If A is not B, then C is not D, (iv)

is termed the obverse of the typical Theorem (i). 6. Sometimes the hypothesis of a Theorem is complex, i.e., consists of several distinct hypotheses; in this case every Theorem formed by interchanging the conclusion and one of the hypotheses is a converse of the original Theorem.

7. The truth of a converse is not a logical consequence of the truth of the original Theorem, but requires independent investigation.

8. Hence the four associated Theorems (i) (ii) (iii) (iv)

resolve themselves into two Theorems that are independent of one another, and two others that are always and necessarily true if the former are true; consequently it will never be necessary to demonstrate geometrically more than two of the four Theorems, care being taken that the two selected are not contrapositive each of the other.

9. Rule of Conversion. If the hypotheses of a group of demonstrated Theorems be exhaustive-that is, form a set of alternatives of which one must be true; and if the conclusions be mutually exclusive—that is, be such that no two of them can be true at the same time, then the converse of every Theorem of the group will necessarily be true.

The simplest example of such a group is presented when

a Theorem and its obverse have been demonstrated; and the validity of the rule in this instance is obvious from the circumstance that the converse of each of

two such Theorems is the contrapositive of the other.
Another example, of frequent occurrence in the ele
ments of Geometry, is of the following type :-

If A is greater than B, C is greater than D.
If A is equal to B, C is equal to D.

If A is less than B, C is less than D.

Three such Theorems having been demonstrated the converse of each is necessarily true.

10. Rule of Identity. If there is but one A, and but one B; then from the fact that A is B it necessarily follows that B is A.

This rule may be frequently applied with great advantage in the demonstration of the converse of an established Theorem.

THE

ELEMENTS OF PLANE GEOMETRY

BOOK I

THE STRAIGHT LINE

DEFINITIONS

DEF. 1. A point has position, but it has no magnitude.

DEF. 2. A line has position, and it has length, but neither breadth nor thickness.

The extremities of a line are points, and the intersection of two lines is a point.

DEF. 3. A surface has position, and it has length and breadth, but not thickness.

The boundaries of a surface, and the intersection of two surfaces, are lines.

DEF. 4. A solid has position, and it has length, breadth and thickness.

The boundaries of a solid are surfaces.

DEF. 5. A straight line is such that any part will, however placed, lie wholly on any other part, if its extremities are made to fall on that other part.

DEF. 6. A plane surface, or plane, is a surface in which any

two points being taken the straight line that joins them lies wholly in that surface.

GEOMETRICAL AXIOMS

1. Magnitudes that can be made to coincide are equal.

2. Through two points there can be made to pass one, and only one, straight line: and this may be indefinitely prolonged either way.

Hence,

a. Any straight line may be made to fall on any other straight line with any given point on the one on any given point on the other;

B. Two straight lines which meet in one point cannot meet again unless they coincide.

SECTION I

ANGLES AT A POINT

[An angle is a simple concept incapable of definition, properly so-called, but the nature of the concept may be explained as follows, and for convenience of reference the explanation may be reckoned among the definitions.] DEF. 7. When two straight lines are drawn from the same point, they are said to contain, or to make with each other, a plane angle. The point is called the vertex, and the straight lines are called the arms, of the angle. A line drawn from the vertex and turning about the vertex in the plane of the angle from the position of coincidence with one arm to that of coincidence with the other is said to turn through the angle: and the angle is greater as the quantity of turning is greater.

Since the line may turn from the one position to the other in either of two ways, two angles are formed by two straight lines drawn from a point. These angles (which have a common vertex and common arms) are said to be conjugate. The greater of the two is called the major conjugate, and the smaller the minor conjugate, angle.

When the angle contained by two lines is spoken of without qualification, the minor conjugate angle is to be understood. It is seldom requisite to consider major conjugate angles before Book III.

When the arms of an angle are in the same straight line, the conjugate angles are equal, and each is then said to be a straight angle.

DEF. 8. When three straight lines are drawn from a point, if one of them be regarded as lying between the other two, the angles which this one (the mean) makes with the other two (the extremes) are said to be adjacent angles: and the angle between the extremes, through which a line would turn in passing from one extreme through the mean to the other extreme, is the sum of the two adjacent angles.

DEF. 9. The bisector of an angle is the straight line that divides it into two equal angles.

DEF. 10. When one straight line stands upon another straight line and makes the adjacent angles equal, each of the angles is called a right angle.

OBS. Hence a straight angle is equal to two right angles; or, a right angle is half a straight angle.

DEF. 11. A perpendicular to a straight line is a straight line that makes a right angle with it.