Transactions of the 



fractional numbers may be divested of much of the difficulty usually encountered by the 

 teaclier a-s well as the pupil. 



DISTINCTION BETWEEN NUMBER AND QUANTITY. 



Much confusion seems to exist in the minds of many in regard to the notions of num- 

 ber and quantity. This is partly due to the intimate relation of the tilings, and partly 

 due to the failure of mathematical authors and writers to make the proper discrimination. 



The delinition of number is often stated to be " the ratio of two magnitudes or quanti- 

 ties," or, it is said, "number is a unit or collection of units of wliich one is any magnitude 

 assumed as a standard of measure." In making such statements, authors appear to make 

 number depend upon quantity for its existence, and indeed it is sometimes said, quantity 

 includes both space and number ; from which it would follow that number is simply oneof the 

 kinds of quantity. Professor Olney says "quantity is the amount or extent of that which 

 may be measured." It seems, however, a better statement to say, quantity expresses the 

 limit of magnitude or extent; in other words, quantity expresses "how much." 



The dilterence between number and quantity is precisely of the same kind a-s that 

 between the notions of much and many. 



It is noticeable that in the form of words, the comparatives of much and many are 

 identical. Thus we say much, more, most, and we say, too, many, more, most; but the notions 

 of more and most in one case differ from those in the other case in the same way that 

 much differs from many; that is, in the first case more and most relate to quantity, while 

 in the second case they relate to number. 



This use of one word to express two ideas or notions, which though rehiled ai'e yet 

 distinct, indicates some confusion in the minds of those whose use of words has given 

 form to our language ; but this fact scarcely excuses at the present day any failure on the 

 part of authors and instructors to make the proper discrimination. 



In the French and German languages the distinction is less marked than in the English ; 

 and so in the Latin, multus and mxdti (much and many) differ only in singular and plural 

 forms of the same word. The same is true of the Greek. 



NUMBER USED IN THE MEASURE OP QUANTITY. 



It follows from the foregoing that it is incorrect to say "number is quantity," though 

 number is often used in measuring quantity. It is, in fact, seldom that a quantity is 

 measured in any other way than by meaning the number of standard units of quantity 

 contained in the quantity to be measured. For instance, four feet, twelve bushels, twenty 

 acres, illustrate this statement. Yet this fact, though it indicates an important use of 

 number, still does not constitute the basis of a definition, and should not be so misused, 



IS MATHEMATICS THE "SCIENCE OF QUANTITY'?" 



This statement, which has become stereotyped, fails to perform well the office of a 

 definition, even if it does not fail to express the literal truth. The vague use of the term 

 "quantity" unfits it to appear as the essential part of a definition. In algebra the current 

 use of this term makes it almost equivalent to the term number, or symbol of number. It 

 should be remembered that all algebraic operations are upon number. Thus there can 

 be no multiplication of quantity by quantity : strictly speaking, all multiplication is purely 

 by number; and it would seem better in a science, one of whose chief excellences is 

 precision, to call things by their right names. It is believed that a careful consideration 

 will convince any one that all the symbols of algebra are either the symbols of number 

 or of the relations oj and operation upon number. In the application of algebraic prin- 

 ciples to problems of (juaiitity, we deal only with the measure of quantity in the guise of 

 number. In tlie elements of geometry, the symbols of magnitude are presented for dii'ect 

 consideration ; but in the higher geometry the investigations are made chiefly by algebraic 

 methods; that is, by using the symbols of number to represent the measures of quantities. 

 The statement that mathematics is the science of quantity seems then to be an incomplete 

 definition, if in any true sense it can be called a definition. 



