Chelsea Publishing Company, New York, N.Y., Third Edition

The first and second editions of the present work were originally published under the title "An Introductory Account of Certain Modern Ideas and Methods in Plane Analytic Geometry." The present edition differs from the first and second editions in the incorporation into the text of a number of corrections and additions indicated by the author.

**Preface**

In the following pages I have assumed on the part of the reader as much acquaintance with the processes of Cartesion Geometry and the Differential Calculus as can be obtained from any elementary text-books in these subjects; and starting from this, I have endeavored to give a systematic account of certain ideas and methods, a familiarity with which is tacitly assumed in higher mathematics, while no adequate means of acquiring this familiarity is provided in existing English works. Among these ideas, one of the most important is that of Correspondence, and on this, in a few of its many manifestations, I have dwelt at some length. My desire has been to refrain from encroaching on what properly belongs to the theory of Higher Plane Curves—a theory so extensive, and, as is now acknowledged, so much less simple than it appeared some few years ago, that an introductory study of its fundamental conceptions may well be undertaken as a preliminary.

To a certain small extent the field here marked out coincides with that already occupied by the later chapters of Salmon's Conic Sections. Recognizing that every English-speaking student of mathematics must of necessity acquire an intimate knowledge of Dr. Salmon's incomparable treatises, I have gladly refrained from any discussion of this part, and have simply referred the reader to those chapters, adding occasionally the few words of explanation that seemed necessitated by the different order of treatment here adopted.

It has not been my ambition to add another to the many excellent collections of problems already existing, but I trust the examples scattered through the pages will be found sufficient for purposes of illustration. As these, (many of which contain results of independent importance,) are placed in general immediately after the account of the theorems on which they depend, their position sufficiently indicates the process of solution, and I have therefore included a number of them in the index.

Regarding this work as strictly introductory, I have preferred not to give too many references. Those that do appear have been given, some because they are perhaps not just in the line of reading that is usually followed, some because of special felicity of statement, a few for their historical interest. Thus the frequency of references to any one author is not to be interpreted as an attempt to indicate the extent of my indebtedness. Had the references been so adjusted, it would have been alike my duty and my pleasure to write on ever page the name of Professor Cayley.

My hearty thanks are due to various friends; to Miss I. Maddison and Miss H. S. Pearson, for help in seeing the book through the press; to Mr. F. Morley, for valuable suggestions while the work was in progress; and to Mr. J. Harkness, for his great kindness in reading the whole, not only in proof, but also in manuscript.

C. A. Scott

Bryn Mawr, Pennsylvania

May, 1894.

**Table of Contents**

- Point and Line Coordinates
- Infinity. Transformations of Coordinates
- Figures Determined by Four Elements
- The Principle of Duality
- Descriptive Properties of Curves
- Metric Properties of Curves; The Line Infinity
- Metric Properties of Curves; The Circular Points
- Rational Curves. Tracing of Curves
- Cross-Ratio, Homography, and Involution
- Projection and Linear Transformations
- Theory of Correspondence
- The Absolute
- Invariants and Covariants