PREFACE. The first part of the following volume consists of the principal Books of Euclid's ELEMENTS. Of this part Dr. Simson's edition has been made the basis ; and, except in the Definitions, Euclid's arrangement has been strictly adhered to. In many instances I have endeavoured to improve the work, sometimes by abbreviations in the process of reasoning, and more frequently by the omission of needless repetitions in the language of the demonstrations. In doing this, however, I have cautiously avoided making such alterations as would injure the true spirit of the original : I have taken care not to apply too rude a hand to a work that, for more than twenty centuries, has been justly held in the highest estimation in all civilized countries. The Notes and Illustrations at the end of the volume, and those interspersed through the work, will point out some of the more important of the modifications above referred to. In addition to these, it may be here mentioned, that considerable changes, and, it is hoped, improvements, have been made in the demonstrations of the 5th and 35th Propositions of the First Book ; in the 13th of the Second Book; in the 14th, 15th, 21st, 31st, and 32d of the Third Book ; are. and in several in the Fourth, Sixth, Eleventh, and Twelfth Books. It is not pretended that these, and similar improvements, if such they shall be found to be, are all new, though I believe that several of them Much real originality is not to be expected in writing on a subject which has occupied a large portion of the time and attention of many eminent men ; and, as my great object has been utility, I have freely availed myself of whatever appeared to be useful in the writings of those who have preceded me. An APPENDIX has been subjoined, which, besides affording practice to the student in the application of what is contained in the ELEMENTS, will also add much to his stock of geometrical knowledge. It contains a number of miscellaneous propositions, several of which are valuable ; and it treats on the tangencies, on loci, porisms, isoperimetrical figures, and the quadrature of the circle. While, in so narrow limits, these subjects must necessarily be discussed in a brief manner, as much has been given as will make students acquainted with the nature of these curious speculations, and for more than this, the majority of students have neither time nor inclination. In the Notes, also, the nature and application of geometrical analysis are illustrated in such a manner, as will enable the student to employ that ancient instrument of investigation in other cases. The Third Book of this APPENDIX contains a short treatise on plane trigonometry, designed for the use of junior students, and of those who intend to apply mathematics to practical purposes. In this are given the best methods at present known for the resolution of plane triangles, and the application of trigonometry in determining heights and distances in some of the simplest and most useful cases. Spherical trigonometry has been omitted, on account of the superior mode of investigating the principles of that important and interesting branch of science afforded by the modern analysis ; that method of investigation requiring a knowledge of algebraic operations, which, in the rest of the volume, the student is not supposed to possess. Any person, therefore, who wishes to study spherical trigonometry, or even plane trigonometry beyond its mere elements, should have recourse to some of the recent treatises on the subject. The volume closes with a number of propositions intended to serve as exercises on the principles established in the ELEMENTS and the APPENDIX. These, which are partly new and partly selected, are in general easy; and they may be proposed to the student from time to time, according to the progress which he has made in the study of geometry, and the ability which he manifests in performing them. In this SECOND EDITION some beneficial changes have been made with respect to arrangement; and, in many instances, illustrations have been introduced, which will render several propositions plainer and easier to the studerit. Some minute errors, also, have been corrected, which escaped notice in the First Edition ; and the volume, as a whole, will be found to be considerably improved. GLASGOW COLLEGE, Oct. 25th, 1836. |