THEORETICAL GEOMETRY

BOOK I

PRELIMINARY DEFINITIONS AND EXPLANATIONS

1. A point is that which has position but no size. The position of a point on the blackboard, or on paper, is represented by a mark. This mark has some small size and therefore only roughly represents the idea of a point.

2. A line is that which has length but neither breadth nor thickness.

Again, the mark that we use to represent a line has breadth and some small thickness, and consequently, only roughly represents the idea.

The intersection of two lines is a point.

3. Lines may be either straight or curved.

The following property distinguishes straight lines from curved lines and may be used as the definition of a straight line :

Two straight lines cannot have any two points of one coincide with two points of the other without the lines coinciding altogether.

This is sometimes stated as follows:-Joining two points there is always one and only one straight line.

It follows from this definition that two straight lines cannot enclose a space.

Can the circumferences of two equal circles coincide in two points without coinciding altogether?

4. A surface is that which has length and breadth but no thickness.

A sheet of tissue paper has length and breadth and very little thickness. It thus roughly represents the idea of a surface. In fact the sheet of paper has two well-defined surfaces separated by the substance of the paper.

The boundary between two parts of space surface.

5. Surfaces may be either plane or curved.

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The following property distinguishes plane surfaces from curved surfaces and may be used as the definition of a plane surface:—

The straight line joining any two points on a plane surface lies wholly on that surface.

Give examples of curved surfaces on which straight lines may be drawn in certain directions. Notice the force of the word "any" in the definition above.

6. A solid is that which has length, breadth and thickness.

7. Any combination of points, lines, surfaces and solids is called a figure.

8. Geometry is the science which investigates the properties of figures and the relations of figures to one another.

9. In Plane Geometry the figure, or figures, considered in each proposition are confined to one plane, while Solid Geometry treats of figures the parts of which are not all in the same plane.

Plane Geometry is also called Geometry of Two Dimensions (length and breadth), and Solid Geometry is called Geometry of Three Dimensions (length, breadth and thickness).

GEOMETRICAL REASONING

10. Two general methods of investigating the properties or relations of figures may be distinguished as the Practical Method and the Theoretical Method.

Some properties may be tested by measurement, paper-folding, etc., while in the same or other cases it may be shown that the property follows as a necessary result from others that are already known to be true.

The Theoretical Method, has certain advantages over the Practical method. Measurements, etc., are never exact, and in many cases cannot be made directly; but in the Theoretical Method, starting from certain simple statements, called axioms, the truth of which is selfevident, or, it may be in some cases, assumed, the consequent statements follow with absolute certainty.

The Practical Method is also known as the Inductive Method of Reasoning, and the Theoretical Method as the Deductive Method.

11. Figures may be compared by making a tracing of one of them and fitting the tracing on the other. In many cases the process may be made a mental operation and the comparison made with absolute certainty by means of the following axiom:

A figure may be, actually or mentally, transferred from one position to another without change of form or size.

When two figures are shown to be exactly equal in all respects by supposing one to be made to fit exactly on the other, the proof is said to be by the method of superposition.

Figures which exactly fill the same space are said to coincide with each other.

12. In general a proposition is that which is stated or affirmed for discussion.

In mathematics a proposition is a statement of either a truth to be demonstrated or of an operation to be performed. It is called a theorem when it is something to be proved, and a problem when it is a construction to be made.

Example of Theorem:-If two straight lines cut each other, the vertically opposite angles are equal. Example of Problem:-It is required to bisect a given straight line.

13. Theorems are commonly stated in two ways:First, the General Enunciation, in which the property is stated as true for all figures of a class, but without naming any particular figure, as in the first example given in § 12; second, the Particular Enunciation, in which the theorem is stated to be true of the particular figure in a certain diagram.