ENUNCIATIONS OF THE PROPOSITIONS IN EUCLID.

BOOK I.

1. To describe an equilateral triangle on a given finite straight line.

2. From a given point to draw a straight line equal to a given straight line.

3. From the greater of two given straight lines to cut off a part equal to the less.

4. If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal, and their other angles shall be equal, each to each, namely those to which the equal sides are opposite.

5. The angles at the base of an isosceles triangle are equal to one another, and, if the equal sides be produced, the angles on the other side of the base shall be equal to one another.

6. If two angles of a triangle be equal, the sides also which subtend, or are opposite to the equal angles, shall be equal to one another.

7. On the same base and on the same side of it there cannot be two triangles having their sides, which are terminated in one extremity of the base, equal to one

another, and likewise those which are terminated at the other extremity, equal to one another.

8. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal, the angle which is contained by the two sides of the one shall be equal to the angle which is contained by the two sides, equal to them, of

the other.

9. To bisect a given rectilineal angle-that is, to divide it into two equal parts.

10. To bisect a given finite straight line—that is, to divide it into two equal parts.

11. To draw a straight line at right angles to a given straight line from a given point in the same.

12. To draw a straight line perpendicular to a given straight line of unlimited length from a given point without it.

13. The angles which one straight line makes with another straight line on one side of it are either two right angles or are together equal to two right angles.

14. If at a point in a straight line two other straight lines on opposite sides of it make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line.

15. If two straight lines cut one another, the vertical or opposite angles are equal.

16. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior and opposite angles.

17. Any two angles of a triangle are together less than two right angles.

18. The greater side of every triangle has the greater angle opposite to it.

19. The greater angle of every triangle is subtended by the greater side or has the greater side opposite it.

20. Any two sides of a triangle are together greater than the third side.

21. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.

22. To make a triangle of which the sides shall be equal to three given straight lines, any two of which are together greater than the third.

23. At a given point in a given straight line to make an angle equal to a given rectilineal angle.

24. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them of the other, the base of that which has the greater angle shall be greater than the base of the other.

25. If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of the one greater than the base of the other, the angle contained by the sides of that which has the greater base, shall be greater than the angle contained by the sides, equal to them, of the other.

26. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, namely, either the sides adjacent to the equal angles or sides which are opposite to equal angles in each, then shall the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other.

27. If a straight line falling on two other straight lines makes the alternate angles equal to one another, the two straight lines shall be parallel.

28. If a straight line falling on two other straight lines makes the exterior angle equal to the interior and opposite angle on the same side of the line, or makes the interior angles on the same side together equal to two right angles, the two straight lines shall be parallel.

29. If a straight line fall on two parallel straight lines, it shall make the alternate angles equal, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles.

30. Straight lines which are parallel to the same straight line are parallel to one another.

31. To draw a straight line through a given point parallel to a given straight line.

32. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of every triangle are together equal to two right angles.

33. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselves equal and parallel.