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It is, therefore, the design of the following performance, to obviate these objections, and to render the subject more familiar and perspicuous, without weakening its evidence, or destroying its elegance and simplicity. 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 essential 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

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 precepts that have yet been invented ; the simplicity of its first principles ; the clearness and certainty of its demonstrations; the regular concatenation of its parts ; and the univerfality of its application being such as no other subject can boasta

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, some 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.

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In the first fix books, every thing has been demonstrated with a scrupulous accuracy ; and it was at first designed that the fame. method should have been observed throughout; but this, in treating of the solids, 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 formance, 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 sufficient to observe that much time and

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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 some propositions in the fifth and fixth books, made it necessary to abridge that part of the subject more considerably than the former ; but it is conceived that what is here given will be fully sufficient to answer all the purposes of the learner.

To avoid critical objections were a vain endeavour : they may be made against every system 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.

E L E.

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1. A Solid is that which has length, breadth and thickness,

2. A Superficies is one of the bounds of a solid, and has length and breadth without thickness.

3. A Line is one of the bounds of a superficies, and has length without breadth or thickness.

4. A Point is one of the extremities of a line, and has neither length, breadth, nor thickness.

5. A right line is that which has all its parts lying in the same direction.

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