34 Colorado College Studies. 



" In fine, we live. We are surrounded by objects; the idea of com- 

 paring these objects, to consider the groups which they form with each 

 other, is natural to man, and has, we are sure, presented itself to him 

 since early times. Attentive observation reveals to us the fact that 

 objects do not exist exactly equal to one another: but, through an opera- 

 tion of the mind which demands no effort, although it involves the 

 entire secret of mathematical abstraction, we consider as alike the 

 bodies which seem to us to resemble each other, and we wave momen- 

 tarily the examination of the ditferences which distinguish the one from 

 the other. Hence arises the origin of calculation; the process of count- 

 ing is in general quite simple, even to those who are the least mathe- 

 matical. Thus, if we count the trees in a park, we know full well that 

 the trees may be of different kind, that they are not the same in shape, 

 that they have not the same age, nor the same number of branches and 

 leaves. Some grains of wheat are placed on a table; we say, here are 

 twenty-five grains of wheat, and if we set ourselves to examine them 

 with a magnifying glass, we perceive that each has its special proper- 

 ties which enable us, if necessary, to distinguish it from all the rest. 

 But by a conventionality that is natural and even instinctive, we have 

 created in the term " tree " or " grain of wheat " an abstraction which 

 is indispensable in the act of counting." — C. A. Laisant, La Math&imit 

 ique, Pliilosopliie-EnHeignemeut. Paris, 1898, pp. 15-16. 



" Primary number is an abstraction from a group of objects which 

 represents their individual exisience.'" — A. Lefevre, Number and Its 

 Algebra. 1896, p. 20. 



"The word which stands at the head of this chaper [ " number " ] 

 contains six letters. In order to find out that there are six, we count 

 them; n one, u two, m three, b four, e five, r six. In this process we 

 have taken the letters one by one, and have put beside them six words 

 which are the first six out of a series of words that we always carry 

 about with us, the names of numbers. After putting these six words 

 one to each of the letters of the word number, we found that the last 

 of the words was six; and accordingly we called that set of letters by 

 the name six. 



" If we counted the letters in the word ' chapter ' in the same way, 

 we should find that the last of the numeral words thus used would be 

 seven; and, accordingly, we say that there are seven letters. But now 



