Ancient Computers, is a paper of mine posted 7/15/2010 on IEEE Global History Network.
It largely supercedes other material on this site. It determines:

1. that the Romans used The Salamis Tablet structure as their line abacus and used it for both duodecimal and decimal calculations;
2. that the Babylonians designed The Salamis Tablet abacus structure to accomodate their cuneform sexagesimal numbers;
3. how the pebbles were manipulated to do all four arithmetic operations in all three bases (10, 12, 60); and
4. that the methods of use are very rapid, the total number of pebbles needed very small, and the line abacus very portable.

Kojima, Takashi, 1954. The Japanese Abacus: Its Use and Theory, Charles E Tuttle Company, Tokyo. [page numbers refer to my 1963 printing. There is an on-line version retrieved July 11, 2007, from http://www.tux.org/~bagleyd/takashikojima1.pdf.]

SKS: My copy is the 16th printing of 1963. I bought it in Tokyo, along with a small Japanese abacus, called a soroban, in the Spring of 1964. It sparked my interest in computing devices, and a 30 year career in the computer field. There are still pockets of use, even outside of Japan; e.g. Abacus for the Blind.

[p.25] "All three types of abacuses were found at some time or other in ancient Rome —the dust abacus, the line abacus, and the grooved abacus [see Roman Hand Abacus below]. Out of this last type yet a fourth form of the abacus was developed—one with beads sliding on rods fixed in a frame. This form, the bead or rod abacus, with which calculations can be made much more quickly than on paper, is still used in China, Japan, and other parts of the world. In Europe, after the introduction of Arabic numerals, instrumental arithmetic ceased to make much progress and finally gave way altogether to the graphical as the supply of writing materials became gradually abundant.
 
As for the Orient, a form of the counting-rod abacus, called ch’eou in China and sangi in Japan, had been used since ancient times as a means of calculation. The Chinese abacus itself seems, according to the best evidence, to have originated in Central or Western Asia. There is a sixth-century Chinese reference to an abacus on which counters were rolled in grooves [see Kojima, 1963, below]. The description of this ancient Chinese abacus and the known intercourse between East and West give us good reason to believe that the Chinese abacus was suggested by the Roman. The Chinese write in vertical columns from above downwards. If they ever are compelled to write in a horizontal line, they write from right to left. But the abacus is worked from left to right. This is another indication that the abacus was not indigenous to China. The present Chinese bead abacus, which is generally called suan-pan (arithmetic board) in Mandarin and soo-pan in the southern dialect, was a later development, probably appearing in the twelfth century, and did not come into common use till the fourteenth century. It is only natural that the people of the Orient, having retained a system of numerical notation unsuited for calculation, should have developed the abacus to a high degree, and its continuous universal use even after the introduction of Arabic numerals is eloquent testimony to the great efficiency achieved in its development.
 
The Japanese word for abacus, soroban, is probably the Japanese rendering of the Chinese suan-pan. Although the soroban did not come into popular use in Japan until the seventeenth century, there is no doubt that it must have been known to Japanese merchants at least a couple of centuries earlier."

SKS: Kojima's drawing of the Roman Hand Abacus is different from the two referenced below. The dual-base column is sandwiched between the last decimal column and a duodecimal column, and seems to be up-side-down. I wonder if this is an actual alternate configuration that the Romans used, or is it a printing error? Kojima may have gotten the drawing from this "Roman Soroban" shown on http://www.soroban.com/index_eng.html.

Kojima, Takashi, 1963. Advanced Abacus: Japanese Theory and Practice, Charles E Tuttle Company, Tokyo. [page numbers refer to an on-line version retrieved July 11, 2007, from http://www.tux.org/~bagleyd/takashikojima2.pdf.]

[pp.5-6] "The only reliable account of the origin of the Oriental abacus is in a book entitled Mathematical Treatises by the Ancients compiled by Hsu Yo toward the close of the Later Han dynasty (A.D. 25—220) at the beginning of the third century and annotated by Chen Luan in the sixth century. This book gives some information about various reckoning devices of those days and was one of the Ten Books on Mathematics (Suan-hwei-shi-chu) which were included among the textbooks to be read for government service examinations in China and Japan for many centuries.
 
Chen Luan in his note gives the following description of the calculating device:
 
'The abacus is divided into three sections. In the uppermost and lowest section, idle counters are kept. In the middle section designating the places of numbers, calculation is performed. Each column in the middle section may have five counters, one uppermost five-unit counter and four differently colored one-unit counters.'
 
The ... figure [to the left] represents the abacus as pictured in accordance with the foregoing description. The board represents the number 37,295.
 
The extent to which the counting board was used may be told by Hsu Yo’s poetical description of the board. The verse, which is highly figurative and difficult to decipher, may read: 'It controls the four seasons, and coordinates the three orders, heaven, earth, and man.' This means that it was used in astronomical or calendar calculations, in geodetic surveys, and in calculations concerning human affairs.
 
The reader will notice a close similarity between this original Oriental abacus and the Roman grooved abacus [i.e. The Roman Hand Abacus, see below], except for the difference that counters were laid down in the former while they were moved along the grooves in the latter. Because of this and other evidence, many leading Japanese historians of mathematics and the abacus have advanced the theory that the above-mentioned prototype of the abacus was the result of the introduction into the East of the Roman grooved abacus.

The following corroborative pieces of evidence in favor of this theory are cited in the latest works by Prof. Yoemon Yamazaki and Prof. Hisao Suzuki of Nihon University.
 
(1) The original Chinese abacus has a striking resemblance in construction to the Roman grooved abacus, as is evident in the foregoing quotation from Hsu Yo’s book, e.g., four one-unit counters and one five-unit counter in each column.
 
(2) The method of operation of the ancient Chinese abacus was remarkably similar to the ancient Roman method.

 (3) Traces of reckoning by 5’s may be found in the Chinese pictorial representation of reckoning-block calculation as in the Roman numerals, as:  

(4) Trade was carried on between China and Rome. Chinese historical documents written in the Han dynasty (206 B.C.-A.D. 220) furnish descriptions of two land routes, called silk roads, connecting the two great empires.
 
Inasmuch as even in olden days valuable products or devices made in one country were transmitted to others with astonishing rapidity, the above facts may well substantiate this theory.

Among the dozen other reckoning devices mentioned in this book are the reckoning boards pictured below [called Chu Pan]. These boards are presumed to date back to the days of the Chou dynasty, which ended in 249 B.C.

(The number on the board [to the left] is 23 957)

([On the board to the right] When yellow counters were used, the squares in each column represented 1, 2, 3, and 4 respectively. When blue ones were used, they represented 5, 6, 7, 8, and 9 respectively. The black balls in the figure stand for blue counters. The number on the board represents 3581.)"

SKS: Note how very different in structure these Chu Pan boards are from the abacus above. The Chu Pan boards define or record a number, the abacus above counts the number. How would you add two numbers on the Chu Pan? We certainly know how we'd do it on the Roman Hand Abacus.

[p.7] "These and other reckoning devices are believed to have gone out of use as the previously mentioned abacus developed and gained popularity.
 
... The chief calculating devices which are known to have been used in China from before 1,000 B.C. to the days when the abacus came into wide use are reckoning blocks called ch’eou in China and sangi in Japan and slender bamboo sticks called chanchu in China and zeichiku in Japan. The former device continued to be used in the East for calculation until not many years ago, and the latter device, which was more awkward, was largely replaced by the former for calculating purposes and is presently used only by fortunetellers for purposes of divination.
 
Until the introduction of Western mathematics, mathematicians in China and Japan utilized reckoning-block calculation, which had not only been developed to the point of performing basic arithmetic operations but was also used to solve quadratic, cubic, and even simultaneous equations. It is presumed that they did not think it worth while to concern themselves with the other reckoning devices, including the abacus, which was, in their eyes, an inferior calculator barely capable of performing multiplication and division by means of the primitive cumulative method of addition and subtraction. Probably another reason which alienated mathematicians from these reckoning devices was that these instruments gave only the result of calculation, and were incapable of showing either the process of calculation or the original problem.
 
In ancient times China was primarily a nomadic and agricultural country, and business in those days had little need of instruments of rapid calculation. Anyway a millennium after the Han dynasty there was no record of the abacus. During the dozen centuries beginning with its first mention in the Han dynasty until its development, this primitive calculator remained in the background.

However, with the gradual rise of commerce and industry, the need for rapid calculation grew. The modern, highly efficient abacus, which probably appeared late in the Sung dynasty (906—1279), came into common use in the fourteenth century. The great rise and prosperity of free commerce and industry during the Ming dynasty (1368 - 1636) are presumed to have promoted the use and development of the abacus. A number of books on mathematics brought out in those days give descriptions of the modern Chinese abacus and give accounts of the modern methods of abacus operation, including those of multiplication and division.

BIBLIOGRAPHY

Yamazaki, Yoemon: A Collection of Eastern and Western Literature on the Abacus, two volumes, Morikita Publishing Company, Tokyo (in Japanese).
Yamazaki, Yoemon: The Origin of the Chinese Abacus.
Yamazaki, Yoemon; Suzuki, Hisao; and Toyo, Sei-ichi: A History of Abacus Calculation, Morikita Publishing Company, Tokyo (in Japanese)."

SKS: So, at the time when the Romans were using their Hand Abacus to run their already urban civilization, the Chinese were still in their mainly nomadic and agricultural stage and did not recognize any value to the abacus introduced to them by the Romans, and ignored it for over a thousand years!

Here are Asian scholars saying that the Chinese Abacus, the suan-pan, was actually a modified copy of the Roman Hand Abacus, confirming my earlier conjecture. How then can so many web pages claim that the Chinese invented the abacus? How many other things have they claimed to invent but did NOT? (Here's one that writes, "the Chinese invented paper around the end of the first century AD", even though papyrus paper was being used by the Egyptians since 4000 B.C.) To repeat, the Chinese DID NOT invent the abacus! (And they DID NOT invent paper!) The Roman Hand Abacus is the earliest known constrained-bead abacus. The earliest extant counting-board style abacus is The Salamis Tablet, which was used about 300 B.C. by the ancient Greeks and contemporaneous Babylonians in Mesopotamia, present day Iraq. Precursors of the Babylonians, the Akkadians, also Mesopotamian, are thought to have invented the earliest counting-board type abacus about 2300 B.C.

Li, Shu-T'ien, 1959. Origin and Development of the Chinese Abacus, Journal of the ACM Volume 6, Issue 1, January 1959. Retrieved July 19, 2007, from http://webhome.idirect.com/~totton/soroban/.

SKS: Mr. Li tries to establish continuity of development from the Chu Pan, the two bead boards above, to the Roman style abacus on pages 105-106. But there are only statements claiming the fact, not a description of how the fundamental structure was changed by the Chinese from number definition or recording to number counting, and why. And there is no mention of the contact with the Romans over the silk roads at the time of the first appearance of the Roman style abacus in China, nor why there was a thousand year delay in further development of the abacus into the suan-pan. So I'd say Mr. Li's arguments are not as illuminating nor as convincing as Mr. Kojima's.

It is certainly true that China had calculating devices well before 1000 B.C., it's just that theirs were not abaci, neither constrained-bead nor counting-board forms. And the origins of the Roman Hand Abacus, which China copied, can be traced through The Salamis Tablet back to the Akkadian's invention of the counting-board form abacus in Mesopotamia about 2300 B.C.

In a July 25, 2007, message on SorobanAbacus@yahoogroups.com, Fernando Ota provided this insight:

"I have noticed that at least in Japanese the name soroban seems to be given to almost any counting device, be it a Roman hand abacus, a Russian schoty, a Chinese suan pan, or a Persian/Iranian chortke. If the same happens in Chinese, then this may explain why Li saw a continuity in the evolution of the Chinese abacus."

So we might forgive Mr. Li for his culturally motivated oversight, but it is still a mistake nonetheless. And the Chinese DID NOT INVENT THE ABACUS.

Images of replicas of The Roman Hand-Abacus:

[1] Showing one rightmost column slot, Dave King, Dorling Kindersley, Courtesy of The Science Museum, London.	[2] Showing 3 rightmost column slots, Prof. Dr. Jörn Lütjens, Hamburg, Germany, May 2004.
[3] Showing reversed rightmost columns, with 3-slot column upside-down, Computer Museum, University of Tokyo.	[4] is the source for [2] and is the image on p.819 of Opera historica et philologica, 1682, by Marcus Welser. (referenced on p.305 of Number Words and Number Symbols, A Cultural History of Numbers, by Karl Menninger, MIT and Dover Publications.

SKS: There are 3 slots in the rightmost column in [2] to allow representation of multiple bases. In this Excel workbook there are sheets modeling various Roman abaci, including two versions of The Roman Hand-Abacus using different bases in the last digit. [3] and Kojima's erroneous drawing above may both have come from the photo, attributed to the British Museum, on p.19(?) of Pullan, J.M., 1968, The History of the Abacus (noted below). 

SKS: The image in [4] is the one that is used to definitively show that the Romans used the Salamis Tablet as their heavy duty counting board; and what methods to use.

SKS: The possible configurations for the three slot rightmost column of [2] are:

		Slot Values
Units	Base	Bottom	Middle	Top
halves	2	0	0	1/2
thirds	3	1/3	0	0
fourths	4	0	1/4	1/2
fifths	5	1/5	1/5	1/5	(all as one 4 bead slot)
sixths	6	1/6	1/2
eighths	8	1/8	1/8	1/2	(B & M as one 3 bead slot)
ninths	9	1/9	1/3	1/3	(M & T as one 2 bead slot)
twelfths	12	1/12	1/4	1/2

Since the second column from the right is a base-12 duodecimal column, the Romans could thus count multiples of the unit fractions 1/12, 1/24, 1/36, 1/48, 1/60, 1/72, 1/96, 1/108, and 1/144 on the Hand Abacus. If they ignored the duodecimal column, just using the multiple base column, they could also count multiples of the unit fractions 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, and 1/9.

Even more possible fractions here.

Ifrah, Georges, 2000. The Universal History of Numbers: From Prehistory to the Invention of the Computer, French to English translation by Bellos, Harding, Wood, and Monk, Wiley.

SKS: Not well regarded by some scholars, yet it is a monumental work of great scope. It represents one man's 10 year quest to find and explain the history of numbers to us non-scholars.

[p.189] "Fig. 16.31. Latin inscriptions from the Republican era showing the use of the principle of subtraction. Use of this principle (which undoubtedly reflects the influence of the popular system on the monumental system) was nevertheless unusual on well-styled inscriptions.

[p.viii] "CIL = Corpus inscriptionum latinarum, Leipzig & Berlin, 1861-1943"

SKS: These inscriptions are strong evidence that the Romans registered numbers with negative parts on their abacii. So 8 would be represented as 10-2, 9 as 10-1, and 4 as 5-1, etc.

Stephenson, Steve, Sep. 1, 2002. The Roman Hand-Abacus, http://www.ee.ryerson.ca/%7Eelf/abacus/roman-hand-abacus.html.

SKS: The Roman Hand Abacus is the progenitor of all the non-American constrained counter/bead abaci (see Kojima, 1954 & 1963 above). But I believe the Romans developed it as a portable computing device, and that THE ROMAN HAND ABACUS WAS LESS CAPABLE THAN THEIR COUNTING-BOARD ABACI.

For one thing, the counting-board abaci could easily be drawn with as many duodecimal, hexadecimal, or octal digits as needed for the problem at hand. Certainly Frontius (below) MUST have used a counting-board abacus to do his pipe calculations (two squares and a division) because he refers to 2/3 of 1/288, which can't even be registered on a Hand Abacus, let alone calculated there (click here to see how Frontius probably calculated this).

Frontinus, Sextus Julius. The Aqueducts of Rome. Retrieved July 6, 2007, from http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Frontinus/De_Aquis/text*.html.

[§24] "... Now the digit, by common understanding, is 1/16 part of a foot; the inch 1/12 part. ..."

[§25] "... The most probable explanation is that the quinaria received its name from having a diameter of 5/4 of a digit ..."

[§26] "... Now the inch ajutage, has a diameter of 1 1/3 digits. Its capacity is [slightly] more than 1 1/8 quinariae, i.e., 1 1/2 twelfths of a quinaria plus 3/288 plus 2/3 of 1/288 more. ..."

SKS: Click here to see how Frontius probably calculated this on an abacus.

Frontinus, Sextus Julius. The Aqueducts of Rome. Retrieved July 6, 2007, from http://www.uvm.edu/~rrodgers/Frontinus.html.

[§26] "... The inch pipe has a diameter of l  l/3 digits; its capacity is a little more than l  l/8 quinariae, the fraction being l/8 plus 3/288 plus 2/3 of another l/288.[97] ..."

[footnote 97] "Normally a scripulum (1/288) is the smallest fraction used by the Romans."

SKS: Perhaps Romans didn't use a smaller fraction as a direct measure, they certainly took fractional parts of the one they named the smallest, i.e. "2/3 of another l/288".

Pullan, J.M., 1968. The History of the Abacus, Frederick A. Praeger, Inc., New York.

[pp.3-4] Containing a table of numbers and their squares, "The Senkereh Tablet, 2300-1600 B.C., ... is proof that some method of multiplying numbers was known in Babylon about" 2000 B.C.

[pp.17-18] "The Latin 'abacus', derived from the earlier Greek word 'abax', meant, simply, 'a flat surface'."  The word 'abacus' did not derive from the Hebrew word 'abaq', dust, "and there is little evidence to support a common idea that a table strewn with dust, or sand, was at one time widely used for reckoning." On the other hand, "Sanded tables certainly seem to have been used for the drawing of geometrical figures, ... But it is not so easy to imagine counters being moved easily from place to place on a sandy surface. "

[p.89] "It is, strictly speaking, a misuse of the word [abacus] to apply it to a bead-frame calculating device."

[p.94] " ... it is rare to find a teacher, or anyone else for that matter, who understands how calculations were made and accounts kept before the introduction of Arabic figures. Too often it is stated, even in authoritative books, that when Roman figures were used there must have been difficulty because of the absence of a sign for zero. This would have been true only if people of the time had been foolish enough to try to do their 'sums' in the same manner that we now use Arabic figures, i.e. by writing on paper. Instead, it was the practice to write the amounts concerned, using cursive Roman numerals, to perform the actual calculation with counters on the abacus or counting-board, and to read off the result from the counters as they lay on the board. It could then be recorded on paper in the written notation."

[p.95] "Much attention has been given to ways in which 'number concepts' develop in a child's mind, and one result of this has been the appearance of a variety of types of 'structural apparatus'. Properly used, they can undoubtedly be of great value, and many teachers have implicit trust in the particular type of apparatus they have chosen. There is, however, a fundamental objection to most of these devices. They give a different picture to each dimension, units, tens, hundreds, etc., whereas it is a basic principle of the Arabic system of notation that the same figures are used in each position. The noughts in such numbers as 10, 100, and 1000 are not intended to give a different appearance but to put the significant figure (in this case 1) in its proper place. They could be omitted if the position of the figure were known ... This difficulty does not arise on the abacus where the same counters 'sometimes stand for more, sometimes for less'; indeed, it is difficult to see what advantages the newly invented types of apparatus have over the old method of counter-casting."

Barnard, Francis Pierrepont, 1916. The Casting-Counter and the Counting-Board, A Chapter in the History of Numismatics and Early Arithmetic, Oxford University Press, London.

[pp.313-314] Prof. Barnard (late professor of medieval archaeology, Univ. of Liverpool) describes and quotes from

Legendre, François, 1753. L'Arithmétique en sa perfection, Paris, pp. 497-528, Traité de l'arithmetiqué par les jetons.

"It was permissible to set and to work the jettons of the sum without using the spaces [between lines], ... But it was much more convenient to anyone who was expert at the practice, and less confusing to the eye, to reduce the number of jettons by using the spaces." [SKS: for counters of value 5 · 10ⁿ, n whole]

SKS: Barnard's scope is medieval Europe, and his emphasis is more on the counters per se than on counting-boards. Much knowledge of ancient science and mathematics was lost to Europeans during and before the dark ages; e.g., the suppression of knowledge and the persecution and execution of scholars by the early Christian Church, the burnings of the Library of Alexandria, etc. So the medieval and later use of the abacus in Europe, as described by Barnard, is primitive compared to what it must have been in and during the Late Babylonian and comparable Egyptian and Greek periods, when, I think, the Salamis Tablet was used in a manner like The Stephenson Abacus™; probably even well into the Roman period, for the practical Roman engineers would well have used an efficient calculating device. 

Lang, Mabel, 1957. Herodotos and the Abacus, Hesperia, Vol. 26, No. 3. (Jul. - Sep., 1957), pp. 271-288.

[p.271-272] "It is perhaps natural to assume that To, like his contemporaries, used an abacus for calculation. When his calculations are correct, however, it is impossible to trace the steps by which they were made and so to determine the means used in the calculation. But where he gives us both an incorrect solution and the material for the correct solution, it is possible not only to check his results but also to show the means he used to arrive at them.

Of six such calculations where Herodotos provides an apparently incorrect solution, only three show arithmetical error ... our material is almost certainly complete, the text appears to be sound, and the errors can be shown to be arithmetical. It is to these that we must look for enlightenment on Herodotos' arithmetical means and methods.

II, 142: the problem is how many years are represented by the 341 kings of Egypt. Herodotos calculates as follows : '300 generations of men make 10,000 years, for three generations of men are 100 years. Of the other 41 generations, which are over and above the 300, the number of years is 1340 [incorrect].' ... The material is given for the correct solution (11,366 2/3 years) both in formula and by example"

SKS: Sheet 5 of this Excel workbook shows the point where Herodotos made his error. He left the result of 100/3 in place and used it as the multiplier of 41. But the result of a division is recorded as the negative of the quotient, so Herodotos carefully entered the opposite of each partial product for –30 and –3. He resolved the counters, and then multiplied by –0.3, forgetting to enter the –12.3 as an opposite. So his erroneous answer was 1230 + 123 – 12.3 = 1340.7, rounded down to 1340.

[p.273] "VII, 187: the problem is how many medimnoi, 48 choinikes each, will 5,283,220 men [in the Persian army], each consuming a choinix a day, eat in one day. Herodotos' [incorrect] solution is 110,340 medimnoi. ... Having correctly divided the 528 myriads by 48, he set down the [partial] solution: 110,000. ... [Then, in calculating 3220 / 48] Herodotos has mistaken the penultimate [next to last] remainder (340) for the true quotient."

SKS: Sheet 5 of this Excel workbook shows the point where Herodotos made his error. The same arrangement of three side-by-side abaci is used for multiplication and division. The answer for multiplication, the product, is read from the rightmost abacus.The answer for division, the quotient, is read from the leftmost abacus. Perhaps distracted, Herodotos forgot he was dividing instead of multiplying and copied the 340 appearing in the rightmost abacus as his answer.    

[pp.281-282] "Now let us try the same multiplication on the same abacus (No. I), but treat it as if it were a modern abacus with the middle line dividing purely decimal columns into fives and units. ... Again, as with the example of addition, it is seen that the purely decimal columns with the middle line allow a technique that makes of the abacus a true calculating machine, whereas the quinary-decimal labeled columns can be used only as a scoreboard to record results or as a simple counting board. ... It is neither right nor necessary to assume that the Salamis abacus with its unlabeled and transected columns would be operated in the same way as the Minoa countingboard (No. 9) with its labeled columns. The latter is a scoreboard which is completely adequate only for straightforward addition and subtraction of pebbles, but the former is a stream-lined machine for more complicated operation. It is wrong, again, to assume that because the Greeks recorded numbers with a combination of the quinary and decimal systems that their abaci must have alternating quinary and decimal columns."

SKS: Prof. Lang has discovered some of the power of the Salamis Tablet. The Stephenson Abacus™ adds subtractive notation to the technique to make it a super-calculating machine! Note that the space on both sides of the dividing line is the same, hardly necessary if you're only putting one counter in one side. The Stephenson Abacus™ uses both sides for positive and negative numbers, thereby effectively utilizing the space on both sides of the dividing line equally.

Høyrup, Jens, 2002. A Note On Old Babylonian Computational Techniques, Historia Mathematica, Volume 29, Issue 2, May 2002, Pages 193-198. Preprint (PP) retrieved June 7, 2007 from http://akira.ruc.dk/~jensh/Publications/index_B.htm.

[PP-Abstract] "Analysis of the errors in two Old Babylonian “algebraic” problems show, firstly, that the computations were performed on a device where at least additive contributions were no longer identifiable once they had entered the computation; secondly, that this device must have been some kind of counting board or abacus where numbers were represented as collections of calculi; and, thirdly, that units and tens were represented in distinct ways, perhaps by means of different calculi."

[PP-p.1] "It has been known for more than a century that Babylonian calculators made use of tables of multiplication, reciprocals, squares and cubes. It is also an old insight that such tables alone could not do the job – for instance, a multiplication like that of 2 24 and 2 36 (performed in the text VAT 7532, obv. 15, ed. [Neugebauer 1935: I, 294]) would by necessity require the addition of more partial products than could be kept track of mentally, even if simplified by means of clever factorizations. It has therefore been a recurrent guess that the Babylonians might have used for this purpose some kind of abacus ..."

SKS: See a generalized example of the partial product problem at the beginning of this paper. See 2 24 times 2 36 performed on The Stephenson Abacus™ here.

[PP-p.1] "... calculations whose result could not be found by mental calculation (after adequate training) were performed in a different medium; thus, the tablet UET VI/2 222 states directly (and correctly) that 1 03 45 times 1 03 45 (expressed by the writing of one number above the other) is 1 07 44 03 45. Since none of the round tablets discussed by Robson contains the details of such calculations, we must presume that they were not made in clay. How they were then made remains an open question."

SKS: See 1 03 45 times 1 03 45 performed on The Stephenson Abacus™ here.

[PP-p.1-2] "Errors contained in two (equally Old Babylonian[, from 2000 BCE to 1600 BCE in the currently used 'middle
chronology']) texts belonging to the so-called 'algebraic' genre turn out to shed some light on the nature of the devices of which their authors made use.

The first is problem no. 12 of the tablet BM 13901 (obv. II, lines 27–34, ed. [Neugebauer 1935: III, 3]). Line 29 asks for the multiplication of 10 50 by 10 50, and line 30 states the result as 1 57 46 40 – wrongly, indeed, the true answer being 1 57 21 40. Since the erroneous result is used further on, it must be due to the author of the text, not to a copyist."

SKS: See how the scribe possibly made his mistake here.

Friberg, Jöran, 2005. Unexpected Links Between Egyptian and Babylonian Mathematics, World Scientific Publishing Co. Pte. Ltd., Singapore.

[p.27] " ... in OB [Old Babylonian] cuneiform texts special cuneiform signs are used for the Babylonian basic fractions 3' (= 1/3), 2' (= 1/2), 3" (= 2/3), and 6" (= 5/6). In a similar way, special notations are used in [Egyptian] hieratic mathematical papyri for the hieratic basic fractions 6' (= 1/6), 4' (= 1/4), 3' (= 1/3), 2' (= 1/2), and 3" (= 2/3). All other fractions, not counting fractions of measures, are written as "parts" (also called "unit fractions") with dots over the numbers."

SKS: 6" and 3" are strong indicators of the use of subtractive notation on a counting board like The Stephenson Abacus™. Here's how they would be represented:

6" = 5/6 = 1 – 1/6	3" = 2/3 = 1 – 1/3 = 1 – 2/6

[p.32] "Hence, if the quantity is 1 – 1/10 times 10 = 10 – 1 = 9. ... (Note that, apparently, in this exercise 9 divided by 10 is not represented by 3" + 1/5 + 1/30, a sum of parts, as in the 1/10 · n table in P.Rhind and in P.Rhind # 6, but by 1 – 1/10, a difference!)"

SKS: Assume the scribe was using a counting-board like The Stephenson Abacus™, then the representation 1 – 1/10 uses far fewer counters and is easier to grasp visually than the representation 3" + 1/5 + 1/30.  It's also easier to multiply by ten: 6 · 1/60 · 10 = 1, and 1 · 10 = 10.  I.e.:

1 – 1/10	3" + 1/5 + 1/30	(1 – 1/10) · 10 = 10 – 1 = 9

[p.128] "In P.Cairo § 11 a (DMP # 36), ... Apparently, all calculations are carried out by use of sexagesimal arithmetic, although the results of the computations are expressed in terms of sums of parts."

[p.137] Referencing "m. P.Cairo §§ 15-16 (## "32-33")":  "(Without the use of covert counting with sexagesimal numbers, the answer[,] w = 8 3' 1/10 1/60[,] would have been instead that w = 8 3' 1/14 1/21.)"

[pp.165-166] Referencing P.British Museum 10794: "This small fragment contains only the first ten lines of two multiplication tables, one for 1/90, the other for 1/150. Note that 90 and 150 both can be expressed as regular sexagesimal numbers, 90 = 1 30 and 150 = 2 30. Their reciprocals are the sexagesimal fractions ;00 40 and ;00 24. The computation of the two tables probably made use of sexagesimal arithmetic. In his paper about Greek and Egyptian techniques of counting with fractions, Knorr, HM 9 (1982), 156, confessed that he (like Parker before him) was puzzled by 'idiosyncrasies' in the computational procedure ... Why was 2 · 1/150 given as 1/90 + 1/450 and not as 1/75, and why was 3 · 1/150 given as 1/60 + 1/300 and not as 1/50, and so on? ...The assumption that the scribe used sexagesimal arithmetic leads to a much simpler explanation ... . The ... slightly puzzling feature is why, in line 4, the sexagesimal fraction ;01 36 was not simply resolved as ;01 + ;00 36 = 1/60 + 1/100, and why similarly, in line 9, the sexagesimal fraction ;03 36 was not resolved as ;03 + ;00 36 = 1/20 + 1/100."

SKS: Assume the scribe was using a counting-board like The Stephenson Abacus™, then the representations are as follows:

;01 36	;01 20 + ;00 16
;03 36	;02 + ;01 20 + ;00 16

[pp.189-190] "The rather detailed analysis ... above of the contents of P.Cairo clearly shows that in the third century BCE, if not sooner, Egyptian mathematics had become deeply influenced by Babylonian mathematics. ... Above all, the Babylonian influence is evident in the hidden use of sexagesimal fractions as a convenient computational tool, side by side with the Egyptian traditional use of sums of parts ..."

[p.192] "... there can be little doubt that there were no significant differences between the general level and extent of the knowledge of mathematics in Egyptian demotic mathematical texts and in Mesopotamian cuneiform mathematical texts towards the end of the first millennium BCE, and that there are no signs of influence on either from high-level Greek mathematics."

[pp.269-270] " ... the initial development of mathematical ideas started at a very early date with the invention of words for sexagesimal or decimal numbers in various ancient languages, and with the widespread use of number tokens in the Middle East. A major step forward was then the invention of an integrated family of number and measure systems, in connection with the inventions of writing in Mesopotamia and neighboring areas of Iran in the late fourth millennium BCE. There must have been a similar development in Egypt, about which not much is known at present. A small number of known examples of proto-Sumerian, Old Sumerian, Old Akkadian, and Eblaite mathematical exercises and table texts are witnesses of the continuing important role played by education in mathematics in the scribe schools of Mesopotamia throughout the third millennium [BCE].

Then there is a strange gap in the documentation, with almost no mathematical texts known from the Ur III period in Mesopotamia towards the end of the third millennium BCE. nevertheless, at some time in the Ur III period a new major step in the development of mathematics was taken with the invention of sexagesimal place value notation. To a large part as a result of that invention, mathematics flourished in the Old Babylonian scribe schools in Mesopotamia. Simultaneously, mathematics may have reached a comparable level in Egypt, and, in spite of the fundamentally different ways of counting in the two regions, there was clearly some communication of mathematical ideas between Egypt and Mesopotamia.

A few late Kassite mathematical texts seem to indicate that the Old Babylonian mathematical tradition was still operative, although reduced to a small trickle, in the second half of the second millennium BCE.

Then follows a new strange gap in the documentation. 

SKS: Perhaps because of the eruption of Thera that destroyed the Minoan civilization, and much of the normal life in the greater eastern Mediterranean area.

When mathematics flourished again in Mesopotamia in the Late Babylonian and Seleucid periods in the second half of the first millennium BCE, possibly in connection with the rise of mathematical astronomy, a great part of the Old Babylonian corpus of mathematical knowledge had been taken over relatively intact. However, for some reason, the transmission of knowledge cannot have been direct, which is shown by an almost complete transformation of the mathematical vocabulary.

Similarly in Egypt, after a comparable gap in the documentation, there was a new flourishing of mathematics, documented by demotic and Greek mathematical papyri and ostraca from the Ptolemaic and Roman periods. Some of the Greek mathematical texts are associated with the Euclidean type of high-level mathematics. Except for those, the remainder of the demotic and Greek mathematical texts show clear signs of having been influenced both by Egyptian traditions, principally the counting with sum of parts, and by Babylonian traditions. An interesting new development was the experimentation with new kinds of representations of fractions, first sexagesimally adapted sums of parts, soon to be abandoned in favor of binomial fractions, the predecessors of our common fractions.

The observation that Greek ostraca and papyri with Euclidean style mathematics existed side by side with demotic and Greek papyri with Babylonian style mathematics is important for the reason that this surprising circumstance is an indication that when the Greeks themselves claimed that they got their mathematical inspiration from Egypt, they can really have meant that they got their mathematical inspiration from Egyptian texts with mathematics of the Babylonian type."

Menninger, Karl, 1992. Number Words and Number Symbols: A Cultural History of Numbers, German to English translation, M.I.T., 1969, Dover Publications.

SKS: A readable,  excellent, and scholarly book. The Sand Reckoner and Archimedes' large number scheme are described on pp. 139-140. Back- counting, what I've called subtractive notation, is described for many cultures starting on p. 74.

Burton, David M., 1999. The History of Mathematics: An Introduction, 4th Edition, McGraw-Hill, Boston.

SKS: My Spring 2001 graduate history of math course used this book. In this course I made connections between ancient number representations and counting-boards. The professor challenged me to do division with a counting board. As you can see from these web pages, I figured it out.

Hawking, Stephen (Ed.), 2005. God Created the Integers: The Mathematical Breakthroughs That Changed History, Running Press, Philadelphia.

SKS: Archimedes' The Sand Reckoner is included in this book edited by Stephen Hawking; plus other interesting stuff.

© 2006-2008 Stephen Kent Stephenson. Some Rights Reserved.
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