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Euclid, Elements, Book XI, Definitions©
translated by Henry Mendell

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Propositions

Definitions Brief comments

1. A solid is what has length and width and depth,

2. and a limit of a solid is a surface.

3. A straight-line is at right angles to a plane whenever it makes right angles to all the lines touching the straight line itself and which are in the plane.

4. A plane is at right angles to a plane whenever the straight-lines drawn at right angle to the common section of the planes in one of the planes are at right angles to the other plane.

5. An inclination of a straight-line to a plane is, whenever a perpendicular is drawn from the limit of the line above the plane to the plane and a straight-line is joined from the point which comes about to the limit of the line on the plane, the angle enclosed by the drawn line and the standing line.

6. An inclination of a plane to a plane is the acute angle enclosed by the lines drawn at right angles to the common section at the same point in each of the planes.

7. A plane is said to be similarly inclined to a plane, i.e. one to another, whenever the just mentioned angles of the inclinations are equal to one another.

8. Parallel planes are those which do not meet.

9. Similar solid figures are those enclosed by similar planes equal in number.

10. Equal and similar solid figures are those enclosed by similar planes equal in number and in magnitude.

11. A solid angle is the inclination by more than two lines touching one another and which are not in the same surface as all the lines. Alternatively, a solid angle is that enclosed by more than two plane angles which are not in the same plane and are constructed together at one point.

12. A pyramid is a solid figure enclosed by planes constructed from one plane to one point.

13. Prism is a solid figure enclosed by planes constructed from one plane at one point.

14. A sphere is the figure which results when with the diameter of the semicircle staying in place the semicircle is rotated and arrives at the same spot from which it began to be moved, The definiiton of a sphere is a generative definition

15. and an axis of the sphere is the straight-line which remains in place about which the semicircle is turned,

16. and a center of the sphere is the same as the center of the semicircle,

17. and a diameter of the sphere is a certain straight-line drawn through the center and having its limits on each side of the surface of the sphere.

18. A cone is the figure which results when with one side of a right-angled triangle staying in place (of the sides of about the right angle) the triangle is rotated and arrives at the same spot from which it began to be moved, The definiiton of a cone is a generative definition

and if the straight-line which stays in place is equal to the other side about the right angle which is rotated, the cone is right-angled, while if it is less the cone is obtuse-angled, while if it is more the cone is acute-angled, Note that the definition of the three species is in terms of the legs of the generating triangle and not the angles.

19. and an axis of the cone is the line which states in place about which the triangle is turned,

20. and a base is the circle which is inscribed by the rotating line.

21. A cylinder is the figure which results when with one side of a rectangular parallelogram staying in place (of the sides of about the right angle) the parallelogram is rotated and arrives at the same spot from which it began to be moved, The definiiton of a cylinder is a generative definition

22. and axis of the cylinder is the straight line which stays in place about which the parallelogram turns,

23. and bases are the circles which are inscribed by the two opposite lines drawn around.

24. Cones and cylinders are similar whose axes and diameters of their bases are proportional.

25. A cube is a solid figure enlosed by six equal squares.

26. An octahedron is a solid figure enclosed by eight triangles that are equal and equal-sided.

27. An icosohedron is a solid figure enclosed by twenty triangles that are equal and equal-sided.

28. A dodecahedron is a solid figure encosed by twelve pentagons that are equal and equal-sided and equal-angled.

Propositions:

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	Definitions	Brief comments
1.	A solid is what has length and width and depth,
2.	and a limit of a solid is a surface.
3.	A straight-line is at right angles to a plane whenever it makes right angles to all the lines touching the straight line itself and which are in the plane.
4.	A plane is at right angles to a plane whenever the straight-lines drawn at right angle to the common section of the planes in one of the planes are at right angles to the other plane.
5.	An inclination of a straight-line to a plane is, whenever a perpendicular is drawn from the limit of the line above the plane to the plane and a straight-line is joined from the point which comes about to the limit of the line on the plane, the angle enclosed by the drawn line and the standing line.
6.	An inclination of a plane to a plane is the acute angle enclosed by the lines drawn at right angles to the common section at the same point in each of the planes.
7.	A plane is said to be similarly inclined to a plane, i.e. one to another, whenever the just mentioned angles of the inclinations are equal to one another.
8.	Parallel planes are those which do not meet.
9.	Similar solid figures are those enclosed by similar planes equal in number.
10.	Equal and similar solid figures are those enclosed by similar planes equal in number and in magnitude.
11.	A solid angle is the inclination by more than two lines touching one another and which are not in the same surface as all the lines. Alternatively, a solid angle is that enclosed by more than two plane angles which are not in the same plane and are constructed together at one point.
12.	A pyramid is a solid figure enclosed by planes constructed from one plane to one point.
13.	Prism is a solid figure enclosed by planes constructed from one plane at one point.
14.	A sphere is the figure which results when with the diameter of the semicircle staying in place the semicircle is rotated and arrives at the same spot from which it began to be moved,	The definiiton of a sphere is a generative definition
15.	and an axis of the sphere is the straight-line which remains in place about which the semicircle is turned,
16.	and a center of the sphere is the same as the center of the semicircle,
17.	and a diameter of the sphere is a certain straight-line drawn through the center and having its limits on each side of the surface of the sphere.
18.	A cone is the figure which results when with one side of a right-angled triangle staying in place (of the sides of about the right angle) the triangle is rotated and arrives at the same spot from which it began to be moved,	The definiiton of a cone is a generative definition
	and if the straight-line which stays in place is equal to the other side about the right angle which is rotated, the cone is right-angled, while if it is less the cone is obtuse-angled, while if it is more the cone is acute-angled,	Note that the definition of the three species is in terms of the legs of the generating triangle and not the angles.
19.	and an axis of the cone is the line which states in place about which the triangle is turned,
20.	and a base is the circle which is inscribed by the rotating line.
21.	A cylinder is the figure which results when with one side of a rectangular parallelogram staying in place (of the sides of about the right angle) the parallelogram is rotated and arrives at the same spot from which it began to be moved,	The definiiton of a cylinder is a generative definition
22.	and axis of the cylinder is the straight line which stays in place about which the parallelogram turns,
23.	and bases are the circles which are inscribed by the two opposite lines drawn around.
24.	Cones and cylinders are similar whose axes and diameters of their bases are proportional.
25.	A cube is a solid figure enlosed by six equal squares.
26.	An octahedron is a solid figure enclosed by eight triangles that are equal and equal-sided.
27.	An icosohedron is a solid figure enclosed by twenty triangles that are equal and equal-sided.
28.	A dodecahedron is a solid figure encosed by twelve pentagons that are equal and equal-sided and equal-angled.