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ISBN: 1418165832

[SR: 17895676], Hardcover, [EAN: 9781418165833], University of Michigan Library, University of Michigan Library, Book, [PU: University of Michigan Library], University of Michigan Library, Excerpt from Principles of Geometry, Vol. 2There are several matters, readily understood by the reader of Volume I, in regard to which -sve have not there entered into the detail which may be desirable for the purposes of the present volume.Related ranges on the same line. With the purpose of avoiding the use of points whose existence could only be assumed after the consideration of the so-called imaginary points, we have (Vol. I, pp. 18, 25) defined two ranges on the same line as being related when, one of them is in perspective with a range on a second line which is related to the other range of the first line. From this definition we have shewn (Vol. I, p. 160) that in the abstract geometry two such related ranges on the same line have two corresponding points in common, though these may coincide. Assuming this, we may now formally prove that two such ranges also satisfy the general definition, namely that they are both in perspective with the same other range on another line, from different centres.Let the ranges (a), (b), on the same line, l be such that (a) is in perspective with a range (c), while (c) is related to (b). Let O be a point of the line l which corresponds to itself whether regarded as belonging to the range (a) or to the range (b); let A1, A., A be other points of the range (a), respectively corresponding to the points B1, B2, B of the range (b). Let H, K be any two points in line with 0; let A1H, A.K meet in P, and B1H, B2K meet in Q, and let Pa meet the line Ohk in A'. Then the range 0, H, K, A' is in fact related to 0, B1, B2, B. For the former is in perspective, from P, with the range 0, A1, A2, A; this is, by hypothesis, in perspective with a range (c), which is itself related to O, B1, B2, B; so that the result follows from Vol. l, pp. 22-24. Thence, as the ranges, 0, H, K, A' and 0, B1, B2, B, have the point 0 in common, they are in perspective (Vol. I, p. 58, Ex. 2 (c) ). Thus the line Qb passes through A' and the two ranges(a), (b) are in perspective with the same range on the line Ohk, respectively from P and Q. This is what we were to prove.About the PublisherForgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.comThis book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works., 226700, Geometry & Topology, 13928, Algebraic Geometry, 13930, Analytic Geometry, 13932, Differential Geometry, 13936, Non-Euclidean Geometries, 13987, Topology, 13884, Mathematics, 75, Science & Math, 1000, Subjects, 283155, Books

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ISBN: 1418165832

[SR: 17895676], Hardcover, [EAN: 9781418165833], University of Michigan Library, University of Michigan Library, Book, [PU: University of Michigan Library], University of Michigan Library, Excerpt from Principles of Geometry, Vol. 2There are several matters, readily understood by the reader of Volume I, in regard to which -sve have not there entered into the detail which may be desirable for the purposes of the present volume.Related ranges on the same line. With the purpose of avoiding the use of points whose existence could only be assumed after the consideration of the so-called imaginary points, we have (Vol. I, pp. 18, 25) defined two ranges on the same line as being related when, one of them is in perspective with a range on a second line which is related to the other range of the first line. From this definition we have shewn (Vol. I, p. 160) that in the abstract geometry two such related ranges on the same line have two corresponding points in common, though these may coincide. Assuming this, we may now formally prove that two such ranges also satisfy the general definition, namely that they are both in perspective with the same other range on another line, from different centres.Let the ranges (a), (b), on the same line, l be such that (a) is in perspective with a range (c), while (c) is related to (b). Let O be a point of the line l which corresponds to itself whether regarded as belonging to the range (a) or to the range (b); let A1, A., A be other points of the range (a), respectively corresponding to the points B1, B2, B of the range (b). Let H, K be any two points in line with 0; let A1H, A.K meet in P, and B1H, B2K meet in Q, and let Pa meet the line Ohk in A'. Then the range 0, H, K, A' is in fact related to 0, B1, B2, B. For the former is in perspective, from P, with the range 0, A1, A2, A; this is, by hypothesis, in perspective with a range (c), which is itself related to O, B1, B2, B; so that the result follows from Vol. l, pp. 22-24. Thence, as the ranges, 0, H, K, A' and 0, B1, B2, B, have the point 0 in common, they are in perspective (Vol. I, p. 58, Ex. 2 (c) ). Thus the line Qb passes through A' and the two ranges(a), (b) are in perspective with the same range on the line Ohk, respectively from P and Q. This is what we were to prove.About the PublisherForgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.comThis book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works., 226700, Geometry & Topology, 13928, Algebraic Geometry, 13930, Analytic Geometry, 13932, Differential Geometry, 13936, Non-Euclidean Geometries, 13987, Topology, 13884, Mathematics, 75, Science & Math, 1000, Subjects, 283155, Books

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ISBN: 1418165832

[SR: 14251790], Hardcover, [EAN: 9781418165833], University of Michigan Library, University of Michigan Library, Book, [PU: University of Michigan Library], University of Michigan Library, There are several matters, readily understood by the reader of Volume I, in regard to which -sve have not there entered into the detail which may be desirable for the purposes of the present volume. Related ranges on the same line. With the purpose of avoiding the use of points whose existence could onlv be assumed after the consideration of the so-called imaginarv points, we have (V ol. I, pp. 18, 25) defined two ranges on the same line as being related when, one of them is in perspective with a range on a second line which is related to the other range of the first line. From this definition we have shewn (V ol. i, p. 160) that in the abstract geometry two such related ranges on the same line have two corresponding points in common, though these may coincide. Assuming this, we may now formally prove that two such ranges also satisfy the general definition, namely that they are both in perspective with the same other range on another line, from difte9f rent centres. Let the ranges (a), (b), on the same line, ,be such that (a) is in perspective with a range (c), while (c) is related to {b). Let Obe a point of the line I which corresponds to itself whether regarded as belonging to the range (a) or to the range (b); let A, A o, A be other points of the range (a), respectively corresponding to the points Bj, B2, Bof the range {b). Let H, Kbe any two points in line with 0; let AH, A.K meet in P, and BH, B.K meet in Q, and let PA meet the line OHK in A .T hen the range 0, H, K,A is in fact related to 0, B, B2, B. For the former is in perspective, from P, with the range 0, A-A A ;this is, by hypothesis, in perspective with a range (c), which is itself related to O, jB j, B.2,B; so that the result follows from Vol. i, pp. 22-24. Thence, as the ranges, 0, H, K, A and 0, B5.,, B, have the point 0in common, they are in perspective (V ol. i, p. 58, Ex. 2(c)). Thus the (Typographical errors above are due to OCR software and don't occur in the book.), 226700, Geometry & Topology, 13928, Algebraic Geometry, 13930, Analytic Geometry, 13932, Differential Geometry, 13936, Non-Euclidean Geometries, 13987, Topology, 13884, Mathematics, 75, Science & Math, 1000, Subjects, 283155, Books

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ISBN: 1418165832

Hardcover, [EAN: 9781418165833], University of Michigan Library, University of Michigan Library, Book, [PU: University of Michigan Library], 1922-01-28, University of Michigan Library, 349803011, Science & Nature, 349778011, By Subject, 349777011, Antiquarian, Rare & Collectable, 1025612, Subjects, 266239, Books, 278353, Geometry & Topology, 278320, Mathematics, 57, Science & Nature, 1025612, Subjects, 266239, Books, 922942, Maths, 922868, Popular Science, 57, Science & Nature, 1025612, Subjects, 266239, Books, 570936, Geometry, 564352, Mathematics, 564334, Scientific, Technical & Medical, 1025612, Subjects, 266239, Books

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ISBN: 9781418165833

Hardback, [PU: University of Michigan Library], Geometry

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There are several matters, readily understood by the reader of Volume I, in regard to which -sve have not there entered into the detail which may be desirable for the purposes of the present volume.

Related ranges on the same line. With the purpose of avoiding the use of points whose existence could only be assumed after the consideration of the so-called imaginary points, we have (Vol. I, pp. 18, 25) defined two ranges on the same line as being related when, one of them is in perspective with a range on a second line which is related to the other range of the first line. From this definition we have shewn (Vol. I, p. 160) that in the abstract geometry two such related ranges on the same line have two corresponding points in common, though these may coincide. Assuming this, we may now formally prove that two such ranges also satisfy the general definition, namely that they are both in perspective with the same other range on another line, from different centres.

Let the ranges (a), (b), on the same line, l be such that (a) is in perspective with a range (c), while (c) is related to (b). Let O be a point of the line l which corresponds to itself whether regarded as belonging to the range (a) or to the range (b); let A1, A., A be other points of the range (a), respectively corresponding to the points B1, B2, B of the range (b). Let H, K be any two points in line with 0; let A1H, A.K meet in P, and B1H, B2K meet in Q, and let Pa meet the line Ohk in A'. Then the range 0, H, K, A' is in fact related to 0, B1, B2, B. For the former is in perspective, from P, with the range 0, A1, A2, A; this is, by hypothesis, in perspective with a range (c), which is itself related to O, B1, B2, B; so that the result follows from Vol. l, pp. 22-24. Thence, as the ranges, 0, H, K, A' and 0, B1, B2, B, have the point 0 in common, they are in perspective (Vol. I, p. 58, Ex. 2 (c) ). Thus the line Qb passes through A' and the two ranges(a), (b) are in perspective with the same range on the line Ohk, respectively from P and Q. This is what we were to prove.

About the Publisher

Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com

This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

** Detailangaben zum Buch - Principles of Geometry Volume 2**

EAN (ISBN-13): 9781418165833

ISBN (ISBN-10): 1418165832

Gebundene Ausgabe

Erscheinungsjahr: 1922

Herausgeber: UNIV OF MICHIGAN LIB

264 Seiten

Gewicht: 0,526 kg

Sprache: eng/Englisch

Buch in der Datenbank seit 27.10.2007 04:11:27

Buch zuletzt gefunden am 18.03.2017 14:23:43

ISBN/EAN: 1418165832

ISBN - alternative Schreibweisen:

1-4181-6583-2, 978-1-4181-6583-3

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