the Elements of Euclid
1. | A point is that which has no parts. | |
2. | A line is length without breadth. | |
3. | The extremities of a line are points. | |
4. | A straight or right line is that which lies evenly between its extremities. | |
5. | A surface is that which has length and breadth only. | |
6. | The extremities of a surface are lines. | |
7. | A plane surface is that which lies evenly between its extremities. | |
8. | A plane angle is the inclination of two lines to one another, in a plane, which meet together, but are not in the same direction. | |
9. | A plane rectilinear angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. | |
10. | When one straight line standing on another straight line makes the adjacent angles equal, each of these angles is called a right angle, and each of these lines is said to be perpendicular to the other. | |
11. | An obtuse angle is an angle greater than a right angle. | |
12. | An acute angle is less than a right angle. | |
13. | A term or boundary is the extremity of any thing. | |
14. | A figure is a surface enclosed on all sides by a line or lines. | |
15. | A circle is a plane figure, bounded by one continued line, called its circumference or periphery; and having a certain point within it, from which all straight lines drawn to its circumference are equal. | |
16. | This point (from which the equal lines are drawn) is called the centre of the circle. | |
17. | A diameter of a circle is a straight line drawn through the centre, terminated both ways in the circumference. | |
18. | A semicircle is the figure contained by the diameter, and the part of the circle cut off by the diameter. | |
19. | A segment of a circle is a figure contained by a straight line, and the part of the circumference which it cuts off. | |
20. | A figure contained by straight lines only, is called a rectilinear figure. | |
21. | A triangle is a rectilinear figure included by three sides. | |
22. | A quadrilateral figure is one which is bounded by four sides. The straight lines | |
23. | A polygon is a rectilinear figure bounded by more than four sides. | |
24. | A triangle whose three sides are equal, is said to be equilateral. | |
25. | A triangle which has only two sides equal is called an isosceles triangle. | |
26. | A scalene triangle is one which has no two sides equal. | |
27. | A right angled triangle is that which has a right angle. | |
28. | An obtuse angled triangle is that which has an obtuse angle. | |
29. | An acute angled triangle is that which has three acute angles. | |
30. | Of foursided figures, a square is that which has all its sides equal, and all its angles right angles. | |
31. | A rhombus is that which has all its sides equal, but its angles are not right angles. | |
32. | An oblong is that which has all its angles right angles, but has not all its sides equal. | |
33. | A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles. | |
34. | All other quadrilateral figures are called trapeziums. | |
35. | Parallel straight lines are such as are in the same plane, and which being produced continually in both directions, would never meet. |
I. | Let it be granted that a straight line may be drawn from any one point to any other point. |
II. | Let it be granted that a finite straight line may be produced to any length in a straight line. |
III. | Let it be granted that a circle may be described with any centre at any distance from that centre. |
I. | Magnitudes which are equal to the same are equal to each other. | |
II. | If equals be added to equals the sums will be equal. | |
III. | If equals be taken away from equals the remainders will be equal. | |
IV. | If equals be added to unequals the sums will be unequal. | |
V. | If equals be taken away from unequals the remainders will be unequal. | |
VI. | The doubles of the same or equal magnitudes are equal. | |
VII. | The halves of the same or equal magnitudes are equal. | |
VIII. | The magnitudes which coincide with one another, or exactly fill the same space, are equal. | |
IX. | The whole is greater than its part. | |
X. | Two straight lines cannot include a space. | |
XI. | All right angles are equal. | |
XII. | If two straight lines |
)
Describe and
(post. III) ;
draw and
(post. I).
Then will be equilateral.
For | (def. 15) ; | |||
and | (def. 15) ; | |||
(ax. I) ; |
and therefore is the equilateral triangle required.
Q. E. D.
),
)
Draw (post. I), describe
(pr. I.I),
produce (post. II),
describe (post. III),
(post. III) ;
produce (post. II), then
is the line required.
For | ||||
and | ||||
but (def. 15) |
||||
is equal to the given line |
Q. E. D.
),
)
Draw
(pr. 2.) ;
describe (post. 3.),
then
.
For | (def. 15.), | |||
and | (const.) ; | |||
(ax. I.). |
Q. E. D.