# Outline of Principles and Aims

## THE PAST

In early times, numbers did not merit the importance they have acquired today, but there is no doubt that all interactions with our material environment now depend on them for measuring, arranging and recording results. They should, of course, carry out their duties in an efficient unobtrusive manner, fulfilling our needs without imposing any conditions of their own. Written and spoken forms have developed with time, starting with the original one, two (using words for man and woman in early Chaldean) through fours and eights by counting on the fingers, to reach ten when thumbs were included in the process. This method thereby became a barrier to any further progress.

The first use of numbers would have been no more than simple counting, with values less than one inconceivable. However, fractions as pan of a complete group would have been used for an equitable distribution of the results of a hunt or gather, or an issue of rations from the communal store. These would have been the elementary and understandable proportions we still use today. All early systems of weights and measures contained fractions of halves, thirds and quarters within the use range, implying that bulk quantities were arranged so that these could be selected from a range of main divisions.

A use of such everyday fractions is inhibited by a numbering scale that does not contain them as whole parts. Halves, thirds and quarters also form the framework of mathematical operations, hence, at the inception of modem calculating methods mathematicians became aware of the need for divisibility in their basis. This was once achieved by the Sumerian sixty based numbering system for their measures and astronomical calculations, from which we have inherited our advantageous means of measuring time and angle. By containing factors of five and ten it was compatible with finger counting methods also in use, but would be too difficult for modem man. Consideration has been given to eight or sixteen, which allow a presentation in the simplest forms of important binary sequences but little else. Only a scale of twelve numbers contains the close-spaced small primary factors essential for a concise representation by single figures of social practices and scientific concepts.

## PLACE VALUE

An early understanding of numbers as such was vague, and they needed tactile aids for operations with them ('calculate' is derived from calculii or pebbles) results being recorded by assigning letters to fixed quantities. Arithmetic was freed from this limitation by the advent of place-value notation, which originated in India and came to Europe via the Arab world in the tenth century. With this method, a small range of figure symbols defining a series of unit values only, acquire increased value according to their place in a row.

The present convention is that at each place to the left from the units position, the value of a figure symbol is increased by a constant factor known as the base or radix of the particular system. The principle is valid for any number of figure symbols, but all have to include a means of denoting a nil value place, which is again of Indian origin from the Buddhist symbol 'Sunya', a circular space that is empty but full of meaning. Thus, the first place shift is given by 1 0, which defines the totality of figures in use, but not necessarily ten as is commonly read.

It was shown later that values less than unity can be represented by figures placed to the right of the units position, which are considered to be successively divided by the base number. Place-value fractions are obtained by dividing out a common fraction according to the particular base rules, remainders being increased for further division by appending a zero (multiplying by the base) but the result is corrected when put in the lower value place. A mark is required to distinguish between values greater or less than unity, and in the denary - decimal system this has been variously a point on the line or midway, or a comma on the line or as a suffix. A row of figures can, therefore, be multiplied or divided by the base number by moving the separatrix to the right or left, a property which is the same for all place-value arithmetical systems.

With a place-value method of numbering, an unending series of values may be recorded, which can be considered as sums of twos, threes, fours or fives, etc. Hence, half of all numbers, regardless of the base in which they are 'spelt', are divisible by two, one third by three, a quarter by four and so on. It is obvious that groups formed as a product of the more frequent divisors will allow the simplest manipulation of a series of numbers, and this must be 2 x 2, or 4, x 3 = Twelve. Ten was settled upon for the sole purpose of counting long before the properties of numbers and their arithmetical iinportance could have been recognised.

## THE PRESENT

Mathematicians would not now consider a scale of ten numbers relevant for their work since it has only two whole divisors - two and five - with the latter of esoteric significance only and of infrequent occurrence. Apart from the representation of these by single figures, 0·5 and 0·2, decimalised division results in rows of figures for much used and important ratios; thirds and sixths in particular do not terminate. Since most scientific co-efficients are the result of divisions - so much per something it follows that one third of these must be approximations when in decimal form.

In so far as one number is able to serve the multitude of duties that will be required of it, only a scale of twelve can effectively represent structures of the material world in all their practical and theoretical forms. Dozenal Societies accordingly hold that all calculations should be in base twelve so that operations involving time, spatial relationships, weights, volumes and coinage could comply with natural laws and their constraints. They would then be in harmony with common practices and allow important relationships such as three-quarters, halves, thirds, quarters and sixths, which occur as often in scientific work as they do in every-day life, to be expressed exactly by single figures.

We affirm the opinion of many mathematicians and philosophers in the past that a twelve-based structure for calculation will be of great benefit to all users. Our longterm objective therefore is a reform of numeration and its arithmetic to a Duodecimal or Dozenal basis ie, to work in groups of twelves rather than tens. To this end we foster discussions on all aspects of the Dozen principle and work towards agreement on such matters as single symbols for the values now referred to as ten and eleven, nomenclature, and arithmetic conventions.

To facilitate publication of ideas the DSGB augments the present Indo-Arabic numerals with adaptations of existing forms once used by Sir Isaac Pitman in his Phonetic Journal the figures 2 and 3 inverted - regarded as cursive forms of T(en) and E(Ieven).

We also encourage constructive thought and research into both common and scientific measures, with the intention of developing a coherent system that will be flexible enough to provide unifying links between all levels of use instead of the barrier that contemporary scientific methods have become. Recognition must be given to our practical, convenient, traditional units, with their relationships, evolved through historical experience and common sense. These are well-suited to a more concise and intelligible science and mathematics; and the needs of manufacturing, commerce and domestic life. A true measurement system to suit all purposes must continue to be of human proportions appropriate to our perceptions and bodily sensibilities, thereby providing a perspective to daily affairs.

## THE FUTURE

The requirement of divisibility in measurement, and the disposition of objects for use, as distinct from accounting overall quantities, has always been at variance with primitive counting in groups of ten, which is adequate enough for trading and administrative purposes. The debate over measurement is, at the core, one over arithmetical notation. Its complete resolution can be only by a reform of this since the laws of geometric proportions and arithmetic processes are immutable. Traditional measures and methods can be given a rational basis with a dozenal placevalue system of calculation, science can retain its size-order ranges, often as hnportant as size, and mathematicians will acquire a precise representation of their fundamental relationships. Social metrology would be compatible with both in a manner that is not possible with the present conflict between mutually exclusive patterns of thought.

The Society considers that to learn from and conserve the best features of our heritage from the past is the correct way to development in the future. Radical changes for doctrinaire reasons usually result in wasteful periods of confusion followed by flawed compromises. It has been well observed in several ways that those who cut themselves off from their past, do so also from their future. Other commentators on the 20th century scene have noted the promotion of ideas intended to make us disdain, even forget, our cultural antecedents. Decimal-metrication by its abstract detachment from reality, has proved to be a useful tool in this process, which is an additional reason for our rejection of it as a vehicle for social and scientific metrology besides its inability to comply with requirements outlined above. We therefore seek to offer informed comment on the many attempts that are made to extend decimalised methods into areas where they are not appropriate.

There are many who resent on cultural grounds the efforts being made to extirpate our historically founded ergonomic measures in order to replace these with some arbitrary alien system, but without conceding the need to put them into a sound arithmetic structure. We do consider that the cultural aspects of measurement are of equal importance to the technical, and, indeed, complement one another. It is essential at this stage that the age-old means of conducting our every-day affairs should be available for use alongside whatever artificial arrangement is in vogue. We invite support from all who are in agreement with this. Common measures perform the same function that common laws do in other spheres, providing safeguards against encroachments on our liberties of thought and action by technical trickery.

The DSGB was formed in 1959 in response to the proposed conversion of our highly divisible currency to the less flexible decimal base, and a clear intention to impose metric methods on all aspects of public life. We work in close co-operation with the DSA, who are now facing similar problems, exchanging information and ideas with them. We now have a Forum on the Internet in addition to our Dozenal Journal to promote members' ideas and to review the latest developments. Articles, correspondence, illustrations and helpful criticisms are welcome from all who are concerned with or by the topics mentioned; these should be sent to the Secretary.