
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 straightline is upright to a plane whenever it makes
right angles to all the lines touching the straight line itself and whichare on the plane. 

4. 
A plane is upright to a plane whenever the straightlines
drawn on one of the
planes at rightangles to the common section of the planes are at rightangles to the remaining plane. 

5. 
Whenever a perpendicular is drawn from the limit of the line highup the plane to the
plane and a straightline is joined from the point whichcomesabout to
the limit of the line on the plane, an inclination of a straightline to a plane is the angle enclosed by the drawnline and the standingline. 
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 whichdonotmeet. 
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 straightline 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 straightline 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 rightangled
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 straightline which stays in place is equal to the other
side about the right angle which is rotated, the cone is rightangled,
while if it is less the cone is obtuseangled, while if it is more
the cone is acuteangled, 
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 equalsided. 

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

28. 
A dodecahedron is a solid figure encosed by twelve pentagons
that are equal and equalsided and equalangled. 
