# Quinquennial, decimal, duodecimal, vigesimal, sexagesimal

**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 duode*narius 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*

*Note*: *actus *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* ...

No comment published yet.