Rithmomachia Het spel der filosofen Inhoud Contents ............................................................................................................................................. 1 Dutch Translation Rithmomachia, the Philosophers' Game A Mediaeval Battle of Numbers ..... 1 Commentary on Rithmomachia ........................................................................................................ 7 http://jducoeur.org/game-hist/mebben.ryth.html............................................................................ 14 Rithmomachia, the Philosophers' Game A Mediaeval Battle of Numbers .................................. 14
Dutch Translation Rithmomachia, the Philosophers' Game A Mediaeval Battle of Numbers
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Rithmomachia, het Spel der Filosofen (7de correctie fin)
Een middeleeuwse strijd van nummers Peter Mebben Vertaald door H.Th.Frenkel, Stichting Metapontum Geselecteerd op internet door: Plato harmony.
[N.V. = notitie en aantekening van de vertaler] 1.Introductie “Een kennis van de strijd van nummers is een bron van vreugde en van voordeel”. Met deze lovende woorden prees John of Salisbury het gebruik en misbruik van spelen toen hij een verslag uitbracht in zijn Policratus (1,5) in 1180. Toen in die tijd de middeleeuwse scholasticus sprak over de competitie van nummers, bedoelde hij dat hij Rithmomachia leerde kennen als een verdienstelijk en plezierig leermiddel in rekenkunde of arithmeticales. Dit spel had zich verspreid vanuit de kloosterscholen in Zuid-Duitsland naar Engeland (Evans 1976). Wat voor soort spel is het dat John of Salisbury hier zo hoog aanprijst? Wat bewoog hem en andere mensen te spreken over Rithmomachia, zo lang geleden, nadat het bijna was vergeten na een hoogtepunt van ongeveer 700 jaren? Roger Bacon, [n.v. Engelse geleerde ca. 1245, universiteitsdocent te Oxford en Parijs, eerste voorstander van de wetenschappelijke methode: het empirisme, later werd hij Franciscaans monnik] beval Rithmomachia eveneens aan bij zijn studenten in zijn gemeenschap van wiskundigen ‘communio mathematics’[I,3,4] in de 13de eeuw. Hij noteerde 7 punten waarmee zijn studenten arithmetica konden leren volgens de methode van Boethius [N.V. Romeins filosoof en politicus ca 525, vertaler van Plato & Aristoteles’ rekenkundige verhandelingen ‘Institutio Mathematica’ in het Latijn], en hij adviseerde tenslotte dat zij het spel Rithmomachia moesten gebruiken als leermiddel. Thomas More, [n.v. 1478-1535, filosoof, advocaat, staatsman, auteur, lord kanselier c.q. hoogste rechter van Koning Henry VIII van Engeland, was een Rooms katholiek humanist, werd martelaar en is heiligverklaard door Paus PiusXI in 1935]. Hij was overtuigd van de deugdzaamheid van het spel. More liet de denkbeeldige inwoners in Utopia een beroemd traktaat over de ideale staat, dit spel ‘s avonds spelen voor hun ontspanning (1516 II,5). Robert Burton, beschouwde het gebruik van Rithmomachia eveneens als een efficiënt geneesmiddel voor melancholie, omdat het een goede oefening was voor de menselijke geest [Anatomy of Melancholy 1651. II,4]. Betekenis van de naam Rithmomachia De naam van het spel Rithmomachia heeft een Griekse oorsprong. Het eerste gedeelte ‘Rithmo’ is afkomstig van een combinatie van de woorden arithmos en rhythmos. Arithmos betekent nummer en rhythmos had naast ritme ook de betekenis van proportie van nummers in de middeleeuwen, omdat in het spel het niet alleen om de nummers op de stukken gaat, maar ook om de relatie van de nummers onderling. Het tweede gedeelte van de naam ‘machia‘ komt van machos, dat oorlog, slag of slaan betekent. Rithmomachia kan dus omschreven worden als de slag of de oorlog van nummers. Daarom kan Rithmomachia beschreven worden als de oorlog van nummers. In Engeland stond het spel ook bekend als ‘het Filosofenspel’. Rithmomachia is een strategisch spel voor twee spelers. Een zwarte en een witte partij van nummers die tegenover elkaar staan vergelijkbaar met schaak. 2
Er was een tijd dat Rithmomachia in competitie was met schaak en zelfs meer gerespecteerd werd dan schaak, bijvoorbeeld in sommige verhandelingen was Rithmomachia in de middeleeuwen de favoriet (Folkerts 1989). De reden was dat Rithmomachia het enige spel was in het studiepakket der middeleeuwse scholen en universiteiten -een eer welke schaak nooit had gekregen, omdat het slechts werd gespeeld door de adel als een tactisch oorlogsspel voor hun pure ontspanning-, maar dit gebruik paste niet in de kerkelijke regel van de 7 liberale kunsten. [N.V. De 7 liberale kunsten: 1. Grammatica; 2. Retoriek; 3. Logica; 4. Geometrie; 5. Rekenkunde; 6. Muziek; 7. Astronomie. De moderne universiteiten voegden eraan toe: 1. Theologie; 2.Filosofie; 3. Literatuur; 4. Talenkennis; 5. Geschiedenis; 6. Wiskunde; 7. Natuurwetenschappen].
Doelstelling van Rithmomachia In Rithmomachia is de doestelling niet om tegen elkaar te vechten met legers van nummers, maar wel om deel te nemen in een wedstrijd waar de spelers sommige van hun stukken in een harmonieuze orde moeten opstellen. De tegenstellingen tussen zwart en wit, even en oneven, gelijkheid en ongelijkheid ontwikkelen zich en worden op het einde opgelost in harmonie. In het bijzonder verschijnen de laatste twee paren in de nummerfilosofie van Boethius, welke een selectie van nummers voorschrijft op de stukken [Borst 1990]. De Boethius-nummertheorie is gebaseerd op de Pyhthagorese filosofie betreffende zijn aanwijzingen over classificaties, volgorden en beeldvorming van nummers en hun overdrachtelijke voorstellingen en hun onderlinge harmonische proporties. Al die kenmerken van de Boethius-nummertheorie komen terug in het spel Rithmomachia. Pythagoras´nummersymboliek, als deel van Boethius-nummerfilosofie was van bijzondere interesse gedurende de periode van het ontstaan van Rithmomachia. De gehele wereldorde werd hier van binnenuit onderzocht en voorgesteld door deze nummersymboliek (Coughtrie 1984). Rithmomachia was een geestige wijze om zich de Boethius-nummertheorie te herinneren. Fundamenteel gezien was het een amusant spel, plezierig om Rithmomachia te spelen, het enige spel dat door de Christelijke wetenschappelijke gemeenschap in de middeleeuwen werd aanvaard, omdat in tegenstelling met schaak en dobbelspelen, het van groot nut was. 2. Geschiedenis van Rithmomachia In vele oude registers werd Boethius of Pythagoras verondersteld de uitvinder te zijn van Rithmomachia, echter, zij creëerden slechts de wiskundige basis voor dit spel. Het is zeker dat de oudste geschreven bewijsstukken van Rithmomachia gevonden zijn in Wurzburg rond 1030. Bij een competitie tussen de kathedraalscholen van Worms en Wurzburg, beide goed bekend voor hun leidinggevende positie in arithmetica, werd een argumenttekst geschreven met arithmetica volgordes van nummers gebaseerd op de `DE INSTITUTIONE ARITHMETICA van Boethius. Gebaseerd op dit schrijven, creëerde een monnik Asilo geheten, een spel Rithmomachia dat de nummertheorie van Boethius illustreerde voor de studenten van de kloosterscholen. Deze eerste omschrijving werd aangepast door andere geleerden. Hermannus Contractus, een gerespecteerde monnik in Reichau, onderzocht de regels van het spel geschreven door Asilo, breidde deze verder uit en voegde er muziektheoretische opmerkingen aan toe. In een school te Liège ontwikkelde zij een manier om het spel praktisch te realiseren en, niet alleen om het spel zelf te corrigeren, maar ook om de training van studenten in arithmetica te verbeteren (Borst 1987). In de 11de en 12de eeuw verspreidde Rithmomachia door middel van de kloosterscholen in Zuid-Duitsland en Frankrijk. Daar werden de regels verzameld, geordend en samengevat. De regels werden vermeerderd en uitgebreid, voldoende en genoeg om zonder leraar te worden gespeeld. Rithmomachia was een voortreffelijk leermiddel. Geleidelijk aan werd het ook gespeeld door de intellectuelen voor hun genoegen. In de 13de eeuw verspreidde Rithmomachia zich door Frankrijk en van daaruit naar Engeland. De wiskundige Bradwardine en enige van zijn collega’s schreven een tekst over Rithmomachia en zelfs in een pseudo Ovidian gedicht ”De vetula”, werd Rithmomachia 3
hoog aangeprezen. Rithmomachia bereikte zijn grootste uitbreiding in de tijd van de boekdrukkunst. De boeken geschreven over Rithmomachia bezaten gevarieerde bedoelingen. Faber (1496) en Boissirie (1554-56) beiden professoren van wiskunde, schreven hun verhandelingen voor hun studenten aan de universiteit van Parijs. Faber en een latere Italiaan pasten het aan in het Florentijnse dialect met een dialoog in de vorm van de Griekse didactiek en de Pythagoraanse traditie volgens hun tijd. Shirwood (1474) en Fulke/Lever (1563) schreven hun boek over Rithmomachia voor hun vorsten en broodheren. Het handgeschreven manuscript van Abraham Ries (1562) was met dezelfde intentie geschreven. Abraham Ries was de tweede zoon en erfgenaam van het wiskundige talent van de meest bekende Duitse rekenkundige Adam Ries. Snellius (1616) [N.V. Nederlands wiskundige wiens echte naam hertog Augustus I van Brunswijk Lymburg is], publiceerde zijn Rithmomachia als een aanhangsel van zijn schaakboek. Al deze teksten werden gekarakteriseerd door het feit dat Rithmomachia slechts gespeeld werd door intellectuelen voor puur genot en geestelijke ontspanning Illmer (1987). Rithmomachia was toentertijd vooral bekend in Engeland, Frankrijk, Italië en Oost Duitsland. Aan het einde van de 17de eeuw verloor Rithmomachia zijn grote populariteit. De wiskundige en filosoof Leibniz kende alleen de naam, niet de regels van het spel. Het voornaamste onderwerp van wiskunde veranderde in die tijd. De introductie van het getal -nul-, de integratie en de differentiatie van integralen, de berekeningen met decimalen en de kleinste eenheden paste niet in de nummertheorie van Boethius. Wiskunde bewoog zich naar het berekenen van kansen met de waarschijnlijkheids- berekeningen (Folkerts 1989). Schaak werd het grote spel van die tijd en beschermde de tradities van Rithmomachia voornamelijk in Duitsland ondanks zijn grote onpopulariteit van die tijd. Omdat Snellius als een groot schaakenthousiast een versie drukte van Rithmomachia in een aanhangsel van zijn boekwerk van schaak, hebben latere schrijvers van schaakboeken Rithmomachia bijgesloten als het rekenkundige schaak –arithmetica -schaak in het Duitssprekende gebied (Allgaier 1796, Waidder 1873, Koch 1803). Op een gelijke wijze paste Zimmermann (1821) Rithmomachia aan tot dammen (Dame in het Duits als het Getal-Damenspel) of nummerisch-schaak. Tot nu toe wordt Rithmomachia in het bijzonder beschreven in spelboeken (Archief 1819, Jahn 1917, Strutt 1801). Twee Duitse leraren werden ook geïnspireerd door Selenus om Rithmomachia wederom aan te kondigen: Adler, een gepassioneerde wiskundige en schaakspeler ontdekte de didactische voordelen van Rithmomachia en publiceerde een tekst van de regels in zijn schoolprogramma in 1852, maar hij ontving geen grotere aandacht hiermee. Jahn, in 1917 65 jaar later, die een parochiepriester was en rector van de Zullshower Anstalten bij Stettin, probeerde ook om het een grotere populariteit te doen verkrijgen maar ook hij moest met magere resultaten kampen (1917-1929). Academisch onderzoek bewees dat er onafhankelijke tradities van Rithmomachia in de middeleeuwse geschiedenis werden ontwikkeld gedurende een periode van meer dan 100 jaar. In 1986 ontstond door dit academisch onderzoek met “het mittelalterliche Zahlenkampfspiel” door Borst, een basisch werk waarin de oudste bronteksten werden bewerkt. De middeleeuwse tradities van Rithmomachia zijn zeker. Illmer (1987) echter vermoedt dat Rithmomachia ouder is. Er zijn opvallende parallellen tussen de verhoging en de zetten der stukken en de verhoging en bewegelijkheid van de Romeinse legers. Reeds in ±1070 in Liège verschafte dit Romeinse model de spelers een gemakkelijker manier van spelen (Borst 1986). Er zijn echter geen schriftelijke getuigenissen, maar in het algemeen zijn de teksten van antieke bordspelen zeer kort, zoals bijvoorbeeld in verschillende werken van Plato. Een exacte beschrijving of zelfs een regel van het spel is moeilijk te reconstrueren. Ook zijn geen archeologische bewijzen tot nog toe gevonden. Er zijn geen antieke noch middeleeuwse stukken 4
gevonden. 3. De regels van Rithmomachia Er bestaat geen eenduidig stelsel van regels voor Rithmomachia. Gedurende de 1000 jarige geschiedenis van het spel zijn de regels vaak veranderd. De omvang hiervan vermeerderde van wat handgeschreven pagina’s tot meer dan 100 gedrukte pagina’s waarin gedetailleerde wiskundige en harmonische achtergronden worden beschreven. Doch de regels hebben de volgende dingen gemeen: • Het aantal stukken met de gedrukte nummers erop, de twee piramides en een rechthoekig bord. • Bovendien is het doel van het spel gemeenschappelijkheid. • Twee spelers proberen door vastgestelde zetten een opstelling te bouwen van drie of vier stukken aan de kant van zijn tegenspeler. • Het aantal van de stukken behoort in een speciale verhouding te zijn met de andere; met de opstelling van een van die groepen behaalt de speler zijn overwinning. • In het spelproces mogen de tegenstanders stukken gevangen nemen of veroveren volgens bepaalde regels. Afhankelijk of men een perfect spel zoekt of een gemakkelijker vorm ervan, de grootte van het bord en andere details van de regels kunnen variëren. De regels die hier zijn weergegeven komen meestal overeen met de manier waarop Rithmomachia werd gespeeld in de 17de eeuw, voordat het zich terugtrok in een schaduwachtig bestaan. Deze regels zijn geschikt om het vandaag weer te spelen. A. Voorbereiding • Rithmomachia wordt gespeeld op een bord van 16 x 8 vierkanten. • De zwarte en witte stukken zijn genummerd volgens de nummertheorie van Boethius. (Zie aan het einde van het spel) • De tweede en volgende rijen van nummers worden verkregen van de eerste. • De witte stukken worden de even en de zwarte de oneven genoemd, maar er zijn oneven nummers in de partij van de even en omgekeerd. • Op de ronde stukken zijn de veelvouden geschreven. • De basisrij is opgebouwd uit hoeveelheden van 1. • In de tweede rij worden de basisnummers vermenigvuldigd met zichzelf. • De nummers op de driehoeken zijn zeer speciaal of superparticular. Zij bevatten het volgende nummer en een fractie ervan ([ n+1] : n). • De nummers op de vierkanten worden opgebouwd met de voorafgaande nummers en een veelvoud van de fractie ervan dus ([n + 2] : [ n + 1]. Deze heten de superpartientes. Er zijn vele wiskundige relaties tussen de nummers. Boethius gaf bepaalde procedures voor de afleiding van de nummers. Een van die wiskundige relaties is dat de eerste rij van driehoeken kan worden opgebouwd door de som van de nummers van de twee voorgaande ronde stukken of cirkels. Op dezelfde manier wordt de 1ste rij van kwadraten van de 2 rijen van driehoeken afgeleid. Op de positie van wit 91 staat een piramide. De kwadraatnummers 36 en 25 op de vierkantige, 16 en 9 op de driehoekige en 4 en 1 op de ronde stukken zijn opgeteld tot de som van 91 op de witte piramide. In overeenstemming hiermee wordt het zwarte 190 vervangen door een piramide met de totale som van 190, bestaande uit de kwadraatsnummers 64 en 49, 36 en 25 op driehoeken, en 16 op een cirkel. De stukken worden geplaatst als in figuur nummer 2. De tafels van harmonieën, waarin alle combinaties van de stukken zijn genoteerd, zijn zeer nuttig in het spel.(2) 5
Het doel van het spel • Door tactische zetten, moeten de spelers trachten een harmonie te organiseren uit drie of vier stukken in een bepaalde proportie in het gebied van zijn tegenspeler. • De spelers echter moeten letten op hun tegenspeler om hem te beletten deze poging tot harmonie te blokkeren. • De eerste speler die deze harmonie ordent is de winnaar. C. De Beweging der Stukken • De spelers zetten (hun stukken) om beurten in de lege ruimte (N.V. onbezet veld) • Geen stuk mag over een ander heen springen. • Zwart begint, omdat wit betere mogelijkheden heeft om te veroveren en harmonieën te arrangeren. • Deze ongelijkheid of dit verschil is Rithmomachia’s speciale aantrekkingskracht omdat daardoor een gebalanceerde manier van spelen wordt mogelijk gemaakt tussen ongelijke spelers. • De cirkels bewegen in het tweede veld, vooruit, achteruit, zijwaarts, maar niet diagonaal. • De driehoeken bewegen in het derde veld, alleen diagonaal. • De kwadraten of vierhoekjes bewegen in het vierde veld, in alle richtingen, inclusief diagonaal. • Wanneer er wordt gezet worden beide (getallen, n.v.) van de begin en eindpositie geteld. • De 5 of 6 piramiden bewegen volgens hun individuele onderdelen. *** fig.2 Het bord van Rithmomachia met de beginopstelling van de stukken. D. Het Slaan der Stukken • Stukken kunnen andere veroveren die in de weg staan van hun bewegingszet, maar zij blijven op hun plaats en nemen het veld van het stuk van de tegenspeler niet in. • Door ontmoeting: Wanneer een stuk zo wordt geplaatst dat het op zijn volgende reguliere zet de plaats van het stuk van de opponent kan innemen met hetzelfde nummer, wordt het tegenspelers’ stuk verwijderd. • Door hinderlaag: Wanneer twee of meer stukken van eigen partij in een positie zijn waarin zij met de volgende zet in het veld van een tegenstander zetten, en de optelling/som of het verschil is gelijk aan het nummer van het tegenstanders’ stuk, wordt de tegenstander verwijderd. • Door aanval: Als in zijn gewone richting een stuk een stuk van de tegenstander kan ontmoeten, en het nummer is gelijk aan de vermenigvuldiging of deling van het nummer van de velden van de twee stukken, wordt het tegenstanders’ stuk verwijderd van het bord. De velden van de verovering en dat van het veroverde stuk worden geteld. • Door belegering: Wanneer een tegenspelers stuk wordt omcirkeld door de stukken van de andere partij op zo’n manier dat het noch kan bewegen noch kan worden vrij gezet door een stuk van zijn partij, wordt het belegerde stuk van het bord verwijderd. • De individuele onderdelen van de piramide kunnen beide veroveren en veroverd worden. • •
Wanneer enkelvoudige onderdelen ontbreken, kunnen de piramides worden veroverd door hun totale som, maar zij kunnen andere stukken niet slaan met hun totale som in dit geval. Gedeeltelijke optellingen worden niet toegestaan.
De Overwinning: • Het spel is beëindigd wanneer een speler een harmonie van 3 of 4 stukken heeft opgebouwd in het gebied van de tegenspeler. • Derhalve behoren de stukken te worden gerangschikt in een opklimmende of stijgende rij, in een rechte hoek, of 4 stukken ook in een kwadraat en zij moeten gelijke afstanden hebben. • De veroverde stukken van de tegenspeler mogen ook worden gebruikt om een harmonie te creëren, echter, zij mogen niet het laatste van de harmonie zijn. 6
Drie manieren om een harmonie te creëren met 3 stukken: 1. In een rekenkundige harmonie is het verschil tussen de 2 kleinere nummers gelijk aan het verschil tussen de grotere, d.w.z. 2, 4, 6 = > b-a = c-b. 2. Een geometrische harmonie ontstaat wanneer de verhouding tussen de twee kleinere nummers gelijk is aan de verhouding tussen de 2 grotere d.w.z. 5, 10, 20 = > (a:b) = (b:c). 3. Door de muzikale harmonie is de ratio van het kleinste en het grootste nummer = aan de verhouding tussen het verschil van de 2 kleinere nummers en de 2 grotere nummers, dus 6, 8, 12 => (a:c) = [(b-a) : (c-b)]. Om een snel spel mogelijk te maken behoren de harmonieën berekend te worden; zij kunnen ook worden opgezocht in een tafel van harmonieën. Drie (3) verschillende graden van overwinning kunnen worden behaald door de verschillende harmonieën: • Een kleine overwinning wordt behaald door een rekenkundige, geometrische of muzikale harmonie van 3 stukken. • Een grote overwinning wordt behaald door het bouwen van 2 (maar niet meer dan 2) verschillende harmonieën met 4 stukken. • De grootste overwinning wordt behaald door 4 stukken die alle 3 harmonieën bevatten. De spelers behoren overeenstemming te bereiken betreffende welke overwinning of overwinningen zij willen behalen. Het is mogelijk om zelfs met meer eenvoudige doelen te spelen. Wanneer de spelers een eenvoudiger spel wensen, zou een overwinning mogelijk zijn wanneer een vooraf bepaald aantal van de tegenspelers’ stukken zijn veroverd, of een een bepaalde som van cijfers van de veroverde stukken wordt bereikt of overstegen. De meest essentiële karakteristieken van Rithmomachia zijn hier weergegeven. Aangezien de beknoptheid, ontbreken enige kleinere details hier; echter spelers kunnen zeker zelf vanuit deze omschrijving deze nodige details aanvullen. Een aantal variaties van Rithmomachia zijn hier weergegeven die spelers kunnen proberen. Men moet, ongelukkigerwijze, om Rithmomachia te spelen maar zijn eigen spel maken wanneer men niet geïnteresseerd is om een van de twee computerversies uit Italië of van de VS te gebruiken. In de 16de eeuw was het gemakkelijker omdat het spel in Parijs en Londen kon worden gekocht, zoals Boissire en Fulke @ Lever schreven in hun boeken over Rithmomachia. Vermoedelijk verkocht Jahn in 1929 een spel. In het verleden verschenen er verhandelingen over Rithmomachia wat vaker, en het verscheen vaker in spellenboeken. Daarom is er hoop dat Rithmomachia weer eens meer bekendheid zal krijgen. Deze wens werd geuit in het speudo-ovidian gedicht ‘De vetula’ in de 13de eeuw: “O, als er alleen maar meer mensen van de strijd van de getallen hadden genoten! Als het alleen maar bekend zou zijn, zou het meer door zijn eigen akkoord hoog gerespecteerd worden”. Het is te hopen en zelfs wenselijk dat Rithmomachia weer gespeeld wordt en dat dit een werkelijkheid wordt! Xxxxxxxxxxxxxxxxxxxx
Commentary on Rithmomachia And Dutch Translation, Last visited 04 jan. 2008
Rithmomachia is an ancient board game. It shares information with the recently developed peace game Metapontum (1977). Both games draw from the same historic sources: Pythagoras & Plato; both games relate to chess. Rithmomachia & Metapontum base their game rules & teaching tenets on: 1. Philosophy and ethics; 7
2. Theology; 3. Theory of harmonics; 4: Mathematics; 5; Natural sciences. Plato´s philosophy introduces harmony in games & game playing Plato´s Laws book V, 739 tells of a ´sacred kind of chess´, a game with the best possible rules for a happy life. This inspired the maker of the Metapontum peace game to model its rules after Plato’s ideal State with laws based on friendship, capturing the essence of an ideal community or City State resembling heaven. Plato’s game is not difficult to reconstruct when the rules of chess and the chess mentality are inverted. When we accept his advice as lawgiver for an ideal constitution and attempt to recreate a perfect State to live according to its laws, when this attitude of friendship will be embraced by all, he says, will this State become real and shall its citizens live as sons of god(s) and be in all joyousness as gods themselves, dwelling as close as possible to immortality. The pieces are the people, the board is the world. The pieces that move first in Plato’s game move from the ‘sacred base line’ of the king, queen and nobility. His game represents the very first and very best kind of society, a community without war. This may refer to a future New Eden or as it was before the man’s fall? The one simple rule in Plato’s game is: ‘THE PROPERTY OF FRIENDS IS INDEED COMMON PROPERTY’ The moving of the man (piece) from the sacred base line in Plato’s game, means ‘sharing property, land and power with the men (pieces) from the other side’. In the Metapontum peace game this ideal translates in giving up property, elevating others´ pioneers (pawns) to laureates, giving them the power of a magister [queen]. By Reciprocal power sharing of property and power the whole community benefits and by transferring authority from the highest to the lowest, analogous to elevating a commoner to the rank of spiritual peerage all become happy. Pieces on the sacred line in Metapontum, templars (castles) or magisters (queens) are able to elevate pioneers (pawns) from the other side/colour on their very first move. This is made possible because the squares and ‘square thinking’ are transformed to hexagons and symbiotic thinking, like honeybees in a beehive. This transformation opens up chess’ closed black & white checker board structure. The first act and gesture of goodwill, promoting others pioneers to laureates, strengthens the entire community, making all members from both sides live happily. Possessing property in common signifies this: all what is meant by the word ownership: knowledge; power, honour; land; etc., is shared with the pioneers from the other side. This universal spirit of friendship eliminates poverty. It is (economic) exclusion and lack of means that causes fear, envy and war. Sharing all forges a bond of amity and peace in the entire community. This changes hate to fraternity, division to parity and bondage to freedom while the mentality of greed, selfishness, profit seeking and dominance is transcended by the spirit of goodwill and cooperation. All Metapontum pieces remain therefore alive on the board and this is analogous to ’immortality’. Relevant quotation from Plato’s Laws V,739 translated Translation by E.Taylor. “Our next move in the business of legislation must be -like the moving of a man on the draughtboard from the ‘sacred line’- so singular that it may well surprise you (…).The first-best society, then, that with the best constitution and code of law, is one where the old saying is most universally true of the whole society. I mean the saying that ‘friend’s property is indeed common property´. If there is now on earth, or ever should be such a society, (…) if all means have been taken to eliminate everything we mean by the word ownership from life, (…) if there is anywhere such a city with a number of gods, or sons of gods for its inhabitants, they dwell there thus in all joyousness of life (…) and come as near to the state of immortality under heavens favour”. [Jowett’s translation of Plato’s Laws V,739:’like the withdrawal of the stone from the holy line in the game of draughts (…).”Friends have all things in common”. Whether such a state is governed by Gods or sons of Gods (…), happy are the men who live after this manner. The state which we have now in hand, when created, will be nearest to immortality. Let the citizens at once distribute their lands and houses since a community of goods goes beyond their proposed origin, nurture and education. The earth as he is formed is sacred to the Gods (…) and next priests will offer up prayers]. Comparison between Plato’s game and Metapontum. In this ‘Platonic ‘chess’ and in Metapontum, beating or acts of violence against others of a different colour, even in thinking, do not exist! There are no opponents or antagonists; on the contrary, all participants have become allies, friends, and children of one father God (read the quotations of Plato 8
here under). A genuine bond of filial affection prevails amongst them; a feeling of love lives between friends united in a common cause and purpose: to live under heavens grace. This they express by building structures of social harmony and beauty in their game. From Plato’s Timaues 29. Let me tell you then why the creator made this world of generation. He was good, and the good can never have any jealousy of anything (…)and he desired that all things should be as like himself as they could be. Timaues 32c. The world was created and harmonized by proportion and therefore it has the spirit of friendship. Timaues 37. The soul of man is invisible and partakes of reason and harmony, and being made by the best of the intellectual and everlasting natures, it is the best of things created. Timaues 37 When the father and creator saw the creature which he had made moving and living, the created image of the eternal gods, he rejoiced (…). Timaues 38c-41. The creator proceeded to fashion in the ideal living creature the mind, which perceives ideas of a certain nature and number. God spoke:” gods and children of the gods, who are my works and of whom I am the artificer and father, my creations are indissoluble, if so I will (…) but only an evil being would wish to undo that which is harmonious and happy. Wherefore since ye are but creatures, ye are not altogether immortal and indissoluble, but ye shall certainly not be dissolved, nor be liable to the fate of death, having in my will a greater and mightier bond than those (things i.e. your bodies) to which ye were bound at the time of your birth. Now listen to my instructions:The part worthy of the nature of the name immortal and eternal, that which is called divine, is the guiding principle of those who are willing to follow justice and in you- in that divine part- I, myself will sow the seed, and having made the beginning, I will hand the work over to you. Do ye then interweave the mortal with the immortal, make and beget living creatures, give them food, make them grow, and receive them again in death. When the creator had made all these ordinances he remained in his accustomed nature, and his children heard and were obedient to their father’s word, and receiving from him the immortal principle of a mortal creature, in imitation of their own creator they borrowed portions of fire and earth and water and air from the world which were here after to be restored [at the death of their body]. Timaues 47. Sight in man, God gave as the greatest source of benefit; the sight of day and night (…) and the revolutions of the years have created number to give us a conception of time and the power of inquiring about the nature of the universe. From this source we have derived philosophy (…)which no greater good ever (…) given by the gods to men. (…) All other senses are given that we (…) might imitate the absolutely unerring courses of God (…). Music, adapted to sound and hearing, is granted for the sake of harmony. And harmony has motions akin to the revolutions of our souls,(…) to correct any discord in the soul to be our ally(…).
Plato’s academy, precursor of early Christian societies Plato’s eschatological vision we see 400 years later realised in Christian societies where all possessed things in common. [Acts 2.44 ‘And all who believed were together and had all things in common and they sold all their possessions and goods and distributed them to all as any had need].
All Platonists long for ‘Plato’s game’ to become real. They cherish identical feelings and thoughts, share great reverence for God, the Creator & Father; religion nature; culture; art and science. All this is contained in an analogous heaven, a spiritual, social Utopian society, where they- the players of this game, live in all joyousness as children of (the) god(s). Plato’s game interprets this divine existent harmony and this ideal is also approximated in the rules of Metapontum and made visible and tangible in our game as far as a poor instrument of this type is capable of interpreting such sublime ideals. We discover accordingly relevant Biblical texts of a universe created by God with a humanity that follows the laws of love and goodwill because mankind does not originate out of blind, dark forces. In Plato’s mystical work Timaues we find references to Pythagoras’ theory of a cosmological creation and a sacred mathematics. “The
eternal being of number shapes the all providential source of all the heavens, and the earth and nature in between and is moreover the foundation of the survival of deified man, gods and spirits”. Pythagoras, 6th B.C. PYTHAGOREAN SCIENCE IN 7 STEPS Pythagoras is the first priest-scientist who speaks of the “harmony of the spheres”. He describes the universe as an undivided, living Cosmos whose laws of operation are vested in a holy living science of mathematics; his teachings are summarised in 7 points. 9
1. The fundamental realities of the world and universe are structural and mathematical. 2. These structures constitute what is fundamental and normatively better as a rule; that what displays greater simplicity, regularity and coherence in its mathematical proportions and parameters is aesthetic and therefore more beautiful. 3. Structures in superficially dissimilar context can be basically similar. 4. There is a pervasive affinity or sympathy between the inanimate and the animate, between man’s psyche or soul and the cosmos or universe as a whole. 5. This cosmic sympathy affords the possibility of moral improvement through patterning or modelling of the individual psyche/soul/person on the cosmos or heaven. 6. Beyond moral improvement, this cosmic sympathy affords the prospects of ascent into a trans-human level of existence, of immortality, (becoming a child of God) through a process of purification. Correlatively, it also poses a threat of descent into an infra-human level of existence to even the domain of demons. 7. True knowledge or understanding is inherently mystical and can be attained through disciplined study and purity of life. The study of mathematics is indispensable for intellectual and spiritual progress. Pythagoras’ doctrine of the harmony of the spheres asserts that certain parameters, characterizing the celestial bodies (their number and distance from the earth or periods of revolution) relate to each other “harmoniously” by mathematical rule. Moreover, the revolution of each celestial body produces a distinct tone, inaudible to human ears but influencing man’s soul; while the whole set of tones correspond to the notes in a scale. Author’s commentary: The above is by some philosophers interpreted to mean the eternal Hymn of angels praising God. Extracts from encyclopaedia Americana. This theory was mathematically confirmed by Kepler’s astronomical observations and later verified and reproduced by modern computer analysis at the University in Stamford Con. US around 1980. The universe is harmonic, replete with cosmic or heavenly music. [See the Silent pulse on Kepler]. About pedagogical and philosophical board games Chess is a game of war, a game of booty. It takes its rules from crude historical reality. Life without control of spiritual, moral and ethical principles is a life unguarded, unexamined and unpracticed in virtue; it lacks culture and reverence for a merciful, benign God who gives moral laws for the good of mankind. Therefore this fragile, vulnerable human life without ethics balances precariously between a loose equilibrium of hostile forces, rivals and inimical parties whose sole rule is the ‘right of the strongest, the law of the jungle’. With this kind of ‘morality’, the ‘winner’ takes all, exploiting the weak and trampling the poor. In realpolitik this means confiscation of goods & lands, depopulation, destruction of culture and property. Unfortunately this is still true to day; social chaos, human misery, war and revolutions are still created. Chess and realpolitik have no ear for moral or cultural improvement. Chess as a teaching aid or model of virtue was therefore not appreciated in the early Christian schools of learning; it was regarded as board game barbarism. From the Rithmomachia site: Rithmomachia was the only game that was accepted by the Christian scientific community in the Middle Ages because in contrast to chess and dice it was of great use.
Rithmomachia however, is a hybrid type game; it lies in between war games and peace games. Its aim is to establish order, balance and harmony in other’s territory. Yes, sometimes one beats a piece of the other player. This may be analogous to colonization where natives work for the occupier. But this type of game inclines also to a social structure by introducing culture by means of the science of number given to the pieces. Though players still think in terms of adversaries and battle; but one also yearns for peace. However, is not war waged for the sake peace? Metapontum as a pure and simple peace game is a radical conversion of chess and realpolitik 10
Metapontum creates a totally integrated state of harmony amongst all moving parts (pieces) on the board. Players design symmetrical variations in the begin- and end positions, games that express beauty of composition, order and balance in thinking and doing. In order to achieve this integrated state of harmony they search for the least number or the most efficient moves in each composition. Beauty and efficiency are practically synonymous as shown in the analysis of the games. ¨see game variations’. One could say: ‘Metapontum is the apotheosis of the board games’ because it is totally in tune with harmony and it accords with the Christian ideals of love, grace, respect for life and law; it is also consonant with the ideal of a New Jerusalem, the transcendent spiritual State without violence, with total peace and justice without the death of the pieces.
Parallelism between Rithmomachia en Metapontum From the Rithmomachia site: Rithmomachia is a strategic game for two players with a black and a white party of numbers positioned opposite to each other, comparable to chess. • Metapontum is also a strategic game for two or more players where a white and black party, standing on numbered hexagonal fields are positioned opposite to each other, comparable to chess, but here the superficial comparison ends. • From the Rithmomachia site: The aim in Rithmomachia is not to fight each other with armies of numbers, but rather to participate in a race in which players with some of their pieces create a harmonious order and configuration. There are no written testimonies, but in general is the text of ancient board games very short, like for instance in several works of Plato”. • In Metapontum one does not fight each other either, but rather, players compete to achieve the prearranged end position in harmony in each others territory, doing so in as few moves as possible. The aim in Metapontum is to build a harmony with the pieces from the very first move. The pieces represent the individuals in the community; they act for a common goal: lasting peace for each other. Remarkable is that Rithmomachia’s point of departure from regular games is the creation of harmony in the territory of the opponent. • From the Rithmomachia site: 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 harmonic 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 (Coughtrie 1984)]. • Metapontum shows that Plato and Pythagoras knew ideal games. They created the preconditions for it. Pythagoras as father of mathematics connects the game with number as a divine gift through which all is created and by which man becomes deified when he lives conform the standards of a true philosopher, when he is without violence, respecting his fellow creatures and the whole of creation. Metapontum is played with the intention of building harmony in each others territory. The distance of the most efficient reversal of all pieces is the number of harmony and peace. This number is 1448 and it is found in the foundation stones in the atomic weights and natural constants. (See Theomathesis) •
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From the Rithmomachia site: Contrasts between black and white, even and odd, equality and inequality and are in the end resolved into harmony (Borst 1990).
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In the first move Metapontum players try to build a harmonious relationship with each other. They cooperate so that differences between them are resolved. They take each others handicaps so that the mutual goal of building an ideal society, a collective and individual life of peace and prosperity is realised. The players touch their fellow players through the pieces, advancing and helping them to overcome their difficulties. They exchanging place with them. Head and heart, reason and faith unite in honest thinking without hidden agendas, sanctioned by goodwill. They create a human symbiosis, using this simple game as a tool of cooperation. 11
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From the Rithmomachia site: (Coughtrie 1984). 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).
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The mathematics that issued forth playfully from Metapontum- the Theomathesis- considers the world order from within as well as the concordance between the world order, the exact and natural sciences, and creation in the light of metaphysics and theology.
The 7 ancient liberal arts are: 1. Grammar;2.Rhetoric;Logic; 4. Geometry;5.Arithmatic;6.Music; 7. Astronomy.
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The 7 modern liberal arts are: 1. Theology; 2. Literature;3, Languages; 4. Philosophy;5,.History; 6. Mathematics; 7. Natural sciences.
From the Rithmomachia site: 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 Liege, they worked out a way of realising the game practically (…) (Borst 1987).
• Metapontum also has a musical dimension. The moves of a perfect game of two players playing in harmony to rearrange the pieces in each others territory, are translated to music by a musicologist in the US in 1982. (The music of Metapontum represents the notations of the moves of the game that was translated by prof. G.Lloy of the State University of the Music department of San Diego in Southern California. This music can be listened to on this site’ The arts and sciences in Metapontum are used for the ennoblement and enrichment of human nature, never to destroy ones fellow men ‘for fun’, as is done in the majority of games modern, computer horror simulation games. Rules of Rithmomachia as part of a wider consideration From the Rithmomachia site: 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 Metapontum game board is also rectangular; it has a 8x8 numbered honeycomb structure.
hexagonal
From the Rithmomachia site: In Rithmomachia the begin and end positions are determined: In Metapontum the begin- and end position are also determined. In Metapontum the players rearrange their pieces in a predetermined position in each others territory. In Metapontum we call the double victory an Olympic victory because the players help each other in coming home in each others territory. Preparation Metapontum´s game board has 8x8 hexagonal fields and the board can be enlarged to 13x13 = 169 hexagons. (See the “New Eden” game). AIM OF THE GAME In Metapontum the 2x 16 pieces, [or the 6x 16 pieces in the New Eden game], are regrouped harmoniously on the other side. One enhances the harmony of the pieces, one never blocks each other out of malice, however, this may happen sometimes out of incompetence. In order to prevent this one discusses and monitors each others moves because these have to be mutually advantageous, based on the principle of reciprocity. The good is after all communicative! In Metapontum players become victors by building harmony simultaneously in each 12
others
territory; this demands goodwill so that one can maintain a balance. This requires much practice. •
From the Rithmomachia site: C. The Moves of the Pieces. The players move alternately into an empty space. No piece is allowed to be jumped over.
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In Metapontum jumping is not permitted either. From the Rithmomachia site: 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. In Metapontum white makes the first move, but this privilege is offered by one player to the other who chooses freely. The slight advantage of white is used however on behalf of the black player so that both sides are able to end simultaneously. In Metapontum this difference becomes the stimulance to magnanimity, altruism and mutual assistance in order to maintain the delicate balance for creating harmony.
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From the Rithmomachia site:
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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.
Victory in Metapontum • Metapontum has a major resemblance here which has been mentioned: the total harmony of all pieces. Metapontum has a likeness because all pieces are reorganized in a special way in the last move, the key move of friendship or the peace move, when all pieces have been moved in their proper position. No pieces are conquered or lost, they are promoted! The end.
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Rithmomachia, the Philosophers' Game A Mediaeval Battle of Numbers Peter Mebben
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 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)
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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 fictitious 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.
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 Würzburg around 1030. At a competition between the cathedral schools of Worms and Würzburg, 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 Liège, 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). 15
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 Boissière (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-Lüneburg, 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 Züllchower 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 Liège 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.
3. The Rules of Rithmomachia 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 16
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. ***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)
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. ***fig. 2: Board of Rithmomachia with the arrangement of the pieces at the beginning*** 17
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: •
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 Boissière and Fulke/Lever wrote in their books on Rithmomachia. Presumably Jahn (1929?) offered a set of the game for sale. 18
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. References
Referenties (Ongewijzigd en gekopieerd van Internet) Borst, A. 1986. Das mittelalterliche Zahlenkampfspiel. Supplemente zu den Sitzungsberichten der Heidelberger Akademie der Wissenschaften, Philosophisch-historische Klasse, vol. 5. Heidelberg: Carl Winter Universitätsverlag 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 Mass, Zahl und Gewicht: Mathematik als Schlüssel zu Weltverständnis 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 überliefertes Zahlenspiel: Neu entdeckt für die Schule. Master's thesis, Pädagogische Hochschule Freiburg (available by the author upon request) Richards, J. F. C. 1946. "Boissière'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 Lefèvre 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 Boissière. 1554. Le très 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. 19
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-Lüneburg).1616. Rythmomachia. Ein vortrefflich und uhraltes Spiel desz Pythagorae. In Das Schach= oder König=Spiel. 443-495. Leipzig: Henning Gross jun. Reprint 1978. Zürich: Olms
Appendix II Texts of modern era with a description or rules of Rithmomachia until 1940 - The special German tradition Johannes Allgaier. 1796. Das pythagorische oder arithmetische Schachspiel. In Neue theoretisch-praktische Anweisung zum Schachspiel. Vol. 2, p. 73-97. Wien: Franz Joseph Rötzel. Johann Friedrich Wilhelm Koch. 1803. Die Rythmomachie. In Die Schachspielkunst nach den Regeln und Musterspielen der grössten 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 Volständiger Codex der Damenbrett-Spielkunst. p. 365-404. Köln, 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 Königlichen und Städtischen Gymnasiums in Sorau. Sorau. Fritz Jahn. 1917. Rythmomachia. In Alte deutsche Spiele. p.1-4, 15. Berlin. [Fritz] Jahn. 1929(?). Zahlenschach für Mathematiker. In Verzeichnis Weihnachtskrippen und Spiele der Züllchower Anstalten 1929/30. Züllchow. Joseph Strutt. 1801. The Sports and Pastimes of the People of England. p. 313-316. London.
Appendix III: The Rithmomachia site: 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).
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