By JOHN BONNY CASTLE, OF THE ROYAL MILITARY ACADEMY, WOOLWICH. THE THIRD EDITION. LONDON: PRINTED FOR J. JOHNSON, NO.72, ST. PAUL'S BY BYE AND LAW, ST. JOHN's-SQUARE, CLERKENWELL. PREFACE. Of all the works of antiquity which have been tranfmitted to the prefent times, none are more univerfally and defervedly esteemed than the Elements of Geometry which go under the name of EUCLID. In many other branches of fcience the moderns have far furpaffed their masters; but, after a lapse of more than two thousand years, this performance still maintains its original preeminence, and has even acquired additional celebrity from the fruitless attempts which have been made to establish a different fyftem. It is, however, generally allowed, that the Elements, as they now ftand, are attended with many difficulties, which greatly retard the progrefs of learners, on their first entrance upon this study, and prevent them from applying to other branches of knowledge, which, in the prefent advanced state of the fciences, are equally useful and im portant. Among other obftacles of this kind A 2 may may be mentioned the theory of parallel lines, the doctrine of proportion, and many things in the eleventh and twelfth books, relating to folids, which are ufually found extremely embarraffing; and notwithstanding the numberlefs efforts which have been made to elucidate and explain them, are still liable to many objections. On this account, it has been found neceffary, in most of our academical institutions, to have recourfe to fome of the more compendious rudiments of later writers, who, by means of a different arrangement, have endeavoured to new-model the subject, and to render it lefs complex and elaborate, But the greater part of them are for ill digefted that they serve rather to mislead the learner than to afford him any affiftance. For, befides being deficient in order and method, fome of thefe authors have treated the fubjet algebraically and others, by introducing a number of exceptionable principles, and a vague unfatisfactory mode of demonftration, have degraded the fcience, and deprived it of fome of its moft ftriking advantages.iud cigo! Juelles It is, therefore, the defign of the following performance, to obviate these objections, and to render the fubject more familiar and perfpicuous, without weakening its evidence, or deftroying its elegance and fimplicity. For this purpose, many propofitions in EUCLID, which are of little or no ufe in their application, and were only introduced into the Elements as neceffary links in the chain of reafoning, are here omitted; and others fubftituted in their place, which are equally conducive to that end, and at the fame time more useful and concife. By this means all the most effential principles of the science have been brought into a fhorter compafsy and the demonftrations, which lead to its fublimer truths, fo continued, as to render their connection as obvious and comprehenfive as poffible, be Great care has also been taken to preferve that methodical precifion and rigour of proof, which, in treating of this fubject, are requi fites of nearly equal importance with the fcience itfelf. For independently of its other advantages, Geometry has always been con fidered as an excellent logic, which in form |