COUNTING BY THE AID OF THE FINGERS. 431 



accenting every second number it is not difficult to run up to six or 

 eight, and still keep the count. In reflecting upon the answers to my 

 interrogatories, I was led to believe that the possession of ten fingers 

 was not the only cause of our counting by fives and tens, but that a 

 certain rhythm in a system of counting by twos enabled us to over- 

 come a resistance to memory. 



This point can be elucidated in the following manner : If we desire 

 to keep the count of the letters of the alphabet while we repeat their 



names, we can 



arrange 



them 



B 



l-l 



Fig. 1. 



advantageously in a system of 

 squares separated by a clamp 

 of two, as in Fig. 1. Here we A 

 have a system of twos counting 

 up to ten. A system of mental squares, so to speak, is formed, which 

 enables us to hold the numbers apart, and to form a distinct classifi- 

 cation. This system is capable of much extension : for instance, we 



FIG. I 



FlG.I 



FI6.I 



FlG.I 



0- 



Fig. 2. 



Fig. 3. 



can readily form another square in which a mental diagram like Fig. 

 1 is placed again at the four corners of a square, giving us forty ; and 

 the system of squares is capable of much further extension before the 

 mind becomes confused and loses its count. In repeating these dia- 

 grams in the mind, a certain rhythm will be perceived which is want- 

 ing when we use the system of triangles which is represented in Fig. 

 3, or a system of pentagons or hexagons. Indeed, with the last-named 

 figures great mental confusion speedily arises ; the mental resistance 

 to holding a clear image of a square or triangle in the mind is much 

 less than that which arises when we wish to behold mentally a penta- 

 gon or a hexagon. 



It would not be difficult to prove a close relation between the 

 forms of verse and the instruments by which a mathematician mounts 

 to the expression of thought. The commonest forms of verse are 

 written in four or five feet. In reading such lines the memory retains 

 the rhythm and the words of each line without effort. When, how- 

 ever, we increase the number of feet in the verse, their length becomes 

 cumbrous and the memory flags. No system of squares or triangles 

 can obviate this difficulty. A system of geometrical mnemonics could 

 undoubtedly be based upon the preceding exemplifications. 



In the early dawn of human knowledge the arrangement of points 

 in squares and triangles, and the further conception of areas by their 

 subdivision into triangles, undoubtedly arose from the inability of the 



