A Mediaeval Battle of Numbers

'A knowledge of the battle of numbers is a source of enjoyment and of profit.' John of Salisbury stated this praise in his Policraticus (I,5) in 1180, when reporting about the use and abuse of games. When the mediaeval scholar talked about this competition of numbers, he meant Rithmomachia, which he got to know as a useful and pleasant teaching aid for arithmetical lessons. This game had spread from monastery schools in southern Germany to England. (Evans 1976)

What kind of game is it that John of Salisbury highly praises? What made him and other people talk about Rithmomachia so long ago, after it had been almost forgotten after a prime of about 700 years?

Roger Bacon also recommended Rithmomachia to his students in his 'communio mathematica' (I, 3,4) in the 13th century. He listed seven points on how his students should learn their arithmetics according to Boethius, and at the end he advised that they use the game Rithmomachia as a teaching aid. As Thomas More was convinced of the good character of the game, he let the fictious inhabitants of 'Utopia' (1516. II,5) play it for recreation in the evening hours. As well Robert Burton regarded the use of Rithmomachia as an efficient cure for melancholy, because it is a good exercise for the human spirit. (The Anatomy of Melancholy. 1651. II,4)

The name of the game is of Greek origin. The first part 'Rithmo-' is derived from a combination of arithmos and rhythmos. Arithmos means number and rhythmos had, besides rhythm, also the meaning number and proportion of numbers in the Middle Ages, because not only is the game about the numbers on the pieces, but also about the relation between numbers. The second part of the name '-machia' comes from machos, which means battle. Therefore Rithmomachia can be described as a 'battle of numbers'. In England the game was also known as the 'Philosophers' Game'.

Rithmomachia is a strategy game for two players. A black and a white party of numbers face each other, similar to chess. There was a time when Rithmomachia was in competition with chess and was even more respected than chess, for example in some mediaeval treatises Rithmomachia was favoured. (Folkerts 1989) The reason was, that Rithmomachia was the only game in the curriculum of the mediaeval schools and universities - an honour which chess had never received, because it was played as a tactical war game in the nobility for pure entertainment, but it did not suit the canon of the seven liberal arts. In Rithmomachia the aim is not to fight against each other with armies of numbers, rather to take part in a contest, where the players must bring some of their pieces into a harmonious order.

Contrasts between black and white, even and odd, equality and inequality develop and are in the end resolved into harmony. Especially the latter two pairs appear in the philosophy of numbers of Boethius, which dictates a selection of numbers on the pieces. (Borst 1990). The Boethius number theory is based on the Pythagorean philosophy of numbers, which deals with classification, sequences, and figured presentation of numbers (figurative numbers), and the harmonical proportion between the numbers. All of these features of Boethius' number theory recur in the game of Rithmomachia. Pythagoras' number symbolism, as a part of Boethius' philosophy of numbers, was of particular interest during the period of origin of Rithmomachia. The complete world order was searched for within and represented by this number symbolism. (Coughtrie 1984). Rithmomachia was an entertaining way to memorize the number theory of Boethius. Basically, it was a pleasure to play Rithmomachia, the only game accepted by the Christian scientific community of the Middle Ages, because, unlike chess and dicing games, it was of great use.

In many old records Boethius or Pythagoras were presumed as the inventors of Rithmomachia, however, they only created the mathematical basis of this game. It is certain, that the oldest written evidence of Rithmomachia was found in W?rzburg around 1030. At a competition between the cathedral schools of Worms and W?rzburg, both well-known for their leading position at arithmetics, a disputational text was written with arithmetical sequences of numbers based on 'De institutione arithmetica' of Boethius. On the basis of these writings a monk by the name Asilo created a game - Rithmomachia - which illustrated the number theory of Boethius for the students of monastery schools.

The first outline was adapted by other scholars. Hermannus Contractus, respected monk in Reichenau, checked the rules of the game written by Asilo, enlarged them and added music theoretical remarks. At a school in Li?ge, they worked out a way of realising the game practically not only to enhance the game itself, but also to improve the training of the students in arithmetics. (Borst 1987).

In the 11th and 12th century Rithmomachia spread through monastery schools in southern Germany and France. There the rules were collected, ordered and summarised. The rules became more extensive, and sufficient enough to be played without a teacher. Rithmomachia was an excellent teaching aid. Gradually it was also played by intellectuals just for pleasure. In the 13th century Rithmomachia spread through France and swept over into England. The mathematician Bradwardine and some of his colleagues wrote a text about Rithmomachia, and even in the pseudo-ovidian poem 'De vetula', Rithmomachia was highly praised.

Rithmomachia reached the greatest expansion at the time of book printing. The books written about Rithmomachia had various intentions. Faber (1496) and Boissi?re (1554/56), both professors of mathematics, wrote their treatises for their students at the university of Paris. Faber and a later Italian adapter, whose text is called 'Florentine dialogue' (1539) adopted even the form of the Greek didactic dialogue and the Pythagorean tradition again according to their times. Shirwood (1474) and Fulke/Lever (1563) wrote their book about Rithmomachia for their sovereigns or patrons. The hand-written manuscript by Abraham Ries (1562) was written with the same intention. Abraham Ries was the second son and heir of the mathematical talents of the most well-known German Rechenmeister (arithmetic teacher) Adam Ries. Selenus (1616), whose real name is duke August II of Brunswick-L?neburg, published his Rithmomachia as an appendix to his book about chess. All these texts were characterised by the fact that Rithmomachia was merely played by intellectuals for pure pleasure and mental recreation. (Illmer 1987) Rithmomachia was known at this time mainly in England, France, Italy and eastern Germany.

At the end of the 17th century Rithmomachia lost its great popularity. The mathematician and philosopher Leibniz knew only the name, not the rules of the game. The main subject of mathematics changed during that time. The introduction of the zero, the integration and differentiation of integrals, the calculation with fractions and smallest units did not fit into the number theory of Boethius. Mathematics moved towards the calculation of chance with probability calculus.(Folkerts 1989)

Chess became the great game of that time, and protected the traditions of Rithmomachia mainly in Germany despite its unpopularity of the time. Because Selenus, as a great enthusiast of chess printed his version of Rithmomachia in the appendix of his book of chess, later writers of chess books included Rithmomachia as 'arithmetical chess' in German speaking area. (Allgaier 1796, Waidder 1837, also Koch 1803) In a similar way Zimmermann (1821) adapted Rithmomachia to checkers (in German, Dame) as 'Zahl-Damenspiel' (numerical checkers). Until now Rithmomachia is described particularly in game books. (Archiv 1819, Jahn 1917, Strutt 1801) Two German teachers were also inspired by Selenus to announce Rithmomachia again. Adler, a passionate mathematician and chess player, discovered the didactical profit of Rithmomachia and published a text with the rules in his school programme in 1852, but he received no greater attention. (Jahn 1917). 65 years later Jahn, parish priest and rector of the Z?llchower Anstalten near Stettin, took up the game in effort to contribute to a greater popularity, but he suffered the same meager results. (1917, 1929?)

For more than 100 years the academic research of the origin of Rithmomachia and the mediaeval history of it developed independently to the traditions of the game. In 1986 this academic research obtained with 'Das mittelalterliche Zahlenkampfspiel' by Borst a basic work, in which the oldest source texts are edited.

The mediaeval traditions of Rithmomachia are certain. Illmer (1987) however suspects, that Rithmomachia is older. There are conspicuous parallels between the raising and the moves of the pieces and the raising and the mobility of Roman armies. Already in approximately 1070 in Li?ge this Roman model provided the players with an easier way of playing. (Borst 1986) There are, however, no testimonies of texts, but generally the sources of texts about ancient board games are very short, like, for example, in different works by Plato. An exact description or even a rule of the game is difficult to reconstruct. Also no archaeological evidence has been hitherto found. There have been no pieces found neither ancient nor mediaeval.

There is no one set of rules for Rithmomachia. During the 1000-year history of the game the rules have changed often. The extent increased from few hand-written pages to more than 100 printed pages, in which detailed the mathematical and harmonic backgrounds are described. But the rules have the following things in common: the number of pieces with the numbers printed on them, the two pyramids and a rectangular board. In addition the goal of the game is common: Two players try to build through fixed moves an arrangement of three or four pieces on the opponent's side of the board. The numbers of the pieces must be in a specific proportion to each other and with the arrangement of one of these groups the player gains victory. In the process the opponent's pieces can be captured according to certain rules. Depending on whether one seeks a perfect game or an easier version of it, the size of the board and other details of the rules may vary.

The rules presented here correspond mostly to the way Rithmomachia was played during the 17th century, before it retreated in a shadowy existence. (1) These rules are suitable for playing today.

Rithmomachia is played on a board of 16 by 8 squares. The white and black pieces have numbers written on them according to the number theory of Boethius.

***fig. 1: The numbers on the pieces according to the number theory of Boethius*** ***see at the end***

The second and proceeding rows of numbers are derived from the first. The white pieces are called the even and the black are called the odd, but there are odd numbers in the even party and vice versa.

On the round pieces the multiples (multiplices) are placed. The base row is built from multiples of 1. In the second row the base numbers are multiplied with themselves. The numbers on the triangles are the superparticulares. They contain the preceding number and one fraction of it ([n + 1] / n). The numbers on the squares are built with the preceding number and a multiple fraction of it ([n + 2] / [n + 1]). They are the superpartientes. There are many mathematical relations between the numbers. Boethius gave several procedures for derivation of the numbers. One of these mathematical relations is that the first row of triangles can be built by adding the numbers of the two preceding circles. In the same way the first row of the squares is obtained from the two rows of triangles.

At the position of the white 91 a pyramid is located. The square numbers 36 and 25 on square, 16 and 9 on triangular, and 4 and 1 on round pieces add up to the total sum of 91 of the white pyramid. Corresponding to this the black 190 is replaced by a pyramid with the total sum 190, consisting of the square numbers 64 and 49 on squares, 36 and 25 on triangles, and 16 on a circle. The pieces are set up as illustrated in fig. 2. The tables of harmonies, in which all combinations of pieces for harmonies are recorded, are very helpful in playing. (2)

Through tactical moves, the players should attempt to arrange a harmony out of three or four pieces in a specific proportion in the opponent's field. The players should however pay attention to the opponent and prevent him from blocking his harmony. The first player who arranges a harmony is the winner.

The players move alternately into an empty space. No piece is allowed to be jumped over. Black starts, because white has better possibilities for capturing and arranging harmonies. This inequality is a special attraction of Rithmomachia, because through this a balanced playing is possible between unequal players.

The circles move into the second field, forwards, backwards or sideways, but not diagonally. The triangles move into the third field, only diagonally. The squares move into the fourth field, in all directions (including diagonally). When moving, both the starting and finishing field are counted. The 5 or 6 piece pyramids move according to their individual components.

***fig. 2: Board of Rithmomachia with the arrangement of the pieces at the beginning***

Pieces can capture others that stand in the way of their movement, but they remain at their place and do not take the field of the opponent's captured piece.

By meeting: If a piece is so placed, that in its next regular move it could take the place of an opponent's piece with the same number, the opponent's piece is taken away.

By ambush: If two or more pieces of ones party are in a position in which in their next move they could move into the field of an opponent's piece, and the sum or difference equals the number of the opponent's piece, the opponent's piece is taken away.

By assault: If in its ordinary direction a piece could meet an opponent piece, and its number equals by multiplication or by division the number of fields between the two pieces, the opponent's piece is taken away from the board. The fields of the capture and the captured piece are counted.

By siege: If an opponent's piece is encircled by pieces of the other party in such a way that it could neither move nor be set free by one piece of its party, the besieged opponent's piece is taken away from the board.

The individual components of the pyramids can both capture and be captured. If single components are missing, the pyramids can be captured by their total sum, but they cannot capture other pieces with their total sum in this case. Partial sums are inadmissible.

The game is finished, when one player has built up a harmony of three or four pieces in the opponent's field. Therefore the pieces must be arranged in an ascending row, in a right angle, or four pieces also in a square, and must be equidistant. The captured opponent's pieces may also be used in creating a harmony, however, they may not be the last piece of a harmony. There are three ways of creating harmonies with three pieces:

In an arithmetical harmony the difference between the two smaller numbers equals the difference between the two bigger ones, e. g. 2, 4, 6 => b - a = c - b. A geometrical harmony exists, when the ratio between the two smaller numbers equals the ratio between the two bigger ones, e. g. 5, 10, 20 => (a / b) = (b / c). By the musical harmony the ratio of the smallest and the biggest number equals the ration between the difference of the two smaller numbers and of the two bigger ones, e. g. 6, 8, 12 => (a / c) = [ (b - a) / (c - b) ]. To enable a rapid game, the harmonies need not be calculated; rather, they can be looked up in the tables of harmonies. Three different grades of victories can be gained from different harmonies:

- A small victory is reached by an arithmetical, geometrical or musical harmony of three pieces.
- A big victory is gained by building two (but not more than two) different harmonies with 4 pieces.
- A great victory is reached by 4 pieces containing all three harmonies.

The players agree on which victory or victories they are aiming for. It is possible to play with even simpler goals. If the players desire a simpler game, a victory could be possible when a predetermined number of opponent's pieces are captured, or a certain sum or number of digits of the captured pieces is reached or exceeded.

The most essential features of Rithmomachia have been represented. Because of the briefness some smaller details are missing; but players can certainly work with this outline and work out smaller details as necessary. A few variations of Rithmomachia have been presented, which players can try.

Unfortunately, to play Rithmomachia today, one must build a game for oneself, if one is not interested in using one of the two computer games from Italy or from the USA. In the 16th century it was easier, because the game could be bought in Paris and London, as Boissi?re and Fulke/Lever wrote in their books on Rithmomachia. Presumably Jahn (1929?) offered a set of the game for sale.

In the past treatises about Rithmomachia were published more often, and it also appeared in game books. So there is still hope, that Rithmomachia will be known better again. This desire was expressed in the pseudo-ovidian poem 'De vetula' in the 13th century: 'Oh, if only more people had enjoyed the battle of numbers! If it was only known, it would on its own accord be highly respected.' Hopefully this wish, that Rithmomachia be played again, will come true.

Borst, A. 1986. Das mittelalterliche Zahlenkampfspiel. Supplemente zu den Sitzungsberichten der Heidelberger Akademie der Wissenschaften, Philosophisch-historische Klasse, vol. 5. Heidelberg: Carl Winter Universit?tsverlag

Borst, A. 1990. Rithmimachie und Musiktheorie. In Geschichte der Musiktheorie. Vol. 3, Rezeption des antiken Fachs im Mittelalter. edited by Frieder Zaminer, 253-288. Darmstadt: Wissenschaftliche Buchgesellschaft

Coughtrie, M. E. 1984. Rhythmomachia: A Propaedeutic Game of the Middle Ages. Ph.D. diss., University of Cape Town (in typewriting)

Evans, G. R. 1976. "The Rithmomachia: A Mediaeval Mathematical Teaching Aid?" Janus 63:257-273

Folkerts, M. 1989. Rithmimachie. In Ma?, Zahl und Gewicht: Mathematik als Schl?ssel zu Weltverst?ndnis und Weltbeherrschung. edited by M. Folkerts and others, 331-344, Ausstellungkataloge der Herzog August Bibliothek, no. 60. Weinheim: VCH, Acta humanoria

Illmer, D. and others. 1987. Rhythomomachia: Ein uraltes Zahlenspiel neu entdeckt von --. Munich: Hugendubel

Mebben, P. 1996. Rithmomachie - Ein aus dem Mittelalter ?berliefertes Zahlenspiel: Neu entdeckt f?r die Schule. Master's thesis, P?dagogische Hochschule Freiburg (available by the author upon request)

Richards, J. F. C. 1946. "Boissi?re's Pythagorean Game". Scripta Mathematica 12:177-217

Stigter, J. 199?. The History and Rules of Rithmomachia, the Philosophers' Game: An Introduction. London: (will be published soon)

Some old, famous and well-known printed books about Rithmomachia

John Shirwood. 1480. Ad reverendissimum religiosissimumque in Christo patrem ac amplissimum dominum Marcum cardinalem Sancti Marci vougariter nuncupatum Johannis Shirvuod quod latine interpretatur Limpida Silva sedis Apostolicae protonotarii Anglici, praefatio in Epitomen de ludo arithmomachiae feliciter incipit. Rome: Ulrich Han.

Jacobus Faber Stapulensis (Jacques Lef?vre d'Etaples). 1496. Rithmimachie ludus qui pugna numerorum appellare. In Jordanus Nemorarius. Arithmetica decem libris demonstrata. edited by Jacobus Faber Stapulensis. Paris: David Lauxius of Edinburgh.

Claude de Boissi?re. 1554. Le tr?s excellent et ancien Jeu Pythagoriqhe, dit Rhythmomachie. Paris: Amet Breire. Or the latin translation: Claudius Buxerius. 1556. Nobilissimus et antiquissimus ludus Pythagoreus (qui Rythmomachia nominatur). Paris: Guilielmum Canellat. (Translated into English by Richards 1946)

Rafe Lever and William Fulke. 1563. The Most Noble Ancient, and Learned Playe, Called the Philosophers Game. London: Iames Rowbothum.

Francesco Barozzi. 1572. Il nobilissimo et antiquissimo Givocco Pythagorea nominato Rythmomachia cioe Battaglia de Consonantie de Numeri. Venice: Gratioso Perchacino.

Gustavus Selenus (Duke August II of Brunswick-L?neburg).1616. Rythmomachia. Ein vortrefflich und uhraltes Spiel desz Pythagorae. In Das Schach= oder K/nig=Spiel. 443-495. Leipzig: Henning Gro? jun. Reprint 1978. Z?rich: Olms

Texts of modern era with a description or rules of Rithmomachia until 1940 - The special German tradition

Johannes Allgaier. 1796. Das pythagor?ische oder arithmetische Schachspiel. In Neue theoretisch-praktische Anweisung zum Schachspiel. Vol. 2, p. 73-97. Wien: Franz Joseph R/tzel.

Johann Friedrich Wilhelm Koch. 1803. Die Rythmomachie. In Die Schachspielkunst nach den Regeln und Musterspielen der gr/?ten Meister. Part 2, p.V-VI, 127-154. Magdeburg: Georg Christian Keil.

Archiv der Spiele. 1819. Das Zahlenspiel (Rythmomachie). In --. vol. 1, sect. 2, 11., p. 94-106. Berlin: Ludwig Wilhelm Wittich.

Ferd. Zimmermann. 1821. Zahl-Damenspiel. In Volst?ndiger Codex der Damenbrett-Spielkunst. p. 365-404. K/ln, Rommerskirchen.

S. Waidder. 1837. Das arithmetische Schachspiel. In Das Schachspiel in seinem ganzen Umfange. Vol. 2, sect. 2,C., p. 118-142. Wien: Mich. Lechner.

Karl-Friedrich Adler. 1852. Beschreibung eines uralten, angeblich von Pythagoras erfundenen, mathematischen Spieles. Schulprogramm des K/niglichen und St?dtischen Gymnasiums in Sorau. Sorau.

Fritz Jahn. 1917. Rythmomachia. In Alte deutsche Spiele. p.1-4, 15. Berlin.

[Fritz] Jahn. 1929(?). Zahlenschach f?r Mathematiker. In Verzeichnis Weihnachtskrippen und Spiele der Z?llchower Anstalten 1929/30. Z?llchow.

Joseph Strutt. 1801. The Sports and Pastimes of the People of England. p. 313-316. London.

Table of harmonies 1. Harmonies for a small victory Geometrical harmony: 2 4 8 2 12 72 3 6 12 4 6 9 4 8 16 4 12 36 4 16 64 4 20 100 4 30 225 5 15 45 9 12 16 9 15 25 9 30 100 9 45 225 16 20 25 16 28 49 16 36 81 20 30 45 25 30 36 25 45 81 36 42 49 36 66 121 36 90 225 49 56 64 49 91 169 64 72 81 64 120 225 81 90 100 81 153 289 100 190 361 Arithmetical harmony: 2 3 4 2 4 6 2 5 8 2 7 12 2 9 16 2 15 28 2 16 30 3 4 5 3 5 7 3 6 9 3 9 15 3 42 81 4 5 6 4 6 8 4 8 12 4 12 20 4 16 28 4 20 36 4 30 56 5 6 7 5 7 9 5 15 25 5 25 45 6 7 8 6 9 12 6 36 66 7 8 9 7 16 25 7 28 49 7 49 91 7 64 121 8 12 16 8 25 42 8 36 64 8 49 90 8 64 120 9 12 15 9 45 81 9 81 153 12 16 20 12 20 28 12 42 72 12 56 100 12 66 120 15 20 25 15 30 45 15 120 225 16 36 56 20 25 30 20 28 36 20 42 64 28 42 56 28 64 100 30 36 42 42 49 56 42 66 90 42 81 120 49 169 289 56 64 72 72 81 90 81 153 225 91 190 289 Musical harmony: 2 3 6 3 4 6 3 5 15 4 6 12 4 7 28 5 8 20 5 9 45 6 8 12 7 12 42 8 15 120 9 15 45 9 16 72 12 15 20 15 20 30 25 45 225 30 36 45 30 45 90 72 90 120 2. Harmonies for a big victory Arithmetical and musical harmony: 3 4 5 6 3 4 5 15 3 5 7 15 4 5 6 12 4 6 12 20 4 12 15 20 5 7 9 45 6 7 8 12 9 12 15 45 9 15 30 45 9 15 45 81 12 15 20 28 15 20 25 30 15 30 36 45 30 36 42 45 72 81 90 120 Geometrical and musical Harmony: 2 3 6 12 3 4 6 12 3 5 15 45 3 6 8 12 4 6 12 36 5 9 15 45 5 9 45 225 9 12 16 72 9 15 25 45 9 15 45 225 9 25 45 225 20 30 36 45 20 30 45 90 25 30 36 45 25 45 81 225 Arithmetical and geometrical harmony: 2 3 4 8 2 4 5 8 2 4 6 8 2 4 6 9 2 4 8 12 2 7 12 72 2 9 12 16 2 12 42 72 3 6 9 12 3 9 15 25 4 5 6 9 4 6 8 9 4 6 8 16 4 8 12 16 4 8 12 36 4 8 16 28 4 12 20 36 4 12 20 100 4 16 28 49 4 16 28 64 4 20 36 100 4 30 56 225 5 9 15 25 5 15 25 45 5 15 30 45 5 25 45 81 6 9 12 16 6 36 66 121 7 16 20 25 7 16 28 49 7 49 91 169 8 9 12 16 8 64 120 225 9 12 15 16 9 12 15 25 9 12 16 20 9 15 20 25 9 25 45 81 9 45 81 225 9 81 153 289 12 16 20 25 15 16 20 25 15 64 120 225 16 20 25 30 16 36 56 81 20 25 30 36 20 25 30 45 25 30 36 42 30 36 42 49 36 42 49 56 42 49 56 64 49 56 64 72 49 91 169 289 56 64 72 81 64 72 81 90 72 81 90 100 81 153 225 289 3. Harmonies for a great victory Arithmetical, geometrical and musical harmony: 2 3 4 6 2 3 6 9 2 4 6 12 2 5 8 20 2 7 12 42 2 9 16 72 3 4 6 8 3 4 6 9 3 5 9 15 3 5 15 25 3 9 15 45 4 6 8 12 4 6 9 12 4 7 16 28 4 7 28 49 5 9 25 45 5 9 45 81 5 25 45 225 6 8 9 12 6 8 12 16 7 12 42 72 8 15 64 120 8 15 120 225 9 12 15 20 12 15 16 20 12 15 20 25 15 20 30 45 15 30 45 90

(1) This description of the rules is based for the most part on Illmer (1987), who used the rules of Selenus (1616) as basis. There are some differences in Stigter's version (199?). His rules include many details with a good structure and the mathematical basis, they could not be represented here because of the necessary briefness. It contains a more extensive derivation of the numbers. More extended descriptions of the rules of the Rithmomachia can be found in Coughtrie (1984) and Richards (1946). In my master's thesis (1996) I have also listed many different variations for playing.

(2) More extensive tables of harmonies can be found in Richards (1946), Illmer (1987) and Mebben (1996).