O ECLECTIC EDUCATIONAL SERIES. A TREATISE ON ANALYTIC GEOMETRY, ESPECIALLY AS APPLIED TO THE PROPERTIES OF CONICS: INCLUDING THE MODERN METHODS OF ABRIDGED NOTATION. WRITTEN FOR THE MATHEMATICAL COURSE OF JOSEPH RAY, M.D., BY GEORGE H. HOWISON, M. A., PROFESSOR IN WASHINGTON UNIVERSITY. CINCINNATI: WILSON, HINKLE & CO. PHIL'A: CLAXTON, REMSEN & HAFFELFINGER. NEW YORK: CLARK & MAYNARD. A 1871, Srug. 24. Minot Fund. $1.88 BOUND DEC 27 1909 Entered, according to Act of Congress, in the year 1869, by WILSON, HINKLE & CO., In the Clerk's Office of the District Court of the United States, for the Southern District of Ohio. ELECTROTYPED AT THE CINCINNATI. PREFACE. IN preparing the present treatise on Analytic Geometry, I have had in view two principal objects: to furnish an adequate introduction to the writings of the great masters; and to produce a book from which the topics of first importance may readily be selected by those who can not spare the time required for reading the whole work. I have therefore presented a somewhat extended account of the science in its latest form, as applied to Loci of the First and Second Orders; and have endeavored to perfect in the subject-matter that natural and scientific arrangement which alone can facilitate a judicious selection. Accordingly, not only have the equations to the Right Line, the Conics, the Plane, and the Quadrics been given in a greater variety of forms than usual, but the properties of Conics have been discussed with fullness; and the Abridged Notation has been introduced, with its cognate systems of Trilinear and Tangential Co-ordinates. On the other hand, to facilitate selection, these modern methods have been treated in separate chapters; and, in the discussion of properties, distinct statement, as well as natural grouping, has been constantly kept in view. It is to be hoped, however, that omissions will be avoided rather than sought, and that the modern methods, which are here for the first time presented to the American student, may awaken a fresh interest in the subject, and lead to a wider study of it, in the remarkable properties and elegant forms with which (iii) it has been enriched in the last fifty years. The labors of PONCELET, STEINER, MÖBIUS, and PLÜCKER have well-nigh wrought a revolution in the science; and though the new properties which they and their followers have brought to light, have not yet received any sufficient application, nevertheless, in connection with the elegant and powerful methods of notation belonging to them, they constitute the chief beauties of the subject, and have very much heightened its value as an instrument of liberal culture. To render the book useful as a work of reference, has also been an object. In the Table of Contents, a very full synopsis of properties and constructions will be found, which it is hoped will meet the wants, not only of the student in reviewing, but of the practical workman as well. In the demonstrations, convenience and elegance have been aimed at, rather than novelty. When it has seemed preferable to do so, I have followed the lines of proof already indicated by the leading writers, instead of striking out upon fresh ones. My chief indebtedness in this respect, is to the admirable works of Dr. GEORGE SALMON. The treatise of Mr. Todhunter has furnished some important hints; while those of O'Brien and Hymers have been often referred to. For examples, I have drawn upon the collections of Walton, Todhunter, and Salmon. Of American works, those of Peirce and Church have been consulted with advantage. To Professor William Chauvenet, Chancellor of Washington University, formerly Head of the Department of Mathematics in the United States Naval Academy, I am indebted for many valuable suggestions. WASHINGTON UNIVERSITY, ST. LOUIS, Sept., 1869. } H. |