2012, ISBN: 1130253848

[EAN: 9781130253849], Neubuch, [PU: Rarebooksclub.com, United States], Language: English Brand New Book ***** Print on Demand *****. This historic book may have numerous typos and missing… Mehr…

[EAN: 9781130253849], Neubuch, [PU: Rarebooksclub.com, United States], Language: English Brand New Book ***** Print on Demand *****. 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. 1850 Excerpt: .same law, will always converge to equality; for the arcs AB, Ab, are diminished without limit, and therefore their, ., difference, the arc Bb, will be diminished without limit; thus the angle B O b must be diminished without limit; but this angle = L AdO--L ADO. Thus these angles, AdO and ADO, are ultimately equal, and this third case comes under the second case. Cor. 1. Hence by Lemma VII. the limiting ratio of the subtenses will equal that of the squares of the arcs, chords, and tangents. Theorem. The subtense of an arc is ultimately equal to four times the parallel sagitta. Def. The sagitta of an arc is a line drawn at a finite angle to the chord from its middle point to meet the arc. Let B D be a subtense of the arc A B, E C the sagitta parallel to it, bisecting the chord in E, and produced to meet the tangent in F. Fig. 23. Then by similar triangles, AF = AD, andEF = BD. Also by the Lemma, L.R. CF: BD = L.R. AF2: AD2 = 1:4.-. L.R. CE: BD = 1: 4. Cor. 2. The limiting ratio of the sagittae, which bisect the chords and con-A i. verge to a given point, equals that of the squares of the arcs, chords, and tangents. Fig 24 Let EC, ec, be the sagittae of the arcs AEB, Aeb, bisecting the chords AB, Kb in C, c; draw the subtenses BD, bd respectively parallel to them; Then L.R. EC: BD = 1: 4 = L. R. e c: b d;.L.R.EC: ec = L.R. BD: bd = L.R. (arc AB)2: (arc Ab)2 = L.R. (chord AB)2: (chord Ab)2 = L. R. (tangent AD)2: (tangent Ad)2. Cor. 3. Hence if a body describe the arcs A B, A b, with any given velocity, the limiting ratio of the sagittae will be that of the squares of the times, in which they are described. Cor. 4. If the subtenses D B, d b, be perpendicular to the tangent, as in the first case of the Lemma, Aadb: AAdb = AD.DB: Ad. db;.: L.R. Aadb: AAdb = L.<

2012, ISBN: 1130253848

[EAN: 9781130253849], Neubuch, [PU: Rarebooksclub.com, United States], Language: English Brand New Book ***** Print on Demand *****.This historic book may have numerous typos and missing … Mehr…

[EAN: 9781130253849], Neubuch, [PU: Rarebooksclub.com, United States], Language: English Brand New Book ***** Print on Demand *****.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. 1850 Excerpt: .same law, will always converge to equality; for the arcs AB, Ab, are diminished without limit, and therefore their, ., difference, the arc Bb, will be diminished without limit; thus the angle B O b must be diminished without limit; but this angle = L AdO--L ADO. Thus these angles, AdO and ADO, are ultimately equal, and this third case comes under the second case. Cor. 1. Hence by Lemma VII. the limiting ratio of the subtenses will equal that of the squares of the arcs, chords, and tangents. Theorem. The subtense of an arc is ultimately equal to four times the parallel sagitta. Def. The sagitta of an arc is a line drawn at a finite angle to the chord from its middle point to meet the arc. Let B D be a subtense of the arc A B, E C the sagitta parallel to it, bisecting the chord in E, and produced to meet the tangent in F. Fig. 23. Then by similar triangles, AF = AD, andEF = BD. Also by the Lemma, L.R. CF: BD = L.R. AF2: AD2 = 1:4.-. L.R. CE: BD = 1: 4. Cor. 2. The limiting ratio of the sagittae, which bisect the chords and con-A i. verge to a given point, equals that of the squares of the arcs, chords, and tangents. Fig 24 Let EC, ec, be the sagittae of the arcs AEB, Aeb, bisecting the chords AB, Kb in C, c; draw the subtenses BD, bd respectively parallel to them; Then L.R. EC: BD = 1: 4 = L. R. e c: b d;.L.R.EC: ec = L.R. BD: bd = L.R. (arc AB)2: (arc Ab)2 = L.R. (chord AB)2: (chord Ab)2 = L. R. (tangent AD)2: (tangent Ad)2. Cor. 3. Hence if a body describe the arcs A B, A b, with any given velocity, the limiting ratio of the sagittae will be that of the squares of the times, in which they are described. Cor. 4. If the subtenses D B, d b, be perpendicular to the tangent, as in the first case of the Lemma, Aadb: AAdb = AD.DB: Ad. db;.: L.R. Aadb: AAdb = L.<

ISBN: 9781130253849

Sir Isaac Newton, Paperback, English-language edition, Pub by General Books LLC Books, , The-Three-First-Sections-and-Part-of-the-Seventh-Section-of-Newtons-Principia-with-a-Preface-Recom… Mehr…

Sir Isaac Newton, Paperback, English-language edition, Pub by General Books LLC Books, , The-Three-First-Sections-and-Part-of-the-Seventh-Section-of-Newtons-Principia-with-a-Preface-Recommending-a-Geometrical-Course-of-Mathematical-Readi~~Sir-Isaac-Newton, 999999999, The Three First Sections and Part of the Seventh Section of Newton's Principia; with a Preface Recommending a Geometrical Course, Sir Isaac Newton, 1130253848, General Books LLC, , , , , General Books LLC<

ISBN: 9781130253849

Paperback, [PU: Rarebooksclub.com]