PREFACE.

THE following are some of the distinguishing features of this work:

A line is considered the path of a moving point; and a surface is supposed to be generated by a moving line. These lead to the introduction of the term "locus," which is frequently used to shorten and generalize a demonstration. An angle is supposed to be generated by the revolution of a side about the vertex; and is defined as the difference of direction of two lines drawn from the same point. It is taken in this sense in surveying, astronomy, and all applied mathematics. As to the propriety of introducing the idea of motion in these fundamental definitions, I find but one sentiment among those who sympathize with the modern methods of geometry. Indeed, the demand for it induced me, more than any thing else, to commence the preparation of a new text-book.

The theories of LIMITS, HARMONIC PROPORTION, TRANSVERSALS, and POLARS, and the PRINCIPLE OF CONTINUITY, which play so important a part in modern geometry, are briefly stated and applied in connection with the propositions from which they naturally follow. They are so interwoven into higher mathematics, that it has become necessary for the student's progress to introduce them into our textbooks of elementary geometry. Book IX. is a brief introduction to modern geometry.

The articles marked with a star may be omitted, without destroying the connection; but I have endeavored to give them in so simple and attractive a manner, that few will desire to do so.

A complete classification has been made; and each subject is treated by itself as distinctly as possible. It has been my object to make the

definitions and demonstrations short and concise, without sacrificing perspicuity. My own views in regard to "condensing the demonstration by means of symbols" were confirmed by Prof. J. R. French of Syracuse University, who kindly examined the first pages of my manuscript, and the plan of the work. A student will often give a demonstration correctly, without being able to state afterwards the hypothesis, or what he has proved. These are therefore given in a distinct form. Exercises and numerical examples are inserted in numbers sufficiently great to fix the truth of the proposition in the mind of the student, without materially increasing the amount of work to be done. In solid geometry I have shown the relation of spherical polygons to polyhedral angles whose vertices are at the centre of the sphere; so that, when any proposition is proved in reference to the one, we may immediately infer a corresponding proposition in reference to the other, without a separate demonstration.

My constant aim has been, first, to give the leading features of the latest and most approved text-books of this country and Europe; second, to put every proposition in the simplest form possible, and at the same time to make suggestions that will lead the student to investigate for himself. More discipline is acquired by working out a single demonstration without aid than by learning several proposítions that are fully given in the text-book. Most students depend too much upon books, and too little upon their own mental powers, error which ought to be avoided, especially in the study of geometry. This work is not designed for students who learn merely by rote. Every reference must be quoted, and its application shown.

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Besides several foreign works, that by Prof. Chauvenet has been my constant companion.

To my friend and former teacher, Prof. J. M. Van Vleck of Wesleyan University, I am especially indebted for the invaluable aid he has kindly rendered me during the preparation of this work.

WM. H. H. PHILLIPS.

WILBRAHAM, MASS., July 1, 1874.