# Arithmetic calculation before the 17th Century

Apparently Dijkstra wrote in an article in Datamation1 in 1977:

It's very illuminating to think about the fact that some – at most four hundred – years ago, professors at European universities would tell the brilliant students that if they were very diligent, it was not impossible to learn how to do long division. You see, the poor guys had to do it in Roman numerals.

Is there any evidence for Dijkstra's claim? I had always thought that people had been happily doing all the usual arithmetic operations using abacuses (i.e., using decimal notation) or on clay/slate/parchment/paper using decimal or sexagesimal notation since the time of the Babylonians.

1 M.W. Cashman, An Interview with Prof. Edsger W. Dijkstra. Datamation, 23 (5) (1977), pp. 164-166.

• Dijstra is roughly right regarding time frame, with the switch to modern long-hand arithmetic algorithms taking place in the 16th century. There is a famous woodcut by Gregor Reisch from 1504 that shows a competition between people using an abacus vs doing long-hand operations. The influential German mathematician Adam Ries published multiple text books about arithmetic, only the earliest of which (1518) deals with the abacus. All later ones are about long-hand methods, including for square root – njuffa Apr 12 at 23:35
• Thanks for the pointer to that great woodcut, but neither party is calculating with Roman numerals in that illustration. – Rob Arthan Apr 13 at 0:05
• The Romans performed computations by means of an abacus or a reckoning board, like the one shown in the woodcut. Our word calculus derives from the small pebbles used in that mode of computation. – njuffa Apr 13 at 0:44
• @njuffa, "to calculate" comes directly from there, "calculus" is derived from the last one. – vonbrand Apr 15 at 16:23
• @vonbrand source: calculus (n.) mathematical method of treating problems by the use of a system of algebraic notation, 1660s, from Latin calculus "reckoning, account," originally "pebble used as a reckoning counter," diminutive of calx (genitive calcis) "limestone" – njuffa Apr 15 at 20:18

TL;DR Dijkstra basically got it right.

The primary means for computing with Roman numerals were the abacus and the reckoning board. The use of small pebbles in this manner of computation is the origin of our word calculus.

Roger Cooke, "The History of Mathematics. A Brief Course 2nd. ed.", Wiley 2005, p. 144, gives a brief but useful overview how and when the switch took place to our current decimal system using Arabic numerals employing pen-and-paper methods of computation:

For computations these cumbersome numerals were supplanted centuries ago by the Hindu-Arabic place-value decimal system. Before that time, computation had been carried out using common fractions, although for geometric and astronomical computations, the sexagesimal system inherited from the Middle East was also used. It was through contact with the Muslim culture that Europeans became familiar with the decimal place-value system, and such mathematicians as Gerbert of Aurillac encouraged the use of the new numbers in connection with the abacus. In the thirteenth century Leonardo of Pisa also helped to introduce this system of calculation into Europe, and in 1478 an arithmetic was published in Treviso, Italy, explaining the use of Hindu-Arabic numerals [...] In the sixteenth century many scholars, including Robert Recorde (1510-1558) in Britain and Adam Ries (1492-1559) in Germany, advocated the use of the Hindu-Arabic system and established it as a universal standard.

Leonardo of Pisa (c. 1170 – 1240), called Fibonacci, popularized the use of Arabic numerals in his book Liber Abaci published in 1202. It already contained early variants of our modern long-hand methods for performing basic arithmetic as well as extracting square roots. However, a wide-spread switch to these new methods of doing arithmetic did not take place until roughly the first half of the 16th century.

There is a famous allegorical woodcut from a work by Gregor Reisch from 1503 that shows a competition between Pythagoras, computing in the traditional method with an abacus, and Boethius, computing with pen and paper using Arabic numerals. From their respective facial expressions it is clear that the latter won. We also note that the goddess Arithmetica in the background smiles favorably upon Boethius. This is a clear expression that the superiority of paper-and-pen computations with Arabic numerals had been recognized.

The mathematician Adam Ries published several popular textbooks in German. It is interesting to note that his first book Rechnung auff der linihen (1518) explains the use of the abacus, while his second book Rechenung auff der linihen vnd federn (1522) explains the use of both abacus and pen-and-paper computation, indicating a shift toward the latter manner of computation.

Google provides a full scan of the book "Die Coss Christoffs Rudolffs, Mit schönen Exempeln der Cosz. Durch Michael Stifel Gebessert vnd sehr gemehrt" of 1571. This is a book on algebra written originally by Christoph Rudolff (1499-1545) that was improved and expanded by Stifel. Beyond basic arithmetic it covers the extraction of square roots and cube roots. The methods demonstrated are basically identical with the long-hand methods in use today.

• For the international audience: The book tiltes by Ries mean "Computation on the line[s of an abacud or a calculi board]" and "Computation on the line or [with] the quill". It is still common to state the result of a straightforward simple computation (colloquially/tongue in cheek) as "das gibt nach Adam Riese" ("according to Adam Ries, this results in") – Hagen von Eitzen Apr 13 at 9:50
• Thanks, that's very interesting. It does look as if Dijkstra was about right on the timescales. What I still don't know is whether people ever tried to do calculations like what we now call long division using Roman numerals directly (which Dijkstra suggests they did). If so, how did they do it? – Rob Arthan Apr 13 at 19:51
• @RobArthan I think it is important to note that the quotation from Dijkstra in the question is from an interview (I have added the reference to the question; the edit is currently pending approval). It is plausible to assume that Dijkstra spoke more loosely than if he had written a formal paper and referred to Roman numerals pars pro toto for computations with the abacus, as their use was in fact closely linked as I point out. I do not know how division was performed with the (Roman) abacus, that looks like an excellent follow-up question. – njuffa Apr 13 at 20:05
• The Wikipedia article you link writes:"What can be deduced from these Roman abacuses, is the undeniable proof that Romans were using a device that exhibited a decimal, place-value system, and the inferred knowledge of a zero value as represented by a column with no beads in a counted position." This is a stretch, but so is Dijkstra's "in Roman numerals". The results were in Roman numerals, but not the abacus calculations. The reason for the speed up, I think, was not so much the switch from Roman to decimal numerals as the development of better decimal division algorithms. – Conifold Apr 14 at 8:07
• For example, Liber Abaci already uses decimals, but its division algorithm consisted in factoring the divisor and dividing by the factors successively, and repeated subtraction was primarily used on abacus. Something remotely resembling "long division" only spread in Europe at the end of 15th century, soon after the printing press and along with decimals, see Windsor-Booker. So there are important parts of the story that Dijkstra got quite wrong. – Conifold Apr 14 at 8:24

There were at least two very different kinds of calculations performed in medieval Europe: a) calculation with integers, for example in accounting, and b) calculation with long fractional numbers in astronomy. Positional, sexagesimal system was invented in Babylon for astronomy and other theoretical purposes, and it was not used in Europe, in ordinary daily life, for example for accounting. Since Dijkstra speaks of "university professors" he probably means astronomy. Accounting was not taught in the universities at those times.

Decimal system was introduced by Fibonacci 1170-1275 for integers, and for accounting, and only in the end of 16th century Simon Stevin 1548-1620 popularized decimal fractions.

Division using Roman numerals is indeed awkward. But even in a positional (say, decimal or sexagesimal) system, division and multiplication of long numbers are tiresome. Have you actually ever tried to multiply two random 6 digit numbers in decimal system?

You do not have to be very brilliant for this, but once you try, you will probably agree that this is a time consuming process requiring great attention. This problem was solved in the late 16th century by Napier. The previous solutions of this problem Prosthapheresis were much more complicated.

To conclude, one can say that "brilliant" is an exaggeration, one should probably say "assiduous".

• Thanks, but that doesn't really give any evidence for or against Dijkstra's claim. Multiplying two 6 digit decimal numbers would have been considered a trivial task 100 years ago - and I could do it now. Napier's invention of logarithms and tools like the slide rule is irrelevant - my question is about precise arithmetic calculations. – Rob Arthan Apr 12 at 22:12
• Sexagesimal system was not invented in Babylon, Babylonians inherited it from Sumerians, and it was not for astronomy or theory, most clay tablets contain land size and weight/density calculations, see e.g. Land surveying in ancient Mesopotamia. Astronomy specialization only came much later. – Conifold Apr 13 at 3:06