Elements of Conic Sections Deduced from the Cone
Format: Paperback
ISBN13: 9780217799799
Paperback|9780217799799
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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. 1825 edition. Excerpt: ... the parabola; then, by the above, and article 48, DL: GM:: t. L S or s. LS': t. MR2 or s. MR':: t. LP2 or s.LP': t.MN2 or s. MNr. 131. A diameter (B G) of a conic section (A B C) PROP. II. bisects any secant (A C) it meets in the same section or FlBopposite hyperbola, parallel to a tangent (D B E) passing through its vertex (B); and ordinates to a diameter (B G) and a tangent passing through its vertex are parallel to one another. In the ellipse and hyperbola this has been proved in article 64; and in the parabola it may be proved in the following manner. Part I. Through A, C, let AD, CE be drawn parallel to B G, and let A D meet the tangent in D, and C E meet it in E. Then, by article 121, A D, C E are diameters; and, by article 130, A D: C E:: D B2: B E2; and, (34. i.) as AD = CE, DB=BE, and therefore AG= GC. Part II. If it be possible, let the straight line HR in the parabola A B C be bisected by the diameter B G in N, and not be parallel to A C or D E. Through R draw R M parallel to the diameter B G, and through H draw H L parallel to A C or D E, and let it meet B G in K, the curve again in L, and R M in M. Then as H R is bisected in N, and as K N, M R are parallel, (2. vi.) HN: NR:: HK: KM, andHK = KM. But, by Part 1. HK = KL, and therefore BOOK KL = KM; which is absurd. Consequently ordinates to the diameter B G must be parallel to one another, and to the tangent D E, passing through the vertex. 132. From the last, and article 64, it is evident that if in any conic section a secant be an ordinate to a diameter, any other secant in the section or opposite hyperbola, and parallel to it, will be an ordinate to the same diameter. 133. From the last article it is evident, that if a straight line bisect two parallel secants in a conic sectio
| ISBN-13 | 9780217799799 |
|---|---|
| ISBN-10 | 0217799795 |
| Weight | 0.29 Pounds |
| Dimensions | 9.00 x 6.00 x 0.19 In |
| List Price | $14.14 |
| Format | Paperback |
|---|---|
| Pages | 80 pages |
| Publisher | |
| Published On | 2009-12-01 |
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