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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 lefs complex and elaborate. But the greater part of them are fo ill digested that they ferve rather to mislead the learner than to afford him any affistance, 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 moft ftriking advantages.

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It is, therefore, the defign of the following performance, to obviate thefe objections, and to render the fubject 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 reafoning, are here omitted; and others fubstituted 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 compafs, 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 alfo been taken to preferve that methodical precifion and rigour of proof, which, in treating of this fubject, are requifites of nearly equal importance with the fcience itself. For independently of its other advantages, Geometry has always been confidered 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 fuperior to all the dialectical principles that have yet been invented; the fimplicity of its firft principles; the clearness and certainty of its demonftrations; the regular concatenation of its parts; and the univerfality of its application being fuch as no other subject can boaft.

For thefe reafons, it was judged neceffary to adhere as clofely as poffible to the plan of the original Elements; this being, in many refpects, much more natural and judicious than any of thofe 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 subject would admit.

In the first fix books, every thing has been demonftrated with a fcrupulous accuracy; and it was at firft defigned that the fame method fhould have been obferved throughout; but this, in treating of the folids, was found incompatible with the plan of the work, it being here fcarcely poffible to follow the strict principles of EUCLID without becoming prolix and obfcure. 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 perfpicuous than the former, and equally fatisfactory and convincing to the mind; especially in the way it is here given, which is fomething lefs 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 confift chiefly of improvements and emendations which are too obvious to efcape the notice of the reader, any further account of them would be unneceffary. It is fufficient to observe that much time and attention

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attention have been bestowed upon the work; and that nothing which was judged effential to the science, or useful in facilitating its attainment, has been omitted. The acknowledged intricacy of fome propofitions in the fifth and fixth books, made it neceffary to abridge that part of the fubject more confiderably than the former; but it is conceived that what is here given will be fully fufficient to anfwer all the purposes of the learner.

To avoid critical objections were a vain endeavour: they may be made against every fyftem 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 prefent work, where many things of this kind are pointed out which have hitherto escaped notice. These were added chiefly for the information of young ftudents, and ought to be carefully confulted by thofe who wish to obtain a just idea of the science, and the principles upon which it is founded.

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