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This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
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Excerpt from A Course of Plane Geometry for Advanced Students, Vol. 2 The author ))olds very strongly that the educational value of higher geometry lies in that novelty of idea and generality of con ception which is more characteristic of this subject than of any other branch of elementary mathematics. A course of geometrical conics, based on the focus-directrix definition and developed by Euclidean methods, provides excellent practice in rider-work but although the student may gain from it a systematic knowledge of the metrical properties of the conic, yet he will assimilate few, if any, really new ideas. On the other hand a projective treatment introduces the student to a region of geometrical thought, unlike anything he has seen in the past - a transition as abrupt and fertile as the crossing from algebra to the calculus. There are constant surprises, apparent contradictions, features of absorbing interest, and principles which, by the generality of their application and the variety of their expression, cannot fail to fascinate the reader and incite him to investigate their developments for himself. Experience proves incontestably that analytical methods frequently elucidate difficult geometrical conceptions. The theory of ideal elements in pure geometry, the notion of one-to - one correspondence and its application to homography and involution, the principles of conical projection are undoubtedly illuminated by a joint use of geometry and analysis. 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.
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. 1910 edition. Excerpt: ... is the envelope of x. Now the pole of x lies on b;.'. the pole of b lies on x;.'. x envelopes the pole of b;.'. the reciprocal of a, viz. A, is the pole of b. q.e.d. THEOREM 101. (1) If P, Q are conjugate points w.r.t. Sv then p, q are conjugate lines w.r.t. S2; and conversely. (2) If PQR is a self-conjugate triangle w.r.t. Sx, then pqr is a self-conjugate triangle w.r.t. S2. The proof is left to the reader. THEOREM 102. (1) If O is the centre of the base-conic S, the centre of Sx reciprocates into the polar of O w.r.t. S2, and the polar of O w.r.t. Sx reciprocates into the centre of SQ. (2) The reciprocal of O is the line at infinity; and conversely. (3) The asymptotes of Sj reciprocate into the meets of S2 with the polar of O w.r.t. S2. (4) is an ellipse, parabola, or hyperbola, according as O lies inside, on, or outside Sx. The proof is left to the reader. 1. Prove Theorem 95. 2. Prove Theorem 96. 3. Prove Theorem 99. 4. Prove Theorem 101. 5. Prove Theorem 102. 6. If two conics touch each other, prove that their reciprocals touch each other. 7. What is the reciprocal of a common chord of two conics? 8. What is the reciprocal of a common tangent of two conics? 9. Two conics cut at P; what is the reciprocal of the tangents at P to the conics? 10. What is the reciprocal of two parallel tangents to a conic? 11. What is the reciprocal of a pair of conjugate diameters of a conic? 12. What is the reciprocal of two parallel chords of a conic? 13. P, Q are two points on a tangent // to a conic 5i; the other tangents from P, Q to Sx meet at R; construct the reciprocal figure. 14. PQR is a triangle, self-conjugate to a conic Su and circumscribing a conic S; what is the reciprocal figure? Attention has already been directed to the fact that all...
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