Chapters on the Modern Geometry of the Point, Line, and Circle (Volume 2); Being the Substance of Lectures Delivered in the University of Dublin to the Candidates for Honors of the First Year in Arts - Softcover
Synopsis
This historic book may have numerous typos, missing text or index. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. 1865. Not illustrated. Excerpt: ... When either anharmonic ratio = 0, the other = cc; and, conversely, when either = cc, the other = 0; in both those extreme cases it is evident that one or other of the two points or lines of section, C and D, coincides with one or other of the two extremities of the line or angle, A and B. When either anharmonic ratio =±1, the other also = + 1; these are the only two cases in which the two anharmonic ratios of the section of a line or angle are equal, +1 and -- 1 being the only two numbers which are equal to their reciprocals; in the latter case the section of the line or angle is, as already noticed, harmonic; and in the former case it is evident that either the two points or lines of section, C and D, or the two extremities of the line or angle, A and B, coincide with each other. For the three particular values of either anharmonic ratio of the section of a line or angle AB by two points or lines C and D, 0, oo , and + 1, some two of the four points or lines A, B, C, D, therefore, coincide. For every other value of either, however, they are all four distinct from each other. 270. When for one of the two points or lines of section, D suppose, the two simple ratios for the single section each = 1, that is, when D is the point or line of external bisection of the segment or angle AB; then, whatever be the position of the other point or line of section G, the two anharmonic ratios for the double section by G and D combined, become in that case the two simple ratios for the single section by G alone. This particular case is deserving of special attention, not only on account of its comparative simplicity, but because, as we shall presently see, every other case of anharmonic section of a line or angle, whatever be the positions of the two points or lines of sec...
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