. And mak••ng also here as in the hyper∣bola MC=d, and the Latus Transversum DE=ob, so that DI will be ob−eb; then will (by reason of the simili∣tude of the ▵ ▵ BMC, DEP and DIO) EP be=od, and IO=od−ed. Therefore ▭ of NIO will be = oecd−eecd=□IK. But this square divided by the abscissa EI=eb gives 〈 math 〉〈 math 〉 or 〈 math 〉〈 math 〉 for that line IS which with the abscissa would make the rectangle ES= to the said square IK. Now therefore if we call the Latus Rectum a right line found after the same way as in the parabola, by making ac∣cording to Cons. 1. Prop. 1.

as b to c− so od− to a fourth 〈 math 〉〈 math 〉 i. e. as the line pa∣rallel to the section — to the intercepted diameter — so the Latus Primarium, &c. It is manifest that the Latus Re∣ctum is one part of the line just now found; and the other part 〈 math 〉〈 math 〉 is a fourth proportional to b, c and ed, or (to speak with Apollonius as we have done in the Prop.) to 〈 math 〉〈 math 〉 and eb (for there will come out the same quantity 〈 math 〉〈 math 〉) where∣fore now it is evident that the □ of the semiordinate IK is e∣qual to the ▭ IL (of the Latus Rectum 〈 math 〉〈 math 〉 and the ab∣scissa eb=oecd) having first taken out thence the ▭ LS, or eecd out of that fourth proportional 〈 math 〉〈 math 〉 by the same abscissa eb; which was to be found and demonstrated.