EUCLID'S ELEMENTS.

BOOK I.

DEFINITIONS.

1. A POINT is that which has no magnitude, or is no part of any thing.

2. A line is length without breadth.

3. The extremities of a line are points.

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

5. A superficies is that which has only length and breadth.

6. The extremities of a superficies are lines.

7. A plane superficies is that which lies evenly between its lines.

8. A plane angle is the mutual inclination of two lines to one another in the same plane, so touching each other as not both to lie in the same right line.

9. When the lines containing the said angle are right lines, it is called a rectilineal angle.

10. When a right line standing on another right line, makes the adjacent angles equal to one another, each of the equal angles is a right angle, and the right line standing on the other is called a perpendicular.

11. An obtuse angle is that which is greater than a right angle.

12. An acute angle is that which is less than a right angle.

PART I.

B

13. A term is the extremity of any thing.

14. A figure is that which is contained under one or

more terms.

15. A circle is a plane figure contained by one line, which is called the circumference, to which all right lines drawn from one point within the figure are equal to one another.

16. And this point is called the centre of the circle. 17. A diameter of a circle is a certain right line drawn through the centre, and terminated both ways by

the circumference of the circle, and divides the circle into two equal parts.

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

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

20. Triangles are such as are contained by three right lines.

21. Quadrilateral, by four right lines.

22. Multilateral figures, or polygons, by more than four right lines.

23. Of trilateral figures, an equilateral triangle is that which has three equal sides.

24. An isosceles triangle is that which has only two equal sides.

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

26. Of three sided figures, a right angled triangle is that which has a right angle.

* The segment of a circle which is defined in this place, I have purposely omitted, as being of no use, until the third book, where the definition is repeated; instead of this Proclus has given in his Commentaries the following. The centre of the semicircle is the same with that of the circle; but as this is never used in the Elements, I have thought proper to reject it like

27. An obtuse angled triangle is that which has an obtuse angle.

28. An acute angled triangle is that which has three acute angles.

29. Of four sided figures, a square is that which has its sides equal, and its angles right

angles.

30. An oblong is that which has its angles right angles, but all its sides not equal.

31. A rhombus has its sides equal, but its angles not right angles.

32. A rhomboid has its opposite sides and
angles equal to one another, but all

its sides are not equal, nor its angles
right angles.

33. All other four sided figures besides these are called trapeziums.

34. Parallel right lines are those which are in the same plane, and being infinitely produced either way, do not meet one another."

POSTULATES.

1. Grant, that a right line may be drawn from any one point to any other point.

2. That a finite right line may be produced directly forwards.

3. That a circle may be described with any distance and from any centre.

4. That all right angles are equal to one another.† 5. That if a right line falling on two right lines make the interior angles at the same parts less than two right angles; these right lines being continually produced shall meet on that side where the angles are less than two right angles.

6. That two right lines cannot inclose a space.

Newton in lemma 22, book 1, of his Principia, says, that parallels are such lines as tend to a point infinitely distant.

+ For a demonstration of this, see Legendre's Geometry, proposition 1, book 1.

13. A term is the extremity of any thing.

14. A figure is that which is contained under one or

more terms.

15. A circle is a plane figure contained by one line, which is called the circumference, to which all right lines drawn from one point within the figure are equal to one another.

16. And this point is called the centre of the circle. 17. A diameter of a circle is a certain right line drawn through the centre, and terminated both ways by the circumference of the circle, and divides the circle into two equal parts.

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

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

20. Triangles are such as are contained by three right lines.

21. Quadrilateral, by four right lines.

22. Multilateral figures, or polygons, by more than four right lines.

23. Of trilateral figures, an equilateral triangle is that which has three equal sides.

24. An isosceles triangle is that which has only two equal sides.

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

26. Of three sided figures, a right angled triangle is that which has a right angle.

The segment of a circle which is defined in this place, I have purposely omitted, as being of no use, until the third book, where the definition is repeated; instead of this Proclus has given in his Commentaries the following. The centre of the semicircle is the same with that of the circle; but as this is never used in the Elements, I have thought proper to reject it like

27. An obtuse angled triangle is that which has an obtuse angle.

28. An acute angled triangle is that which has three acute angles.

29. Of four sided figures, a square is that which has its sides equal, and its angles right

angles.

30. An oblong is that which has its angles right angles, but all its sides not equal.

31. A rhombus has its sides equal, but its angles not right angles.

32. A rhomboid has its opposite sides and angles equal to one another, but all its sides are not equal, nor its angles right angles.

33. All other four sided figures besides these are called trapeziums.

34. Parallel right lines are those which are in the same plane, and being infinitely produced either way, do not meet one another.*

POSTULATES.

1. Grant, that a right line may be drawn from any one point to any other point.

2. That a finite right line may be produced directly forwards.

3. That a circle may be described with any distance and from any centre.

4. That all right angles are equal to one another.† 5. That if a right line falling on two right lines make the interior angles at the same parts less than two right angles; these right lines being continually produced shall meet on that side where the angles are less than two right angles.

6. That two right lines cannot inclose a space.

Newton in lemma 22, book 1, of his Principia, says, that parallels are such lines as tend to a point infinitely distant.

+ For a demonstration of this, see Legendre's Geometry, proposition 1, book 1.