GEOMETRY.

INTRODUCTION.

1. If a block of wood or stone is cut in the shape represented in Fig. 1, it will have six flat faces.

Each face of the block is called a surface; and if the faces are made smooth by polishing, so that, when a straight edge is applied to any one of them, the straight edge in every part will touch the surface, the faces are called plane surfaces, or planes.

FIG. 1.

2. The intersection of any two of these surfaces is called a line.

3. The intersection of any three of these lines is called a point.

4. The block extends in three principal directions:

From left to right, A to B.

From front to back, A to C.

From top to bottom, A to D.

These are called the dimensions of the block, and are named in the order given, length, breadth (or width), and thickness (height or depth).

5. A solid, in common language, is a limited portion of space filled with matter; but in Geometry we have nothing to do with the matter of which a body is composed; we study simply its shape and size; that is, we regard a solid as a limited portion of space which may be occupied by a physical body, or marked out in some other way. Hence,

A geometrical solid is a limited portion of space.

6. The surface of a solid is simply the boundary of the solid, that which separates it from surrounding space. The surface is no part of a solid and has no thickness. Hence,

A surface has only two dimensions, length and breadth.

7. A line is simply a boundary of a surface, or the intersection of two surfaces. Since the surfaces have no thickness, a line has no thickness. Moreover, a line is no part of a

surface and has no width. Hence,

A line has only one dimension, length.

8. A point is simply the extremity of a line, or the intersection of two lines. A point, therefore, has no thickness, width, or length; therefore, no magnitude. Hence,

A point has no dimension, but denotes position simply.

9. It must be distinctly understood at the outset that the points, lines, surfaces, and solids of Geometry are purely ideal, though they are represented to the eye in a material way. Lines, for example, drawn on paper or on the blackboard, will have some width and some thickness, and will so far fail of being true lines; yet, when they are used to help the mind in reasoning, it is assumed that they represent true lines, without breadth and without thickness.

10. A point is represented to the eye by a fine dot, and named by a letter, as A (Fig. 2). A line is named by two letters, placed one at each end, as BF.

A surface is represented and named by the lines which bound it, as BCDF.

Α

solid is represented by the faces which bound it.

B

A

FIG. 2.

11. A point in space may be considered by itself, without reference to a line.

12. If a point moves in space, its path is a line. This line may be considered apart from the idea of a surface.

13. If a line moves in space, it generates, in general, a surface. A surface can then be considered apart from the idea of a solid.

14. If a surface moves in space, it generates, in general, a solid.

B

D

H

E

G

F

FIG. 3.

Thus, let the upright surface ABCD (Fig. 3) move to the right to the position EFGH, the points A, B, C, and D generating the lines AE, BF, CG, and DH, respectively. The lines AB, BC, CD, and DA will generate the surfaces AF, BG, CH, and DE, respectively. The surface ABCD will generate the solid AG.

15. Geometry is the science which treats of position, form, and magnitude.

16. A geometrical figure is a combination of points, lines, surfaces, or solids.

17. Plane Geometry treats of figures all points of which are in the same plane.

Solid Geometry treats of figures all points of which are not in the same plane.

GENERAL TERMS.

18. A proof is a course of reasoning by which the truth or falsity of any statement is logically established.

19. An axiom is a statement admitted to be true without proof.

20. A theorem is a statement to be proved.

21. A construction is the representation of a required figure by means of points and lines.

22. A postulate is a construction admitted to be possible.

23. A problem is a construction to be made so that it shall satisfy certain given conditions.

24. A proposition is an axiom, a theorem, a postulate, or a problem.

25. A corollary is a truth that is easily deduced from known truths.

26. A scholium is a remark upon some particular feature of a proposition.

27. The solution of a problem consists of four parts:

1. The analysis, or course of thought by which the construction of the required figure is discovered.

2. The construction of the figure with the aid of ruler and compasses.

3. The proof that the figure satisfies all the conditions.

4. The discussion of the limitations, if any, within which the solution is possible.

28. A theorem consists of two parts: the hypothesis, or that which is assumed; and the conclusion, or that which is asserted to follow from the hypothesis.

29. The contradictory of a theorem is a theorem which must be true if the given theorem is false, and must be false if the given theorem is true. Thus,

Its contradictory: If A is B, then C is not D.

30. The opposite of a theorem is obtained by making both the hypothesis and the conclusion negative. Thus,

A theorem:

Its opposite:

If A is B, then C is D. ·

If A is not B, then C is not D.

31. The converse of a theorem is obtained by interchanging the hypothesis and conclusion. Thus,

A theorem:

Its converse:

If A is B, then C is D.

If C is D, then A is B.

32. The converse of a truth is not necessarily true.

Thus, Every horse is a quadruped is true, but the converse, Every quadruped is a horse, is not true.

33. If a direct proposition and its opposite are true, the converse proposition is true; and if a direct proposition and its converse are true, the opposite proposition is true.

Thus, if it were true that

1. If an animal is a horse, the animal is a quadruped;

2. If an animal is not a horse, the animal is not a quadruped; it would follow that

3. If an animal is a quadruped, the animal is a horse.

Moreover, if 1 and 3 were true, then 2 would be true,