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1001 deeds, sayings, curiosities and anecdotes of the ancient world

Quinquennial, decimal, duodecimal, vigesimal, sexagesimal

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Samuel Noah Kramer (1897-1990) published his work "The story begins in Sumer", worldwide famous, in 1956. Certainly, many things started before, but the first written records are found in Sumer and so that's where we first talk about "History".

The man began to count very soon; Paleolithic marks on bone, stone or other materials testify to this. No doubt on this task he used the help of his hands, fingers of his hands, his feet ...

Soon he needed some figurative representation; he had need of the figures, because visually with a stroke of the eye he sees only up to four units; to recognize more he needs to count. And for counting large numbers he must have different orders or hierarchical levels, so that  he can represent large quantities with few figures.

In Sumer there are traces of a numeral system with five as the base (quinary) (five lower units amount to a higher level), certainly in relation to the five fingers.

The decimal system is certainly in connection with the fingers of both hands.

Also the Sumerian used a duodecimal system, a dozen. Probably they counted the phalanges of the four fingers (three phalanges in each finger) pointing with the thumb. Probably a system of this type is in relation to the twelve months of the year or 12 lunar months that occur in the solar year and the twelve signs of the zodiac; number 12, by the way, could serve to divide the day into twelve spaces, the twelve hours.

And also they used a sexagesimal system, in which sixty lower units amount to a higher order.

We do not really know why this sexagesimal system was invented, a tiresome system. Many hypotheses have been formulated. One of the most plausible is that, coexisting decimal and duodecimal system, the least common multiple to facilitate equivalence was sought. In fact 60 it is divisible by numerous divisors: 1, 2, 3, 4, 5, 6, 10, 12, 20, 30, 60.

The Sumerians were provided with a numeric notation circa 3200 BC on the base 60; and they attributed a special sign to the following units: 1, 10, 60, 600, 3600, 36,000, 216,000

Sumerians  raised on this basis  the division of time into hours, minutes and seconds and the division of the circle into three hundred sixty degrees; they divided these into sixty minutes and these in sixty seconds. We used still this system.

The Egyptians and everyone on the Middle East, including the Mediterranean peoples and the Greeks at the time, learned certainly from Sumerians.

Today, the decimal system is the most widespread, but the duodecimal system has been remarkably successful, because there are still remains in modern societies and languages. We continue demanding on Spanish "a dozen eggs ...".

Dozen derives from the Latin Duodecim (duo=two and decim=ten), (two + ten)

The term remains buoyant in many languages, which testify the historical importance of the duodecimal system. So on French "douzaine" ("à la douzaine); on English “by the dozen” and twelve; on German Dutzend; on Italian dozzina (1 dozzina di uova); on Portuguese dúzia (uma dúzia de ovos); on Catalan dotzena (una dotzena d'ous).

Naturally the term and the concept existed in the ancient world. On ancient Greek it is δυωδεκάς, duódekas, from δώδεκα, Dodeka (two and ten).

On Latin it is used generally the expression  Numerus duodenarius and later, since Tertullian (II-III century BC), the term duodecas also appears.

Duodenarius is built from numeral duodecim, twelve (two and ten) as I stated above and duodecas is constructed from the Greek (dŭōdĕcas, ădis.  f., = δυωδεκάς).

The term duodecas apears on the “Adversus omnes haereses, IV, 1, Against All Heresies,
work attributed to Tertullian, considered spurious; it sometimes is seen following his De Praescriptione haereticorum and  therefore also its unknown author is called the "Pseudo-Tertullian"

for of these last also proceeded twelve Aeons; from Speech, moreover, and Life proceeded other ten Aeons: such is the Triacontad of Aeons, which is made up in the Pleroma of an ogdoad, a decad, and a duodecad. (Translated by the Rev. S. Thelwall.])

Sed enim ex his quoque processisse duodecim Aenoas; de sermone autem et vita Aeonas alios decem: hanc esse Aeonum triacontada, quae fit in pleromate ex Ogdodae et Decade ac duodecade.

Then from Renaissance and Humanism the term "duodecadas" (a dozen) was used frequently in the title of several books and treatises, for example: Philippi: Duodecas thematum, 1612, “A dozen themes”; Hermann Witsius, Exercitationum academicarum Duodecas -, Duodecas dissertationum exegeticarum et apologeticarum ... etc.

The oldest and most important in the early days of Rome statute, legal text, the Twelve Tables, developed in the V century BC, warns us  the importance of the number 12th. Naturally the wooden support is not retained but the reference to the text are very abundant in Latin authors and historians.

There is a text of Marcus Terentius Varro in his "De lingua Latina" (On the Latin language) that warns us of the old system in Rome.

Varro Says in De Lingua Latina, Book V, (6), 34:

There are many things which the ancients delimited with a multiple of twelve, like the actus of twelve ten-foot measures. (Translation by Roland G. Kent. Ph.D.Loeb Classical Library.1938)

Multa antiqui duodenario numero finierunt ut duodecim decuriis actum

Noteactus is the way in which livestock can be driven, as it is told earlier in the same paragraph:

As an ager “field” is that to which driving can be done, so that whereby driving can be done is an actus “driveway”. (Translation by Roland G. Kent. Ph.D.Loeb Classical Library.1938)

“Ut ager quo agi poterat, sic qua agi actus”:

Duodecaiugum is twelve  horses team...; duodecennium is the period of twelve years; duodecemviri is the college of twelve magistrates.

The duodecimal system is also applied  to establish their system of weights. This system is more practical than the decimal because it supports more divisions (1, 2,3,4,6) and therefore more fractions can be expressed.

The duodecimal system was well suited to the application of interest on loans. At first it seems that the rule was charged twelve ounces of annual interest on a loan of one hundred ounces; ie, it is charged  12% pa (per annum), equivalent to an as (Roman coin), because one as was worth 12 ounces. If the account is made for months, six months interest was called semis, which means half an as, four months interest was called triens, which is the third part of an as, to three months interest was called quadrans, which is the fourth part, and a month interest is called unciaria, which is the twelfth  part of as or one part; because that the interest of 1% per mensem, per month, is called hundredth (centesimal) interest.

Anyway, all this numerically and terminological precision is possible precisely because the duodecimal system is used.

In Anglo-Saxon countries this system has endured even more in the system of weights and measures; and a foot is twelve inches, a pound is 12 ounces, ...

The sexagesimal system had also and continues to have an amazing success, because it is still used to measure the space in "degrees",minutes and seconds and to measure time in hours, minutes and seconds.
The word "gradus", "degree" means step.

To know the origin of the words hour, minute and second , I refer to the article published in this blog  http://en.antiquitatem.com/hours-minutes-seconds-almagest-sumer

We can also refer to other systems such as the vigesimal; we have  a remainder, for example, on French, when eighty, 80,  is called quatre vingt (four twenties); the existence of twenty fingers in the four limbs or human extremities would lead to it.

It is worth  make a reference to binary numbering system, or base two,  which is the basis of all current computer and digital development.(Latin bini-ae-a means two by two.  But experts use the hexadecimal, that is, base 16 ...


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