The young student of Mathematics should not attempt to read the whole of this volume at once, but should peruse each portion of it in connexion with the temporary subject of his special study: the first chapter of the first book, for example, while he is studying Algebra; the first chapter of the second book, when he has made some progress in Geometry; and so with the rest. Passages which are obscure at the first reading will brighten up at the second; and as his own studies cover a larger portion of the field of Mathematics, he will see more and more clearly their relations to one another, and to those which he is next to take up. For this end he is urgently recommended to obtain a perfect familiarity with the "Analytical Table of Contents," which maps out the whole subject, the grand divisions of which are also indicated in the Tabular View facing the title-page. Corresponding heads will be found in the body of the work, the principal divisions being in SMALL CAPITALS, and the subdivisions in _Italics_. For these details the translator alone is responsible. ANALYTICAL TABLE OF CONTENTS. INTRODUCTION. Page GENERAL CONSIDERATIONS ON MATHEMATICAL SCIENCE 17 THE OBJECT OF MATHEMATICS 18 Measuring Magnitudes 18 Difficulties 19 General Method 20 Illustrations 21 1. Falling Bodies 21 2. Inaccessible Distances 23 3. Astronomical Facts 24 TRUE DEFINITION OF MATHEMATICS 25 A Science, not an Art 25 ITS TWO FUNDAMENTAL DIVISIONS 26 Their different Objects 27 Their different Natures 29 _Concrete Mathematics_ 31 Geometry and Mechanics 32 _Abstract Mathematics_ 33 The Calculus, or Analysis 33 EXTENT OF ITS FIELD 35 Its Universality 36 Its Limitations 37 BOOK I. ANALYSIS. CHAPTER I. Page GENERAL VIEW OF MATHEMATICAL ANALYSIS 45 THE TRUE IDEA OF AN EQUATION 46 Division of Functions into Abstract and Concrete 47 Enumeration of Abstract Functions 50