The Difficulties Of Elementary Geometry Especially Those Which Concern The Straight Line The Plane And The Theory Of Parallels

Author : Francis William Newman
Publisher : Rarebooksclub.com
ISBN 13 : 9781230165356
Total Pages : 40 pages
Book Rating : 4.1/5 (653 download)

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Download or read book The Difficulties of Elementary Geometry, Especially Those Which Concern the Straight Line, the Plane, and the Theory of Parallels written by Francis William Newman and published by Rarebooksclub.com. This book was released on 2013-09 with total page 40 pages. Available in PDF, EPUB and Kindle. Book excerpt: This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1841 edition. Excerpt: ...with the rectilinear angle, (because the former is indefinitely less, ) we may probably in many cases compare two curvilinear angles with each other. For, drawing the chord A R, we have: Rect. angle BAQ: Rect. angle BAR = PQ: PB, the radius being here quite immaterial, while it is the same for both arcs. Let the radius perpetually diminish; in which case the rect. angles tend to confound themselves with the curvilinear ones; so that we get: Angle BAC: angle BAD = limit of P Q: PR. Now a priori it is impossible to foresee what will be the limit of the last ratio, which must differ exceedingly in different curves. It is not difficult, however, to invent curves in which the limit may he (1: 1) or (2: 1) or any other finite limit: or again, in which it shall be zero, that is, in which the ratio shall diminish below all limit. 128. To exemplify this, it will somewhat simplify the matter to confine ourselves to curves drawn on a plane. Let A be the centre of a plane, upon which are drawn an Fie-indefinite number of circles, whose common centre is A, as in Art. 81. Let AB be a generatrix of the plane, and A C any curve drawn upon the plane, so as to be touched by A B in A; and of course A C, like A B, will cross the circles. Let P P P..". be points in which A B cuts them, and Q Q OF the corresponding points of intersection for A C. We will first show, that a new curve is conceivable, which shall have its curvilinear angle (or curvature) at A just one half that of A C. Bisect the arc B Cin D, the arc P Q in R, the arc P" Q in R, P" Q' in R'; and so on; then the series of points D, R, R, R..". lie between the curve A G, and tangent A B. And they are indefinite in number, as are the circles which lie on the plane. By increasing...