This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1822 Excerpt:...applied to the axis AB Cor. 2. 11. 1 Sup.), and the rectangle OLR is to the rectangle ALB, as the rectangle PHQ is to the rectangle AHB (14. 1 Sup.); but PQ and OR being bisected in H and L by part 1), and OL and PH being, because of tVe parallelogram OH, equal, the rectangles OLR and PHQ are equal, therefore the rectangles ALB and AHB are equal, (14. 5 Eu.), and therefore AL is equal to HB Cor. 1 and 2. 7. g Eu.); but AC is equal to CB (l. 1 Sup.), therefore LC and CS are equal Ax. 2 and 3. 1 Eu.); but, because of the paral Iclograms OC and CP, the right lines OK and KP arc equal to LC and CH (34-l Eu.), therefore OK and KP are equal, and so OP is bisected in K. Scholium. A perpendicular, let fall from any point of a conick section, on an axis, which is notthe second axis of a hyperbola, meets the axis within the section, for it is parallel to a tangent, drawn through the nearer vertex of the axis Cor. 1. 11. 1 Sup. and 28. 1 Eu), which tangent falling wholly without the section Def. 10. 1 Sup.), if the perpendicular did not meet the axis within the section, it would meet the tangent, contrary to the definition of parallel right lines. And this perpendicular is an ordinate to the axis (Cor. 2. 11. 1 and Def. 12. 1 Sup.), and if it be produced beyond the axis, so that the part produced may be equal to the ordinate, its other extreme is in the section, for otherwise, a right line ordinately applied to the axis, and terminated by the section, would not be bisected by the axis, contrary to this proposition. Cor. A tangent to a conick section, which is perpendicular to an axis, which is not the second axis of a hyperbola, touches the section in a vertex of that axis; for a perpendicular to the axis, drawn from any other point of the section, meets the axis with...