MATHEMATICAL DEFINITIONS 463 
Evidently in the introduction of such matter the New Interna- 
tional 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 cer- 
tainly 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.t 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 defini- 
tion 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 
*See Encyclopedia Britannica, article “ Geometry.” 
