# Quotations

### What people have said about dozenals

Comments and extracts in favour of base twelve made by writers ancient and modern - and the modern ones are not all Anglo-Saxons, jealously guarding their ancient traditions, either ... see "Douze notre Dix Futur" by Jean Essig (Dunod, Paris 1955) for example.

(The following quotations were printed in the "Dozenal Review" (Spring 1978), the official magazine of the Dozenal Society of Great Britain (DSGB). They are reproduced here to show that the idea of replacing base ten with base twelve is not a recent one, and that it has been advocated by many famous people in the past.)

In Ancient Italy the duodecimal system of division predominated to the South of the Appenines, while the decimal division was in use to the Northward. In Sicily the two systems were confused together. China has had a purely decimal system from an unknown epoch in antiquity. In England duodecimal and binary divisions have existed from very early times. The duodecimal system has marked advantages, because it allows of division into several aliquot parts, involving the factor 2 twice over and the next higher factor 3 once...

The decimal system is far less simple and in some ways less convenient. Ten admits of only two factors superior to unity, namely 2 and 5, and 5 is a more complex prime factor than appears in either the binary or duodecimal systems. Decimal numeration is fixed among the institutions of the human race, as an heriditary habit, derived from the early practice of counting on the digits of both hands. (Jevons, "Money" 1885).

The number ten has therefore rightly been given preference over all below it. But we shall see that it should not be thought to have the same advantages over all the numbers above it. An arithmetical system which had been based on a progression of twelve would have been much more useful; the large numbers would have taken up less space, and at the same time fractions would have represented larger proportions. Man has so felt this to be true that, having decided on decimal arithmetic, this does not prevent him from using the duodecimal progression: we often count by dozens, by dozens of dozens, or grosses; the foot is the third denomination of the line, and the inch the second denomination in a duodecimal progression.

The number twelve represents a unit: the year is divided into twelve months, the day into twelve hours, the Zodiac into twelve signs, the sou into twelve deniers. All the small measurements make use of the number twelve, because it can be divided by two, three, four and six; whilst ten can only be divided by two and five, which makes in practice an essential difference in the case of calculating and measuring. This new scale would need only two new characters, one to represent ten and one to represent eleven, by means of of which one would have a system of arithmetic which is far easier in its way than our ordinary arithmetic. (Georges Louis Leclerc, "Essai d'Arithmetique Morale", 1760.)

... the duodecimal system of arithmetic.. is superior even to the decimal system for simplicity of expression; and the only additional burden to the memory is two characters for representing ten and eleven, for the multiplication table in our ordinary arithmetic is generally carried as far as twelve times twelve, although its natural limit is only 9 times 9, which is clear proof that the mind is capable of working with the duodenary system, without any inconvenience or embarassment... (Peter Barlow, "An elementary investigation at the theory of numbers", 1811.)

Of all the numbers capable of use as a system base, 12 presents the greatest number of practical advantages. We have, through the familiarity which custom has produced, become so accustomed to the use of ten in that capacity that the assertion just made seems unwarrantable. But a moment's reflection will show that the ten fingers of the human species have entailed upon us a number decidedly inferior to twelve. In the simple business affairs of life we deal most extensively with the three familar fractions one-half, one-third and one-quarter and the auxiliary fractions two-thirds and three-quarters.

Such being the case it needs no argument to prove that the most convenient base is that which will admit of division without a reminder by the numbers 2, 3 and 4. Ten can be divided by but one of these numbers without a remainder; hence the confusion of fractions is at once introduced. Twelve, on the other hand, is an exact multiple of each of the three numbers. It offers, then, to the mass of mankind an enormous advantage over ten or any other number as a base for computation. With the growth of business in its many forms, the civilised world has long since come to recognise this fact, and in many ways to make practical use of it.

The word 'dozen' and its equivalent in other languages, has been coined as a noun to express the number twelve, and in a very great number of the commercial transactions of the world the dozen and its square, the gross, are the common units of measure.

So palpable are the advantages of 12 from this point of view that some writers have gone so far as to advocate the entire abolition of the decimal system and the substitution of a duodecimal one in its place. (Levi, L. Conant, "Primitive Number Systems", 1892)

In planning a rational future for human communication, we must always ask whether there is not a grain of truth in the conservatism which has obstructed progress. For the purpose of calculation ten is a bad number, however holy its devotional associations and however venerable its biological antecedents. It has only three exact divisors: 1, 2, 5. The number twelve has 1, 2, 3, 4, 6 as its divisors and the number 60 has 1,2,3,4,5,6,10,12,15,20,30.

A large number of factors is a great advantage in rapid calculation. So it would be an improvement on our present standards to make a Hegelian compromise of the English and French systems by adding two numbers to the Hindu number script, and making a positional notation based on the twelve-fingered abacus with weights and measures adjusted accordingly. (L.Hogben, "Mathematics for the Million", 1945)

Duodecimal arithmetic, once adopted, would bring the calculations of the scientist to the utmost simplicity. (Lobkowitz, "Mathesis biceps", 1644)

School Arithmetic, Bozman, 1938
There remains a practical disadvantage; although numbers are simply dealt with in decimal fractions, quantities are not. Consider the division of a piece of butter, a quantitiy of milk, a length of material, a space of ground; these can be divided more easily and accurately into halves, quarters, eighths, even into thirds, sixths and twelfths, than into tenths.