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

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. What is defined here, the limit or the surface or neither, in which case it states a property of solids? See I defs. 3, 6. 3. A straight-line is upright to a plane whenever it makes right angles to all the lines touching the straight line itself and which-are on the plane. 4. A plane is upright to a plane whenever the straight-lines drawn on one of the planes at right-angles to the common section of the planes are at right-angles to the remaining plane. 5. Whenever a perpendicular is drawn from the limit of the line high-up 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, an inclination of a straight-line to a plane is the angle enclosed by the drawn-line and the standing-line. The inclination is an acute angle, since it is an angle in a right triangle. If the line were perpendicular, a line could not be drawn from its perpendicular to its other end point, as they would be the same. See xi 13. 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 on each of the planes. There is a presumption that all such angles will be equal. See Heath. 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. This is much more terse (one word) than the definition of parallel lines in I def. 23. 9. Similar solid figures are those enclosed by similar planes equal in plêthos. The definition is inadequate. Also, no definition of solid figure has been provided. But see I def. 14 and note. 10. Equal and similar solid figures are those enclosed by similar planes equal in plêthos 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 definition 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. In the context of Elements, the only reason to include these species is that they will form the basis for the theory of conic sections. 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:

Prop. 1: There is not any part of a straight line that is on the underlying plane and any part in a region higher-up.

Prop. 2: If two straight-lines cut one another, they are on one plane, and every triangle is on one plane.

Prop. 3: If two planes cut one another, the common section of them is straight.

Prop. 4: If to two straight-lines cutting one another a straight-line stands-upon their common section at right-angles, it will also be at right angles to the plane through them.

Prop. 5: If a straight-line stands at right-angles on the common cut of three lines touching one another, the three lines are on one plane.

Prop. 6: If two straight-lines are at right-angles to the same plane, the straight-lines will be parallel.

Prop. 7: If two straight-lines are parallel, but points, as happen, are taken on each of them, the straight-line joining the points is on the same plane as the parallels.

Prop. 8: If two straight-lines are parallel, but one of them is at right-angles to some plane, the remaining line will also be at right-angles to the same plane.

Prop. 18: If a straight-line is at right-angles to some plane, all the planes through it will also be at right-angles to the same plane.

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