The doctrine of limits with its applications; namely, conic sections, the first three sections of Newton, the differential calculus. A portion of a course of university education

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. 1838 Excerpt: ...Prop. VII. The parallelograms made by drawing tangents at the extremities of two conjugate diameters of an ellipse are all equal in area. Let APDZ be an ellipse, ApdZ its circular projection, k CP, CD two semi-diameters conjugate to each other: KL a parallelogram made by drawing tangents at the extremities of the diameters PC, DC. Draw HK perpendicular to AZ, meeting in k the tangent of the circular projection at p. Therefore since the tangent of the ellipse and of its circular projection meet AZ in the same point T, we have HK: Hk:: MP: Mp, that is, HK: Hk:: BC: AC. For the same reason the tangent at D will meet Hk in a point determined by the same proportion. Therefore the two tangents at p and d meet HK in the same point k. And Cp is at right angles to Cd and equal to it; therefore Cpkd is a square. Now the triangles THK, THk are as their bases HK: Hk; that is, THK: THk:: BC: AC. Also TMP: TMp:: BC: AC; hence the differences are in the same proportion; that is, trapezium MPKH: MpkH:: BC: AC. In like manner, trapezium EDKH: EdkH:: BC: AC. Also triangle CPM: CpM:: BC: AC, and triangle CDE: CdE:: BC: AC. Add together the two former of these four sets of proportionals, and subtract the two latter, and we have CPKD: Cpkd:: BC: AC. Whence, CPKD: Cpkd:: AC. BC: AC. But Cpkd is equal to Cp1 or AC2. Therefore CPKD is equal to AC. BC. The parallelogram KL is four times the parallelogram CPKD. Therefore the parallelogram KL = 4AC.BC, and is constant. Cor. If PF be drawn perpendicular on DC, the parallelogram CPKD is equal to CD. PF. Therefore CD.PFAC. BC. THE HYPERBOLA. The Hyperbola is a line in which every point has the difference of its distances from two given points (the foci), equal to a constant quantity. It has appeared (B. i. Arts. 19, 20, 23) that the dif...