PREFACE. HESE Elements of Geometry were compiled, about seventy years ago, by Abbé Darré, a French refugee, for the use of his pupils in Maynooth College. They were subsequently enlarged and very much improved by the late lamented Doctor Callan, and were always used by him as a text-book in his annual course of Mathematical Lectures. The present edition, though for the most part rewritten, differs not so much in substance as in form from those that have preceded it. A new theorem, or a more comprehensive form of an old one, has been introduced occasionally; but, except in a few instances, the order has been preserved. With a view to greater accuracy, many of the definitions and some of the demonstrations have been considerably modified; and, instead of the method of indivisibles, that of limits, or some equivalent method, has been substituted throughout. In most of the changes that have been made, I adopted as models the best handbooks on the subject at present in use in France and Germany since Euclid, however, notwithstanding its many defects, is still retained in nearly all the schools and colleges in these countries, I deemed it advisable to borrow from it a few problems and one or two theorems, not given in former editions of this treatise ; and I think it will be found that every proposition of importance usually read in the first six and the eleventh and twelfth books of Euclid is contained, in substance at least, in the text or exercises. 6 The introduction of 'proportion,' with all the difficulties of incommensurables,' so early as pp. 10 and 60, will appear anomalous to persons not acquainted with the Maynooth mathematical course; but it must be borne in mind that the Continental practice of reading Algebra before Geometry, initiated by Darré at the end of the last century, is still kept up, and will continue, I trust, for years to come. Slight imperfections in a few of the diagrams are to be ascribed to the hurry in which I was obliged to make some of the drawings, and not to the engraver, who has executed his task with the utmost exactness and despatch. The treatises on Trigonometry which in former editions accompanied the Elements of Geometry are omitted in the present, but will be ready in a separate form, it is hoped, in a few months. MAYNOOTH COLLEGE, Sept. 20, 1872. F. L. |