iv

In overcoming the difficulties of establishing this new principle, which will appear consider. able 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.

THOMAS ELRINGTON.

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.

THE

ELEMENTS OF EUCLID.

BOOK I.

DEFINITIONS.

1. A point is that which hath no parts. 2. A line is length without breadth.

3. The extremities of a line are points.

See Notes.

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 breadth.

6. The extremities of a surface are lines.

Fig. 1.

7. A plane surface is that which lies evenly between See N. 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 not Fig. 2. 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;

B

Fig. 4.

Fig. 4.

See N.

Fig. 5.

See N.

Fig. 6.

Fig. 7.

Fig. 8.

Fig. 8.

Fig. 9.

Fig. 7.

Fig. 10.

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.

30. A parallelogram is a quadrilateral figure, whose Fig. 11. opposite sides are parallel.

31. A square is a quadrilateral figure, which has all

its sides equal, and all its angles right angles.

32. Rectilineal figures, which have more than four sides, are called polygons.

POSTULATES.

Let it be granted,

1. That a right line may be drawn from any one point to any other.

2. That a terminated right line may be produced to any length in a right line.

3. That a circle may be described from any centre, See N. at any distance from that centre.

AXIOMS.

1. Things, which are equal to the same, are equal to one another.

2. If equals be added to equals, the wholes are equal. 3. If equals be taken from equals, the remainders are equal.

4. If equals be added to unequals, the wholes are unequal.

5. If equals be taken from unequals, the remainders are unequal.

6. Things, which are double of the same or of equals, are equal to one another.

7. Things, which are halves of the same or of equals, are equal to one another.

8. Magnitudes, which coincide with one another, are equal to one another.

9. The whole is greater than its part.

10. Two right lines cannot inclose a space.

11. All right angles are equal to one another.

12. If a right line meet two right lines, so as to make See N. the two internal angles on the same side of it taken together less than two right angles; these right lines, being continually produced, shall at length meet upon that side on which are the angles, which are less than two right angles.

B 2

Fig. 12.
See N.

PROPOSITION I. PROBLEM.

To describe an equilateral triangle upon a given finite right line (AB).

From the centre A, with the radius AB, describe the (1) Post. 3. circle BCD (1), and from the centre B, with the radius BA, describe the circle ACE. From the point of intersection C, draw the right lines CA and CB to the ex

(2) Post. 1. tremities of the given right line (2).

It is evident that ACB is a triangle constructed upon the given line; but it is also equilateral: for the right line AC is equal to AB, as they are radii of the same circle (3) Def. 13. DCB (3); and the right line BC is equal to BA, as they are radii of the same circle ACE. Since then both the lines AC and BC are equal to the same AB, they must (4) Ax. 1. be equal to one another (4), and therefore the triangle ACB is equilateral.

Schol. Draw the lines AG and GB; and in the same manner it can be demonstrated that the triangle AGB is equilateral.

PROP. II. PROB.

Fig. 13.
See N.

(1) Post. 1. (2) Prop. 1. (3) Post. 3.

(4) Def. 15. (5) Con

struct.

(6) Ax. 3.

From a given point (A) to draw a right line equal to a given finite right line (BC).

From the given point A draw a right line AB to either extremity B of the given line (1). Upon AB construct an equilateral triangle ADB (2). From the centre B, with the radius BC, describe a circle GCF (3), and produce the right line DB, until it meets the circumference in G. From the centre D, with the radius DG, describe the circle GLO; and produce the right line DA, until it meets the circumference in L. The right line AL is equal to the given line BC.

For the right line DL is equal to DG, because they are radii of the same circle GLO (4); and if the equals DA and DB (5) be taken away from both, the remainders AL and BG are also equal (6); but BG is equal to BC, as they are radii of the same circle GCF; therefore the right lines AL and BC, which are equal