Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/192



Becaue the figure EFLC is given in kind the three right lines EF, EL and EC, that is GD, HK and EC will have given ratio's to each other.

If three right lines, two whereof are parallel, and given by poition, touch any conic ection; I ay, that the emidiameter of the ection which is parallel to thoe two is a mean proportional between the egments of thoe two, that are intercepted between the points of contact and the third tangent. Pl. 11. Fig. 3.

Let AF, GB be the two parallels touching the conic ection ADB in A and B; EF the third right line touching the conic ection in I, and meeting the two former tangents in F and G, and let CD he the emi-diameter of the figure parallel to thoe tangents; I ay, that AE, CD, BG are continually proportional.

For if the conjugate diameters AB, DM meet the tangent FG in E and H, and cut one the other in C, and the parallelogram IKCL be compleated; from the nature of the conic ections, EC will be to CA as CA to CL, and o by diviion, EC - CA to CA - CL or EA to AL; and by compoition, EA to EA + AL or EL, as EC to EC + CA or EB; and therefore (becaue of