Principles of Geometry Volume 2 - edition reliée, livre de poche

Excerpt from Principles of Geometry, Vol. 2

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.

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