by A.E.H.Campbell

(The following formed the text of an introductory leaflet by the Duodecimal Association before its amalgamation with the Dozenal Society of Great Britain.)

It is not generally realised that everyone constantly makes arithmetical calculations every day. This is because we confine ourselves to very simple (and usually approximate) answers, which we achieve largely by sub-conscious thought. Almost all our actions are governed by question of time and value, which matters are hardly capable of consideration with out the mathematical symbols marked on our clocks and our money. However any symbol of quantity (whether of time or money or of anything else) only has effective meaning in relation to other symbols denoting different quantities of the same thing. The relationship is necessarily mathematical. Thus, although are may not realise it, we are all constantly doing sums in our heads.

It follows that the mathematics basis on which we work matters very much, not just to mathematicians and scientists, but to every one of us ordinary people. Since we form the bulk of the community, anything which contributes (even in a very small degree) to the efficiency of our daily lives is enormously important. If we can compute comparative quantities of time, money, or anything else more easily or more quickly, then we shall all be much better off.

We ordinary people, mainly using figures subconsciously, can only compare them in very simple fractions of halves, quarters and (perhaps) thirds. For practical everyday purposes , we form an instantaneous mental image of the relative values presented to us by different figures. Consequently, and because our minds boggle at anything but the simplest sums, we automatically make a mental approximation of "difficult" figures. Thus, if for example we have to compare 19 with 43, we consider 19 (vaguely and inaccurately - but quickly) as being half of 43. We may sometimes stop and calculate deliberately; but normally our considerations of comparative values are simple, rapid and largely subconscious.

For the purpose of making these mental comparisons, we require systems of measurement based on the most convenient unit - which is not ten. That is why no monetary system has ever naturally evolved with basic unit of ten. Units of 8 and sixteen form the basis of some systems; but since most people can imagine thirds (in addition to halves and quarters), twelve is the basic unit, because twelve is the smallest number divisible by 2, 3 and 4. It is for this reason that the commercial world operates in dozens. This is why we learn our "tables" up to "twelve twelves". If we think of dozens, then only the figures 5, 7 and 11 fail to conform at once to a known fraction; so that we do much less mental approximation and compute our values quicker and more accurately.

However it is often necessary just to add figures up, which is easiest done in tens. That is of course because our mathematical system operates on ten single digit figures. Addition, as opposed to computation, is thus normally decimal. There is a further complication that single units of 10, 20, 30 etc. - when considered by themselves without division- are easier to visualise for comparative purposes than units of 12, 24, 36 etc. Again, this is because of our system of mathematics, which interpolates 0 in units of ten.

Consequently, in considering the best system of measuring and comparing time, money or anything else, we are faced with the problem of combining the convenience of adding and visualising in units of ten with that of calculating in fractions, for which the most convenient unit is twelve. Thus we find for example that, although our time scale starts with 12 hours in the day, it operates below that on a unit of 60, which is the first number divisible by both ten and twelve. This allows the hour to be divided into 12 units of 5 minutes, so that hours are easily fractionalised but minutes can be added in tens. Similarly, British currency, before it was thrown entirely out of gear by inflation, used the penny-shilling scale for everyday calculation of normal values and reverted to adecimal system of addition on reaching what used to be the large sum of 20 shillings. Hence the simultaneous circulation of the florin and the half-crown, comprising respectively two dozen pence and one-eighth of a pound, but capable of rapid mental comparison as 2 to 2 ½.

Present day problems are accentuated by the comparatively new technique of decimal calculation with machines. However decimals (being units of ten) are wholly unsuited to mental use; and we must avoid the fatal mistake of ordering our mathematics to suit our machines - became it was thus that all our troubles began. The real problem arises because, when ancient Indian and Arab mathematicians evolved a system of single figure digits to replace the existing Roman numerals, they worked on the number of ten. If only they had invented twelve single figures, repeating with 0 in dozens instead of in tens, all our present complications would disappear. The dozenal system (thus created) would automatically have become the basis of all measurement and comparison. Ordinary citizens, expert egg-heads, and the most complex machines could all use variations of the same system to suit themselves and yet all compute in harmony.

Our trouble arose because for thousands of years (and to this very day) every man carried with him his own calculating machine in the form of his ten fingers. Unhappily, although Nature evolved these admirably for physical purposes, she never intended them to have mathematical significance. We can forgive the Arabs for missing this point, when they passed on to future generations in the West the new system of mathematical notation, without which our present day world would hardly be possible. However, more than 1000 years later, we have no excuse for perpetuating their error.

The object of the Association is therefore to devise and have adopted two additional symbols to take the place of the present double figures representing ten and eleven, so that "10" will in future denote one dozen digits and mathematics will be converted fron a decimal to a dozenal bests. It is not suggested that mankind, naturally conservative, will readily agree to this change; but it seems reasonable to suppose that it will come about sooner or later. Our object is to make this sooner rather than later.

A note from Alan Beecham - re the penultimate paragraph:

In fact if you look more closely at human hands you will see that they are uniquely suited for dozenal counting.

Each hand has four fingers, each with three joints. Using the thumb of one hand as a counter, point in turn to each joint of the four fingers, starting at the tip of the little finger and ending at the third joint of the index finger. Use the other hand in the same way to record each completed dozen.

You can count up to a gross (144), all the time keeping a visible and tangible record so that you do not lose count.

Alternatively, you could use the five digits of the other hand to record your dozens, giving you a range of 60, which is the base of Sumerian counting system (and the number of minutes in an hour)./

Has anyone else noticed this natural abacus built into the human hands?