Page:Popular Science Monthly Volume 77.djvu/469

Rh Evidently in the introduction of such matter the New International has broken away from the long established custom followed by dictionaries and popular cyclopedias, of inserting only what will be fairly intelligible to any well-informed person. This rule probably holds still in other fields of knowledge in the dictionary, but it certainly does not hold in the fields of pure and applied mathematics.

The definitions of ratio and proportion as given by lexicographers in times past strike the present-day reader as curious, and thereby hangs a tale. The old writers following Euclid looked upon ratio and proportion as expressing the relation of quantities, such as lines, and were not ready to admit that ratio could be always a number, since two lines taken at random in general are incommensurable. The old Euclid definition of a proportion (given at the beginning of his Book V.) avoided entirely the question as to whether the ratio of two numbers would always give rise to a number. Whether Euclid's ratios are or are not always numbers, it certainly is true that Euclid cuts irrationals out of his theory of proportion. The modern tendency is to teach that ratios and quantities in algebra are numbers. Certainly in elementary mathematics it is highly desirable for pedagogical reasons that the ratio of two quantities be defined specifically as the quotient of the first divided by the second, a proportion as an equality of ratios, and a quantity in algebra as a number.

Oddly enough, the old writers did not distinguish between ratio and proportion, using the two interchangeably. To understand how this probably came about, it must be observed that proportion is applied to two or more quantities that are thought of as changing, or assuming different values. Thus if two bushels of wheat cost $2.10, five bushels will cost $5.25. Here there are only two kinds of quantity, bushels and dollars. With this view of proportion in mind, we can see what Dr. Johnson meant when he said proportion is the comparative relation of one thing to another; ratio; settled relation of comparative quantity. Ash (1775), Fenning (1761) and Webster (1806) give practically the same definition. Bailey (1736), whose work shows he was something of a mathematician, gives the definitions for ratio and proportion now in use, except that he adds at the end that a ratio is a proportion. Webster in his first large dictionary (that of 1828) gives for the definition of proportion "the comparative relation of one thing for another; the identity of two ratios."

It is interesting to find some of the old lexicographers falling into errors of pupils at the present time. Thus Ash defines an angle as the point or corner where two lines meet. Bailey says an angle is the space comprehended between the meeting of two lines, but he hastens to add "which is either greater or less, as those lines incline towards