SYNTHETIC GEOMETRY OF THE POINT, LINE AND CIRCLE IN THE PLANE BY N. F. DUPUIS, M.A., F.R.S.C. PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF QUEEN'S New York THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., LTD. 1896 All rights reserved HARVARD COLLEGE LIBRARY Bestons Public Library Norwood Press: PREFACE. THE present work is a result of the Author's experience in teaching Geometry to Junior Classes in the University for a series of years. It is not an edition of "Euclid's Elements," and has in fact little relation to that celebrated ancient work except in the subject matter. The work differs also from the majority of modern treatises on Geometry in several respects. The point, the line, and the curve lying in a common plane are taken as the geometric elements of Plane Geometry, and any one of these or any combination of them is defined as a geometric plane figure. Thus a triangle is not the three-cornered portion of the plane inclosed within its sides, but the combination of the three points and three lines forming what are usually termed its vertices and its sides and sides produced. This mode of considering geometric figures leads naturally to the idea of a figure as a locus, and consequently prepares the way for the study of Cartesian Geometry. It requires, however, that a careful distinction be drawn between figures which are capable of superposition and those which are equal merely in area. The properties of congruence and equality are accordingly carefully distinguished. The principle of motion in the transformation of geometric figures, as recommended by Dr. Sylvester, and as a consequence the principle of continuity are freely employed, and an attempt is made to generalize all theorems which admit of generalization. An endeavour is made to connect Geometry with Algebraic forms and symbols, (1) by an elementary study of the modes of representing geometric ideas in the symbols of Algebra, and (2) by determining the consequent geometric interpretation which is to be given to each interpretable algebraic form. The use of such forms and symbols not only shortens the statements of geometric relations but also conduces to greater generality. In dealing with proportion the method of measures is employed in preference to that of multiples as being equally accurate, easier of comprehension, and more in line with elementary mathematical study. In dealing with ratio I have ventured, when comparing two finite lines, to introduce Hamilton's word tensor as seeming to me to express most clearly what is meant. After treating of proportion I have not hesitated to employ those special ratios known as trigonometric functions in deducing geometric relations. In the earlier parts of the work Constructive Geometry is separated from Descriptive Geometry, and short descriptions are given of the more important geometric drawing-instruments, having special reference to the geometric principles of their actions. Parts IV. and V. contain a synthetic treatment of the theories of the mean centre, of inverse figures, of pole and polar, of harmonic division, etc., as applied to the line and circle; and it is believed that a student who becomes acquainted with these geometric extensions in this their simpler form will be greatly assisted in the wider discussion of them in analytical conics. Throughout the whole work modern terminology and modern processes have been used with the greatest freedom, regard being had in all cases to perspicuity. |