Popular Science Monthly/Volume 12/February 1878/Counting by the Aid of the Fingers
By Prof. JOHN TROWBRIDGE.
ONE cannot with any reason contend that the universal possession of ten fingers argues a natural tendency of the human mind toward the decimal system; it is certainly true, however, that multitudes of men and women find their fingers of great assistance in arithmetical operations. The intelligent school-teacher is apt to discourage the pupil's use of the fingers in addition, and to encourage mental counting without their aid. I have been interested to discover the nature of this mental process which goes on apparently without the aid of the hands. From questioning a large number of persons, I find that five or six is the limit to the numbers of things which one can repeat, and also keep the count. Of course, this limit can be much exceeded by practice; one person who was interrogated could count up to fifty, but he was an astronomer. Most persons reply to the interrogatory, "How do you keep the count?" by saying, "I run up to five, and then again to five, and so on." In most cases it was found that a subdivision into ones and twos preceded this division into fives. The division into twos seemed to be the most common; by 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 overcome 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 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 classification. This system is capable of much extension: for instance, we
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 diagrams in the mind, a certain rhythm will be perceived which is wanting 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 pentagon 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, however, 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 human mind to retain distinct images of figures more complicated than rectangles or squares. In the case of curved lines, the mind has a tendency to refer all arcs to circles, since a circle forms as definite a conception as a square. The fact that it is made up of an infinite number of straight lines has significance only to a geometer.