H. F. (Henry Frederick) Baker

Principles of Geometry Volume 2 H. F. (Henry Frederick) Baker

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
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