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In the first fix books, every thing has been demonftrated with a fcrupulous accuracy; and it was at first designed that the fame. method should 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 ftric 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 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 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 attention

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 students, and ought to be carefully confulted by those who wish to obtain a juft idea of the science, and the principles upon which it is founded.

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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 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.

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 fame direction.

6. A plane fuperficies is that which is everywhere perfectly flat and even.

7. A plane 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.

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 lefs than a right angle.

12. A figure is that which is inclofed by one or more boundaries.

13. A circle is a plane figure, contained by one line, called the circumference, which is every where equally diftant from a point within the figure, called the centre.

14. Rectilineal figures are those which are contained by right lines.

15. All plane figures, bounded by three right lines, are called triangles.

16. An equilateral triangle, is that which has all its fides equal to each other.

17. An ifofceles triangle, is that which has only two of its fides equal to each other.

18. A right-angled triangle, is that which has one right angle; the fide which is oppofite to the right angle being called the hypotenuse.

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