The attempt of an Amateur

For two millennia, especially the last two hundred years, mathematicians have made great efforts to understand the ancient Greek arithmetic. The most meaningful work on this topic is the book T.L. Heath [2]. The most important additional sources of information on ancient Greek arithmetic are, of course, the comments of Eutocius of Ascalon (5th-6th century AD) on the writings of Archimedes and Apollonius. However, they leave it incomprehensible how exactly the ancient Greeks calculated, although the scientists were convinced that they calculated correctly.

Scientists sought to understand the ancient Greek alphabetical arithmetic from the standpoint of the modern Indo-Arabic decimal positional number system. But this approach also contains an error: the alphabetic number system is not periodic, no matter how similar it would be to the periodic system. The alphabet system was progressive and very convenient at the beginning of the commercial and social activities of people, when the numbers did not exceed thousands – tens of thousands. Here the advantage was so obvious (with the help of three characters to display any number less than a thousand) that many nations copied this system into their alphabets. But the transition from a non-periodic numerical system to a periodic one is dangerous in that we may lose something inherent in the non-periodic system. And I show that this “something” is an important function, called a numerical ladder, which is absent in the periodic system, and in a non-periodic system it was actively used for calculations.

I believe that analyzing ancient Greek arithmetic, i.e. logistics, as it was called in ancient times, it is necessary with the help of only those tools that the Greeks of that time had. Such instruments are, first of all, the alphabetic sequence of letters borrowed by the Greeks from the Phoenicians, as well as the abacus for the Ionian alphabetic system, i.e. a counting device that allows to perform arithmetic operations of addition and subtraction of numbers written in this system.

Abacus as a device did not appear immediately. Apparently, at the end of the 9th century BC, the Fayum tables [3] were dated this way, the abacus was not yet invented, and merchants and other accountants had to memorize the abecedarium [4] by heart. And we see that they did not always remember the abecedarium correctly. For example, in an abecedarium from the excavations in Tel Zaiit, waw is placed before he, het is placed before zayin, and lamed is placed before kaph. Also in the reverse abecedarium found in the excavation of the ‘Izbet Sartah of 11 century BC, deviations from the order of the Phoenician letters were also found. That is, in those days it was required to memorize an abecedarium by heart, as indirectly indicated by the artifacts found. And this memorization requirement means that the physical abacus device did not exist yet.

And only the Marsilian tablet, which dates from a later time (mid-8th – early 7th century BC) shows us an example of a physical device of an abacus.

Separately, there is the problem of Greek multiplication. This, undoubtedly, the Greek invention successfully worked during the times of Pythagoras – Archimedes and later with the help of those tools that the Greeks of that time owned. Among these tools I include the multiplication table of pythmens [5] (9 × 9 or 10 × 10), as well as a numerical ladder. The idea of a numerical ladder was first pointed out by Professor D. Psychoyos [6]. I just implemented this idea in the form of a table.

The idea of a numerical ladder makes sense and utility only in a non-periodic numerical system, which was the alphabetic system of Phoenicians, Greeks, etc. This system is good for writing small numbers not exceeding a thousand. After all, each category (units, tens, hundreds) requires nine different signs. The Greeks for units of thousands used their letters for units, just added an apostrophe to them on the left. And the Armenians of Mesrop Mashtots increased the number of letters of their alphabet, and they had 37 letters of the alphabet (36 letters for units, tens, hundreds and units of thousands and one letter for ten thousand).

All these ideas allowed me to examine in detail the entire kitchen of ancient Greek arithmetic and understand how they calculated. All nations that were within the empire of Alexander the Great were also calculated, the empire fell apart, and the calculation technique was preserved. Including and in ancient Israel in the Hellenistic era.

All this is described in detail in my book “The Fundamentals of the Ancient Greek Arithmetics”, 2018 [7]. The book can be ordered through Amazon.com [8].

Mark Tsayger [1]

Notes

1. Mark Tsayger, PhD, Beer Sheva, ISRAEL, email: m_tsayger@hotmail.com

2. T.L. Heath, “A History of GreekMathematics”, Oxford, 1921

3. M. Tsayger, “About the Fayum Tablets”, kontinentusa.com, 05.01.2018, https://kontinentusa.com/about-the-fayum-tablets/

4. Abecedary-the sequence of the alphabet from the first to the last letter. Direct abecedary – when the sequence is built in the direction of writing in a given language. The reverse abecedary is built in the direction opposite to the direction of writing.

5. Pythmens – here are the first nine letters of the Greek alphabet corresponding to numbers from 1 to 9.

6. Dimitris K. Psychoyos, The Forgotten Art of isopsephy and the magic number KZ, Semiotica, vol. 154, April 2005.

7. Mark Tsayger, “The Fundamentals of the Ancient Greek Arithmetics”, 2018.

8. See https://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Daps&field-keywords=tsayger.