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MILLIKEN, Bookseller and Printer in Ordinary to His Majesty.



In preparing this edition, designed solely for the use of the Undergraduates in the University of Dublin, it was the object of the Editor to retain the original strictness of the demonstrations as given by Euclid, and at the same time to throw them into the most simple form. For this purpose the syllogistic mode of argument, in which all the direct proofs had usually been drawn up, has been abandoned, and the arrangement approved of by Locke adopted in its stead.. But the theory of Proportion in the fifth book has been entirely altered, for reasons assigned in the notes. To defend the definition of proportional quantities given by Euclid, it has been asserted, that there is no other principle from which the doctrine can be demonstrated; but that defence, it is hoped, must now be abandoned, as the demonstrations here given are derived from a more simple and natural definition.


In overcoming the difficulties of establishing this new principle, which will appear considerable to any person who examines the propositions of the fifth book, the Editor has to acknowledge the assistance afforded by his predecessor in the Professorship of Mathematics, the venerable Dr. Murray, who communicated to him the elegant and comparatively easy demonstrations of the 20th and 38th propositions; but such was his averseness from being known, that this obligation could not be acknowledged till after his death, which unfortunately followed but too soon his much desired and universally applauded promotion to the Provostship.

The corollaries and scholia have been selected. from the best commentators, with anxious care to retain whatever was useful, and to avoid encumbering the work with any thing superfluous. In the notes the motives for any changes deserving of notice are briefly assigned.

Trin. Coll. Dub.


N. B. This work has been translated from the last Latin Edition, with the consent of the Editor, and at the request of the Masters of several English schools in Ireland.







1. A point is that which hath no parts.
2. A line is length without breadth.
3. The extremities of a line are points.

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4. A right line is that which lies evenly between its Plate 1. extreme points.

5. A surface is that which hath only length and


Fig. 1.

6. The extremities of a surface are lines. 7. A plane surface is that which lies evenly be- See N. tween its extreme right lines.

8. A rectilineal angle is the inclination of two right See N. lines to one another, which meet together, but are Fig. 2. not in the same right line.

9. The sides of an angle are the lines which form See N. the angle.

10. The vertex of an angle is the point in which the sides meet one another.

An angle is expressed either by one letter placed at the vertex, or by three letters, of which the middle one is at the vertex, the others any where along the sides.

11. When a right line standing on another makes Fig 3. the adjacent angles (ABC and ABD) equal to one another, each of these angles is called a right angle;


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and the right line, which stands on the other, is called a perpendicular to it.

12. The angle (ABC) which is greater than a right angle, is called obtuse.

13. The angle (ABD) which is less than a right angle, is called acute.


14. A plane figure is a plane surface, which is bounded on all sides by one or more lines.

15. A circle is a plane figure bounded by one line, which is called the circumference, and is such that all right lines drawn from a certain point within the figure, to the circumference, are equal to one another. 16. And this point is called the centre of the circle. 17. A diameter of a circle is a right line drawn through the centre, and terminated both ways by the circumference.

18. A radius of a circle is a right line drawn from the centre to the circumference.

19. A semicircle is the figure contained by a diameter, and the part of the circumference cut off by the diameter.

20. A rectilineal figure is a plane surface bounded; by right lines.

21. A triangle is a rectilineal figure bounded by three right lines.

22 An equilateral triangle is that which has three equal sides.

23. An isosceles triangle is that which has two sides equal.

24. A scalene triangle is that which has three unequal sides.

25. A right angled triangle is that in which one of the angles is right.

26. An obtuse angled triangle is that in which one of the angles is obtuse.

27. An acute angled triangle is that in which the three angles are acute. >

28. Parallel right lines are those which, lying in the same plane, never meet on either side, though, indefinitely produced.

29. A quadrilateral figure is a rectilineal figure, which is bounded by four right lines.


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