Page:Popular Science Monthly Volume 77.djvu/468

462 "volume is the space occupied as measured by cubic units, i. e., cu. ft., cu. in., etc.

The new dictionary, unlike the old, defines a right angle, not by a synonym, but by saying it is an angle included between two radii subtended by a quarter circle. Under angle, "right angle" is not given. Thus the new dictionary can not be criticized as the old was. But it does seem a pity that the dictionaries can not give the simple, plain essential definition found in almost all geometry text-books.

Under "straight line," the first definition, a line having an invariable direction, is credited to Newcomb, thus retaining the old weak idea of direction, used in defining parallel lines. Next, Euclid's definition is given, and then the Hilbert axiom as a definition. Legendre's definition is criticized. Of course the reader of the dictionary will not learn the meaning of a straight line from the Newcomb definition, but will learn the meaning "of the same direction" from his knowledge of a straight line. It would have been far wiser to have told what Euclid meant by his definition.

In defining angle the same discredited definition of the difference in direction of two lines appears again. Curiously enough the generalized, or trigonometrical, definition of an angle is found not under the word angle at all, but from a cross reference to "Mathematical angle." It is only at this place that the essential quality of an angle as a magnitude is given.

The word congruent, for some unknown reason is not given the meaning applied to it in foreign and recent American text-books on elementary geometry.

Under "Parallel lines" we find "lying evenly everywhere in the same direction, however far extended; in all parts equally distant." This is said to be the Euclid idea of the term! Then there is given the modern geometry conception of a point and line at infinity in which parallel lines and parallel planes meet.

Under "Parallel Postulate" is presented a good idea of the difference between Euclidean, Lobachevskian and Elliptical space. Such features as this go far to show that the dictionary is up-to-date in dealing with important ideas of mathematics which the general public has not had a chance to know about heretofore. While this explanation of the parallel postulate deals with one of the most abstruse matters in modern mathematics it is still reasonably intelligible to the ordinary reader. But there have been introduced into the New International the definitions of numerous highly technical mathematical terms whose meaning is quite beyond the ken of all except a very limited number of technical school graduates. Thus, under Dirichlet's theorem is found a triple integral involving perhaps a dozen elements. Similar technical matter will be found under numerous headings.