Page:First six books of the elements of Euclid 1847 Byrne.djvu/28

xxiv The twelfth axiom may be expressed in any of the following ways:

1. Two diverging straight lines cannot be both parallel to the same straight line.

2. If a straight line intersect one of the two parallel straight lines it must also intersect the other.

3. Only one straight line can be drawn through a given point, parallel to a given straight line. Geometry has for its principal objects the exposition and explanation of the properties of figure, and figure is defined to be the relation which subsists between the boundaries of space. Space or magnitude is of three kinds, linear, superficial, and solid.

Angles might properly be considered as a fourth species of magnitude. Angular magnitude evidently consists of parts, and must therefore be admitted to be a species of quantity The student must not suppose that the magnitude of an angle is affected by the length of the straight lines which include it, and of whole mutual divergence it is the measure. The vertex of an angle is the point where the sides or the legs of the angle meet, as A.

An angle is often designated by a single letter when its legs are the only lines which meet together at its vertex. Thus the red and blue lines form the yellow angle, which in other systems would be called the angle A. But when more than two lines meet in the same point, it was necessary by former methods, in order to avoid confusion, to employ three letters to designate an angle about that point,