PREFACE. PROFESSOR DE MORGAN, in speaking of Simson's Edition of the Elements (Dictionary of Greek and Roman Biography and Mythology, s. v. Eucleides), says, "with the exception of the editorial fancy about the perfect restoration of Euclid, there is little to object to in this celebrated edition. It might indeed have been expected that some notice would have been taken of various points on which Euclid has evidently fallen short of that formality of rigor which is tacitly claimed for him." In preparing an edition for the use of schools and those commencing the study of geometry, it has been the Editor's aim to restore such "formality of rigor" in all places where it seemed wanting, and to render both text and figures as accurate as he could. No step has accordingly been omitted in the propositions, or left implied; the text has been made clearer and more a symmetrical by marking the divisions into cases, and stating similar pieces of reasoning as far as may be in the same words; all looseness of expression (in the enunciations, for example, or the determining of points, etc. in the figures) has been carefully corrected; and a new set of figures drawn, the thick lines of which are those that are given in the enunciation of a proposition, and the thin such as are afterwards made use of in the construction or proof. GREAT DEAN'S YARD, 23rd Feb. 1853. THE ELEMENTS OF EUCLID. BOOK I. DEFINITIONS. I. A POINT is that which has neither length, breadth, nor thickness, but position only. OBS. A point is usually denoted by a single capital letter of the alphabet. II. A line is that which has neither breadth nor thickness, but length only. III. The extremities of a line are points. OBS. A line is usually denoted by two letters, these two letters denoting the points which are the extremities of the line. lines are sometimes denoted by a single letter. But B с X Ex. We speak of the lines AB, CD; A, B denoting the points which are the extremities of one line, and C, D those which are the extremities of the other. We may also speak of the line E. E C B D |