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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 mafters; but, after a lapse of more than two thousand years, this performance ftill 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 progress 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 important. Among other obftacles of this kind

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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 fubject, and to render it less complex and elaborate. But the greater part of them are fo ill digefted that they ferve rather to mislead the learner than to afford him any assistance. For, befides being deficient in order and method, fome of these authors have treated the fubject 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 most striking advantages.

It is, therefore, the defign of the following performance, to obviate these objections, and to render the subject more familiar and perfpicuous, without weakening its evidence, or destroying 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 reasoning, are here omitted; and others substituted 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 compass, and the demonftrations, which lead to its fublimer truths, fo continued, as to render their connection as obvious and comprehenfive as poffible.

Great care has also been taken to preserve that methodical precifion and rigour of proof, which, in treating of this fubject, are requifites of nearly equal importance with the science itself. For independently of its other advantages, Geometry has always been confidered as an excellent logic, which in form

ing the mind, and establishing a habit of close thinking and juft reasoning, in every enquiry after truth, is far fuperior to all the dialectical precepts that have yet been invented; the fimplicity of its firft principles; the clearness and certainty of its demonstrations; the regular concatenation of its parts; and the univerfality of its application being fuch as no other fubject can boast.

For these reasons, it was judged neceffary to adhere as closely as poffible to the plan of the original Elements; this being, in many refpects, much more natural and judicious than any of those which have fince been propofed by other writers. But as the work was rather defigned as a regular Inftitution of the most useful principles of the fcience, than a ftrict abridgment of EUCLID, fome alterations have been made, both in the arrangement of the propofitions and the mode of demonstration; the latter of which, in particular, it is prefumed, will be found confiderably improved, being here delivered in a more convenient form, and rendered as clear and explicit as the nature of the fubject would admit.

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