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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 solids, which are usually found extremely embarrassing; and notwithstanding the numberless efforts which have been made to elucidate and explain them, are still liable to many objections.

On this account, it has been found necesfary, in most of our academical institutions, to have recourse to some 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 less complex and elaborate. But the greater part of them are fo ill digested that they serve rather to mislead the learner than to afford him any assistance, For, besides 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 unsatisfactory mode of demonstration, have degraded the science, and deprived it of some of its most striking advantages.

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PREFACE, 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 propositions in Euclid, which are of little or no use in their application, and were only introduced into the Elements as necessary 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 same time more useful and concise. By this means all the most effential principles of the science have been brought into a shorter compass, and the demonstrations, which lead to its sublimer truths, fo continued, as to render their connection as obvious and comprehenfive as possible.

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


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ing the mind, and establishing a habit of close thinking and just reasoning, in every enquiry after truth, is far superior to all the dialectical principles that have yet been invented; the fimplicity of its first principles; the clearnefs and certainty of its demonftrations; the regular concatenation of its parts ; and the univerfality of its application being fuch as no other subject can boast.

For these reasons, it was judged necessary to adhere as closely as possible to the plan of the original Elements ; this being, in many respects, much more natural and judicious than any of those which have since been proposed by other writers. But as the work was rather designed as a regular Institution of the most useful principles of the science, than a strict abridgment of EUCLID, fome alterations have been made, both in the arrangement of the propositions and the mode of demonstration; the latter of which, in particular, it is presumed, will be found considerably improved, being here delivered in a more convenient form, and rendered as clear and explicit as the nature of the subject would admit.

In the first fix books, every thing has been demonstrated with a fcrupulous accuracy; and it was at first designed that the fame method should have been observed throughout; but this, in treating of the folids, was found incompatible with the plan of the work, it being here scarcely possible to follow the strict principles of Euclid without becoming prolix and obscure. It was therefore thought proper, in this part of the performance, to adopt a mode of proof, which though not geometrically exact, is far more perspicuous than the former, and equally satisfactory and convincing to the mind; especially in the way it is here given, which is something less exceptionable than that of CAVALERIUS, by whom it was first introduced.

Many other particulars might be mentioned, in which this performance will be found to differ from most others of the like nature; but as they consist chiefly of improvements and emendations which are too obvious to escape the notice of the reader, any

further account of them would be unnecessary. It is fufficient to observe that much time and



attention have been bestowed


the work; and that nothing which was judged essential to the science, or useful in facilitating its attainment, has been omitted. The acknowledged intricacy of fome propositions in the fifth and sixth books, made it neceffary to abridge that part of the subject more considerably than the former; but it is conceived that what is here given will be fully fufficient to answer all the purposes of the learner.

To avoid critical objections were a vain endeavour: they may be made against every fystem of Geometry now extant; and to EUCLID as well as to other writers. Of this abundant proofs are given by the Commentators; and in the Notes at the end of the present work, where many things of this kind are pointed out which have hitherto escaped notice. These were added chiefly for the information of young students, and ought to be carefully consulted by those who wish to obtain a just idea of the science, and the principles upon

which it is founded.

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