Author: Knew'n'Tell

I like improvements.

Dozenification of Contemporary Weights

Previously (Jan 21 2017, 09:21 PM, DozensOnline, Applications, “All Your Base Are Belong To Us”; Apr 28 2017, 08:47 PM, “Requirements For Implementation Of Uncial”) was proposed how using division of the day by a power of the dozen, the gravity of the Earth, and the density of water can produce a unit of mass equal to the metric gram. All metric weights can thereby be converted into a coherent dozenal metrology by moving the decimal point to write the weight in grams, followed by changing the notation of the number from decimal to base twelve.

In choosing units of measurement for a dozenal metrological system, for minimum change and least disruption it is preferable to retain existing units wherever it is possible to incorporate these into a dozenal framework. Retention of the maximum number of existing units while allowing dozenal scales of magnitude would suggest that existing units of measurement still in use that already have dozenal ratios of their magnitudes should be chosen. Retention of existing measures acts as a celebration of the people’s choice of dozenal in the history of measurement and evidence for the benefits of twelve as a division. For this reason, I suggested retention of the inch as a unit of length, which is multiplied by twelve to produce the foot, and has been divided dozenally to the pica and point in typography or graphical design.

As mentioned previously (“Base Twelve in Measurement”, Knew’n’Tell, Ideas & Observations, Thought Views, WordPress, 17th Dec 2018), there are “for weight twelve ounces to a pound troy”. The troy system of weights was used in Britain for weighing of pharmaceuticals of medical prescriptions until it was made illegal since the year 1976 for such use as part of decimal metrication. The troy ounce is, however, still legal for use in the trade of investment precious metal bullion. Another pre-metric system of weights still used is the avoirdupois, which forms part of the American Customary and British Imperial systems of measurement. The troy and avoirdupois systems share as a common unit the weight called a grain, which is about 0.0648 grams, but differ in the multiples of this unit in the formation of larger named units. In the troy system, 480 grains form an ounce troy, so 480 * 12 = 5760 grains form a pound troy, 5760 * ~0.0648 g = ~373.25 g. In the avoirdupois system, on the other hand, seven thousand grains form a pound avoirdupois, 7000 * ~0.0648 g = ~453.6 g. But the avoirdupois pound has sixteen ounces avoirdupois rather than twelve, so the avoirdupois ounce is about 453.6 g / 16 = ~28.35 grams, nearly the same as an ounce troy of about ~373.25 g / 12 = ~31.1 grams.

For unification and dozenification of these current and related systems of weight, I propose that their ounces be dozenalised by being rounded to the same thirty grams, which dozenally is two-and-a-half dozen grams. This is the same as a metric ounce that already exists, so it is not a new unit. Additionally, I suggest that the multiples and divisions of this ounce to form the other named units of weight in the systems should as much as possible be the same as or similar to the ratios of units in the troy and avoirdupois systems, but should be changed only where such ratios involve awkward prime numbers in their factorisations, such as the prime number seven seen in the 7000 grains avoirdupois per pound avoirdupois, or where multiples or divisions are overly decimal (Conversion of decimally rounded numbers to dozenally rounded numbers was described earlier at DozensOnline, Number Bases, Dozenal, “Rounding, Surrogates, And Auxiliaries”, May-June 2017). This procedure allows the adjusted units to resemble in magnitude and relations the existing units, and is thereby along the lines of minimum change while at the same time being dozenalised. After conversion of the sizes of the units, it remains only for the numbers to be written dozenally using dozenal numerical notation of twelve phalangeal numerals.

The dozenified troy, which I call the trade weight system, would then be
0.0625 grams = 1 trade grain
1.5 grams = 1 trade pennyweight = 24 trade grains
30 grams = 1 trade ounce = 20 trade pennyweights = 480 trade grains
360 grams = 1 trade pound weight = 12 trade ounces = 240 trade pennyweights = 5760 trade grains

The dozenalised avoirdupois system or unified American Customary and British Imperial weight systems, which I call the system of civil weights, would become
0.0625 grams = 1 trade grain
1.875 grams = 1 civil dram = 30 trade grains
30 grams = 1 trade ounce = 16 civil drams = 480 trade grains
480 grams = 1 civil pound weight = 16 trade ounces = 256 civil drams = 7680 trade grains
5760 grams = 1 civil stone = 12 civil pounds weight
11520 grams = 1 civil quarter weight = 24 civil pounds weight
46080 grams = 1 civil “hundredweight” = 4 civil quarters weight = 96 civil pounds weight
829440 grams = 1 civil ton = 18 civil “hundredweights” = 144 civil stone = 1728 civil pounds weight

Common to both these trade and civil systems are the thirty gram metric ounce and the trade grain of a sixteenth of a gram. Sixteenths and other binary powers are very convenient for weights and are much simpler in dozenal than decimal, as mentioned before (“The Trouble With Base Thirty”, Knew’n’Tell, Ideas & Observations, Thought Views, WordPress, 26th Jan 2019). At first it might have seemed that the old grain being common to both the troy and avoirdupois systems should have been maintained unchanged. However, in effect that would be no change or no dozenalisation, as the value of the old grain would not be a convenient fraction of the dozenal basic unit of weight, and, with the old grain, only one unit would be common to both the troy and avoirdupois, whereas the sixteenth of a gram allows both the grain and ounce to be the same in both systems dozenalised.

Notice too that in the civil weight scheme, the number of civil pounds weight per civil ton is a cubic dozen, which is near the two thousand pounds per ton of the American Customary and is at the same time the dozenal analogue of the thousand or ten cubed of kilograms per tonne of the metric system.

A weight currently used for gemstones is the carat, which was formerly metricated to 0.205 grams and latterly downgraded to the more decimalised 0.200 grams. The carat is said to be derived from a 24th part of a solidus, where a solidus was a 72nd part of a Roman pound weight. Unifying the Roman pound weight or libra with the trade pound of 360 grams, this would lead to 360 g / 72 = 5 grams for the gem solidus, and 360 g / (24 * 72) = 360/12^3 = ~0.2083 grams for the gem carat weight. I therefore propose replacement of the metric carat by this dozenalised carat of 5/24 grams. However, in earlier times, the number of the gold solidus making up a Roman pound was sixty rather than 72, which would produce a gold solidus weight of 360 / 60 = 6 grams, and 360 g / (24 * 60) = 0.25 g or a convenient quarter of a gram for the gold carat weight. This value for the carat is necessary if the number of trade grains making up a carat weight is to be four. However, since the quarter gram value differs so much from the metric carat, I propose that the quarter gram carat be limited to being a unit of the trade system for precious metals, while the 5/24 gram carat would replace the metric carat used for gemstones.

Hence, the system of trade weights can be supplemented with these further weights, where:
5/24 grams = ~0.2083 grams = 1 gem carat = 1 square dozenth trade ounce = 1 cubic dozenth trade pound
0.25 grams = 1 gold carat weight = 1.2 gem carats = 4 trade grains
1.25 grams = 1 trade scruple = 5 gold carats weight = 6 gem carats = 20 trade grains = 1/4 gem solidus
5 grams = 1 gem solidus = 24 gem carats = 1/72 trade pounds weight
6 grams = 1 gold solidus = 24 gold carats weight = 96 trade grains
360 grams = 1 libra = 60 gold solidi = 1440 gold carats weight = 1728 gem carats

The Latin word siliqua can be used for a carat. The Greek word nomisma can be used for a solidus.

References:
https://en.wikipedia.org/wiki/Troy_weight
https://en.wikipedia.org/wiki/Apothecaries%27_system
https://en.wikipedia.org/wiki/Avoirdupois_system
https://en.wikipedia.org/wiki/Ounce
https://en.wikipedia.org/wiki/Carat_(mass)
https://en.wikipedia.org/wiki/Solidus_(coin)

A Reply to Dozensonline, Number Bases, Dozenal, Request for Help with mnemonics for Dozenal

Replying here in WordPress because Dissenter is not working right now: “500: Failed To Dissent
No workers available to handle the requested job.” 

To “SenaryThe12th 2:54 PM – 1 day ago#1”, at

https://www.tapatalk.com/groups/dozensonline/request-for-help-with-mnemonics-for-dozenal-t2020.html

The simplest option would probably be to split apart from each other the voiced and unvoiced consonants. Eight groups could then be: P, F; B, V; T, Th, Ch; D, Dh, J or Jh; S, Sh; Z, Zh; K, Q, X, H; G. The four remaining groups could be chosen from N and Ng, M, L, R. Y and W or Wh might be considered. The sounds of the letters are to be understood as from “Appendix B: Consonants and Phones” at “SenaryThe12th 6:04 AM – Apr 25#1”, Dozensonline, Number Bases, Non-Dozenal bases, Senary and {2, 3} Bases, Mnemonics for Senary, at https://www.tapatalk.com/groups/dozensonline/mnemonics-for-senary-t1991.html.

A Reply to https://www.tapatalk.com/groups/dozensonline/duodecimal-myriad-system-t1970.html#p40018012

Dozensonline > Tools > Number Names > Duodecimal Myriad System
https://www.tapatalk.com/groups/dozensonline/duodecimal-myriad-system-t1970.html#p40018012
“sunny
8:48 PM – Mar 01 #3″:

“I always liked groupings in fours, replacing a bit unwanted terms like ‘great gross’ or ‘grand’ or any such others by the simple word ‘dozen gross’ that seems straightforward and more helpful. Why should we bother to say in decimal, ‘one thousand five hundred’ when we could do it with the more simpler ‘fifteen hundred’?”

I also rejected the terms “great” and “grand”. Grand means a thousand in decimal. I would go further and discourage “gross” because I have never seen it used for a dozen dozen by anyone other than dozenists and it is not likely to be understood by others. Nevertheless, I do use gross sometimes for brevity.

For grouping of dozenal digits,

[Citation reference:
Dozensonline > Tools > Number Names > Uncial Nomenclature
https://www.tapatalk.com/groups/dozensonline/uncial-nomenclature-t1570.html#p40009563
“DavidKennedy
May 08, 2017 #14″
“Five or six digits are too many to subitise. There is decimal notation that groups digits into threes by commas. From the point of view of visual subitisation of numerical symbols alone, figures could be grouped in threes or fours, but by twos would probably be too frequent. However, from the language point of view, grouping in twos is probably better,”]

I also preferred grouping of dozenal digits by fours, as demonstrated in the tables at

Dozensonline > Number Bases > Dozenal > Rounding, Surrogates, And Auxiliaries
https://www.tapatalk.com/groups/dozensonline/rounding-surrogates-and-auxiliaries-t1748.html

In some browsers the tables may not display anymore. If so, right click on the table and select “inspect element” and read through the pane for the table cell entries.

A Reply to https://www.tapatalk.com/groups/dozensonline/hexcalibur-date-time-and-angles-part-2-t1969.html#p40018104

Dozensonline > Applications > Calendar Reform > Hexcalibur: Date, Time and Angles | Part 2

https://www.tapatalk.com/groups/dozensonline/hexcalibur-date-time-and-angles-part-2-t1969.html#p40018104

Kodegadulo: “11:28 PM – 1 day ago #8

Silvano wrote:
http://www.intuitor.com/hex/hexclock.html

Who ever heard of any kind of “minute” that is only broken up into sixteen of any kind of “second”? Really, can anyone get any more desperate trying to shoehorn a system into the old and moldy sexagesimal pattern? Leave the “hour”, “minute”, and “second” alone. Reserve those terms just for the sexagesimal units we all know and love, uh tolerate.”

To subdivide the period of a hand of a clock into so small a number of tick marks as twelve or forezeen which is sixteen decimally wastes energy through such a large movement in the clockwork mechanism. Currently, my opinion is that the most efficient system for a clock would be to have as many ticks marks as are possible while being still readily distinguishable. The square of forezeen is too many tick marks for the face of a wrist watch because of the difficulty of distinguishing them as a result of parallax [Citation reference: Dozensonline > Number Bases > Non-Dozenal bases > All Your Base Are Belong To Us “:
https://www.tapatalk.com/groups/dozensonline/all-your-base-are-belong-to-us-t1615.html#p40005801
” DavidKennedy
Jan 22, 2017 #17″:
“With as many tick marks as for hexadecimal, parallax would be a huge problem in reading which mark the hand points to.”].

Hence, for a dozenal clock, the number of tick marks should not be as few as twelve, but rather should be as many as the square of twelve which is perhaps just about not too many for a clock or even the face of a wrist watch as long as there is a ticking mechanism which improves distinguishability. However, the most useful number of tick marks for small watch faces is likely to be fewer, which is a reason for preferring half a gross or six dozen in civil analogue clocks. The clock is only a measurement device and does not constrain the consistently dozenal notation and divisions for time for scientific purposes [Citation reference: Dozensonline > Tools > Applications and appliances > Requirements For Implementation Of Uncial
https://www.tapatalk.com/groups/dozensonline/requirements-for-implementation-of-uncial-t1591.html#p40009356
“DavidKennedy
Apr 28, 2017 #2″:
“Time
Hours should be converted to semi nullry semiduors. Minutes should be converted to quick primry units of fifty seconds. Seconds should be converted to quick bisecondry units of the square of five sixths of a second. (Multibase clock)”].

The system of subdivisions for scientific purposes, as opposed to that for civil purposes which is so closely related, thereby becomes one not of base twelve but rather of twelve squared, and consistently so, yet encoded dozenally by pairing of dozenal figures, which is a natural way to group the “digits” linguistically [Citation reference: Dozensonline > Tools > Number Names > Uncial Nomenclature
https://www.tapatalk.com/groups/dozensonline/uncial-nomenclature-t1570.html#p40009563
“DavidKennedy
May 08, 2017 #14″:
“from the language point of view, grouping in twos is probably better, and this also makes sense from the viewpoint of the hands necessary for a clock with tick marks, as I have already explained.”].

I point out and emphasize that this scheme of subdivisions or series of magnitudes in the scale is consistently of the square dozen base, by choosing a period of twelve days and subdividing this into successive levels of subdivisions or ordinately. It is not a mixture of different bases at various subdivisional levels as the sexagesimal system is because of the number of hours per day differing from the number of minutes per hour or seconds per minute. Thus, the number of *secondries per *primery or minette is a square dozen, the number of minettes per *nullry or duor or double hour is a square dozen, and the number of duors per twelve days is a square dozen.

However, in the civil scheme, the divisions of the day become hemiduors, the *primeries are minettes as for the scientific scheme, and then *dysecondries from the viewpoint of the scientific scheme. The *dysecondry can be abbreviated to *dyse. The *trisecondry may be abbreviated to *tryse which is near to a conventional sexagesimal second in duration.

The Trouble With Base Thirty

Whereas the importance of a prime number in metrology is influenced by such objective features as the ease with which a division into that prime number of pieces can be constructed, determined by laws of mathematical geometry, the optimally efficient bases for certain metrological procedures including measurement and the specification of values to the greatest accuracy balanced with the minimum number of steps or symbols, and subjective or human cognitive capabilities such as limits of subitisation, the importance of a prime number as a factor can be quantified mathematically as the expected number of times that the prime number appears in a randomly selected number.

For example, the prime number two appears at least once in all of the even numbers or half of all whole numbers. Furthermore, the prime factor two appears again in numbers divisible by four or a quarter of all whole numbers, and yet more frequently in numbers divisible by eight, and so on. The sum of these expected frequencies is the converging limit of the infinite series

1/2 + 1/4 + 1/8 + … + 1/2^n

which is a geometric series of common ratio 1/2 between adjacent terms. The sum is given by the first term 1/2 divided by the subtraction of the common ratio 1/2 from unity, which works out to be one. This means that if numbers were selected at random, the expected average number of times the prime number two would appear as a factor would be once.

Similarly, the expected number of times the prime number three occurs in randomly selected numbers is given by the sum of the infinite series

1/3 + 1/3^2 + 1/3^3 + … 1/3^n

and this works out to be a half or 1/2, so the expected frequency of the prime number three in randomly selected numbers is a half.

In general, the sum of the infinite series for a prime number p is (1/p)/(1 – 1/p) = 1/(p – 1). So, the expected frequency of the prime number five would be 1/(5 – 1) = 1/4, a quarter. Likewise, the expected frequency for the prime number seven turns out to be a sixth. [The author completed this reasoning and derived this formula for this purpose of the quantification of the importance of prime numbers by the month June of the year 2017]

From these calculations, and specifying the importance of a prime number as being directly proportional to its expected frequency of occurance in randomly selected numbers, we see that the prime number two is twice as important as the prime number three, which in turn is twice as important as the prime number five.

To choose or design a base of numeration to have in its prime factorisation an appropriate balance of prime factors according to their importances so defined mathematically, we ought to select the prime factorisation to be a power of the form

[2^1 * 3^(1/2) * 5^(1/4) * 7^(1/6) * … * p^(1/(1-p))]^n,

choosing some value of n such that the exponents of the prime numbers are whole numbers.

Since the number of prime numbers is infinite and they cannot all be included as prime factors in a usable base, the infinite product must be truncated after the first few prime numbers, such that only the most important prime numbers in order are prioritised. Truncation after just the smallest prime number produces the binary-type bases of the form 2^n, such as two, four, and the square of four.

Truncation after the second smallest prime number three produces the number twelve or its powers as the base.

If the prime factor five is insisted upon, truncation to exclude the prime number seven and larger primes gives the number 2^4 * 3^2 * 5^1 = 720, which happens to be six factorial or the product of the first six whole numbers, as the smallest possible example that maintains the correct frequencies of the three smallest prime factors in accordance with their mathematically expected occurrances.

For a practical base of numeration, the number 720 is far too large. Those who desired the prime number five at any cost would therefore have to decrease the frequency in the prime factorisation of the smaller and objectively more important prime numbers two or three, while preferably still maintaining that the exponent of a less important prime number will not be greater than that of a more important prime number, resulting in such numbers as

2^3 * 3^2 * 5^1 = 360
2^4 * 3^1 * 5^1 = 240
2^2 * 3^2 * 5^1 = 180
2^3 * 3^1 * 5^1 = 120
2^2 * 3^1 * 5^1 = 60
2^1 * 3^1 * 5^1 = 30.

The number 360 was used as the number of angular degrees in a full circular turn. The number 60 was used as the base of the sexagesimal system of numeration, now retained in the numbers of minutes in an hour, seconds in a minute, and for subdivisions of angular degrees. Others have emphasised that these numbers as bases remain too large for convenient computations from the viewpoint of the limitations of the long-term human memory for multiplication tables. However, the inconvenience of large numbers as bases of numeration can also be argued from less subjective or more objective reasons of mathematics and consequently metrology.

Consider the smallest of the above numbers 30 as a base to include the prime factor five. Since this number has a disproportionately large exponent of the prime factor five, the smaller and more important prime factors two and three are under-represented. Consequently, numbers as the denominators of unit fractions which are expected to contain more of the prime number two which occur more often would be represented by the same number of digital value places as much less frequent unit fractions having much larger denominators as powers of the less important prime number five. This is because the number of digital value places in the digital form of a fraction written in a base B is determined by the prime factors of that base B and the prime factors in the denominator of the fraction in its simplest form. So, in the case of base thirty, the fraction 1/8, an eighth, where eight is two to the power of three, which is expected to appear frequently compared to fractions with larger denominators, would have three digital value places, because the power of the prime number two in the base thirty is only one. In base thirty, the digital form of a fraction such as 1/5^3, which is decimally 1/125, or 1/3^3, which is decimally 1/27, would also have three digits after the fractional point. However, we should have preferred for rather the numbers that appear more frequently to be represented by fewer digital value places or significant figures at the expense of those that are less frequent. In contrast, a base such as dozenal having a larger and more proportionate exponent of the smallest prime numbers would represent the fraction 1/8 in digital form by fewer than three digital value places.

This mathematical reasoning becomes relevant to the representation of numbers and their measurement in metrology. In a base such as thirty, where the number of figures after the fractional point required to represent the fraction 1/8 digitally is three, the amount of precision by obligation to attain this number of significant figures is phenomenal, because thirty to the power of three is twenty-seven thousand. A requirement to reach such precision or one part in so many thousands in order to express a fraction so simple and frequent as an eighth or one part in eight is unwarranted and heavily burdensome. A result of such excessive precision is many irrelevant steps or degrees of the scale at each value position or place in the digital representation or on the ruler or implement of measurement. [This reasoning by the author on the base thirty was done on Tuesday 18th December 2018]

A base such as the square of four, being a binary power, would represent a commonplace fraction such as an eighth using fewer digits. However, a base with only the prime number two in its factorisation would then not represent the frequent ternary fractions with a judicious amount of concision. Only the base twelve and its square represent the fractions of the most important prime numbers in the optimal way, until the number 720 which is too large as a base for the reason that it would demand too much precision at each positional digit even for the representation of the simplest and most commonly used fractions.

In order to discriminate a unit fraction such as an eighth, 1/8, at its level of precision it is required to be discernable as distinct from a seventh, 1/7, and a ninth, 1/9. The difference to be discriminated or resolved between 1/8 and 1/7 is their difference 1/56. The resolution needed to separate 1/8 from 1/9 is their difference 1/72. The average of these differences 1/56 and 1/72 is 1/63, which is near 1/64. In general, for resolution of a unit fraction 1/n, it must be distinguished from 1/(n – 1) and 1/(n + 1), by differences of 1/(n(n – 1)) and 1/(n(n + 1)). The average of n(n – 1) and n(n + 1) is n^2, so the required precision is about 1/n^2. This means that if the fraction to be resolved is 1/n and the base is n, then there is a requirement to go to the next digital value place after those needed to just specify the fraction to ensure that there is a following zero.

Fractions whose denominators have prime factors that are not factors of the base of numeration by which those fractions are represented do not have terminating digital forms. Their number of significant figures would be endless. Such non-terminating digital expressions are not convenient either for mathematical calculations where their necessary truncation would cause the accumulation of rounding errors or for metrological measurement where they would require an unattainable infinite number of steps in procedure. Numbers such as a third in decimal as 0.333… are of this type, where the digital representation never ends but does repeat. The number of digits between repetitions depends on whether the denominator of the fraction shares a factor with one of the numbers beside or adjacent to the base or a power of the base. For a fraction to have fewer digits before they repeat is slightly better than having a longer stretch of digits. However, non-terminating digital representations remain a nuisance no matter how short the cycle of repetition because of the errors and inexactness that they introduce into calculations and recorded values, such that their use is preferably avoided altogether. In the metrological sense, satisfactorily representating fractions with long periods of repetition, such as a seventh has in decimal, does not practically occur, especially for large bases, because the necessary precision is physically unrealistic. This means that, for example, in ordinary circumstances it is not practically feasible for a quantity to be measured with the decimal base in order to recognise whether it was intended to be exactly a seventh.

From the viewpoint of metrology, there is an inconvenience in a base having too many different kinds of prime factor. Before the metric system, haphazard agglomerations of units of measurement gave rise to irregular and unpredictable multiples of units to make up to the next named unit in the same type of measureable quantity. Irregularities such as these make calculations with them quite inconvenient. For example, the number of feet in a mile has an erratic prime factor, and there is no predictable way to extend the system of units to larger or smaller extremes or incorporate this into a sensible pattern. Having three different prime factors as the number thirty has would produce a similar effect of irregularity within its subdivisions and multiples between its powers. Subdivisions and multiples between powers become practically necessary where the base is too large to be subitised.

While historically, powers of two were used for convenience in metrology such as for denominations of weights and coins of currency, in the modern world with the advent of digital electronic computers and their storage of information in binary format, powers of base two have become even more important. In the decimal base to which society became accustomed, a fraction such as 1/2^6 = 1/64 having its digital decimal form as 0.015625 requires a ludicrous number of significant figures and consequent amount of precision. In contrast, the base twelve in representing the same fraction would require only half as many digits after the fractional point, because the power of two in the prime factorisation of the number twelve is twice that in the base ten. This means that base twelve is twice as suitable in the modern era than decimal for the representation of binary powers.

In summary, base twelve is the only practically sized base suitable for optimally representing fractions according to their mathematical distributions.

Base Twelve in Measurement

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.