The Number Concept. 35 



a question arises. Let us suppose that the letters of the word number 

 are printed upon separate small pieces of wood belonginfj to a box of 

 letters; that we put these into a bag and shake them up and bring 

 them out, putting them down in any other order, and then count them 

 again ; we shall still find that there are six of them. For example, if 

 they come out in the alphabetical order, b, e, m, n, r, «, and we put to 

 each of these one of the names of numbers that we have before used, 

 we shall still find that the last name will be six. In the assertion that 

 any group of things consists of six things, it is implied that the word 

 six will be the last of the ordinal words used, in whatever order we take 

 up this group of things to count them. That is to say, the number of 

 any set of things is the same in ichatever order we count them. 



'•Upon this fact, which we have observed withregard to the particular 

 number six, and which is true of all numbers whatever, the whole of 

 the science of number is based.'' — W. K. Clifford, Common Sense of 

 the Exact Sciences. 1891, pp. 1, 2. 



" The oldest calculations were probably achieved by a certain ar- 

 rangement, either of the objects themselves, which were the subject of 

 calculation, or of other things more easily handled. Pebbles, small 

 shells, may have served as representatives, as they still do at the pres- 

 ent time among certain tribes, and these marks . . . , when brought 

 into smaller or larger heaps, arranged in rows, will have facilitated 

 materially the adding together or the division of a collection of objects. 

 As long as only small numbers had to be dealt with, man carried the 

 simplest mode of visualization with him; namely, the fingers of his 

 hands and the toes of his feet. To be sure, he could not thereby advance 

 very far without some new device. Certain tribes of South Africa still 

 show us to-day how friendly cooperation may be used to overcome the 

 difficulty of visualizing larger numbers by using the fingers only. ' In 

 counting beyond one hundred the difficult task must, as a rule, be per- 

 formed by three men. One of them counts the units on his fingers, by 

 raising one finger after the other and pointing out the object counted 

 or, if possible, touching it. The second man raises a finger (always 

 beginning with the little finger of the left hand and proceeding continu- 

 ously toward the little finger of the right hand) for every ten, as soon as 

 it is completed. The third man counts the hundreds.' * 



'' Whatever explanation may be offered for the fixed order of using 

 the fingers, the fact of its existence remains, and in the course of our 

 researches we shall repeatedly encounter this fixed order as the founda- 

 tion of the so called finger counting."' — M. Cantor, of Heidelberg, in 

 Vorlesungen uber Geschichte tier Mathematik. Vol. I, p. G. 1894. 



♦SCHISUMPF, in Zi-itschri/t d. dculsch. iiuirgcntuiul. lii-.scllnchti/l Wl, MtX 



