PREFACE. In issuing this New Edition of Euclid, the Publishers believe that they have succeeded in introducing several very important improvements never before attained. These improvements may be shortly summed up as follows: The particular Enunciation of each Proposition has been changed and simplified, so as to enable the Pupil to distinguish at a glance between what is given and what has to be proved in each--thus making the Propositions much more easily understood and sooner mastered. As a further aid to the Pupil, a blank space has been left between each step or clause of the reasoning in every Proposition throughout the book. Particular care has been taken to render the language clear both in the Enunciations and Demonstrations; and in addition to Euclid's Demonstrations, which have been retained for the sake of those who prefer them, several of his Propositions have been also demonstrated in a different way, which, it is believed, will be found to be both more clear and elegant. In addition to the First Six Books of Euclid's Elements, there are given a Book on the Quadrature of the Circle, a Book on Maxima and Minima, and a Treatise on Plane Trigonometry The Fifth Book, as amended by the late Professor Playfair, is given, with the addition of a Supplementary Fifth Book, of a simpler but not less logical nature. To render the whole more complete, several important Corollaries have been added; and, lastly, a number of instructive Exercises are annexed to each Book, to enable the Teacher to test the skill and inventive powers of the Pupil. Edinburgh, 1860. CONTENTS DEFINITIONS OF MAGNITUDES, . . EXPLANATION OF SIGNS, ABBREVIATIONS, PROPOSITIONS XLI. TO XLV., . . PROPOSITIONS XLVI. TO XLVIII., . . PROPOSITIONS I. TO XIV., . PROPOSITIONS A TO D, . . . PROPOSITIONS I, TO XXXVII., . . EXERCIS . . . . |