With the age of Enlightenment, dogmas that had persisted for millenia were being questioned and replaced; as was the astronomy of Ptolemy modelled on epicycles and the philosophy of Aristotle held and enforced as doctrine by the Christian Roman Church. The theory of Copernicus and the discoveries of Galileo through testing and verifying theory against reality with construction of instruments revised the knowledge of the heavens. Whereas the opinions of Aristotle were supported mainly by thought alone and persisted by the insistence of belief under authority, the enhanced understanding that was being developed arose through experimentation.
In France at this time, there was an accompanying redistribution of the ruling structure, where monarchy and aristocracy were threatened by the oppressed. While in the past, autocrats had endorsed official units of measurement for consistency in trade and commerce, now a society governed with the insight of thinkers and scientists bore the responsibility once restricted to royalty and endeavoured to design a system of measures suitable not just for the longstanding accounting and exchange of resources, but also for the greater precision and cross-national applicability required for scientific measurement and dissemination for reproduceability of experimental findings.
Whereas before, we hear that the ancient Egyptian unit of length the cubit was decided upon by a distance between the tip of the nose and outstretched arm of a Pharaoh, and the imperial foot was named for its length being that of the anatomical part of some king, with the revision of units of measurement, the basis for them was sought not in the divinity of a hereditary ruler, but rather in some more universal and global phenomenon or condition shared by all humanity.
Furthermore, this opportunity was used to attempt to unify the units of measurement for different kinds of quantity, so that those for lengths, areas, volumes, weight, and time could be related to each other and replicated anywhere in the world without reference to an arbitrary artifact. Originally, therefore, the unit of length was defined as a precise division of the distance between the pole and equator of the common Earth, just as the unit of time had been specified by an agreed division of the day the span of which was likewise the same for all dwelling on this planet. The unit of weight was defined by a reproduceable volume of standard water.
However, during this process of designing a unified system of measurement or metrology, the scheme of division chosen was based on the decimal system of numeration, which then was mainly how numbers were recorded and calculations conducted, as by the digits derived from the positional Indian or Arabic figures and before them the Roman numerals. Even earlier, numbers were written using a decimally encoded sexagesimal system preserved from the ancient civilisations of Sumeria and Babylonia through the Greeks. The Greeks and Hebrews too had a further non-positional decimal system of notation which likewise for Roman numerals in calculation has been superseded by the positional kind not requiring the aid of an abacus. Nevertheless, the sexagesimal notation persisted in the tabulation of astronomical data relating to co-ordinates of the sky around the Earth and division of the circle into angles, and in division of time from the daily global rotation. The strictly decimal metrology never managed to eradicate the use of the sexagesimal system in division of cycles. Consequently, the resulting and still current metrological system combines a mixture of divisions to units by more than one numerical base.
While people can manage or struggle with using this mixture of disparate bases in metrology, such that with our astronomy it is still possible to deliver astronauts into space, there are problems or difficulties that could be resolved if a single base could be selected and used for all metrological divisions leading to units of measure. The failure of the complete decimalisation of metrology could be attributed to the weakness of decimal as a base for practical purposes, particularly for division of quantities into fractions, in contrast to the strength for such purposes of the sexagesimal base, which is of a class of numbers called by mathematicians highly composite. A highly composite number, say the mathematicians, is a number containing more factors than any smaller number than itself. The number sixty, of which the sexagemsimal system is based, has as factors or divisors all of the smallest whole numbers 1, 2, 3, 4, 5, 6, as well as the larger 60, 30, 20, 15, 12, and 10. Thus, all of the simplest unit fractions, the whole, half, third, fourth, fifth, and sixth, can be easily written and used in calculation by sexagesimal. Sexagesimal is in fact the smallest base containing all of the first five or six factors, because sixty is the lowest common multiple of the first five or six counting or cardinal numbers. However, sixty as a base is still rather larger than would be convenient for easily learnt arithmetic. It is very likely that people would find a smaller number as base much more manageable for calculations.
Decimal evidently has a functional size, but lacks divisibility by the number three. The numbers dividing evenly into ten are the unit one, the base ten, its half five, and two. On the other hand, the number twelve is divisible likewise by the unit one, itself twelve, its half six, as well as two. Additionally, however, twelve is divisible by three and four, which decimal does not have as factors. Clearly therefore, if we were to use twelve as the base of our numeration, we would have these extra factors. Twelve, like sexagesimal, is one of the numbers called highly composite, but unlike sexagesimal is considerably closer to the size of ten with which we have been familiar as base.
While ten for the base is more familiar, the use of twelve as a division is quite known to us in metrology. There are for length twelve inches to the foot, for time twelve hours twice a day, and twelve months in a year, and for weight twelve ounces to a pound troy. The metrological uses are in conjunction with the practice in the marketplace of packaging goods by twelves, such as eggs by the dozen or half dozen. The utility of twelve as a factor in actual measures undoubtedly arises from its practicality by high divisibility.
Not only does twelve have extra factors, but these are more important factors for metrology than the factor five in decimal that twelve lacks. The most important factors are those that are smaller, so the most important factor is two, the next most important is three, and so on. Factors of two or three are especially important in metrology because of how easily their divisions can be cut. Furthermore, halves and thirds produce optimum divisions in terms of efficiency of information in certain metrological applications such as scales of weights and accounting where there is a subtractive component to the specification of values. Larger prime numbers such five, seven, or eleven have little value for metrology in themselves.
Ideas & Observations 