Page:Popular Science Monthly Volume 77.djvu/464

458 lines as lines that are everywhere the same distance apart (Webster); others that they are lines in the same plane that will not meet however far produced; and others still that they are lines having the same direction. In one sense these definitions mean the same thing, but in the development of geometry they give rise to very different series of propositions.

Perhaps the most far-reaching of differences in definitions are found in those for parallel lines. Wentworth, following Chauvenet, says parallel lines are those having the same direction throughout their whole extent. This definition is very objectionable for two reasons; first, because the meaning of the word direction is ambiguous, the word being used to signify either one way or the exactly opposite, or in the sense of the angle a line makes with a standard line; second, because the idea of direction in the sense intended is difficult to explain. The Century Dictionary gives this definition: "The direction of point A from point B is or is not the same as that of another point C from point D, according as a straight line drawn from B to A and continued to infinity would or would not cut the celestial sphere at the same point as the straight line from D to C continued to infinity." Chauvenet and Wentworth thought they had found a way to simplify the definition of parallelism. It is clear from the preceding that what they did was to slur over a very complex concept. As a matter of fact the use of the word direction in trying to define parallels was not new. Thus Dr. Johnson defines parallels "as lines extended in the same direction, and preserving always the same distance." The definition used by Euclid, viz., lines in the same plane that will not meet, however far produced, is practically the best, and has the merit of preparing the student for the non-Euclidean geometry.

Not only have parallel lines been defined differently by different authors, but other important terms have met the same fate. The concept angle has been presented in three or four ways: (1) As the figure formed by two lines meeting, which is essentially a description, not a definition. (2) As the difference in direction of two lines. (3) As the inclination of one line to another. (4) As the amount of divergence of two lines that meet. The objection to the first is that it does not call attention to an angle as a magnitude, but rather as a shape. A recent author gives this definition and then asks on the next line whether increasing the lengths of the lines would increase the size of the angle? Of course it would if the pupil judged by the definition given. The use of the term direction to define angle is as objectionable as for parallels. The third definition, Euclid's, is better than the others, but not as clear as it might be on account of the meaning of the word inclination. Thus we are led to the fourth definition, which is objectionable chiefly on the ground that, following the usual custom in English,