Numbers battle game

Game board after the illustration by Claude de Boissière (1556)

The number battle game ( Latin numerorum conflictus or Greco-Latin rithmomachia , arithmomachia or Rithmomachia , later also philosophy game ) was a medieval board game based on the harmony theories of Boëthius (which were based on the introduction to the arithmetic of Nicomachus of Gerasa ) and was in competition standing for chess .

Initially intended in the early 11th century as a competition between two mathematically trained masters, it quickly spread throughout Europe, especially in France, England and the Holy Roman Empire . Its use for didactic purposes was particularly popular in European cathedral schools to learn arithmetic, proportion theory and music theory through play. Its popularity lasted well into the 16th century, when Thomas More and Gottfried Wilhelm Leibniz still knew it. In the following years it was almost completely forgotten and only rediscovered by historians in the 19th and 20th centuries ( Arno Borst is particularly worthy of mention).

Invention and dissemination

The starting point for the invention of the number fighting game is the so-called Worms school dispute, which took place in the 1030s. This was a dispute between the cathedral schools in Worms and Würzburg , especially since Worms no longer wanted to accept the priority of the Würzburg school. Research assumes that a cousin of Emperor Konrad II , who was close to the two episcopal cities, mediated a scholarship competition (both schools also fought for their school to come from the aristocratic elite).

The number fighting game in the Liège version around 1070

This is the context in which the number fighting game was invented by a Würzburg clergyman named Asilo, who wrote instructions for "rithmimachia" ( Quinque genera inequalitatis ) and which could be identical to Adalbero, who was appointed Bishop of Würzburg in 1045 . At least this is what Arno Borst suspects, since Adalbero had attended the cathedral school in Würzburg in the 1020s, especially since he had the best access to the writings of Boëthius in neighboring Bamberg (especially De institutione arithmetica ), which were decisive for the game of numbers. On the other hand, Enno Bünz suspects, based on philological and name-related considerations and thorough knowledge of the Würzburg cathedral chapter, behind the author of the circular Asilo either the cathedral dean Aselo or the cathedral provost Acelin, who were not only long-term members of the cathedral chapter, but had previously attended the Würzburg cathedral school (whereby Bünz tends towards the cathedral dean Aselo / Asilo as the inventor, even if this can of course never be finally clarified due to the tradition). This would put the two cathedral canons in the context of a possible authorship of the number battle game than, for example, the later Bishop Adalbero (whose membership in the Würzburg cathedral chapter after Bünz is questionable anyway), and this would affect the overall picture of meaning and quality developed in research speak of the Würzburg Cathedral School in the 11th century.

The Worms school dispute may not have been resolved in a game of numbers, but Asilo's circular, which contained the game, quickly spread, initially among other scholars, such as Hermann the Lame , who primarily examined the game theoretically (especially backwards on the music and harmony theories of Boëthius). The game thus gained great prestige, especially in the learned world. It spread rapidly in Europe, especially in the Empire, France and England.

In 1070 it came to the cathedral school at Liege and found there a critical revision. On the one hand, the anonymous reworker called the game for the first time aptly number battle (numerorum conflictus), on the other hand he gave the game board a fixed shape (8 × 16 fields), which Asilo had not done in the original variant (he only spoke of a board with fields). He further arranged the stones as it later became customary, mainly to improve the flow of the game, and he also separated the halves of the game with a dividing line. In addition, the Liege Anonymous added a number of tables with which the game could be won. The number battle game had finally become a didactic tool, especially for cathedral school masters, to teach the pupils arithmetic and the theory of proportions, while Asilo still intended it as a learned competition between two masters. In this streamlined form, the number fighting game spread very quickly; it was rounded off completely after 1090 by Odo von Tournai, who compiled all previous revisions and brought the game into a uniform form, which remained almost unchanged until the 16th century and joined Laity and clergy enjoyed great popularity (always in competition with the much more popular chess).

Boëthius' number theory as the basis of the number battle game

Without the underlying mathematical theories of the Middle Ages, which were particularly related to Boëthius, one cannot understand why and how the respective numbers in a number battle game are distributed one way and not another on the board. An important starting point is the medieval notion of proportions, which are always based on the , since all other numbers only represent multiples of (this can be very confusing nowadays, as we no longer think in terms of proportions). ${\ displaystyle 1}$ ${\ displaystyle 1}$

The starting point in the number battle game was now the numbers to , i.e. the finger numbers , whereby these were left out, because according to Boëthius this was considered a separate value and as a guide number for all others, as was the one , this was the beginning of the so-called joint numbers and therefore did not belong in Boëthius the same row with the numbers from 1 to 9. What remained were the natural numbers from to , which were divided into even and odd so that each player had four basic numbers ( or ). Thus, a player had the even basic values (the side straight ) and his opponent the odd (the side Odd ); however, odd numbers also occur on the straight side and vice versa. Building on these basic values, according to Boëthius, their extensions had to be calculated, which, unlike today, are not based on quantities per se, but on proportions (e.g. ratios of , etc.). Asilo only mentioned the first proportion, the first multiplices ( ), in his circular. Whoever wanted to play had to know the writings of Boëthius and the script of a pupil of the Würzburg cathedral based on it (De aggregatione naturalium numerorum) , because based on the multiplices , the superparticulares and the superpartientes (multiple parts ) were added. ${\ displaystyle 1}$ ${\ displaystyle 10}$ ${\ displaystyle 1}$ ${\ displaystyle 10}$ ${\ displaystyle 2}$ ${\ displaystyle 9}$ ${\ displaystyle 2,4,6,8}$ ${\ displaystyle 3,5,7,9}$ ${\ displaystyle 1: 2.2: 3}$ ${\ displaystyle n ^ {2}}$

Proportions according to Boëthius

According to Boëthius, there was a multiple (multiplex) relationship between two numbers if the larger number contained the smaller number more than once. The simplest form is the second multiplex a base number , so . ${\ displaystyle n}$ ${\ displaystyle n ^ {2}}$

Überteilig (superparticularis) described the relationship between two numbers when the larger number contains not only the smaller number once, but even a single part of it (that is a simple fraction like , etc.). For example, this is for the , for the contains the once whole and also the half ( ). Generally, the first two are superparticulares and . ${\ displaystyle {\ tfrac {1} {2}}, {\ tfrac {1} {3}}, {\ tfrac {1} {4}}}$ ${\ displaystyle 4}$ ${\ displaystyle 6}$ ${\ displaystyle 6}$ ${\ displaystyle 4}$ ${\ displaystyle 4}$ ${\ displaystyle 6 = 4 + {\ tfrac {4} {2}}}$ ${\ displaystyle n \ cdot (n + 1)}$ ${\ displaystyle \ left (n + 1 \ right) ^ {2}}$

In Boëthius' multiple parts (superpartiens) , the ratio was when the larger number not only contained the smaller number in its entirety, but also several parts of the smaller number (i.e. a multiple fraction like etc.). So if you take the (as superparticularis of ), then its first superpartiens is the (as the sum of and of ). The (as the sum of and of ) is related to multipart . In general, the first two are superpartientes and . ${\ displaystyle {\ tfrac {2} {3}}, {\ tfrac {3} {4}}, {\ tfrac {4} {5}}}$ ${\ displaystyle 9}$ ${\ displaystyle 6}$ ${\ displaystyle 15}$ ${\ displaystyle 9}$ ${\ displaystyle {\ tfrac {2} {3}}}$ ${\ displaystyle 9}$ ${\ displaystyle 15}$ ${\ displaystyle 25}$ ${\ displaystyle 15}$ ${\ displaystyle {\ tfrac {2} {3}}}$ ${\ displaystyle 15}$ ${\ displaystyle \ left (n + 1 \ right) \ cdot \ left (2n + 1 \ right)}$ ${\ displaystyle \ left (2n + 1 \ right) ^ {2}}$

Importance to the game

From these rules you got a series of numbers (if you knew Boëthius):

${\ displaystyle n, n ^ {2}, n \ cdot \ left (n + 1 \ right), \ left (n + 1 \ right) ^ {2}, \ left (n + 1 \ right) \ cdot \ left (2n + 1 \ right)}$ and ${\ displaystyle \ left (2n + 1 \ right) ^ {2}}$

Since the game was based on eight basic values ( bis ), you received a total of 48 values, i.e. 24 per player. Due to the calculation with proportions, it also happened that some numbers appeared several times, but in different qualities, e.g. B. sometimes as a basic value, at the same time as superparticularis (such as the or ). ${\ displaystyle 2}$ ${\ displaystyle 9}$ ${\ displaystyle 6}$ ${\ displaystyle 9}$

The pyramids

The figure of the pyramid (similar to the king in chess) was of particular importance . A pyramid appeared once on each side, with the even that , with the odd that . The first pyramid was, again according to Boëthius, the sum of the squares of the first six natural numbers ( ). For Boethius this was a perfect pyramid because it did not leave out a step from the base, here from the square to the top . The pyramid, on the other hand, was the sum of the squares above the natural numbers 4 to 8 ( ). Since the first three square numbers were missing at the top, it was called triple abbreviated. ${\ displaystyle 91}$ ${\ displaystyle 190}$ ${\ displaystyle 1 ^ {2} + 2 ^ {2} + 3 ^ {2} + 4 ^ {2} + 5 ^ {2} + 6 ^ {2} = 91}$ ${\ displaystyle 6}$ ${\ displaystyle 1}$ ${\ displaystyle 190}$ ${\ displaystyle 4 ^ {2} + 5 ^ {2} + 6 ^ {2} + 7 ^ {2} + 8 ^ {2} = 190}$

The game

Structure of the playing field

Excerpt from the number battle game: the proportions are built up along the arrows

The number battle game received its decisive revision (especially in terms of playability) in the Liège version of 1070. Not only did the Liège anonymous stipulate that the playing field should have 8 × 16 fields, he also arranged the various stones in a meaningful way according to their proportions and gave them them different sizes. The multiples are the smallest, the multipartite the larger, the multipartite the largest. The even eight smallest in white, the odd in black, the eight middle in the even in red, in the odd in white, the eight largest in the even in black, in the odd in green.

These were arranged according to their proportions: the basic values in the front center, their multiples behind. This was followed by the overdivisions, namely the two overdivisions each below and above the external basic values (i.e. over the 2 their overdivisions 6 and 9, under the 8 the 72 and 81). To the right behind the 2 was its multiple 4, above it the 6, above this the 9, so that multiplices and superparticulares joined each other at right angles . The remaining superparticulares were also distributed at right angles. The superpartientes came into the two upper and lower corners , these each connected to the respective superparticulares , again at right angles. The proportions of a basic value on the game board were connected with this. The series of proportions for the straight lines were thus designed as follows:

Core value	multiplex	first superparticularis	second superparticularis	first super games	second super games
2	4th	6th	9	15th	25th
4th	16	20th	25th	45	81
6th	36	42	49	91	169
8th	64	72	81	153	289

The color scheme, which was also made in Liège, makes the distribution of the respective proportions very clear.

A line-up next to each other in a row (like in chess) would have hindered the flow of the game enormously, because the front row would only have been occupied by multiples, which on the one hand were very difficult to move and at the same time were very difficult to remove by the opponent.

regulate

Asilo had already set up the five main rules in his circular in the 1030s, which from then on remained almost unchanged - they were simply arranged differently:

A piece of one's own was allowed to take away an opponent's piece if the number of spaces in between, multiplied by the numerical value of one's own piece, exactly reached the value of the opponent's piece. This was a crucial rule, because it was not the largest numbers, but the smallest that were most powerful. For example, a 2 could beat a 16 over eight squares, and so on.
An opposing stone was taken away if it was surrounded by one's own stones, be it in a straight line or at right angles, in such a way that the product or sum of the neighboring stones was equal to the value of the enclosed stone. The even values 15 and 49 beat the odd overdivision 64. The prerequisite was that the stones had got into position by correct moves and were in the immediate vicinity.
With a proper move (i.e. directly on the field of the opponent's piece without multiplication or similar) your own piece took away the opponent's piece whose numerical value was the same as your own, and the opponent's field was occupied, e.g. B. the straight 81 beat the odd 81, this was about the value of the number, not the proportion.
A pyramid could be beaten if its base, which was in the opposing camp, hit it with a proper move. Since the pyramid was intended as a structure of squares stacked one on top of the other, all the squares built over it fell at the same time when the base of the pyramid was hit. So when the odd superparticularis 36 from the third field hit straight pyramid 91, he removed stones 4 (only one of the two), 9, 16, 25 (again only one of the existing ones) and 36 (as squares over 2, 3, 4, 5 and 6). Pyramids were therefore particularly endangered stones.
Prime numbers were then taken away when they were surrounded on all sides and could not escape over any open field. This should offset the seeming invulnerability of the small numbers. But four stones were needed to encircle it, so this was very difficult.

Asilo had also determined that the stones could only be drawn in straight lines or at right angles, and that the way to the opposing stone could not be blocked by other stones. In addition, there was the rule that the stones could move differently in proportion to 1, the basic values only 1 field, the multiplices 2, the superparticulares 3 and the superpartientes 4 fields.

Victory conditions

In order not to entrench yourself in your own half, after Asilo every stone had to have left its starting position. However, this was deleted in later revisions. More important was the actual victory condition, which made the overall character of the game clear: You had to build an arithmetic or harmonic row of three stones in the opponent's camp. First of all, this was supposed to rehearse Boëthius' theories, but at the same time the priority of harmony over conflict was emphasized. As an additional option, Hermann the Lahme brought in the great harmony according to Boëthius, which was not adopted by everyone, as you needed a row of four, so only the arithmetic and harmonic series are mentioned here:

Arithmetic series : The arithmetic mean of Boëthius consisted of three or more terms (termini) between which the same differences prevailed, e.g. from 1, 2, 3 with the difference 1 or from 1, 3, 5 with the difference 2. Asilo was limited on three-part rows, with the middle part being half the sum of the two outer parts, so there were numerous combinations.

Harmonic series : Here the links were connected by a mixture of difference and proportion. The difference between the largest and the middle link, on the one hand, and the difference between the middle and smallest link, on the other hand, had the same relationship to one another as the largest to the smallest link, for example at 3, 4, 6, where 6 - 4 and 4 - 3 so to each other were like 6 to 3 (namely 2: 1). For a harmonious row, however, it was necessary to capture an opponent's stone, which was the attraction of this option.

Whoever placed the first stone in a row in the opposing field had to announce this; of course the opponent could force his way in between, but it was better to build a line in the opponent's camp. The stone that was announced as the beginning of a row could not be removed. Looted stones could initially be removed from the game, later they could be brought back.

The winner was the one who was the first to create harmony in the opponent's camp and thus reconciled the conflict, the unequal, to equality. This was entirely in keeping with the ethics and philosophy of the Middle Ages, as the legacy of Boëthius.

bibliography

Arno Borst : The medieval number battle game . (Supplements to the meeting reports of the Heidelberg Academy of Sciences, Philosophical-Historical Class 5). Winter, Heidelberg 1986.
Arno Borst: What the Middle Ages had to tell us. About science and play , in: Historische Zeitschrift 244 (1987), pp. 537-555, reprinted and expanded in: Arno Borst: Die Welt des Mittelalters. Barbarians, heretics and artists . Nikol, Hamburg 2007, pp. 448-468.
Detlef Illmer and Nora Gädecke, Elisabeth Henge, Helene Pfeiffer and Monika Spicker-Beck : Rhythmomachia. An ancient number game rediscovered , Hugendubel, Munich 1987, ISBN 978-3880343191
Enno Bünz : Did the later Bishop Adalbero von Würzburg invent the number battle game? Reflections on the Würzburg Cathedral School in the first half of the 11th century , in: German Archive for Research into the Middle Ages , No. 49, 1993, pp. 189–199.
Enno Bünz: Haug Abbey in Würzburg. Studies on the history of a Franconian collegiate foundation in the Middle Ages (publications of the Max Planck Institute for History 128 = Studies on Germania Sacra 20), Göttingen 1998. (2 volumes)
Menso Folkerts : Rithmimachia , in: The German Literature of the Middle Ages, Author's Lexicon 8, Sp. 86–94.

Menso Folkerts: Rithmimachie , in: Measure, Number and Weight. Mathematics as the key to understanding the world and dominating the world (exhibition catalog of the Herzog August Library 60), Weinheim 1989, pp. 331–334.

Menso Folkerts: The "Rithmachia" of Werinher von Tegernsee , in: M. Folkerts and JP Hogendijk: Vestigia mathematica: Studies in Medieval and Early Modern Mathematics in Honor of HLL Busard , Amsterdam 1993, pp. 107–142

Alfred Holl: Numbers game - Fight with numbers? The medieval number fighting game Rithmomachie in its Regensburg version around 1090 . (Rapporteur från Växjö Universitet: matematik, naturvetenskap och teknik 3). University, Växjö 2005. Online version of the book (PDF; 742 kB)

David Parlett: The Oxford History of Board Games . Oxford University Press, Oxford and New York 1999, pp. 332-342 ("The Thought that Counts. Rithmomachy - the Philosophers' Game").

Web links

Commons : Numbers battle game - collection of images, videos and audio files

Wiktionary: Rithmimachie - explanations of meanings, word origins , synonyms, translations

Remarks

↑ Enno Bünz: Did the later Bishop Adalbero von Würzburg invent the number fighting game? Reflections on the Würzburg Cathedral School in the first half of the 11th century , in: German Archive for Research into the Middle Ages, No. 49, 1993, pp. 189–199, here: pp. 189f.
^ Asilo of Würzburg. , In: Author's Lexicon . Volume I, p. 508 f.
↑ Arno Borst: The medieval number battle game . (Supplements to the meeting reports of the Heidelberg Academy of Sciences, Philosophical-Historical Class 5). Winter, Heidelberg 1986, pp. 55, 58 f. and 60; Bünz: Considerations , pp. 191–193.
↑ cf. u. a. Enno Bünz: Haug Abbey in Würzburg. Studies on the history of a Franconian collegiate foundation in the Middle Ages (publications by the Max Planck Institute for History 128 = Studies on Germania Sacra 20), Göttingen 1998. (2 volumes)
^ Bünz: Considerations , pp. 193–197 and p. 199.
↑ Excellent editions of Asilo's circulars as well as all other revisions and compilations can be found at Borst: Zahlenkampfspiel , p. 330 u. passim.
↑ Borst: Zahlenkampfspiel , I. The circular Asilos von Würzburg, p. 330f .: “Sit tabula in longitudine et latidudine, ut cernitis, distincta campis, supra quam ex alterutra parte in ultimis campis disponantur usque ad decuplam proportionem omnes praedictorum trium generum species . "
↑ The role that arithmetic played in architecture should also not be neglected, cf. Bünz: Considerations , p. 198.
↑ Arno Borst: What the Middle Ages had to tell us. About science and play , in: Historische Zeitschrift 224 (1987), pp. 537-555, reprinted and expanded in: Arno Borst: Die Welt des Mittelalters. Barbarians, heretics and artists . Nikol, Hamburg 2007, pp. 448-468, especially pp. 460f .; Borst: Numbers Battle Game , pp. 81–97, 101–111 u. 118-130; Bünz: Considerations , p. 189.
↑ De aggregatione naturalium numerorum , ed. v. Maximilian Curtze: The manuscript No. 14 836 of the Royal Court and State Library in Munich . In: Journal of Mathematics and Physics . No. 40, 1895, Supplement pp. 75-142. See also Borst: Numbers Battle Game , p. 77f.
↑ Borst: Zahlenkampfspiel , The circular Asilos von Würzburg, p. 331: " Hinc octo albi minores ex pari denominatas multiplices ostendant proportiones, duplam ut IIII ad II, quadruplam ut XVI ad IIII, sescuplam ut XXXVI ad VI, octuplam ut LXIIII ad VIII His opponantur eiusdem generis octo nigri minores ex impari denominatas habentes proportiones, triplam ut VIIII ad III, quincuplam ut XXV ad V, septuplam ut XLVIIII ad VII, nonuplam ut LXXXI ad VIIII. "
↑ For the calculations, see in detail Borst: Zahlenkampfspiel , pp. 62–65
↑ Borst: Numbers Battle Game , pp. 103-108.
↑ For the rules, see Borst: Zahlenkampfspiel , pp. 69–73.

[1] Enno Bünz: Did the later Bishop Adalbero von Würzburg invent the number fighting game? Reflections on the Würzburg Cathedral School in the first half of the 11th century , in: German Archive for Research into the Middle Ages, No. 49, 1993, pp. 189–199, here: pp. 189f.

[2] Asilo of Würzburg. , In: Author's Lexicon . Volume I, p. 508 f.

[3] Arno Borst: The medieval number battle game . (Supplements to the meeting reports of the Heidelberg Academy of Sciences, Philosophical-Historical Class 5). Winter, Heidelberg 1986, pp. 55, 58 f. and 60; Bünz: Considerations , pp. 191–193.

[4] . u. a. Enno Bünz: Haug Abbey in Würzburg. Studies on the history of a Franconian collegiate foundation in the Middle Ages (publications by the Max Planck Institute for History 128 = Studies on Germania Sacra 20), Göttingen 1998. (2 volumes)

[5] Bünz: Considerations , pp. 193–197 and p. 199.

[6] Excellent editions of Asilo's circulars as well as all other revisions and compilations can be found at Borst: Zahlenkampfspiel , p. 330 u. passim.

[7] Borst: Zahlenkampfspiel , I. The circular Asilos von Würzburg, p. 330f .: “Sit tabula in longitudine et latidudine, ut cernitis, distincta campis, supra quam ex alterutra parte in ultimis campis disponantur usque ad decuplam proportionem omnes praedictorum trium generum species . "

[8] The role that arithmetic played in architecture should also not be neglected, cf. Bünz: Considerations , p. 198.

[9] Arno Borst: What the Middle Ages had to tell us. About science and play , in: Historische Zeitschrift 224 (1987), pp. 537-555, reprinted and expanded in: Arno Borst: Die Welt des Mittelalters. Barbarians, heretics and artists . Nikol, Hamburg 2007, pp. 448-468, especially pp. 460f .; Borst: Numbers Battle Game , pp. 81–97, 101–111 u. 118-130; Bünz: Considerations , p. 189.

[10] De aggregatione naturalium numerorum , ed. v. Maximilian Curtze: The manuscript No. 14 836 of the Royal Court and State Library in Munich . In: Journal of Mathematics and Physics . No. 40, 1895, Supplement pp. 75-142. See also Borst: Numbers Battle Game , p. 77f.

[11] Borst: Zahlenkampfspiel , The circular Asilos von Würzburg, p. 331: " Hinc octo albi minores ex pari denominatas multiplices ostendant proportiones, duplam ut IIII ad II, quadruplam ut XVI ad IIII, sescuplam ut XXXVI ad VI, octuplam ut LXIIII ad VIII His opponantur eiusdem generis octo nigri minores ex impari denominatas habentes proportiones, triplam ut VIIII ad III, quincuplam ut XXV ad V, septuplam ut XLVIIII ad VII, nonuplam ut LXXXI ad VIIII. "

[12] For the calculations, see in detail Borst: Zahlenkampfspiel , pp. 62–65

[13] Borst: Numbers Battle Game , pp. 103-108.

[14] For the rules, see Borst: Zahlenkampfspiel , pp. 69–73.