Wednesday, September 22, 2004



Chronicon Cameracense et Atrebatense, sive Historia utrisque Ecclesiae, III Libris, ab hinc DC sere annis conscripta. Nunc primum in luce edita, & amp;notis illustrata Per G. COLVENERIUS.

Douay, Ioan. Bogarde, 1615. Sm.8vo. Contemp. vellum.

With engraved printer's vignette on title, 9 engraved illustrations of seals and dice in text, and 3 large folding plates, partly in wood cut and partly engraved, of three different tables for a popular lottery game. (40), 601, (1 blank, 16) pp.

Editio princeps of a mediaeval chronicle of Cambray and Arras, containing the earliest known description and representation of a lottery game, which had been invented by Wibold, a French divine from Cambray who died in the year 965.

Inspired by the "Rythmo-machia" or "Philosophical Game" of Pythagoras, the game was referred to by Wibold as "Ludus regularis seu clericalis". However, it was also known as "Alearegularis contra alea secularis".

It was played with a dice with letters instead of numbers, and a board with the names of all 56 virtues arranged in squares around the middle.

At the end of the book three different tables are given to play this game, two with square boards to be played with dice, and one circular board to be played as a wheel of fortune with a turning pointer in the middle.

On verso of two of these tables explanatory text is present, and the game is extensively explained in chapter 88 of the first Book, pp. 143 ff., and is further discussed in the notes at the end, on pp. 461 ff.

The folding tables, size ca. 42 x 37 cm, were meant to be cut and mounted to be played with, including the engraved figures of dice. In text the list of names of the virtues, and the figures of dice were given too.

The chronicle itself is of interest, written by the French historian Balderic the Red, bishop of Noyon andTournay, as it gives numerous accounts of scholarly reseach and curious details.

The book presents the history from Clovis to 1090, as the author died in 1097. The outstanding feature now is the representation of a mediaeval lottery-game, which according to the inventor could be of use at schools or for charity (Redactionairy: Lottery games initially are invented and used to draft new militairy-personnel and create money).

The importance of this chronicle was rediscovered in 1834 by Le Glay,who published a new edition based on three manuascipts, and in the preface discussed and explained the lottery game present in it.

His Latin edition then was also translated into French in 1836, by Faverot and Petit

Good copy of the rare first edition, with the bookplate of Pierre Briffaut.- (Ms. entry on title) Brunet I, 621 Graesse I, 260 cf. Introduction to the new edition by Le Glay, Cambray & Paris, 1834 NUC.

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Monday, September 06, 2004


Aritmomachia will be under construction up to and including December 2005. Please visit again for updates.

Main part of the text is present, however this still needs editing and placement of various pictures.

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Saturday, September 04, 2004

The "Philosopher's Game" or the "Rithmimacia"

From Italian Manuscript Dec 1539 concerning "Pythagoras'Game" or the "Rithmimacia", with an intriguing insert loosely attached to the body of the text with the appropriate numbers and positions.

From the Giannalisa Feltrinelli Library as sold at Christie's in December 1997 (lot 220) with a letterthat explained that the second half of the MS wasfound in the studio of the Aristotelian polymath Jacques Lefevre d'Etaples and sent to Cosimo Rucellaiin Florence.

Introduction Aritmomachia

Friday, September 03, 2004



Thursday, September 02, 2004



The Rhythmomachy board is rectangular.

Generally 8 squares wide by 16 long. Each player begins with his men arrayed on one of the short ends.

In some descriptions, the board is shorter (8 x 14) or even 8 x 9, but the 8 x 16 arrangement seems to be most common.

It is unclear to me whether the board became checkered or not; there is no intrinsic reason why it should be, and most of the historic images don't depict checkering, but a few do.

We assume that both the size and the occasional checkering resulted from the fact that the game is easy to play on two chessboards, set side-by-side; since chess was quite common in pretty much all cultures that had Rhythmomachy, this was likely a common way to play. The checkering with chessboard was introduced during the years 1,00o - 1,300, and also was introduced the longer range of some or the pieces (easier to move long distance).

The black sheckered fields depict the grainfields and the historic chessboard was the city of Babylon.

Thus, the checkering was probably an accident of that, and the board size standardized on the convenient double-chessboard.


Each player has 24 men, eight round, eight triangular, and eight square.

Each man has a number on it; the rationale behind the numbers is discussed below.

Rhythmomachy is unusual in that it is an asymmetrical game -- although each player has the same number of pieces, the numbers written on those pieces differ widely.

We usually think of one side as "even", and the other "odd", but the reality is more complex than that.

The two sides have contrasting colors, usually but not always black and white.

We usually play black as odd and white as even out of habit, but that is a semi-arbitrary choice, based on the illustrations in Fulke.

On average, the odd numbers are significantly higher than the even numbers. The men are usually flat, and double-sided, with the same number in the opposing color on the opposite side.

For my personal sets, we paint the edges of the men with the color of the side they start on, to make them easier to sort for setup.

The Numbers on the Men on each side fall into six "ranks": two ranks of circles, two of triangles, and two of squares. Each rank contains four men on each side. The actual numbers come from straight forward mathematical progressions.

The first rank of rounds can be thought of as the "seeds" for the rest; they are the odd or even numbers less than ten. That is, the evens have 2, 4, 6, and 8; the odds have 3, 5, 7, and 9. Since each rank has four men, each of those flows from a particular seed. (Thus, for the first rank of squares on the even side, one man is based on 2, one on 4, and so on.)

The second rank of rounds are the first rank of rounds squared; thus, on the even side, they are 4, 16, 36, and 64, and on the odd are 9, 25, 49, and 81.

The first rank of triangles are the two rounds added together; thus, on the even side, they are 6 (2 plus 4), 20, 42, and 72, and on the odd are 12, 30, 56, and 90.
The second rank of triangles are the seed plus 1, squared. Thus, on the even side they are 9 (2 plus 1 squared), 25, 49, and 81, and on the odd are 16, 36, 64, and 100.

The first rank of squares are the two triangles added together. Thus, on the even side they are 15 (6 plus 9), 45, 91, and 153, and on the odd are 28, 66, 120, and 190.

The second rank of squares are twice the seed, plus one, squared. Thus, on the even side they are 25 ((2*2)+1, squared), 81, 169, and 289, and on the odd are 49, 121, 225, and 361.

Summarizing all of that into a table, we get::

1st Rounds (x)
2nd Rounds (x2)
1st Triangles (x + x2)
2nd Triangles ((x + 1)2)
1st Squares (x + x2 + (x + 1)2)
2nd Squares ((2x + 1)2)

The Kings::

On each side, one square is replaced by a "King" instead. The king is a large number which happens to be a sum of some square numbers, and takes the form of a pyramid of pieces.

On the even side, the King is the 91, and is composed of a pyramid of six pieces: a square 36, a square 25, a triangular 16, a triangular 9, a round 4, and a round 1. On the odd side, the King is the 190, and is composed of a pyramid of five pieces: a square 64, a square 49, a triangular 36, a triangular 25, and a round 16.

The Kings are generally treated specially: in most variants, they have enhanced powers of movement, and capturing the opponent's King is at least part of the objective of the game. Usually, the King can be captured intact or piecemeal, capturing its component pieces out of the pyramid.

Details vary from version to version, however.


Since each variant is a bit different, I will only speak in broad generalities here. Roughly speaking, movement happens by turns, as in most board games; each player gets to make one move on his turn.

The three shapes move differently; in general, the squares move furthest and the rounds move least.

Movement is never affected by the number on a piece -- the numbers are relevant mainly for capture.

While each version has a couple of degenerate captures (eg, surrounding a piece completely), capture is generally done mathematically.

The details of this vary greatly, but in general you capture opposing pieces by creating mathematical relationships between those pieces and your own. For example, if two of your men are positioned by an enemy man, and they add up to his value, they capture him. Note that capture frequently does not require actually jumping into the enemy's space, as Chess does; you can often capture simply by setting the position up.

In most versions, captured men can then be subverted;
when you capture the man, you flip him over (so that your color shows), and re-enter him on your side. This is essential in some cases, since most numbers exist only on one side or the other.


There are many, many different ways that a Rhythmomachy game can be won, from very simple "basic" victories to the Great Victories.

The latter involve capturing the enemy king, then using your men to form mathematical sequences in enemy territory. Which victories were allowed varies from version to version, but I would generally recommend some practice just getting the basics before trying to win one of the advanced forms.


Mathematical Proportions::

There are many different kinds of mathematical relationships that can figure in Rhythmomachy. we assume that the reader is capable of dealing with basic arithmetic: addition, subtraction, multiplication, and division.

These will serve you fine in most of the basics of the game. But for the more advanced versions of the game, and for the higher victories, you need to understand the three major forms of Proportions:

Arithmetic, Geometric, and Harmonic

Don't worry about learning these upfront; they can come later. Also, all of the proportions available in the game are available as tables in Fullke's book, the period source from which we took these reconstructions. It is common to look up the proportions in tables, especially at first.

In general, all of these proportions are ways of defining a relationship between three numbers. In all of these examples, we will talk about three numbers A, B, and C, where A is the smallest number and C the largest. we will give examples for each, as well.

Arithmetic Proportion::

Three numbers are in arithmetic proportion when the difference between A and B is the same as the difference between B and C. For example, the numbers 2, 4, and 6 form an arithmetic proportion, because (4 - 2) = (6 - 4). Similarly, 5, 25, and 45 form an arithmetic proportion, because (25 - 5) = (45 - 25).

In other words, the numbers are rising in a simple progression, adding the same number each time. 5 plus 20 is 25; 25 plus 20 is 45; so 5, 25, and 45 form an arithmetic proportion. Another way to think of it is that there is a number X, such that (A + X) = B, and (B + X) = C.
There are 49 arithmetic proportions available among the numbers in Rhythmomachy, according to Fulke.

Geometric Proportion::

This is similar to arithmetic proportion, except that instead of adding the same number to go from A to B and B to C, you multiply instead. That is, the ratio between A and B is the same as the ratio between B and C. Similar to the last concept above, there is a number X, such that (A x X) = B, and (B x X) = C.

So for example, 3, 12, and 72 are in geometric proportion, because (3 x 6) = 12, and (12 x 6) = 72. 2, 4, and 8 are in geometric proportion, because (2 x 2) = 4, and (4 x 2) = 8. A more sophisticated example would be 4, 6, and 9, which are in geometric proportion because (4 x 1.5) = 6, and (6 x 1.5) = 9. (Yes, this is a period example; they understood fractions perfectly fine.)

There are 27 geometric proportions available in Rhythmomachy, according to Fulke.

Harmonic Proportion ::

Harmonic, (or Musical in many period sources) proportion refers to the way that musical harmonies relate to each other. Mathematically, it is the relationship (C / A) = ((C - B) / (B - A)) -- the ratio between the largest and smallest numbers equals the ratio of the differences between both of those numbers and the middle one.

So for example, 3, 4, and 6 are in harmonic proportion, because (6 / 3) = 2, and ((6 - 4) / (4 - 3)) = 2 as well. Or 25, 45, and 225, because (225 / 25) = 9, and ((225 - 45) / (45 - 25)) = 9 also.

There are 17 harmonic proportions available in Rhythmomachy, according to Fulke.

Example proportions: 2 3 4 6

Arithmatical 2 3 4
Musical 2 3 6
Geometrical 3/2 = 6/4 (old system, before zerro)


Rithmomachia, the Philosophers' Game ; A Medieval Battle of the Numerical Harmonies

Rithmomachia, the Philosophers' Game::
A Medieval Battle of the Numerical Harmonies

Therefore a Arithmomachia is the "battle of the numbers", or better still the "battle of the numerical harmonies", referring therefore to the substance of the game that is, like we will see, that one to create of the numerical proportions between numbered pawns.

Game privileged from medieval the intellectual class, is known also with the expression of ludus philosophorum ("the game of the intellectuals").

Objective of the game it is to construct with the own pawns one or more proportions (or "harmonies") numerical, therefore to realize a "triumph".

When the triumph happens through a capture (v. over), is worth the classic rule for which the capturing pawns they must be found to the debita distance of movement from that adversary.

Different is the situation if the pawns belong already all to the player.

In the first place, the triumph must be realized in the half opposing chessboard; moreover, not be a matter itself of a capture, the distance of movement of the pawns does not have value.

Then, because a harmony of three pieces is valid, they must find themselves online (horizontal, vertical or diagonal) in adjacent cases or to form three you concern us of a side square three cases.

For the harmonies of four pieces, they must obligatorily form the four concern us of a side square three cases.

1. Introduction

'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 medieval 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 medieval treatises Rithmomachia was favoured. (Folkerts 1989) The reason was, that Rithmomachia was the only game in the curriculum of the mediaval 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.

2. The History of Rithmomachia

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 Wurzburg 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.

( transcription of a 1563 translation by William Fulke (or Fulwood -- the sources disagree) of Boissiere's 1554/56 description of Rythmomachy. It is entry 15542a in the Short Title Catalog of Pollard and Redgrave, and on Reel 806 of the corresponding microfilm collection.)

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 Italy, England, France, 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 medieval.

3. The Rules of Rithmomachia::

The rules have changed over the centuries.

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.

A. Preparations

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.

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). T

he 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 according to the array.

The tables of harmonies, in which all combinations of pieces for harmonies are recorded, are very helpful in playing. (2)

B. The Aim of the Game ::

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.

C. The Moves of the Pieces ::

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.

D. The Capture of Pieces ::

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.

E. The Victory::

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:

1.) A small victory is reached by an arithmetical, geometrical or musical harmony of three pieces.

2.) A big victory is gained by building two (but not more than two) different harmonies with 4 pieces.

3.) 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)
Appendix I
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
Appendix II
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.

Appendix III: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).

TO-DE EDIT !!!!!!!!!!!!!!!

Arithmomachia laughed them to 1030, Wurzburg. In a competition between the schools of the cathedral of Worms and Wurzburg - both many rinomate in the field of the Arithmetic - a text of containing dispute was written up numerical sequences based on the "De arithmetica institutione" of Boezio; on the base of this text, monaco of name an Asylum it created a game - Arithmomachia, exactly - useful to the students of the two monasteri in order to learn the numerical theory of Boezio. The first drawing up was adapted from other scholars. Hermannus Contractus, respected monaco to Reichenau, rimaneggiò the rules of Asylum and added to some notes on the musical theory (a second version wants instead that the author originates them is just Hermannus Contractus [ 5 ]). There is also who thinks that the attribution is gives is given to the bishop of Cambrai, Wibold, than in the 965 it invited the monaci local to stop to play to dice and to play instead "the battle between the virtues and the defects" [ 7]. There are however various hypotheses that rivendicano an origin still more ancient, sights some correspondences between the disposition and the movement of pieces and the disposition and the movement of the armys Roman [ 1][5 ]. Between the XI and XII the Arithmomachia century diffuses po' to the time in all a Germany and France. The rules came ulteriorly extended. Between the XII and XIII the century the game reached also in England. Giovanni di Salisbury writes in its "Policraticus" (1180) that "the acquaintance of the battle of the numbers is one source of divertimento and profit (I,5)". Also Roger Bacon comment the Arithmomachia, in its "mathematica Communio" (I, 3, 4). [ 1 ] To along Arithmomachia it remained in competition with chess and there was a period in which even more it was respected of same chess. The reason of that is in the fact that Arithmomachia was the only game previewed in the programs of the medieval schools and university - I privilege that chess will not never receive, in how much game of military inspiration that did not respect the canoni of the seven liberal limbs. Arithmomachia found the maximum spread in XVI the century, in particular as consequence of the invention of the press. Rules were written from Shirwood (1474), Faber (1496), Boissiere (1554/56), Ries (1562), Fulke/Lever (1563), Selenus (1616). The greater centers of rinascimentale spread were England, France, Italy and the Germany orients them. To the end of XVII the Arithmomachia century lost popolarità and fell in the oblivion. Signals on the numerical theory of Boezio Anicius Manlius Severinus Boethius nacque around to the 480 to (or near) Rome and died to Pavia in the 524. The witnesses matemati us of Boezio, for how much of insufficient quality, were between the best that could be found in the high Middle Ages and were use you for centuries in a rather rear Europe in the field of the mathematics. The Arithmetica di Boezio was based on the job of Nicomaco and for the medieval scholars it was the maximum base of study for the numerical theory of Pitagora. Boezio was one of the greater sources of supplying for the crosss-roads. [ 2 ] the mathematics of Boezio is dominated from the concept of "numerical progression"; there are arithmetical progressions (a+b*n), geometric progressions (a*(b^n)), harmonic progressions (1/(a+(b/n))) and other anchor... [ 6 ] The rules of the game As we have been able to see, an only set of rules for Arithmomachia does not exist. During the history to plurisecolare of the game they they are often changed and in consisting way; their complexity, as an example, is last from the little pages written by hand of the first version until beyond one hundred printed publication pages in the later versions. Probably, in the first phase of spread of the game, every school adopted of the own rules, that is those deductions more adapted for an understanding of the arithmetical bases of the game. Moreover obvious E' that the greater documentary sources regard the rinascimentali versions of the game, while on the first medieval versions it is not equally remained material; the only medieval version available is that one of Asylum (XI sec.). Archaeological evidences do not exist at the moment: no pawn never has been found again, neither medieval neither rinascimentale. The present set of rules, therefore, does not mean to represent a specific version of the game: the objective that is placed me has been that one to create a reasonable set of rules for uses in Italian area in the second half of XIII the century, extrapolating i characters generates them and the rules base of the game, with particular reference to the version of Asylum, and taking part with personal corrections where it is seemed to me not were clarity in the consulted sources here. With the same spirit who - creed - has animated in the time the various authors, in all the doubt situations I have decided to resort to the criterion of the maximum giocabilità, also without to neglect the more fascinating and educational aspect than this game, that is the manipulation of the numbers. The table from game The table from game is composed from 112 cases: 8 cases of width and 14 of depth. The pawns The pawns are in number of 23 more 1 king for every player; the king is formed from 5 or 6 disposed pawns in guisa pyramidal (will see one more ahead detailed description more). The pawns, except that constituent king, are painted on the two faces with contrasting colors (R-bianco.e.nero, red and blue...) and on ciascuna of they they bring back a numerical value; the use of the two colors derives from the fact that the pawns captured to the enemy come introduced in game from the own part. It makes exception the king, than he does not change alignment and, if captured, he comes simply removed: its pawns are of a single color, but they bring back anch' they a numerical value. The pawns have also one various geometric shape: 8 are round, 8 triangular and 7 square ones. The two king has to the base two square pawns, sormontate from two triangular ones and, finally, one or two round pawns. We see hour which numbers must be brought back on ciascuna pawn. The pawns are uniforms in six ranks from 4 elements everyone. The first rank of circles is constituted from the numbers base: 2,4,6,8 for a player (than from hour in then, for this reason, will come defined like equal side) and 3,5,7,9 for the other player (that we will call uneven side). According to rank of circles base is constituted from the square of the numbers: 4,16,36,64 for pars and for the odd number. The first rank of triangles is constituted from the sum of the two previous ranks: 6,20,42,72 for the pars; 12,30,56,90 for the odd number. According to rank of triangles it is obtained adding 1 to every number base and elevating to the square the turning out value: one obtains for pars and 16,36,64,100 for the odd number. The first rank of squares is given from the sum of the two ranks of triangles: 15,45,91(re),153 for the pars and 28,66,120,190(re) for the odd number. According to rank of squares it is obtained adding 1 to the double quantity of every number base and elevating to the square the turning out value: one obtains 25,81,169,289 for pars and for the odd number. The values of the two king are given from the sum of the faces on ciascuna of the constituent pawns the same ones. The equal king is constituted from 6 pawns, that they bring back the following numerical values: 1 on the smaller circle; 4 on according to circle; 9 and 16 on the two triangles; squared 25 and 36 on the two. Analogous, the uneven king is constituted from 5 the following pawns with values: 16 on the only circle, 25 and 36 on the two triangles, squared 49 and 64 on the two. Preparation To the beginning of the game the pawns come disposed in "order of battle". The beginning of the game is up to the uneven player, which it has minors possibility to capture pieces and to realize harmonies. Thanks to this disparity, it is possible to make to meet players of various level. Scope of the game Through tactical movements, every player must per.primo.cosa eliminate the enemy king. Subsequently, he must realize a "triumph" (v. beyond). The realization of a "excellent triumph" door to the definitive conclusion of the game. With the smaller triumphs ("mediocre triumph" and "great triumph") the game continues, but the players can come to an agreement themselves for giving they a partial value of Victoria. The movement of pieces The players alternatively move one pawn to the time in one empty case, according to characteristic rules of ciascuna geometric shape. Also the cases crossed from the pawns in their movement must be empty: it is not possible "to jump" other pawns. The pawns can be moved in whichever direction (horizontal, vertical and diagonal). The circles move exclusively of one case. The triangles move exclusively and exactly of two cases. The squares move exclusively and exactly of three cases. The king moves like the piece that is found to its base. To the beginning of the game, therefore, muoverà of three cases, like a square; but if, in the course of the game, it had to lose both squares, muoverà only more than two cases (like the triangles) and, if it had to lose also both triangles, muoverà only more than one case (like a circle). The king cannot voluntarily be separated in its constituent units. The capture of pieces Five various ways exist in order to capture pieces of the adversary: encounter, I besiege, ambush, onslaught and proportion. In all the cases the king can capture is with its number total (given from the sum of the numbers that constitute it in that moment) are with a single piece; partial sums are not admitted. In any case, the distance for the capture is equivalent to that one of movement, even if the capturing piece is of a geometric shape that normally you would preview a various distance (to es. a still complete king moves and capture to three cases of distance, even if the single used element is a triangle or a circle). Analogous it can be captured in the its totality or a piece to the time, except that in I besiege where can be captured single in its interezza. Every time that a piece is captured, it comes rigirato so as to to change alignment. They make exception constituent pieces the king, than they cannot change alignment and they come therefore simply it eliminates to you from the table from game. In order to capture an opposing piece, it does not have to be entered physically in the case from occupied it: it is necessary only that there is this possibility, that is the piece to capture must be found to a distance that corresponds to the movement of the capturing pawn. In practical, if a pawn before or after just the movement is found to the own distance of movement from an opposing pawn, and exists the conditions for the capture, the opposing pawn is captured. In the case in which the capture it happens before the movement, that replaces the same movement and the turn of the player is ended. The capture are not obligatory


Wednesday, September 01, 2004



The earliest version is attributed to scholar Hermannus Contractus.

There are several descriptions of this, the best of which Written by Asilo, a monk of Wartzburg, has been translated by John Richards, a 20th century writer in a math journal".


RHYTHMOMACHIA (1987) by Illmer, Detlef & Gädeke, Nora, & Henge, Elisabeth & Pfeiffer, Helene & Spickler-Beck, Monika. ISBN 3880343195.

Ann Moyer's xx check

Furthermore, these book titles should be available at your library:

by Robert Charles Bell
3.) DISCOVERING OLD BOARD GAMES by R.C. Bell (Shire Publications LTD)
4.) BOARD GAMES AROUND THE WORLD by Robbie Bell and Michael Cornelius
(Cambridge University Press).
5) Murray History of Boardgames xx
6) Oxford History of Boardgames xx

All these books contain solid information regarding the rules of Rithmomachia.

Historical books::

Selenus, Gustav (d. i. August II. Herzog von Braunschweig-Lüneburg) Das Schach- oder König-Spiel.

(translation UK)

Selenus, Gustav (D i August II. Duke of Braunschweig Lueneburg) chess or king play. Into four distinctive books, with special diligence, creating and properly binded.

This is too end, angefueget, a very old play, genandt, Rythmo Machia. Leipzig, Kober 1617. Fol. (26), 495, (3) sides with 6 stung TitleBroderies, 1 doublesince. Table, 2 Kupfertafeeln, 83 (28 full-page) text-copper and 1 printer mark at the conclusion.

Pgt. of the time with handschriftl. Back title - - Van of the lime tree 1955 265 - Schmid, Schachlit. 118 - Title edition of the first edition 1616, large D J appeared with Henning. - the most important and determining chess text book of its time, a revision from Lopez of Tarsia.

With a listing of the used literature. The attaching work is a pythagoreisches number play, from the Italian of the franc.

Barozzi translates and works on. - the broad Titelbordueren with representation of game of chess scenes.

Chessmen and play positions show the copper.



Monday, August 30, 2004

Number Games ; Rythmomachia / Rithmimachia

Number Games ; Rythmomachia / Rithmimachia


The game is of ancient derivation and involves
pieces moving on a board.

Together with Courier Chess (Germany and
depicted by e.g. Lucas van Leyden), Aritmomachia
(rithmomachia / rithmimachia, latest: rythmomachy),
is considered to be one of the two (dormant) successors of
the ancient game of chess (peaked in popularity at appr.
1,000 ac - 1,800 ac, and was referred to as The Pythagoras Game).


Ritmos = Number
Mache = Battle

Latin: Pugna (certamen) numerorum


The board8*16 (two chessboards; face long side,
right hand is home board). The board as shown above is considered
a practice board seized 8*9.Pieces: Total 24 pieces;
Round, Triange, Square. White is Even (feminine) en
black is Odd (masculine).

Arithmetic / Geometrical / Harmonic progression

The competition board 8 * 16

An Italian Manuscript Dec 1539 concerning "Pythagoras'Game"
or the "Rithmimacia", with an intriguing insert loosely attached
to the body of the text with the appropriate numbers and
positions.From the Giannalisa Feltrinelli Library as sold at
Christie's in December 1997 (lot 220) with a letter that explained
that the second half of the MS was found in the studio of the
Aristotelian polymath Jacques Lefevre d'Etaples and sent to
Cosimo Rucellaiin Florence.

The practice board 8 * 9

from Jordanus and Faber Arithmetica decem librisdemonstrata;
Paris 1496a page from Claude de Boissiere's Rythmomachia,
Paris1556. According to Smith/Plimpton, the book is
profusely illustrated and "was connected with the medieval
number classifications and ratios..."


Pythagoras of Samos, Nicomachus of Gerasa, Anicius Manlius
Severinus Boetius. Six levels of numbers.

Integral and proportional (equal - unequal 5 different ways).


Roman militairy; Vegetius (writter: Flavius Vegettos
Renatus), Belisar's battle formation.