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1. A Geometric Solid is a portion of space. The space occupied by a material or physical solid is a geometric solid.
2. The Surface of a solid is its faces; by its surface the solid is separated from the surrounding space.
3. The intersection of two surfaces is called a line.
4. The intersection of two lines is called a point.
5. A point moving generates a line; a line moving, not in its own direction, generates a surface; a surface moving, not in its own plane or upon itself, generates a solid. 6. The Dimensions of a solid are its length, breadth, and thick
Extent is measured in one or more of these dimensions. 7. The Science of Geometry treats of extension, making use of geometrical figures; the properties, measurement, and construction of these figures are considered.
8. A Point has position, but not extent. 9. A Line has length, without breadth or thickness. 10. A Straight line, or Right line, is one of constant direction, or rather of two directions exactly opposite; as AB or BA.
11. A Curved line is one of continually changing direction; as CD.
с 12. A Surface has length and breadth, but no thickness.
13. A Plane is such a surface that a straight line joining any two of its points lies wholly in the surface.
14. A Solid has length, breadth, and thickness.
15. A Theorem is a statement of a truth to be proved. The hypothesis of a theorem is the conditional part of the statement, that which is assumed to be true; the conclusion is that which is to be proved.
16. A Problem is a statement of something to be done.
17. A Proposition is a general term for either a theorem or a problem.
18. A Corollary is a secondary or derived theorem, suggested by the theorem to which it is attached, and is usually proved after the same manner as the theorem itself.
19. A Scholium is a remark appended to a proposition. 20. An Axiom is a truth assumed as self-evident.
1. Things which are equal to the same thing are equal to each other.
It follows that, a. Things equal to equal things are equal to each other. b. Things that may coincide throughout are equal. c. In an equation, or in any other algebraic expression, equals may be substituted for equals.
2. The whole is equal to the sum of all its parts, and is therefore greater than any of its parts.
3. Equals added to equals give equals.