1 3 ing the mind, and establishing a habit of close thinking and just reafoning, in every enquiry after truth, is far fuperior to all the dialectical principles that have yet been invented; the fimplicity of its first 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 fubject can boast. * For thefe reafons, it was judged neceffary to adhere as closely as poffible to the plan of the original Elements; this being, in many respects, 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 strict abridgment of EUCLID, some 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. 21 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; efpecially in the way it is here given, which is fomething less exceptionable than that of CAVALERIUS, by whom it was first intro'duced. ༢༠ མ་རྟ ~15 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 escape the notice of the reader, any further account of them would be unneceffary. It is fufficient to obferve that much time and 1 attention attention have been bestowed upon the work; and that nothing which was judged effential to the fcience, 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 subject more confiderably 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 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. 1. A Solid is that which has length, breadth and thickness. 2. A Superficies is one of the bounds of a folid, and has length and breadth without thickness. 3. A Line is one of the bounds of a fuperficies, and has length without breadth or thickness. 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 fame direction. 6. A plane fuperficies is that which is every where perfectly flat and even. 7. A plain rectilineal angle is the inclination or opening of two right lines which meet in a point. ? 8. One right line is faid to be perpendicular to another, when it makes the angles on both fides of it equal to each other. I 9. A right angle is that which is made by two right lines that are perpendicular to each other. 10. An obtufe angle is that which is greater than a right angle. 11. An acute angle is that which is less than a right angle. کا |