# Definition:Euclid's Definitions - Book XI

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## Euclid's Definitions: Book $\text{XI}$

These definitions appear at the start of Book $\text{XI}$ of Euclid's *The Elements*.

- A
**solid**is that which has length, breadth, and depth. - An extremity of a solid is a surface.
- A
**straight line**is**at right angles to a plane**when it makes right angles with all the straight lines which meet it and are in the plane. - A
**plane**is**at right angles to a plane**when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane. - The
**inclination of a straight line to a plane**is, assuming a perpendicular drawn from the extremity of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the extremity of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up. - The
**inclination of a plane to a plane**is the acute angle contained by the straight lines drawn at right angles to the common section at the same point, one in each of the planes. - A plane is said to be
**similarly inclined**to a plane as another is to another when the said angles of the inclinations are equal to one another. **Parallel planes**are those which do not meet.**Similar solid figures**are those contained by similar planes equal in multitude.**Equal and similar solid figures**are those contained by similar planes equal in multitude and in magnitude.- A
**solid angle**is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines.

Otherwise: A**solid angle**is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point. - A
**pyramid**is a solid figure, contained by planes, which is constructed from one plane to one point. - A
**prism**is a solid figure contained by planes of which, namely those which are opposite, are equal, similar and parallel, while the rest are parallelograms. - When, the diameter of a semicircle remaining fixed, the semicircle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a
**sphere**. - The
**axis of the sphere**is the straight line which remains fixed about which the semicircle is turned. - The
**center of the sphere**is the same as that of the semicircle. - A
**diameter of the sphere**is any straight line drawn through the centre and terminated in both directions by the surface of the sphere. - When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a
**cone**.

And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be**right-angled**; if less,**obtuse-angled**; and if greater,**acute-angled**. - The
**axis of the cone**is the straight line which remains fixed and about which the triangle is turned. - And the
**base**is the circle described by the straight line which is carried round. - When, one side of those about the right angle in a rectangular parallelogram remaining fixed, the parallelogram is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a
**cylinder**. - The
**axis of the cylinder**is the straight line which remains fixed and about which the parallelogram is turned. - And the
**bases**are the circles described by the two sides opposite to one another which are carried round. **Similar cones and cylinders**are those in which the axes and the diameters of the bases are proportional.- A
**cube**is a solid figure contained by six equal squares. - An
**octahedron**is a solid figure contained by eight equal and equilateral triangles. - An
**icosahedron**is a solid figure contained by twenty equal and equilateral triangles. - A
**dodecahedron**is a solid figure contained by twelve equal, equilateral, and equiangular pentagons.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Definitions