Francisco Cervantes de Salazar (ver M. C. Díaz y Díaz; Aires A. Nascimento; J. M. Díaz Bustamante; M. I. Rebelo Gonçalves; J. E. Lopez Pereira; A. Espírito Santo, Hislampa. Autores latinos peninsulares da época dos Descobrimentos (1350-1560), Lx: INCM, 1993: pp. 271-272) escreveu três obras sobre jogos, suspeito que são todos jogos de bola mas... (as traduções dos títulos que dou podem ser bastante disparatadas porque não faço ideia da descrição do jogo:

O primeiro tem por título Obeliscorum sive lignearum pyramidularum ludus (jogos dos obeliscos ou pirâmides de madeira)

O segundo: Pilae palmariae ludus (joga da bola de palmeira (coco?)

O 3º Saltus, Ludus sphaerae per anulum ferreum (Dança, jogo de meter uma esfera no anel de ferro)

segundo inventário acima citado só se encontram na British Library (cota: 10480.ee.3)

## Sunday, January 25, 2009

## Wednesday, January 14, 2009

### http://chessderivatives.blogspot.com/

RYTHMOMACHIA

The mayor chess variant

through the years 500 to 1,800

During the investigation of the Subject Rythmomachia,

the editor has substantially added new material during the year 2004 to

text as below, and specifically the two first paragraphs

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

Douay, Ioan. Bogarde, 1615. Contemporary 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.

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 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 ( 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

Firs 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.

Geschichte & Literatur des Schachspiels, (prof.) Antonius van der Linde (curator & chess literature collector from NL / DE), 1874 original, reprint in German 1981.

Orthogonal - use Rook - name:

(rook = pyramid)

Array notation on 8*16 board

Based on the *old system* of moves of the chess game.

Translator Barozzi states that chess has been developed from

Rythmomachy, since it is older. However the original Greek

transcripts have never been found.

METROMACHIA ; Metromaxia ; Sive Ludus Geometricus.

Board with 53*33 squares, the figurines are geomatrical.

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 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 1,000 - 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.

Array::

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::

Evens

Odds

1st Rounds (x)

2

4

6

8

3

5

7

9

2nd Rounds (x2)

4

16

36

64

9

25

49

81

1st Triangles (x + x2)

6

20

42

72

12

30

56

90

2nd Triangles ((x + 1)2)

9

25

49

81

16

36

64

100

1st Squares (x + x2 + (x + 1)2)

15

45

91

153

28

66

120

190

2nd Squares ((2x + 1)2)

25

81

169

289

49

121

225

361

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.

Play::

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.

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

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.

***********

The Philosopher's Game

This file is a 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.

Annotation will occur occasionally throughout; they will appear in square brackets and italics, [like this]. Spelling will be erratic; I'm transcribing quickly, so I will often be modernizing the spelling, but will leave original spelling whenever I consider there to be doubt about the meaning. I also will sometimes modernize the punctuation and paragraph breaks in the interests of readability. This is not intended to serve as a definitive critical edition, merely a working copy, good enough to understand the game.

[Title Page]

THE MOST NOBLE

ancient, and learned playe, called the Phi-

losophers game, invented for the honest re-

creation of students, and other [sober?] persons, in

passing the tediousness of time, to the release of

their labours, and the exercise of

their wittes.

Set forth with such playne precepts, rules and ta-

bles, that all men with ease may understande

it, and most men with pleasure practice it.

by Rafe Lever and augmen-

ted by W. F.

[Picture, probably stock, of two men playing at a game on a square 10x10 board.]

Printed at London by James Rowbothum, and are

to be sold at his shop under Bowchurch

in chepe syde.

[Page]

The Lord Robert Duddedlye.

[Picture of Lord Robert, with the motto "Vulnere virescit virtus" alongside.]

The Physiognomie here figured, appeares by Paynters Arte:

But valyant are the vertues that, possesse the inward parte.

Whych in no wise may paynted be, yet playnely so appeare,

& shine abrod in every place with beames most bright & clear.

[Page]

[The Epistle Dedicatory]

TO THE RYGHT HO-

norable, the Lord Robert Dudley, Mai

ster of the Queenes Maiesties horse,

Knight of the most honorable order

of the Garter, and one of the Queenes

maiesties privie Counsell, JAMES

ROUBOTHUM heartelye wisheth long life, with

encrease of godly ho-

nour and eternall

felicitie.

Sith that your honour is full bent,

(right honorable lord)

To wisdom & to godlines

with true faithful accord.

Sith that in deed you do delyte,

in learning and in skyll:

The show wherof doth well expresse

a perfect godly wyll.

Sith that also you have in hand,

affayres of force and waight:

And study do both day and night,

to set all thinges full straight.

[Page]

I thought therfore your honour should

not lacke some godly game:

Whereby you might at vacant times

your self to pastyme frame.

Whereby I say you might release,

such travailes from your mynde:

And in the meane while honest mirth

and prudent pastyme fynde.

Remembring then this auncient play,

where wisdome doth abound:

Called the Philosophers game,

me thinkth I have one found.

Which may your honour recreate,

to read and exercise:

And which to you I here submit,

in rude and homly wise.

Pithagoras did first invent,

this play as it is thought:

And therby after studies great,

his receation sought.

[Page]

Yea therby he would well refreshe,

his studious wery braine:

And still in knowledge further wade

and plye it to his gaine.

Accompting that a wicked play,

wherin a man leudely:

Mispendes his tyme & wit also,

and no good getts thereby.

But grevously offendes the Lord,

and so in steed of rest:

With trouble and vexation great,

on every side is prest.

Most games and playes abused are,

and fewe do now remaine:

In good and godly order as,

they ought to be certaine.

For why? all games should recreat,

the hevy mynde of man:

And eke the body overlayde:

with cares and troubles than.

[Page]

But now in stead of pleasant mirth,

great passions do arise:

In stead of recreation now,

revengings we practise.

In stead of love and amitie,

long discords do appeare:

In stead of trueth and quietnes,

great othes and lyes we heare.

In stead frendship, falshode now,

mixed with cruell hate:

We finde to be in playes & games,

which dayly cause debate.

Pithagoras therfor I saye,

to make redresse herein:

Invented first this godly game,

therby to flye from sinne.

Since which time it continued hath,

in Frenche & Latin eke:

Still exercisde with learned men,

their comforts so to seeke.

[Page]

Wherby without a further prose,

all men may be right sure:

That this game unto gravitie,

and wisdome doth allure.

Els would not that Philosopher,

Pithagoras so wyse:

Have laboured with diligence,

this pastime to devyse.

Els would not so well learned men,

have amplified the same:

From tyme to tyme with travell great,

to bring it into fame.

But let us nerer now proceed,

and come we to theffect:

And then shall we assuredly,

this pastime not neglect.

For it with pleasure doth asswage,

the heavy troubled hart:

And with lyke comforts drives away,

all kynde of sourging smart.

[Page]

The mynde it maketh circumspect,

and heedfull for to bee;

The tyme that theron is bestowd,

is not in vaine trulye.

The body it doth styrre and move,

to lightsomnes and ioye:

The sences and the powers all,

it no wyse doth annoye.

It practiseth Arithmeticke,

and use of number showth:

As he that is conning therein,

assuredly well knowth.

In Geometie it truly wades,

and therein hath to do:

A learned play it is doutlesse,

none can say nay thereto.

Proportion also musicall,

it ioynes with thother twayne:

So that therin three noble artes,

are exercisde certayne.

[Page]

What game therfore lyke unto this,

may gotten be or had?

There is not one that I do know,

the rest are all to bad.

It causeth no contention this,

nor no debate at all,

By this no hatred wrath nor guyle,

in any wise doth fall.

It stirreth not such troubles that,

our frend becomes our foe:

It moveth not to mischiefe this,

as many others do.

Let us avoyde the worst therfore,

and cleve we to the best.

So shall we shunne all wickednes,

and purchase quiet rest.

So shall we serve the living Lorde,

and walke after his will:

So shall we do the thing is good,

and flye that which is yll.

[Page]

So shall we live right christianlyke,

and do our duties well:

So shall we please both god & prince,

none shall us need compell.

And then the Lord of his mercie,

will prosper us alwayes:

And graunt us here to have on earth,

full many godly dayes.

Yea then the Lord of his goodnes,

and grace celestiall:

Will guyde and governe our affaires,

and blesse our doings all.

Which Lord graunt to your honour here,

good dayes & long to have:

with much encrease of helth & welth

and from all hurt you save.

Your honours most humble,

James Roubothum.

[Page]

To the Reader.

I Dout not but some man of severe judgement so soone as he hath one read the title of this boke wyl immediately sai, that I had more need to exhort men to worke, then to teach them to play, which censure if it procede not of such a froward morositie that can be content with nothing but that he doth himself, I do not only well admit, but also willingly submit my self therto. And if I could be persuaded that men at mine exhortation wold be more diligent to labour, I would not only write a treatise twise as long as this, but also thynke my whole time wel bestowed, if I

[Page]

did nothing els, but invent, speake, and write that which might exhort, move, & persuade them to the furtherance of the same. But if after honest labour and travell recreation be requisit, (and that nede no further probation because we favour the cause wel inough) I had rather teach men so to play, as both honestye may be reserved, their wittes exercised, they them selves refreshed, and some profit also attayned, then for lacke of exercise to see them either passe the tyme in idlenes, or els to have pleasure in thyngs fruitles and uncomely. And if great Emperours and mighty Monarches of the world have not bene ashamed by writing bookes to teaches the art of Dyce

[Page]

playing, of all good men abhorred, and by all good lawes condemned: have I not some colour of defence, to teache the game, which so wyse men have invented, so learned men frequented, and no good man hath ever condemned.

The invention is ascribed to Pythagoras, it beareth the name of Philosophers, prudent men do practise it & godly men do praise it. But because many herein (as in a play) have challenged much authoritie, they have filled this game with much diversitie. In which as I could perceive the most differens of playing to consist in thre kindes, so have I playnly and briefly set them forth in Englishe not as though there might not more diversities be espied, but

[Page]

that I thought these to them whom I have written to be sufficient. yet for that I woulde be lothe, from playe & game, to fall to earnest contention, if any man in this doing or any part thereof shall think I have done amisse, and will do better himself, so far am I from envying his good proceding, that I wil be right glad, and geve him heartye thankes therefore.

All things belonging to this game

for reason you may bye:

At the bookeshop under Bochurch

in Chepesyde redilye.

[Page]

The bookes verdicte.

Wanting I have bene long truly,

In english language many a day:

Lo yet at last now here am I,

Your labours great for to delay,

And pleasant pastime you to showe,

Mynding your wits to move I trowe.

For though to mirth I do provoke,

Unto Wisdome yet move I more:

Laying on them a pleasant yoke,

Wisdom I meane, which is the dore,

Of all good things and commendable:

Dout this I thinke no man is able:

CATO

Interpone tuis interdum gaudia curis:

Yt possis animo quemuis sufferre laborem.

[Page]

The diffinition

That moste auncient and learned playe, called the Philosophers game, beinge in Greeke termed [...], is as much to saye in Englishe, as the battell of numbers. Numbers be either even or odde, wherefore the even parte is against the odde, either parte havinge a kyng, whych being taken of the adversaryes part, and a triumphe celebrated within his campe, the game is ended.

Of diverse kyndes of playinge.

Amonge the dyverse kyndes of playing thys game, we shall sette forth three sortes, of which the reader maye chose whether of them he lyketh beste. And of all those three, we shall

[Page]

gyve suche shorte and easye rules, that no man (althoughe he were altogether ignoraunt in Arithmetike) shall fynde the game so hard, but that he may learne to playe it.

Of the partes of thys Game.

He that wyll learne thys game, any of the three waies, muste first be enstructed of these sixe partes.

The table as the fielde

.2. the menne and the numbers of them as the hoste

.3. the placynge of them, as the encampynge

.4. the order of playe and removynge the men, as the marchynge and fyghtynge

.5. the manner and lawes of conqueryng and taking

.6. and last of al the triumphe after the victorye.

Of these partes in the fyrst kynd of playng.

[Page]

The table muste be a playne borde conteynynge .128. squares that is .8. in breadth and .16. in length sette forthe in two dyverse collours. Or for a plainer understandynge, the table is a doble chesse bord, as it were two chessebordes joyned together, the length of twoo, the breadth of one, whereof thys is an example.

[Page]

[Complex picture, showing a 16x8 checkerboard, with various letter and symbols on various parts. This can't be fully represented in text, so I won't try to show the whole thing now. The middle section -- rows 6 through 12 -- have letters on them, and look somewhat like this:]

K L M

X F Z

B S C

R H Y A T I N

E V D W

G

Q P O

[Later sections will refer back to this table frequently, to describe movement.]

[Page]

Of the men.

The men be in number .48. Wherof .24. be of one side & must be knowen by one colour, and .24. on the other syde, whyche also must be marked with a contrarye colour, as White and Blacke, Blew and Redde, or what colours els you lyke best. But in the colering there .3. thinges must be observed, the bottome or lower part of every man (excepte the two kinges) muste by marked wyth his adversaries colour, that when he is taken, he maye chaunge his coate and serve him unto whome he is prisoner.

The seconde thinge considered in the men, is their fashion: for of euther syde .8. are rounds, other .8. are triangles & .7. (the king making .8.) are squares. There fashion is such [small inset showing roundes, triangles, squares].

The kynges because they consist of all three sortes, as it is knowen by the learned speculation of the numbers, beare

[Page]

the fashion of all thre kindes, his foundations are two squares, on which are sette, two triangles & upon them rounds. But this difference is betwene the kinges, the king of the even numbers, hath a pointed toppe, the king of the odde numbers is not pointed, the cause dependeth upon the consideration of these numbers by which they arise into piramidall fashion. The third thing considered in the men, is the number that must be written or graven upon them which to learne plainely for practise marke these short rules.

There be of eche kynde of men, two rankes or orders.

The first ranke or order of roundes be the digites even or odde namely of the even .2.4.6.8. of the odde .3.5.7.9.

The second order of rounds are found by multiplyinge these digites by themselves as .2. times .2. is .4.3. times .3. is .9. Of the even they be .4.16.36.64. of the odde they be .9.25.49.81.

The first order of the triangles are found by addinge two of the roundes together

[Page]

one of the firste order and another of the seconde order, as .2. and .4. make sixe .3. and .9. make twelve, on the even syde they are these .6.20.42.72. on the odde syde .12.30.56.90.

The second order of triangles be made by addynge one to every one of the first order of roundes, and then multiplying that number in hym selfe: as .2. is one of the firste order of roundes, thereto adde one, that is .3. then .3. tymes .3. is .9. a triangle of the seconde order, on the even syde. Likewise to thre a round on the odde side, adde .1. so it is .4. then .4. tymes .4. is .16. On the even parte, they be .9.25.49.81. on the odde parte .16.36.64.100.

The first order of squares (in whyche are contayned the kynges) be made by addynge two triangles together, one of the fyrste order, and another of the secondes, as .6. and .9. make .15. likewyse .12. and .16. make .28. Amonge the even they be .15.45. and .91. the kynge .153. amonge the odde they be .28.66.120. and .190. the kynge.

[Page]

The last order of squares be found, by dobling of every one of the firste order of roundes, and after adding one, last of all be multiplying that number in itself, as twise .2. is .4. and .1. added is .5. so .5. times .5. is .25. likewyse twyse .3. is .6.1. added is .7. then .7. tymes .7. is .49. These be on the even syde .25.81.169.289. And of the odde syde .49.121.225.361.

These numbers must be sette uppon the men both on the upper side, & also on the nether side. Except one of the kynges, which must with the whole number of their pyramid, be marked, onely on the bottome. Because the sydes muste have other numbers, namely the highest point of the even kyng, must have .1. the rounde next under him marke with .4. the uppermost triangle with .9. the nethermost with .16. The uppermost square muste have .25. The nethermost square shall have .36. The king of the odde upon his head, whiche is a rounde, not pointed hath .16. upon his first triangle .25. on the second triangle .36. uppon the fyrste square .49. upon the lowest square .64.

[The numbers in the next paragraph are in circles and triangles, to illustrate.]

Finally it shalbe good for the avoydance of confusion, to drawe a line under every number. Ells may you take one for another, as 6 the even round & 9 the odde rounde, may be taken one for another with oute this lyne or some suche marke, lykewise 6 and 9 Tryangles bothe of one syde. And this sufficient for the men, the fashion, colours and numbers.

The reason of these numbers and the knowledge of their proportione.

For them that seke the speculation of these numbers, rather then the practise for playing, and have some sight in the sciens of Arithmetike, some thyng must be sayde of proportion. For this purpose there be three kyndes of proportion. Multiplex, superparticuler, and superpartiens.

[Page]

Of multiplex.

MULTIPLEX proportion, is when a great number conteyneth a lesse number manye tymes, and leaveth nothing, as .8. conteyneth .2. fower tymes and nothing remaineth .16. conteyneth .4. &, this proportion semeth best to agree with roundes because the one number conteyneth the other and nothynge remaineeth as the fyrste order of roundes be.

[The following table shows the white (even) and black (odd) rounds.]

2 4 6 8

3 5 7 9

[Page]

The second order be these.

[This table shows the first and second orders of rounds for both sides.]

double. quadruple. sextuple. occuple.

2 4 6 8

4 16 36 64 proportion.

triple. quintuple. septupl. nonuple

3 5 7 9

9 25 49 81

[Page]

Of superparticuler proportion.

Superparticuler proportion is when a greater number contayneth a lesser with one part of it, which may measure the whole, as .12. contayneth .9. and .3. whiche is a thyrde parte of nine .6. contayneth .4. and .2. that is one halfe to .4. Thys proportion beinge the cheife, next unto multiplex, is beste figured by a trianguler forme, whyche hathe fewest lynes and angles next unto a circle. For the manner of thys proportion consider thys figure.

[Page]

[This table shows each base number above two rows of triangles, and appears to be badly fouled up. I believe the rows of base numbers are wrong -- the first set should be the evens (2, 4, 6, 8), and the second the odds (3, 5, 7, 9). That would make each Latin header match the number below it, the first row of triangles would be the superparticulates for that base number, and the second row would be the second order of triangles for that base number, which is (x+1)2.]

sesquialter. sesquiquart sesqui.sext sesqu.oct.

3 5 7 9

6 20 42 72

9 25 49 91

sesquiter. sesquiquint. sesquisept. sesquinona.

4 6 8 10

12 30 56 90

16 36 64 100

[Page]

Superpartiens proportion.

The superpartiens proportion is when the greater number conteyneth the lesser and mo partes of it then one as .15. conteyneth .9. and .6. whiche is two thirdes of .9. lyke wyse .28. conteyneth .16. and .12. that is 3/4 of .16. This proportion conteineth divers parts beside the whole number therfore is wel figured in the square, which also conteyneth more corners and sides. For the maner of their proportion consyder thys table.

[Page]

The first order of squares.

[The following table contains the two rows of triangles, with the corresponding square beneath.]

6 20 42 72 supparticulares added

9 25 49 91

15 45 91 153 being the squares.

12 30 56 90

16 36 64 100

28 66 120 190

[Page]

The second order followeth.

third fyft seventh ninth

5. 9. 13. 17. These two rows are just plain numbers.

10. 36. 78. 136.

15 43 91 153 These two rows are the two white orders of squares.

25 81 169 289

superbipartiens tertias supquadrupartiens quintas supsextupartiens septimas supoctupartiens nonas

[Page]

fourth sixth eight tenth

7. 11. 15. 19. These two rows are just plain numbers.

21. 55. 105. 171.

28 66 120 190 These two rows are the two black orders of squares.

49 121 225 361

supertripartiens quartas supquinpartiens sextas supseptupartiens octavas supnonpartiens decimas

[Page]

Of the kings.

[Image of the two kings. Each is a pyramid of numbered pieces, with two squares, two triangles, and a round; one has a little triangular "cap" on top. The images make it look like higher-numbered pieces are actually larger -- I wonder if this is true...]

The kinges conteine in them suche numbers, as beyng all added together, make the whole piramidall number, the lowest square of the even is .36. which riseth of the multiplying of .6. in it selfe. The next square that must be lesse, is .25. arisinge by the multiplyinge of fyve in it self and so followeth .16. of .4. then .9. of .3. laste .4. of .2. and single .1. all these added together make up .91. After the same maner consisteth the king of adde. The lowest square is .64. arisinge of .8. multiplied in himselfe. The next .49. of

[Page]

.7. times .7. then .36. of .6., .25. of .5. and .16. of .4. these numbers make the whole pyramidall number .190. which because it riseth not to the poynct of one, oughte not to be sharpe poyncted, as hathe beene sayde before.

Of the placing, encamping or setting in araie.

To retorne againe to the plaine and easye playing of this game, next to the armie & their armour, follow ether the order of their battel or encamping. Which because it is more playne and easely seen which the eye, then learned by the eare, I referre thee unto the table where the battell is appoynted in suche order as thys kynde of playe requireth.

[Page]

[The table below shows the layout of the board. To illustrate this through text, I put squares into square braces, triangles into angle brackets, and rounds into parentheses. Kings are signified with an asterisk. Note that the top half, the evens, should be upside-down.]

[25] [81] [169] [289]

[15] [45] <25> <20> <42> <49> [91]* [153]

<9> <6> (4) (16) (36) (64) <72> <81>

(2) (4) (6) (8)

(9) (7) (5) (3)

<100> <90> (81) (49) (25) (9) <12> <16> [Error in original: 2 instead of 12]

[190]* [120] <64> <56> <36> <30> [66] [28]

[361] [225] [121] [49]

[Page]

Of the marchinge or removing of the men.

The battell beyng duely placed, it followeth next, to know the maner of marching & removing, for every kynd of men, hath their proper kynde of motion, and fyrste we muste speake of the roundes.

The motyon of the roundes.

The roundes muste move into the space that is next unto them cornerwyse, as in the table, from the space .A. to any of these .B.C.D. or .E.

[Yes, this is clearly saying that rounds only move diagonally.]

Of the triangles.

The triangles passe three spaces counting that in which they stande for one, and that into whych they do remove for another, that is leaping over

[Page]

one space. As from the space .A. he maye remove into any of these space .F.G.H. or .I. this is the motion of the triangle in marchying or takyng. But in flying he maye remove the knyghtes draught of the chesse, as from .A. into .X. or .W. &c.

Of the Squares.

The Squares remove into the fourth place from them, that is leaping over two, right forwarde or sydelong, as from the place of .A. to any of these spaces .L.N.P.R. flyinge they maye remove after the knyghts draught, but that they must passe foure spaces, as from .P. to .Y. or .T. &c. And this for the marchinge and removyng of the men, where note, that with theyr flying draughte they can take no man, but if needed by helpe to besiege a man.

Of the kynge marching.

The kings because thei beare the forme of al the thre kynds, may remove any

[Page]

of all theyr draughts when they list, into the nexte with the rounde, into the thyrde with the triangle, and into the fourth with the square, and finally in all poyntes lyke the Queene at the Chesse, saving that he can not passe above foure spaces at the most.

Of the maner of taking.

The men may be taken sixe wayes, namely by Equalitie, Oblivion, Addition, Subtraction, Multiplication and Division, and also if you wyll, and so agree by

/ Arithmeticall.

Proportion <> <42> <20> <6{r}>

<81> <56> <30> <12>

[Page]

The seconde order of triangles, have all excepts one (whiche is the number of .100.) their cossicall signes, as .9. bothe of the roote and of the quadrate, .25.36. and .49. have the signe of the quadrate .64. of the quadrate and the cube, and also the quadrat of the cube .16. and .81. of the quadrate, and the foure squared quadrate.

<81{24}> <49{2}> <25{2}> <9{r2}>

<100> <64{236}> <36{2}> <16{24}>

In the first order of squares, onely .15. is marked with the roote, all the rest doe want theyr cossicall sygnes in thys game.

[Page]

[A rather odd little illustration here, showing the two kings in some detail, with each constituent pieces marked both with its number and the square root of that number, and the complete number of the king on the top. Also, a couple of odd little images of a round and a square with a peculiar symbol on top; this symbol may be peculiar to king components. Note also that, in the table of squares below, the 91 and 190 have a picture of the king on them.]

[153] [91] [45] [15{r}]

[190] [120] [66] [28]

The seconde order of squares hath .3. numbers marked with cossicall signes, that is .25. and .225. wyth the signe of the quadrate .81. is marked with the sygne of the quadrate and the fouresquared quadrate.

[Page]

[289] [169] [81{24}] [25{2}]

[361] [225{2}] [121] [49]

And thys have you all the men that be marked with cossicall sygnes.

The setting in aray.

The teachers of this kynde of playing, doe not so well allowe, the former kynde of placing or any other, as the naturall placing of every man under him of whome he aryseth. So thei conteyne .6. ranks in length, extending to the furthermoste edge of the Table after this sorte.

[Page]

[Note that, in this table, the odds are colored black, while the evens are left white.]

[361] [225] [121] [49]

King [120] [66] [28]

<100> <64> <36> <16>

<90> <56> <30> <12>

(81) (49) (25) (9)

(9) (7) (5) (3)

(2) (4) (6) (8)

(4) (16) (36) (64)

<6> <20> <42> <72>

<9> <25> <49> <81>

[15] [45] King [153]

[25] [81] [169] [289]

[Page]

The marching or moving.

The men maye remove every way, into voyde places, forwarde, backewarde, towarde both sydes, direct or cornerwyse. So that the rounde men remove into the next space, the triangles into the third place, and the squares into the fourth place, accompting that place in which they stande for one.

Also every man savyng the two kynges to besiege his enemie, or to flye from the siege himself, may remove the knights draught in chesse, but neither take anye man (except it be by siege) nor erect a triumphe by suche motions. The kynges move even as squares, but that they have not the flyinge draughte.

It is compted lawefull amonge suche as wyll to agree, that the Triangles and Squares, maye remove into voyde places, thoughe the spaces betwene be occupyed of other men.

[Page]

The maner of taking.

The men may be taken seven ways by Oblivion, by Equalitie, by Addition, by Subtraction, by Multiplication, by Division, and by Cossicall Sygnes.

Of takynge by Oblivion.

All men maye be taken by Oblivion when by foure men they be letted of theyr ordinarie draughte, as hath bene taught before.

Of takynge by Equalitie.

By Equalitie maye these men take or be taken, as hathe bene sayde before, .9.16.25.36.49.64.81., as yf after you have played your .9. stande in

[Page]

your mans draught, you may take him by not removing into his place, unlesse you espye him standing in your draught before you playe, then muste you take him up and remove into his place.

Of takynge by Addition.

The takyng by Addition is all one with the first kynde of play, in all respectes, saving that some require the men that shoulde take by Addition to stande in the next spaces to him that is taken, either directly, or cornerwyse, but the former waye is better.

Of taking by Subtraction.

That whiche was sayde in the first kinde of subtraction and that whiche was last sayde of Addition may be bothe referred together. For this subtraction

[Page]

differeth not from the former, but for the opinion of them, that would have the two takers stande onelye in the nexte spaces to him that is taken.

Of takyng by Multiplication.

Takyng by multiplication doth differ. For in this kynde of playng, it is thus. When your man standeth so, that beyng lesser than your adversaries man, you may multiplie your man by the voyde spaces betwene them, and the product is all one with the adversarye, you maye take hym upm not removynge into his place, except you espye hym so, before you remove your man.

Of takynge by Division.

[Page]

Lykewise by Division, yf your man beyng greater then the adversarye, stande so, that beyng devyded by the voyde spaces, the quotient is all one with the adversarye, you maye take hym up, not removyng into hys place, unlesse you see hym so standynge before you drawe.

Of taking by Cossicall signes.

By Cossicall sygnes anye man that hath these signes, {2}.{3}.{4}.{6}. meeting with his roote in his ordinary draught that hath this signe {r} taketh him up, or elles is taken of him, without removing into his place, except he maye take him before he remove.

Of the kynges, and their taking.

[Page]

The king of the even must be foursquare, havyng sixe steppes, every one lesser then other, on one syde he muste have on him these rootes .1.2.3.4.5.6. on the other syde the quadrates arising of these rots, that is .1.4.9.16.25.36.

The king of the odde men, muste have but fyve steppes, that is .4.5.6.7.8. lackyng the rootes that he can not ende in .1. The quadrates of hys rootes by these .16.25.36.49.64. These muste be so set on, that the least must be hyghest and the greatest lowest.

The kinges be taken by Oblivion, or yf theyr Pyramidall number, be taken by anye of the aforesayde meanes. Also yf by suche meanes you can take all his quadrates one after another.

The privilege of the king.

[Page]

If anye of the kynges quadrates be taken, he maye redeme it by anye of his men having the same number, and muste remove into his place, whiche redemed hym. But yf he have none of the same number, he maye redeme hym for anye man of hys, that his adversarye wyll chuse, and lykewyse remove into his place by whome he is redemed.

[Page]

A table to take the men by Multiplication and Division.

Even against odd

spaces

6 2 12

8 2 16

15 2 30

45 2 90

4 3 12

4 4 16

9 4 36

16 4 64

6 5 30

20 5 100

2 6 12

15 6 90

20 6 120

4 7 28

8 7 56

2 8 16

8 8 64

4 9 36

9 9 81

25 9 225

9 10 90

21.

odd against even

spaces

3 2 6

36 2 72

3 3 9

5 3 15

12 3 36

5 4 20

9 4 36

16 4 64

3 5 15

5 5 25

9 5 45

12 6 72

7 7 49

5 9 45

9 9 81

3 12 36

3 14 42

17.

[Page]

For Division.

even against odd

spaces

6 2 3

72 2 36

15 3 5

36 3 12

9 3 3

20 4 5

36 4 9

64 4 16

15 5 3

25 5 5

45 5 9

42 6 7

72 6 12

49 7 7

72 7 9

45 9 5

81 9 9

36 12 3

91 13 7

42 14 3

20.

odd against even

spaces

12 2 6

16 2 8

30 2 15

90 2 45

12 3 4

16 4 4

36 4 9

64 4 16

100 4 25

22 5 45 [sic -- 22 should be 225]

30 5 6

100 5 20

12 6 2

36 6 6

90 6 15

120 6 20

28 7 4

56 7 8

16 8 2

64 8 8

120 8 15

3 9 4 [sic -- 3 should be 36]

81 9 9

225 9 25

90 10 9

66 11 6

28 14 2

27.

To take by cossicall signes

2 16{4}

2 64{6}

3 81{4}

3 9{2}

4 16{2}

4 64{3}

5 25{2}

6 36{2}

7 49{2}

8 64{2}

9 81{2}

15 225{2}

[Page]

Of the triumph.

The triumph is after the Kynge be cleane taken away, to be create in the adversaries campe, as well of your owne men as of your adversaries men that be taken, or of both in proportion as hath bene shewed before, and proclaimed that those men ons placed, may not be taken, as it was declared sufficiently, and no difference betwene the triumphes, savyng that some wyll not alowe a triumphe but of foure numbers, and two proportions at the lest. All three for the greater victorie, makynge but two kyndes of triumphes.

Here foloweth the thyrd kynde of playing at the Philosophers game.

There must also in this thyrd kynde be considered the table, the men, their markyng, the order of theyr battell, the motions, their taking, and last of all theyr triumphing.

The table is the same that hath bene twyse already discribed. Yet some wyll not have it so longe, but at the lest is must conteyne .10. squares in length and alwayes .8. in breadth. The longest is best.

Of the men.

The men be .48. as it hath bene told of two contrary collor, the head and bottom all of one collor, because men ons taken be no more occupyed in thys kynde of playing.

The inscription and fashion.

[Page]

The fasion is as hath bene last declared both of the men, and of the kynges, the inscription of numbers the same, but without cossical signes.

Of the order of the battell.

The order of battell is after the firste maner, but not so farre from the bordes end, namely the .4. squares standynge in the plattes nearest to the bordes end the rest accordingly joined to them, as in the first kynde of playing.

[Page]

[Note that, in this diagram, the odds are colored black. Also, it actually shows the odds and evens on reverse ends from this textual representation; I'm not redoing the whole thing right now...]

[25] [81] [169] [289]

[15] [45] <25> <20> <42> <49> [91]* [153]

<9> <6> (4) (16) (36) (64) <72> <81>

(2) (4) (6) (8)

(9) (7) (5) (3)

<100> <90> (81) (49) (25) (9) <12> <16>

[190]* [120] <64> <56> <36> <30> [66] [28]

[361] [225] [121] [49]

[Page]

Of their motions.

The men move frowarde and backward, to the right hand, and to the left hande, but not cornerwise, except the gamesters so agree, the rounds into the next space, the triangles into the thyrde, and the squares into the fourth, the kyngs move as squares. And these be their ordinary draughts in marching.

Of their taking.

They are taken by encountering, bu eruption, by laying wayght, and by Oblivion.

Of takyng by encountering.

To take by encountering is to take by Equalitie, as hath bene twyse before declared.

Of taking by eruption.

To take by eruption is when a lesse number beyng multiplied by the spaces that are betwene him & hys adversary, the product is asmuch as his adversary, he may take his enemie awaye whether he stand directly from him or cornerwise.

[Page]

For men that may be taken by eruption looke in the table of takyng by multiplication in the second kynd of playing.

Of takyng by deceypt or lying weyght.

To take by deceypt or lying weight, is to take by addition, not as before when the adversary standeth within the draught of two men which being added make the juste number of the adversary, but when the .2. numbers that are to be added, stande in the next spaces to the adversarie. For to take by deceipt, looke in the table that was set forth for takyng by addition in the first kynde of playinge.

Of taking by Oblivion.

By Oblivion all men may be taken, when foure men besiege the adversarye, standynge in the foure nexte

[Page]

spaces about him directly, or cornerwise, the man so besieged can not escape, because he can not remove cornerwyse, therefore maye be taken up, so soone as the last of the foure is set in his place.

In all three kyndes of playing no Oblivion can be of any man with some of his fellowes, but all foure muste be hys adversaries.

In this thyrde kynde, these men can be none otherwyse taken but by Oblivion. Namely amonge the even .2.4.4.135. among the odde .3.5.7.190.

In all maner of taking this is to be noted, that we muste not place the man which taketh in place of him that is taken, but when he maye be taken before we drawe, then shall we remove our man into his place.

The privilege of the king.

The king standeth for so many men as he hath steppes, that is the even for .6. the odde for .5. if anye of these

[Page]

(except the lowest and greatest) be taken the king may redeme hym, by any man of his that is of the same number. If he have none of the same number, he maye redeme him be any of his men that hys adversary wyll chuse. But if his lowest square be taken, no ransom will delyver him. Also if the whole kyng at ons that is the whole number of Pyramis be taken, he can not be redemed.

Of the triumphe.

To take awaye the tediousnes of long play from them that be yonge beginners, wryters of this game have invented divers kyndes of shorte victories, wherefore they devide victory into proper and common. Of the proper victory need nothing here be spoken, for all things thereto belonging are sufficientlu set forth in the first kind of playing.

Of the common victory.

The common victorie (they say) is after fyve maners, for men contende either for bodies, goods, quarrelles, honour, or els for both quarels & honor.

[Page]

Victory of bodies.

Victory of bodies is only to take a certain number of men, as if the gamesters agree, that he which first taketh .4. or .5. or .6. or .10. men &c, shall wyn the game.

Victorye of goods.

Victorie of goods, is to take a certain number without respect of the men. As if it be covenanted, that he which first taketh men amounting to the number of .100. or .200. shall have the victorie.

Victory of quarell.

Victorie of quarell is when neither the men, nor the number, but the characters of the number be considered. As if it be determined that he which first taketh .100. in .8. characters not regarding in how many men they standes, shall winne. As .2.4.5.8.24.64. so you have .100. in .8. characters it skilleth not, although there be more then .100. as in this example there is more then .100. by .4.

[Page]

Victorie of honour.

Victorie of honour, is whe a determined number is made in a determined number of men, as if it be determined that he whiche first cometh to .100. in .8. men, shall winne the game. As in these .2.4.6.8.4.16.45.15. And though there were somewhat more then .100. so it be in .8. men, it skilleth not.

Of victorie of honour and quarell.

The victorie of honour and quarell, is when one obteyneth the decreed number, in the decreed number of men and the decreed number of characters: as let .100. be the decreed number .8. the determined number of men, and .9. the determined number of characters. He that obteyneth .2.4.6.8.4.6.9.64. obteineth the victorie of honour and quarell. It shalbe no hinderance though .8.

[Page]

men and .9. characters conteyne somwhat more then .100. so that there be not .100. upon one man, as in the victorie before.

Victorie of standers.

They have invented another victorie, that is of standerdes, by counterfeyting two armies, one of the Christians, another of the Turkes. The whyte men, that is the even hoste, conteyneth .1312. footemen (not compting the rootes of squares expressed in the kynges) let the first and last be captaines and let them devide the whole armye into .10. standerds so every standerd shall have .130. men, besyde the two captaines and the ten standard bearers. The black men, that is the odde armie (except the kings rootes) be .1752. The two captaynes and ten standerd bearers taken out, there remayneth .1740. souldyers, to every standerd .174. He that wynneth more standers have the victorye. If the even hoste

[Page]

wyne .348. men he hath obtayned two standerds if he wynne .522. he hath gotten thre standerds and forth of the rest.

If the odde armye wynne .260. they wyn two standerds .390. three standerds and so of the rest.

Table of the victorye of standerds.

One standerd of the even, conteyneth 130.

Two standerds. 260.

Three standerds. 390.

Foure standerds. 520.

Fyve standerds. 650.

Sixe standerds. 780.

Seven standerds. 910.

Eyght standerds. 1040.

Nyne standerds. 1170.

Tenne standerds. 1300.

One standerd of the odde, conteyneth 174.

Two standers. 348.

Three standerds. 522.

Foure standerds. 696.

Fyve standerds. 870.

Sixe standerds. 1044.

Seven standerds. 1218.

Eyght standerds. 1392.

Nyne standerds. 1566.

Tenne standerds. 1740.

You maye use anye of these syxe kyndes of common victorie, in every one of the three kyndes of playing.

FINIS

Prynted at London by Rouland Hall,

for James Rowbothum, and are to

be solde at his shoppe in

chepeside under Bowe

churche.

1563.

***********

( above 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.

***********

Rithmomachia: The Game for Medieval Geeks (Culture)

In 965 Wibold, bishop of Cambrai, suggested to the local monks that they give up playing dice games and instead play "the battle between virtues and vices". In all likelihood this was the game rithmomachia, otherwise known as the ludus philosophorum. Today we are familiar with a number of ancient board games but unfortunately rithmomachia, as well as some related games Ouranomachia and Metromachia have all but disappeared

Rithomachia was played by the intellectuals of Europe, often Church employees and most likely male. Although it is obscure today, rithomachia played an important part in the lives of this elite group. The rules were complex and convoluted and required considerable computational skill. Its players also claimed that its benefits extended well beyond numerical training - it supposedly improved the moral character of players and even gave deep insights into religious truth.

There are many different versions of the rules of play but there are many common features. There are two players who take turns moving pieces on a board, much like chess or checkers. The board is something like a chess board, though rectangular and not square, with 48 pieces, each inscribed with a number. The size of board varies with rule set but eventually it was standardized on 8 by 16.

But now the complication starts. Rithomachia was inspired by a type of mathematics made popular by Boethius. This work was dominated by various types of numerical progression: in particular arithmetic progressions (where the nth number is a+b*n for constant a and b), geometric progressions (a*b^n) and harmonic progressions (1/(a+b/n)). There were also other more complex types of progression such as multiplex progression or the progression of superparticulars. At the start of the game the initial layout of the numbered pieces, in three ranks for each side, was such that the numbers formed various types of progression. One side had odd numbers in the first rank, the other side even. So even before the game had started quite a bit of mathematical knowledge was required. And don't forget that in earlier times arabic numerals were unknown so any calculations were carried out using Roman numerals. (By the way, these proportions played an important part in medieval music theory. Consider the harmonic progression in particular.)

Pieces from the first rank could move one square, those from the second rank two and those from the third three. As a mnemonic the pieces in these ranks were sometimes circles, triangles and squares respectively (a slightly illogical mnemonic!). Like in chess, pieces could be captured, however there were many ways to capture. One way was to move a piece with a number onto another with the same number. Two pieces could take another if it was the case that if they were simultaneously to make legal moves that would land them both on the captured piece and the sum of the numbers on the two pieces sums to that of the captured one. Another way to capture was to occupy all the spaces that another piece could move to making it unable to move - this was called besieging it. Here is another example of a capture rule taken from Lever and Fulke's "The Most Noble and Auncient, and Learned Playe" published in 1563:

Of taking by cossical signs

By cossical signs: any man that hath these signs, 3, &, 33, 3&, meeting with his root in his ordinary draught that hath this sign z taketh him up, or else is taken of his, without removing into his place; except he may not take him before he remove.

Obviously there has been a little semantic drift over the centuries. There were also some much more complex capture rules requiring the pieces to be arranged in a progression.

Now we've warmed up we can get onto the conditions to achieve victory in the game. Typically a player had to line up a series of pieces in certain arrangements, often as a Boethian progression of a certain length. Whoever did this would win. There were many different choices of victory conditions based on such progressions and players would negotiate before to game to decide which were in play for a particular game. An example simple victory condition follows:

Victory of goods

Victory of goods is to take a certain number without respect of the men. As if it be covenanted that he which first taketh men amounting to the number of 100 or 200 shall have the victory.

There was even a victory condition based on simulating the armies of the Christians and Turks at war.

As you can see, this was no game for weenies and yet it was played all over Europe from Medieval times, through the Renaissance into Elizabethan times. Many rithomachia manuals still exist today and doubtless many more were originally published. It seems to me there is only one explanation: Rithomachia is a geek game and the players were predecessors of today's game playing geeks! They spent their time shut up in dark rooms hunched over books and games believing themselves to be superior to the masses because they were experts in difficult and arcane, but largely useless knowledge.

Today Boethian mathematics is almost unheard of. This is the key to the decline of rithmomachia. Boethian mathematics was highly technical but today it's abstruseness seems completely arbitrary and useless. As it was replaced by more modern approaches to mathematics the rules to rithmomachia came to seem more and more arbitrary until interest in the game completely waned. However it is worth noting that this game did have a lifetime of 500 years and was played by such illustrious luminaries as John Dee and praised by Roger Bacon.

***********

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).

***********

More Rythmomachia and techniques

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).

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". 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. 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".

Also Roger Bacon comment the Arithmomachia, in its "mathematica Communio". 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.

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... 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 9.25.49.81 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 9.25.49.81 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 49.121.225.361 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

***********

Number Games ; Rythmomachia / Rithmimachia

Aritmomachia::

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).

Rythmomachy:

Ritmos = Number

Mache = Battle

Latin: Pugna (certamen) numerorum

Board:

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..."

Origin:

Pythagoras of Samos, Nicomachus of Gerasa, Anicius Manlius

Severinus Boetius. Six levels of numbers.

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

Constillation:

Roman militairy; Vegetius (writter: Flavius Vegettos

Renatus), Belisar's battle formation.

CAVALARY CAVALARY

PHALANX

INFANTRY & CAVALERY

***********

Books::

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

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

1.) OXFORD HISTORY OF BOARD GAMES by James Parlett

2.) BOARD AND TABLE GAMES FROM MANY CIVILIZATIONS

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.

Further References

Borst, A. 1986. Das mittelalterliche Zahlenkampfspiel. Supplemente zu den Sitzungsberichten der Heidelberger Akademie der Wissenschaften, Philosophisch-historische Klasse, vol. 5.

Heidelberg: Carl Winter Universitsverlag

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

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.

***********

THE RULES OF

BOOLEAN RITHMOMACHIA

(A New Twist On An Old Game)

Boolean Rithmomachia

by L. Lynn Smith

The following game is in homage to the Medieval game of numbers. It was thought

that the ideas of this game should be updated and could even be utilized in the

teaching of computer math.

PLAYING FIELD

In order to bring this game into the 21st Century, it was thought that it should

be played upon a 3D field. To make this play as simple as possible, a 4x4x4 field

was selected.

PIECES

The playing pieces consist of two of each of the following. Such pieces, except for the Pyramids, are two sided and have different colors of either side, black or white, but the same value(1 - 14) represented by the binary code. The Pyramids are whole pieces of solid color, two black and two white, one of each particular value(0 and 15) represented by the binary code.

Value Shape

----------------

0000 Pyramid

0001 Circle

0010 Circle

0011 Triangle

0100 Circle

0101 Triangle

0110 Triangle

0111 Square

1000 Circle

1001 Triangle

1010 Triangle

1011 Square

1100 Triangle

1101 Square

1110 Square

1111 Pyramid

SET-UP

The pieces of each player are arranged in two seperate planes, seperated by two planes. Players can occupy opposite sides of the playing field, either horizontally or vertically, in the following manner:

[S][C][C][S]

[T][T][P][T]

[T][P][T][T]

[S][C][C][S]

C=Circle P=Pyramid S=Square T=Triangle

Players have the option of which values may be placed in their respective cells. This form of set-up could be considered one phase of the game as each player may take turns placing a piece upon the field. Or they may agree to a standard form of set-up.

MOVEMENT

Circles move one orthogonal or one diagonal, Triangles move one diagonal or one triagonal, Squares move one orthogonal or one triagonal, Pyramids move one orthogonal or one diagonal or one triagonal. The Pyramid may by landing on any piece then move that piece(regardless of owner) by any normally legal Pyramid step to an adjacent empty cell, such a piece does not change ownership solely by this action but may if it then becomes part of a capture equation. All other pieces must move to empty cells.

[The triagonal movement is a move from one cell to the next cell which is adjacent by

only one of its eight corners. Whereas, the orthogonal move is a change along one axis and the diagonal move is an equal change along two axes, the triagonal move is an equal change along all three axes of the 3D field.]

CAPTURES

The capture of opponent pieces involves flipping the piece to its opposite color. Pyramids are removed from the playing field when captured. The game is over when one player no longer has possession of either Pyramid.

Captures are preformed by legally moving a particular piece then flipping all appropriate enemy pieces.

EQUATIONS

Captures are preformed by applying Boolean equation to and through adjacent pieces.

Capture by the NOT equation. When the appropriate enemy piece is located in a cell which, if empty, would be a legal move for the capturing piece and it is its NOT equivalent, that enemy piece is then captured.

NOT equivalents:

0000 - 1111

0001 - 1110

0010 - 1101

0011 - 1100

0100 - 1011

0101 - 1010

0110 - 1001

0111 - 1000

1000 - 0111

1001 - 0110

1010 - 0101

1011 - 0100

1100 - 0011

1101 - 0010

1110 - 0001

1111 - 0000

This is the simplest take as the attacking piece needs no assistance in capturing its

opponent. Examples: Circle(0001) is able to capture Square(1110) if it becomes

orthogonally or diagonally adjacent. Triangle(0110) is able to capture Triangle(1001) if it is diagonally or triagonally adjacent.

The remaining captures involve the use of two pieces by which to take the target. It is necessary that the second piece used in this form of capture be also owned by the player.

These captures take two forms, Compiling and Filtering.

Compiling, each of the two pieces used to determine the capture must be adjacent to the target piece by their individual legal move.

Filtering, the played piece must be legally adjacent to one which is legally adjacent to the target.

Several takes can be made during a player's turn, if the pieces and positions allow.

There are five forms of operations which can be preformed in order to make such captures; AND, OR, XOR, NAND and NOR.

The following table can be used as a quick reference during the game:

-----------------------------------------------

X Y AND OR XOR NAND NOR

-----------------------------------------------

0 0 0 0 0 1 1

0 1 0 1 1 1 0

1 0 0 1 1 1 0

1 1 1 1 0 0 0

-----------------------------------------------

In such equations, the played piece will be referred to as X, the Compiler or Filter piece will be Y and the target piece will be the answer. If the answer is True then the piece is captured, if False the piece is left as is.

If X = Circle(0100) and the Compiler or Filter = Triangle(1100) then the target must equal the Circle(0100) to be captured using the AND Equation.

Using the OR Equation, the previously stated values could capture Triangle(1100).

Using the XOR Equation, they could capture Circle(1000).

Using the NAND Equation, they could capture Square(1011).

Using the NOR Equation, they could capture Triangle(0011).

The Pyramid is also used to preform such captures. If its moves involves the displacement of a piece, it still may preform the appropriate captures moves. The displaced piece can become either a Compiler, Filter or even a Target.

All captures are mandatory, it is the obligation of either player to assist the other in determining any possible captures which may have resulted from a particular move. Neither player is obliged to suggest any particular move to an opponent.

WINNING THE GAME

When an opponent's last Pyramid is captured, or there are no legal moves or potential captures, victory is determined thus:

Pyramid Victory:

The player who has the most number of Pyramids remaining on the playing field.

Piece Victory:

The player who has the largest number of pieces, including Pyramids, on the playing field.

Value Victory:

The player whose value of pieces, including the Pyramids, on the playing field is the highest total.

The Simple Victory consists of one of the above victories.

The Compound Victory consists of two of the above victories.

The Perfect Victory consists of all three victories.

It is possible that not all of the victories will be available. Example: Both players may have the same value of pieces on the field but not the same number, so neither would be awarded the Value Victory.

Addendum 01.03.2002

It is not necessary to use all formulae in every game. The players may choose to play a NOT-NAND game, a game using the NOT and NAND equations. Or they may play an OR-XOR-NOR game. The desired formulae to be used should be clearly stated before the start of each game.

Addendum 07.11.2002

Allowing the Pyramid to "push" another piece can be optional. Denial of the "push" can increase the opportunity for the game ending with a player having Pyramids but no legal moves. There is no direct penalty for a game ending in such a manner.

--------------

Addendum 02.16.2003

Here is an interesting way to select the equations to be used during the game: Each player secretly writes down two choices, then simultaneously reveals them. This allows for the possibility of either a two-function, three-function or four-function game.

** END **

The mayor chess variant

through the years 500 to 1,800

During the investigation of the Subject Rythmomachia,

the editor has substantially added new material during the year 2004 to

text as below, and specifically the two first paragraphs

**BALDERIC LE ROUGE.**

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

Douay, Ioan. Bogarde, 1615. Contemporary 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.

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 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 ( 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

Firs 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.

**New data pertaining to -integral change txt**

RHYTHMOMACIE

RHYTHMOMACIE

Geschichte & Literatur des Schachspiels, (prof.) Antonius van der Linde (curator & chess literature collector from NL / DE), 1874 original, reprint in German 1981.

**RHYTHMOMACIE, Rhythmomachya, Victoria Ritmachie;**

Notation Parlante / Figurines

Hippogonal (round) - use Pawns - name:Notation Parlante / Figurines

**Rotundi**

Diagonal - use Couriers - name:**Trigoni**Orthogonal - use Rook - name:

**Tetragoni**(rook = pyramid)

Array notation on 8*16 board

**..PPPP..**

BBPPPPBB

QQBBBBQR

QQ....QQBBPPPPBB

QQBBBBQR

QQ....QQ

Based on the *old system* of moves of the chess game.

Translator Barozzi states that chess has been developed from

Rythmomachy, since it is older. However the original Greek

transcripts have never been found.

**Other variant::**

METROMACHIA ; Metromaxia ; Sive Ludus Geometricus.

Board with 53*33 squares, the figurines are geomatrical.

**Rythmomachia**

======================

**Equipment::***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 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 1,000 - 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.

Array::

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::

Evens

Odds

1st Rounds (x)

2

4

6

8

3

5

7

9

2nd Rounds (x2)

4

16

36

64

9

25

49

81

1st Triangles (x + x2)

6

20

42

72

12

30

56

90

2nd Triangles ((x + 1)2)

9

25

49

81

16

36

64

100

1st Squares (x + x2 + (x + 1)2)

15

45

91

153

28

66

120

190

2nd Squares ((2x + 1)2)

25

81

169

289

49

121

225

361

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.

Play::

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.

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

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.

***********

The Philosopher's Game

This file is a 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.

Annotation will occur occasionally throughout; they will appear in square brackets and italics, [like this]. Spelling will be erratic; I'm transcribing quickly, so I will often be modernizing the spelling, but will leave original spelling whenever I consider there to be doubt about the meaning. I also will sometimes modernize the punctuation and paragraph breaks in the interests of readability. This is not intended to serve as a definitive critical edition, merely a working copy, good enough to understand the game.

[Title Page]

THE MOST NOBLE

ancient, and learned playe, called the Phi-

losophers game, invented for the honest re-

creation of students, and other [sober?] persons, in

passing the tediousness of time, to the release of

their labours, and the exercise of

their wittes.

Set forth with such playne precepts, rules and ta-

bles, that all men with ease may understande

it, and most men with pleasure practice it.

by Rafe Lever and augmen-

ted by W. F.

[Picture, probably stock, of two men playing at a game on a square 10x10 board.]

Printed at London by James Rowbothum, and are

to be sold at his shop under Bowchurch

in chepe syde.

[Page]

The Lord Robert Duddedlye.

[Picture of Lord Robert, with the motto "Vulnere virescit virtus" alongside.]

The Physiognomie here figured, appeares by Paynters Arte:

But valyant are the vertues that, possesse the inward parte.

Whych in no wise may paynted be, yet playnely so appeare,

& shine abrod in every place with beames most bright & clear.

[Page]

[The Epistle Dedicatory]

TO THE RYGHT HO-

norable, the Lord Robert Dudley, Mai

ster of the Queenes Maiesties horse,

Knight of the most honorable order

of the Garter, and one of the Queenes

maiesties privie Counsell, JAMES

ROUBOTHUM heartelye wisheth long life, with

encrease of godly ho-

nour and eternall

felicitie.

Sith that your honour is full bent,

(right honorable lord)

To wisdom & to godlines

with true faithful accord.

Sith that in deed you do delyte,

in learning and in skyll:

The show wherof doth well expresse

a perfect godly wyll.

Sith that also you have in hand,

affayres of force and waight:

And study do both day and night,

to set all thinges full straight.

[Page]

I thought therfore your honour should

not lacke some godly game:

Whereby you might at vacant times

your self to pastyme frame.

Whereby I say you might release,

such travailes from your mynde:

And in the meane while honest mirth

and prudent pastyme fynde.

Remembring then this auncient play,

where wisdome doth abound:

Called the Philosophers game,

me thinkth I have one found.

Which may your honour recreate,

to read and exercise:

And which to you I here submit,

in rude and homly wise.

Pithagoras did first invent,

this play as it is thought:

And therby after studies great,

his receation sought.

[Page]

Yea therby he would well refreshe,

his studious wery braine:

And still in knowledge further wade

and plye it to his gaine.

Accompting that a wicked play,

wherin a man leudely:

Mispendes his tyme & wit also,

and no good getts thereby.

But grevously offendes the Lord,

and so in steed of rest:

With trouble and vexation great,

on every side is prest.

Most games and playes abused are,

and fewe do now remaine:

In good and godly order as,

they ought to be certaine.

For why? all games should recreat,

the hevy mynde of man:

And eke the body overlayde:

with cares and troubles than.

[Page]

But now in stead of pleasant mirth,

great passions do arise:

In stead of recreation now,

revengings we practise.

In stead of love and amitie,

long discords do appeare:

In stead of trueth and quietnes,

great othes and lyes we heare.

In stead frendship, falshode now,

mixed with cruell hate:

We finde to be in playes & games,

which dayly cause debate.

Pithagoras therfor I saye,

to make redresse herein:

Invented first this godly game,

therby to flye from sinne.

Since which time it continued hath,

in Frenche & Latin eke:

Still exercisde with learned men,

their comforts so to seeke.

[Page]

Wherby without a further prose,

all men may be right sure:

That this game unto gravitie,

and wisdome doth allure.

Els would not that Philosopher,

Pithagoras so wyse:

Have laboured with diligence,

this pastime to devyse.

Els would not so well learned men,

have amplified the same:

From tyme to tyme with travell great,

to bring it into fame.

But let us nerer now proceed,

and come we to theffect:

And then shall we assuredly,

this pastime not neglect.

For it with pleasure doth asswage,

the heavy troubled hart:

And with lyke comforts drives away,

all kynde of sourging smart.

[Page]

The mynde it maketh circumspect,

and heedfull for to bee;

The tyme that theron is bestowd,

is not in vaine trulye.

The body it doth styrre and move,

to lightsomnes and ioye:

The sences and the powers all,

it no wyse doth annoye.

It practiseth Arithmeticke,

and use of number showth:

As he that is conning therein,

assuredly well knowth.

In Geometie it truly wades,

and therein hath to do:

A learned play it is doutlesse,

none can say nay thereto.

Proportion also musicall,

it ioynes with thother twayne:

So that therin three noble artes,

are exercisde certayne.

[Page]

What game therfore lyke unto this,

may gotten be or had?

There is not one that I do know,

the rest are all to bad.

It causeth no contention this,

nor no debate at all,

By this no hatred wrath nor guyle,

in any wise doth fall.

It stirreth not such troubles that,

our frend becomes our foe:

It moveth not to mischiefe this,

as many others do.

Let us avoyde the worst therfore,

and cleve we to the best.

So shall we shunne all wickednes,

and purchase quiet rest.

So shall we serve the living Lorde,

and walke after his will:

So shall we do the thing is good,

and flye that which is yll.

[Page]

So shall we live right christianlyke,

and do our duties well:

So shall we please both god & prince,

none shall us need compell.

And then the Lord of his mercie,

will prosper us alwayes:

And graunt us here to have on earth,

full many godly dayes.

Yea then the Lord of his goodnes,

and grace celestiall:

Will guyde and governe our affaires,

and blesse our doings all.

Which Lord graunt to your honour here,

good dayes & long to have:

with much encrease of helth & welth

and from all hurt you save.

Your honours most humble,

James Roubothum.

[Page]

To the Reader.

I Dout not but some man of severe judgement so soone as he hath one read the title of this boke wyl immediately sai, that I had more need to exhort men to worke, then to teach them to play, which censure if it procede not of such a froward morositie that can be content with nothing but that he doth himself, I do not only well admit, but also willingly submit my self therto. And if I could be persuaded that men at mine exhortation wold be more diligent to labour, I would not only write a treatise twise as long as this, but also thynke my whole time wel bestowed, if I

[Page]

did nothing els, but invent, speake, and write that which might exhort, move, & persuade them to the furtherance of the same. But if after honest labour and travell recreation be requisit, (and that nede no further probation because we favour the cause wel inough) I had rather teach men so to play, as both honestye may be reserved, their wittes exercised, they them selves refreshed, and some profit also attayned, then for lacke of exercise to see them either passe the tyme in idlenes, or els to have pleasure in thyngs fruitles and uncomely. And if great Emperours and mighty Monarches of the world have not bene ashamed by writing bookes to teaches the art of Dyce

[Page]

playing, of all good men abhorred, and by all good lawes condemned: have I not some colour of defence, to teache the game, which so wyse men have invented, so learned men frequented, and no good man hath ever condemned.

The invention is ascribed to Pythagoras, it beareth the name of Philosophers, prudent men do practise it & godly men do praise it. But because many herein (as in a play) have challenged much authoritie, they have filled this game with much diversitie. In which as I could perceive the most differens of playing to consist in thre kindes, so have I playnly and briefly set them forth in Englishe not as though there might not more diversities be espied, but

[Page]

that I thought these to them whom I have written to be sufficient. yet for that I woulde be lothe, from playe & game, to fall to earnest contention, if any man in this doing or any part thereof shall think I have done amisse, and will do better himself, so far am I from envying his good proceding, that I wil be right glad, and geve him heartye thankes therefore.

All things belonging to this game

for reason you may bye:

At the bookeshop under Bochurch

in Chepesyde redilye.

[Page]

The bookes verdicte.

Wanting I have bene long truly,

In english language many a day:

Lo yet at last now here am I,

Your labours great for to delay,

And pleasant pastime you to showe,

Mynding your wits to move I trowe.

For though to mirth I do provoke,

Unto Wisdome yet move I more:

Laying on them a pleasant yoke,

Wisdom I meane, which is the dore,

Of all good things and commendable:

Dout this I thinke no man is able:

CATO

Interpone tuis interdum gaudia curis:

Yt possis animo quemuis sufferre laborem.

[Page]

The diffinition

That moste auncient and learned playe, called the Philosophers game, beinge in Greeke termed [...], is as much to saye in Englishe, as the battell of numbers. Numbers be either even or odde, wherefore the even parte is against the odde, either parte havinge a kyng, whych being taken of the adversaryes part, and a triumphe celebrated within his campe, the game is ended.

Of diverse kyndes of playinge.

Amonge the dyverse kyndes of playing thys game, we shall sette forth three sortes, of which the reader maye chose whether of them he lyketh beste. And of all those three, we shall

[Page]

gyve suche shorte and easye rules, that no man (althoughe he were altogether ignoraunt in Arithmetike) shall fynde the game so hard, but that he may learne to playe it.

Of the partes of thys Game.

He that wyll learne thys game, any of the three waies, muste first be enstructed of these sixe partes.

The table as the fielde

.2. the menne and the numbers of them as the hoste

.3. the placynge of them, as the encampynge

.4. the order of playe and removynge the men, as the marchynge and fyghtynge

.5. the manner and lawes of conqueryng and taking

.6. and last of al the triumphe after the victorye.

Of these partes in the fyrst kynd of playng.

[Page]

The table muste be a playne borde conteynynge .128. squares that is .8. in breadth and .16. in length sette forthe in two dyverse collours. Or for a plainer understandynge, the table is a doble chesse bord, as it were two chessebordes joyned together, the length of twoo, the breadth of one, whereof thys is an example.

[Page]

[Complex picture, showing a 16x8 checkerboard, with various letter and symbols on various parts. This can't be fully represented in text, so I won't try to show the whole thing now. The middle section -- rows 6 through 12 -- have letters on them, and look somewhat like this:]

K L M

X F Z

B S C

R H Y A T I N

E V D W

G

Q P O

[Later sections will refer back to this table frequently, to describe movement.]

[Page]

Of the men.

The men be in number .48. Wherof .24. be of one side & must be knowen by one colour, and .24. on the other syde, whyche also must be marked with a contrarye colour, as White and Blacke, Blew and Redde, or what colours els you lyke best. But in the colering there .3. thinges must be observed, the bottome or lower part of every man (excepte the two kinges) muste by marked wyth his adversaries colour, that when he is taken, he maye chaunge his coate and serve him unto whome he is prisoner.

The seconde thinge considered in the men, is their fashion: for of euther syde .8. are rounds, other .8. are triangles & .7. (the king making .8.) are squares. There fashion is such [small inset showing roundes, triangles, squares].

The kynges because they consist of all three sortes, as it is knowen by the learned speculation of the numbers, beare

[Page]

the fashion of all thre kindes, his foundations are two squares, on which are sette, two triangles & upon them rounds. But this difference is betwene the kinges, the king of the even numbers, hath a pointed toppe, the king of the odde numbers is not pointed, the cause dependeth upon the consideration of these numbers by which they arise into piramidall fashion. The third thing considered in the men, is the number that must be written or graven upon them which to learne plainely for practise marke these short rules.

There be of eche kynde of men, two rankes or orders.

The first ranke or order of roundes be the digites even or odde namely of the even .2.4.6.8. of the odde .3.5.7.9.

The second order of rounds are found by multiplyinge these digites by themselves as .2. times .2. is .4.3. times .3. is .9. Of the even they be .4.16.36.64. of the odde they be .9.25.49.81.

The first order of the triangles are found by addinge two of the roundes together

[Page]

one of the firste order and another of the seconde order, as .2. and .4. make sixe .3. and .9. make twelve, on the even syde they are these .6.20.42.72. on the odde syde .12.30.56.90.

The second order of triangles be made by addynge one to every one of the first order of roundes, and then multiplying that number in hym selfe: as .2. is one of the firste order of roundes, thereto adde one, that is .3. then .3. tymes .3. is .9. a triangle of the seconde order, on the even syde. Likewise to thre a round on the odde side, adde .1. so it is .4. then .4. tymes .4. is .16. On the even parte, they be .9.25.49.81. on the odde parte .16.36.64.100.

The first order of squares (in whyche are contayned the kynges) be made by addynge two triangles together, one of the fyrste order, and another of the secondes, as .6. and .9. make .15. likewyse .12. and .16. make .28. Amonge the even they be .15.45. and .91. the kynge .153. amonge the odde they be .28.66.120. and .190. the kynge.

[Page]

The last order of squares be found, by dobling of every one of the firste order of roundes, and after adding one, last of all be multiplying that number in itself, as twise .2. is .4. and .1. added is .5. so .5. times .5. is .25. likewyse twyse .3. is .6.1. added is .7. then .7. tymes .7. is .49. These be on the even syde .25.81.169.289. And of the odde syde .49.121.225.361.

These numbers must be sette uppon the men both on the upper side, & also on the nether side. Except one of the kynges, which must with the whole number of their pyramid, be marked, onely on the bottome. Because the sydes muste have other numbers, namely the highest point of the even kyng, must have .1. the rounde next under him marke with .4. the uppermost triangle with .9. the nethermost with .16. The uppermost square muste have .25. The nethermost square shall have .36. The king of the odde upon his head, whiche is a rounde, not pointed hath .16. upon his first triangle .25. on the second triangle .36. uppon the fyrste square .49. upon the lowest square .64.

[The numbers in the next paragraph are in circles and triangles, to illustrate.]

Finally it shalbe good for the avoydance of confusion, to drawe a line under every number. Ells may you take one for another, as 6 the even round & 9 the odde rounde, may be taken one for another with oute this lyne or some suche marke, lykewise 6 and 9 Tryangles bothe of one syde. And this sufficient for the men, the fashion, colours and numbers.

The reason of these numbers and the knowledge of their proportione.

For them that seke the speculation of these numbers, rather then the practise for playing, and have some sight in the sciens of Arithmetike, some thyng must be sayde of proportion. For this purpose there be three kyndes of proportion. Multiplex, superparticuler, and superpartiens.

[Page]

Of multiplex.

MULTIPLEX proportion, is when a great number conteyneth a lesse number manye tymes, and leaveth nothing, as .8. conteyneth .2. fower tymes and nothing remaineth .16. conteyneth .4. &, this proportion semeth best to agree with roundes because the one number conteyneth the other and nothynge remaineeth as the fyrste order of roundes be.

[The following table shows the white (even) and black (odd) rounds.]

2 4 6 8

3 5 7 9

[Page]

The second order be these.

[This table shows the first and second orders of rounds for both sides.]

double. quadruple. sextuple. occuple.

2 4 6 8

4 16 36 64 proportion.

triple. quintuple. septupl. nonuple

3 5 7 9

9 25 49 81

[Page]

Of superparticuler proportion.

Superparticuler proportion is when a greater number contayneth a lesser with one part of it, which may measure the whole, as .12. contayneth .9. and .3. whiche is a thyrde parte of nine .6. contayneth .4. and .2. that is one halfe to .4. Thys proportion beinge the cheife, next unto multiplex, is beste figured by a trianguler forme, whyche hathe fewest lynes and angles next unto a circle. For the manner of thys proportion consider thys figure.

[Page]

[This table shows each base number above two rows of triangles, and appears to be badly fouled up. I believe the rows of base numbers are wrong -- the first set should be the evens (2, 4, 6, 8), and the second the odds (3, 5, 7, 9). That would make each Latin header match the number below it, the first row of triangles would be the superparticulates for that base number, and the second row would be the second order of triangles for that base number, which is (x+1)2.]

sesquialter. sesquiquart sesqui.sext sesqu.oct.

3 5 7 9

6 20 42 72

9 25 49 91

sesquiter. sesquiquint. sesquisept. sesquinona.

4 6 8 10

12 30 56 90

16 36 64 100

[Page]

Superpartiens proportion.

The superpartiens proportion is when the greater number conteyneth the lesser and mo partes of it then one as .15. conteyneth .9. and .6. whiche is two thirdes of .9. lyke wyse .28. conteyneth .16. and .12. that is 3/4 of .16. This proportion conteineth divers parts beside the whole number therfore is wel figured in the square, which also conteyneth more corners and sides. For the maner of their proportion consyder thys table.

[Page]

The first order of squares.

[The following table contains the two rows of triangles, with the corresponding square beneath.]

6 20 42 72 supparticulares added

9 25 49 91

15 45 91 153 being the squares.

12 30 56 90

16 36 64 100

28 66 120 190

[Page]

The second order followeth.

third fyft seventh ninth

5. 9. 13. 17. These two rows are just plain numbers.

10. 36. 78. 136.

15 43 91 153 These two rows are the two white orders of squares.

25 81 169 289

superbipartiens tertias supquadrupartiens quintas supsextupartiens septimas supoctupartiens nonas

[Page]

fourth sixth eight tenth

7. 11. 15. 19. These two rows are just plain numbers.

21. 55. 105. 171.

28 66 120 190 These two rows are the two black orders of squares.

49 121 225 361

supertripartiens quartas supquinpartiens sextas supseptupartiens octavas supnonpartiens decimas

[Page]

Of the kings.

[Image of the two kings. Each is a pyramid of numbered pieces, with two squares, two triangles, and a round; one has a little triangular "cap" on top. The images make it look like higher-numbered pieces are actually larger -- I wonder if this is true...]

The kinges conteine in them suche numbers, as beyng all added together, make the whole piramidall number, the lowest square of the even is .36. which riseth of the multiplying of .6. in it selfe. The next square that must be lesse, is .25. arisinge by the multiplyinge of fyve in it self and so followeth .16. of .4. then .9. of .3. laste .4. of .2. and single .1. all these added together make up .91. After the same maner consisteth the king of adde. The lowest square is .64. arisinge of .8. multiplied in himselfe. The next .49. of

[Page]

.7. times .7. then .36. of .6., .25. of .5. and .16. of .4. these numbers make the whole pyramidall number .190. which because it riseth not to the poynct of one, oughte not to be sharpe poyncted, as hathe beene sayde before.

Of the placing, encamping or setting in araie.

To retorne againe to the plaine and easye playing of this game, next to the armie & their armour, follow ether the order of their battel or encamping. Which because it is more playne and easely seen which the eye, then learned by the eare, I referre thee unto the table where the battell is appoynted in suche order as thys kynde of playe requireth.

[Page]

[The table below shows the layout of the board. To illustrate this through text, I put squares into square braces, triangles into angle brackets, and rounds into parentheses. Kings are signified with an asterisk. Note that the top half, the evens, should be upside-down.]

[25] [81] [169] [289]

[15] [45] <25> <20> <42> <49> [91]* [153]

<9> <6> (4) (16) (36) (64) <72> <81>

(2) (4) (6) (8)

(9) (7) (5) (3)

<100> <90> (81) (49) (25) (9) <12> <16> [Error in original: 2 instead of 12]

[190]* [120] <64> <56> <36> <30> [66] [28]

[361] [225] [121] [49]

[Page]

Of the marchinge or removing of the men.

The battell beyng duely placed, it followeth next, to know the maner of marching & removing, for every kynd of men, hath their proper kynde of motion, and fyrste we muste speake of the roundes.

The motyon of the roundes.

The roundes muste move into the space that is next unto them cornerwyse, as in the table, from the space .A. to any of these .B.C.D. or .E.

[Yes, this is clearly saying that rounds only move diagonally.]

Of the triangles.

The triangles passe three spaces counting that in which they stande for one, and that into whych they do remove for another, that is leaping over

[Page]

one space. As from the space .A. he maye remove into any of these space .F.G.H. or .I. this is the motion of the triangle in marchying or takyng. But in flying he maye remove the knyghtes draught of the chesse, as from .A. into .X. or .W. &c.

Of the Squares.

The Squares remove into the fourth place from them, that is leaping over two, right forwarde or sydelong, as from the place of .A. to any of these spaces .L.N.P.R. flyinge they maye remove after the knyghts draught, but that they must passe foure spaces, as from .P. to .Y. or .T. &c. And this for the marchinge and removyng of the men, where note, that with theyr flying draughte they can take no man, but if needed by helpe to besiege a man.

Of the kynge marching.

The kings because thei beare the forme of al the thre kynds, may remove any

[Page]

of all theyr draughts when they list, into the nexte with the rounde, into the thyrde with the triangle, and into the fourth with the square, and finally in all poyntes lyke the Queene at the Chesse, saving that he can not passe above foure spaces at the most.

Of the maner of taking.

The men may be taken sixe wayes, namely by Equalitie, Oblivion, Addition, Subtraction, Multiplication and Division, and also if you wyll, and so agree by

/ Arithmeticall.

Proportion <> <42> <20> <6{r}>

<81> <56> <30> <12>

[Page]

The seconde order of triangles, have all excepts one (whiche is the number of .100.) their cossicall signes, as .9. bothe of the roote and of the quadrate, .25.36. and .49. have the signe of the quadrate .64. of the quadrate and the cube, and also the quadrat of the cube .16. and .81. of the quadrate, and the foure squared quadrate.

<81{24}> <49{2}> <25{2}> <9{r2}>

<100> <64{236}> <36{2}> <16{24}>

In the first order of squares, onely .15. is marked with the roote, all the rest doe want theyr cossicall sygnes in thys game.

[Page]

[A rather odd little illustration here, showing the two kings in some detail, with each constituent pieces marked both with its number and the square root of that number, and the complete number of the king on the top. Also, a couple of odd little images of a round and a square with a peculiar symbol on top; this symbol may be peculiar to king components. Note also that, in the table of squares below, the 91 and 190 have a picture of the king on them.]

[153] [91] [45] [15{r}]

[190] [120] [66] [28]

The seconde order of squares hath .3. numbers marked with cossicall signes, that is .25. and .225. wyth the signe of the quadrate .81. is marked with the sygne of the quadrate and the fouresquared quadrate.

[Page]

[289] [169] [81{24}] [25{2}]

[361] [225{2}] [121] [49]

And thys have you all the men that be marked with cossicall sygnes.

The setting in aray.

The teachers of this kynde of playing, doe not so well allowe, the former kynde of placing or any other, as the naturall placing of every man under him of whome he aryseth. So thei conteyne .6. ranks in length, extending to the furthermoste edge of the Table after this sorte.

[Page]

[Note that, in this table, the odds are colored black, while the evens are left white.]

[361] [225] [121] [49]

King [120] [66] [28]

<100> <64> <36> <16>

<90> <56> <30> <12>

(81) (49) (25) (9)

(9) (7) (5) (3)

(2) (4) (6) (8)

(4) (16) (36) (64)

<6> <20> <42> <72>

<9> <25> <49> <81>

[15] [45] King [153]

[25] [81] [169] [289]

[Page]

The marching or moving.

The men maye remove every way, into voyde places, forwarde, backewarde, towarde both sydes, direct or cornerwyse. So that the rounde men remove into the next space, the triangles into the third place, and the squares into the fourth place, accompting that place in which they stande for one.

Also every man savyng the two kynges to besiege his enemie, or to flye from the siege himself, may remove the knights draught in chesse, but neither take anye man (except it be by siege) nor erect a triumphe by suche motions. The kynges move even as squares, but that they have not the flyinge draughte.

It is compted lawefull amonge suche as wyll to agree, that the Triangles and Squares, maye remove into voyde places, thoughe the spaces betwene be occupyed of other men.

[Page]

The maner of taking.

The men may be taken seven ways by Oblivion, by Equalitie, by Addition, by Subtraction, by Multiplication, by Division, and by Cossicall Sygnes.

Of takynge by Oblivion.

All men maye be taken by Oblivion when by foure men they be letted of theyr ordinarie draughte, as hath bene taught before.

Of takynge by Equalitie.

By Equalitie maye these men take or be taken, as hathe bene sayde before, .9.16.25.36.49.64.81., as yf after you have played your .9. stande in

[Page]

your mans draught, you may take him by not removing into his place, unlesse you espye him standing in your draught before you playe, then muste you take him up and remove into his place.

Of takynge by Addition.

The takyng by Addition is all one with the first kynde of play, in all respectes, saving that some require the men that shoulde take by Addition to stande in the next spaces to him that is taken, either directly, or cornerwyse, but the former waye is better.

Of taking by Subtraction.

That whiche was sayde in the first kinde of subtraction and that whiche was last sayde of Addition may be bothe referred together. For this subtraction

[Page]

differeth not from the former, but for the opinion of them, that would have the two takers stande onelye in the nexte spaces to him that is taken.

Of takyng by Multiplication.

Takyng by multiplication doth differ. For in this kynde of playng, it is thus. When your man standeth so, that beyng lesser than your adversaries man, you may multiplie your man by the voyde spaces betwene them, and the product is all one with the adversarye, you maye take hym upm not removynge into his place, except you espye hym so, before you remove your man.

Of takynge by Division.

[Page]

Lykewise by Division, yf your man beyng greater then the adversarye, stande so, that beyng devyded by the voyde spaces, the quotient is all one with the adversarye, you maye take hym up, not removyng into hys place, unlesse you see hym so standynge before you drawe.

Of taking by Cossicall signes.

By Cossicall sygnes anye man that hath these signes, {2}.{3}.{4}.{6}. meeting with his roote in his ordinary draught that hath this signe {r} taketh him up, or elles is taken of him, without removing into his place, except he maye take him before he remove.

Of the kynges, and their taking.

[Page]

The king of the even must be foursquare, havyng sixe steppes, every one lesser then other, on one syde he muste have on him these rootes .1.2.3.4.5.6. on the other syde the quadrates arising of these rots, that is .1.4.9.16.25.36.

The king of the odde men, muste have but fyve steppes, that is .4.5.6.7.8. lackyng the rootes that he can not ende in .1. The quadrates of hys rootes by these .16.25.36.49.64. These muste be so set on, that the least must be hyghest and the greatest lowest.

The kinges be taken by Oblivion, or yf theyr Pyramidall number, be taken by anye of the aforesayde meanes. Also yf by suche meanes you can take all his quadrates one after another.

The privilege of the king.

[Page]

If anye of the kynges quadrates be taken, he maye redeme it by anye of his men having the same number, and muste remove into his place, whiche redemed hym. But yf he have none of the same number, he maye redeme hym for anye man of hys, that his adversarye wyll chuse, and lykewyse remove into his place by whome he is redemed.

[Page]

A table to take the men by Multiplication and Division.

Even against odd

spaces

6 2 12

8 2 16

15 2 30

45 2 90

4 3 12

4 4 16

9 4 36

16 4 64

6 5 30

20 5 100

2 6 12

15 6 90

20 6 120

4 7 28

8 7 56

2 8 16

8 8 64

4 9 36

9 9 81

25 9 225

9 10 90

21.

odd against even

spaces

3 2 6

36 2 72

3 3 9

5 3 15

12 3 36

5 4 20

9 4 36

16 4 64

3 5 15

5 5 25

9 5 45

12 6 72

7 7 49

5 9 45

9 9 81

3 12 36

3 14 42

17.

[Page]

For Division.

even against odd

spaces

6 2 3

72 2 36

15 3 5

36 3 12

9 3 3

20 4 5

36 4 9

64 4 16

15 5 3

25 5 5

45 5 9

42 6 7

72 6 12

49 7 7

72 7 9

45 9 5

81 9 9

36 12 3

91 13 7

42 14 3

20.

odd against even

spaces

12 2 6

16 2 8

30 2 15

90 2 45

12 3 4

16 4 4

36 4 9

64 4 16

100 4 25

22 5 45 [sic -- 22 should be 225]

30 5 6

100 5 20

12 6 2

36 6 6

90 6 15

120 6 20

28 7 4

56 7 8

16 8 2

64 8 8

120 8 15

3 9 4 [sic -- 3 should be 36]

81 9 9

225 9 25

90 10 9

66 11 6

28 14 2

27.

To take by cossicall signes

2 16{4}

2 64{6}

3 81{4}

3 9{2}

4 16{2}

4 64{3}

5 25{2}

6 36{2}

7 49{2}

8 64{2}

9 81{2}

15 225{2}

[Page]

Of the triumph.

The triumph is after the Kynge be cleane taken away, to be create in the adversaries campe, as well of your owne men as of your adversaries men that be taken, or of both in proportion as hath bene shewed before, and proclaimed that those men ons placed, may not be taken, as it was declared sufficiently, and no difference betwene the triumphes, savyng that some wyll not alowe a triumphe but of foure numbers, and two proportions at the lest. All three for the greater victorie, makynge but two kyndes of triumphes.

Here foloweth the thyrd kynde of playing at the Philosophers game.

There must also in this thyrd kynde be considered the table, the men, their markyng, the order of theyr battell, the motions, their taking, and last of all theyr triumphing.

The table is the same that hath bene twyse already discribed. Yet some wyll not have it so longe, but at the lest is must conteyne .10. squares in length and alwayes .8. in breadth. The longest is best.

Of the men.

The men be .48. as it hath bene told of two contrary collor, the head and bottom all of one collor, because men ons taken be no more occupyed in thys kynde of playing.

The inscription and fashion.

[Page]

The fasion is as hath bene last declared both of the men, and of the kynges, the inscription of numbers the same, but without cossical signes.

Of the order of the battell.

The order of battell is after the firste maner, but not so farre from the bordes end, namely the .4. squares standynge in the plattes nearest to the bordes end the rest accordingly joined to them, as in the first kynde of playing.

[Page]

[Note that, in this diagram, the odds are colored black. Also, it actually shows the odds and evens on reverse ends from this textual representation; I'm not redoing the whole thing right now...]

[25] [81] [169] [289]

[15] [45] <25> <20> <42> <49> [91]* [153]

<9> <6> (4) (16) (36) (64) <72> <81>

(2) (4) (6) (8)

(9) (7) (5) (3)

<100> <90> (81) (49) (25) (9) <12> <16>

[190]* [120] <64> <56> <36> <30> [66] [28]

[361] [225] [121] [49]

[Page]

Of their motions.

The men move frowarde and backward, to the right hand, and to the left hande, but not cornerwise, except the gamesters so agree, the rounds into the next space, the triangles into the thyrde, and the squares into the fourth, the kyngs move as squares. And these be their ordinary draughts in marching.

Of their taking.

They are taken by encountering, bu eruption, by laying wayght, and by Oblivion.

Of takyng by encountering.

To take by encountering is to take by Equalitie, as hath bene twyse before declared.

Of taking by eruption.

To take by eruption is when a lesse number beyng multiplied by the spaces that are betwene him & hys adversary, the product is asmuch as his adversary, he may take his enemie awaye whether he stand directly from him or cornerwise.

[Page]

For men that may be taken by eruption looke in the table of takyng by multiplication in the second kynd of playing.

Of takyng by deceypt or lying weyght.

To take by deceypt or lying weight, is to take by addition, not as before when the adversary standeth within the draught of two men which being added make the juste number of the adversary, but when the .2. numbers that are to be added, stande in the next spaces to the adversarie. For to take by deceipt, looke in the table that was set forth for takyng by addition in the first kynde of playinge.

Of taking by Oblivion.

By Oblivion all men may be taken, when foure men besiege the adversarye, standynge in the foure nexte

[Page]

spaces about him directly, or cornerwise, the man so besieged can not escape, because he can not remove cornerwyse, therefore maye be taken up, so soone as the last of the foure is set in his place.

In all three kyndes of playing no Oblivion can be of any man with some of his fellowes, but all foure muste be hys adversaries.

In this thyrde kynde, these men can be none otherwyse taken but by Oblivion. Namely amonge the even .2.4.4.135. among the odde .3.5.7.190.

In all maner of taking this is to be noted, that we muste not place the man which taketh in place of him that is taken, but when he maye be taken before we drawe, then shall we remove our man into his place.

The privilege of the king.

The king standeth for so many men as he hath steppes, that is the even for .6. the odde for .5. if anye of these

[Page]

(except the lowest and greatest) be taken the king may redeme hym, by any man of his that is of the same number. If he have none of the same number, he maye redeme him be any of his men that hys adversary wyll chuse. But if his lowest square be taken, no ransom will delyver him. Also if the whole kyng at ons that is the whole number of Pyramis be taken, he can not be redemed.

Of the triumphe.

To take awaye the tediousnes of long play from them that be yonge beginners, wryters of this game have invented divers kyndes of shorte victories, wherefore they devide victory into proper and common. Of the proper victory need nothing here be spoken, for all things thereto belonging are sufficientlu set forth in the first kind of playing.

Of the common victory.

The common victorie (they say) is after fyve maners, for men contende either for bodies, goods, quarrelles, honour, or els for both quarels & honor.

[Page]

Victory of bodies.

Victory of bodies is only to take a certain number of men, as if the gamesters agree, that he which first taketh .4. or .5. or .6. or .10. men &c, shall wyn the game.

Victorye of goods.

Victorie of goods, is to take a certain number without respect of the men. As if it be covenanted, that he which first taketh men amounting to the number of .100. or .200. shall have the victorie.

Victory of quarell.

Victorie of quarell is when neither the men, nor the number, but the characters of the number be considered. As if it be determined that he which first taketh .100. in .8. characters not regarding in how many men they standes, shall winne. As .2.4.5.8.24.64. so you have .100. in .8. characters it skilleth not, although there be more then .100. as in this example there is more then .100. by .4.

[Page]

Victorie of honour.

Victorie of honour, is whe a determined number is made in a determined number of men, as if it be determined that he whiche first cometh to .100. in .8. men, shall winne the game. As in these .2.4.6.8.4.16.45.15. And though there were somewhat more then .100. so it be in .8. men, it skilleth not.

Of victorie of honour and quarell.

The victorie of honour and quarell, is when one obteyneth the decreed number, in the decreed number of men and the decreed number of characters: as let .100. be the decreed number .8. the determined number of men, and .9. the determined number of characters. He that obteyneth .2.4.6.8.4.6.9.64. obteineth the victorie of honour and quarell. It shalbe no hinderance though .8.

[Page]

men and .9. characters conteyne somwhat more then .100. so that there be not .100. upon one man, as in the victorie before.

Victorie of standers.

They have invented another victorie, that is of standerdes, by counterfeyting two armies, one of the Christians, another of the Turkes. The whyte men, that is the even hoste, conteyneth .1312. footemen (not compting the rootes of squares expressed in the kynges) let the first and last be captaines and let them devide the whole armye into .10. standerds so every standerd shall have .130. men, besyde the two captaines and the ten standard bearers. The black men, that is the odde armie (except the kings rootes) be .1752. The two captaynes and ten standerd bearers taken out, there remayneth .1740. souldyers, to every standerd .174. He that wynneth more standers have the victorye. If the even hoste

[Page]

wyne .348. men he hath obtayned two standerds if he wynne .522. he hath gotten thre standerds and forth of the rest.

If the odde armye wynne .260. they wyn two standerds .390. three standerds and so of the rest.

Table of the victorye of standerds.

One standerd of the even, conteyneth 130.

Two standerds. 260.

Three standerds. 390.

Foure standerds. 520.

Fyve standerds. 650.

Sixe standerds. 780.

Seven standerds. 910.

Eyght standerds. 1040.

Nyne standerds. 1170.

Tenne standerds. 1300.

One standerd of the odde, conteyneth 174.

Two standers. 348.

Three standerds. 522.

Foure standerds. 696.

Fyve standerds. 870.

Sixe standerds. 1044.

Seven standerds. 1218.

Eyght standerds. 1392.

Nyne standerds. 1566.

Tenne standerds. 1740.

You maye use anye of these syxe kyndes of common victorie, in every one of the three kyndes of playing.

FINIS

Prynted at London by Rouland Hall,

for James Rowbothum, and are to

be solde at his shoppe in

chepeside under Bowe

churche.

1563.

***********

( above 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.

***********

Rithmomachia: The Game for Medieval Geeks (Culture)

In 965 Wibold, bishop of Cambrai, suggested to the local monks that they give up playing dice games and instead play "the battle between virtues and vices". In all likelihood this was the game rithmomachia, otherwise known as the ludus philosophorum. Today we are familiar with a number of ancient board games but unfortunately rithmomachia, as well as some related games Ouranomachia and Metromachia have all but disappeared

Rithomachia was played by the intellectuals of Europe, often Church employees and most likely male. Although it is obscure today, rithomachia played an important part in the lives of this elite group. The rules were complex and convoluted and required considerable computational skill. Its players also claimed that its benefits extended well beyond numerical training - it supposedly improved the moral character of players and even gave deep insights into religious truth.

There are many different versions of the rules of play but there are many common features. There are two players who take turns moving pieces on a board, much like chess or checkers. The board is something like a chess board, though rectangular and not square, with 48 pieces, each inscribed with a number. The size of board varies with rule set but eventually it was standardized on 8 by 16.

But now the complication starts. Rithomachia was inspired by a type of mathematics made popular by Boethius. This work was dominated by various types of numerical progression: in particular arithmetic progressions (where the nth number is a+b*n for constant a and b), geometric progressions (a*b^n) and harmonic progressions (1/(a+b/n)). There were also other more complex types of progression such as multiplex progression or the progression of superparticulars. At the start of the game the initial layout of the numbered pieces, in three ranks for each side, was such that the numbers formed various types of progression. One side had odd numbers in the first rank, the other side even. So even before the game had started quite a bit of mathematical knowledge was required. And don't forget that in earlier times arabic numerals were unknown so any calculations were carried out using Roman numerals. (By the way, these proportions played an important part in medieval music theory. Consider the harmonic progression in particular.)

Pieces from the first rank could move one square, those from the second rank two and those from the third three. As a mnemonic the pieces in these ranks were sometimes circles, triangles and squares respectively (a slightly illogical mnemonic!). Like in chess, pieces could be captured, however there were many ways to capture. One way was to move a piece with a number onto another with the same number. Two pieces could take another if it was the case that if they were simultaneously to make legal moves that would land them both on the captured piece and the sum of the numbers on the two pieces sums to that of the captured one. Another way to capture was to occupy all the spaces that another piece could move to making it unable to move - this was called besieging it. Here is another example of a capture rule taken from Lever and Fulke's "The Most Noble and Auncient, and Learned Playe" published in 1563:

Of taking by cossical signs

By cossical signs: any man that hath these signs, 3, &, 33, 3&, meeting with his root in his ordinary draught that hath this sign z taketh him up, or else is taken of his, without removing into his place; except he may not take him before he remove.

Obviously there has been a little semantic drift over the centuries. There were also some much more complex capture rules requiring the pieces to be arranged in a progression.

Now we've warmed up we can get onto the conditions to achieve victory in the game. Typically a player had to line up a series of pieces in certain arrangements, often as a Boethian progression of a certain length. Whoever did this would win. There were many different choices of victory conditions based on such progressions and players would negotiate before to game to decide which were in play for a particular game. An example simple victory condition follows:

Victory of goods

Victory of goods is to take a certain number without respect of the men. As if it be covenanted that he which first taketh men amounting to the number of 100 or 200 shall have the victory.

There was even a victory condition based on simulating the armies of the Christians and Turks at war.

As you can see, this was no game for weenies and yet it was played all over Europe from Medieval times, through the Renaissance into Elizabethan times. Many rithomachia manuals still exist today and doubtless many more were originally published. It seems to me there is only one explanation: Rithomachia is a geek game and the players were predecessors of today's game playing geeks! They spent their time shut up in dark rooms hunched over books and games believing themselves to be superior to the masses because they were experts in difficult and arcane, but largely useless knowledge.

Today Boethian mathematics is almost unheard of. This is the key to the decline of rithmomachia. Boethian mathematics was highly technical but today it's abstruseness seems completely arbitrary and useless. As it was replaced by more modern approaches to mathematics the rules to rithmomachia came to seem more and more arbitrary until interest in the game completely waned. However it is worth noting that this game did have a lifetime of 500 years and was played by such illustrious luminaries as John Dee and praised by Roger Bacon.

***********

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).

***********

More Rythmomachia and techniques

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).

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". 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. 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".

Also Roger Bacon comment the Arithmomachia, in its "mathematica Communio". 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.

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... 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 9.25.49.81 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 9.25.49.81 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 49.121.225.361 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

***********

Number Games ; Rythmomachia / Rithmimachia

Aritmomachia::

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).

Rythmomachy:

Ritmos = Number

Mache = Battle

Latin: Pugna (certamen) numerorum

Board:

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..."

Origin:

Pythagoras of Samos, Nicomachus of Gerasa, Anicius Manlius

Severinus Boetius. Six levels of numbers.

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

Constillation:

Roman militairy; Vegetius (writter: Flavius Vegettos

Renatus), Belisar's battle formation.

CAVALARY CAVALARY

PHALANX

INFANTRY & CAVALERY

***********

Books::

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

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

1.) OXFORD HISTORY OF BOARD GAMES by James Parlett

2.) BOARD AND TABLE GAMES FROM MANY CIVILIZATIONS

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.

Further References

Borst, A. 1986. Das mittelalterliche Zahlenkampfspiel. Supplemente zu den Sitzungsberichten der Heidelberger Akademie der Wissenschaften, Philosophisch-historische Klasse, vol. 5.

Heidelberg: Carl Winter Universitsverlag

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

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.

***********

THE RULES OF

BOOLEAN RITHMOMACHIA

(A New Twist On An Old Game)

Boolean Rithmomachia

by L. Lynn Smith

The following game is in homage to the Medieval game of numbers. It was thought

that the ideas of this game should be updated and could even be utilized in the

teaching of computer math.

PLAYING FIELD

In order to bring this game into the 21st Century, it was thought that it should

be played upon a 3D field. To make this play as simple as possible, a 4x4x4 field

was selected.

PIECES

The playing pieces consist of two of each of the following. Such pieces, except for the Pyramids, are two sided and have different colors of either side, black or white, but the same value(1 - 14) represented by the binary code. The Pyramids are whole pieces of solid color, two black and two white, one of each particular value(0 and 15) represented by the binary code.

Value Shape

----------------

0000 Pyramid

0001 Circle

0010 Circle

0011 Triangle

0100 Circle

0101 Triangle

0110 Triangle

0111 Square

1000 Circle

1001 Triangle

1010 Triangle

1011 Square

1100 Triangle

1101 Square

1110 Square

1111 Pyramid

SET-UP

The pieces of each player are arranged in two seperate planes, seperated by two planes. Players can occupy opposite sides of the playing field, either horizontally or vertically, in the following manner:

[S][C][C][S]

[T][T][P][T]

[T][P][T][T]

[S][C][C][S]

C=Circle P=Pyramid S=Square T=Triangle

Players have the option of which values may be placed in their respective cells. This form of set-up could be considered one phase of the game as each player may take turns placing a piece upon the field. Or they may agree to a standard form of set-up.

MOVEMENT

Circles move one orthogonal or one diagonal, Triangles move one diagonal or one triagonal, Squares move one orthogonal or one triagonal, Pyramids move one orthogonal or one diagonal or one triagonal. The Pyramid may by landing on any piece then move that piece(regardless of owner) by any normally legal Pyramid step to an adjacent empty cell, such a piece does not change ownership solely by this action but may if it then becomes part of a capture equation. All other pieces must move to empty cells.

[The triagonal movement is a move from one cell to the next cell which is adjacent by

only one of its eight corners. Whereas, the orthogonal move is a change along one axis and the diagonal move is an equal change along two axes, the triagonal move is an equal change along all three axes of the 3D field.]

CAPTURES

The capture of opponent pieces involves flipping the piece to its opposite color. Pyramids are removed from the playing field when captured. The game is over when one player no longer has possession of either Pyramid.

Captures are preformed by legally moving a particular piece then flipping all appropriate enemy pieces.

EQUATIONS

Captures are preformed by applying Boolean equation to and through adjacent pieces.

Capture by the NOT equation. When the appropriate enemy piece is located in a cell which, if empty, would be a legal move for the capturing piece and it is its NOT equivalent, that enemy piece is then captured.

NOT equivalents:

0000 - 1111

0001 - 1110

0010 - 1101

0011 - 1100

0100 - 1011

0101 - 1010

0110 - 1001

0111 - 1000

1000 - 0111

1001 - 0110

1010 - 0101

1011 - 0100

1100 - 0011

1101 - 0010

1110 - 0001

1111 - 0000

This is the simplest take as the attacking piece needs no assistance in capturing its

opponent. Examples: Circle(0001) is able to capture Square(1110) if it becomes

orthogonally or diagonally adjacent. Triangle(0110) is able to capture Triangle(1001) if it is diagonally or triagonally adjacent.

The remaining captures involve the use of two pieces by which to take the target. It is necessary that the second piece used in this form of capture be also owned by the player.

These captures take two forms, Compiling and Filtering.

Compiling, each of the two pieces used to determine the capture must be adjacent to the target piece by their individual legal move.

Filtering, the played piece must be legally adjacent to one which is legally adjacent to the target.

Several takes can be made during a player's turn, if the pieces and positions allow.

There are five forms of operations which can be preformed in order to make such captures; AND, OR, XOR, NAND and NOR.

The following table can be used as a quick reference during the game:

-----------------------------------------------

X Y AND OR XOR NAND NOR

-----------------------------------------------

0 0 0 0 0 1 1

0 1 0 1 1 1 0

1 0 0 1 1 1 0

1 1 1 1 0 0 0

-----------------------------------------------

In such equations, the played piece will be referred to as X, the Compiler or Filter piece will be Y and the target piece will be the answer. If the answer is True then the piece is captured, if False the piece is left as is.

If X = Circle(0100) and the Compiler or Filter = Triangle(1100) then the target must equal the Circle(0100) to be captured using the AND Equation.

Using the OR Equation, the previously stated values could capture Triangle(1100).

Using the XOR Equation, they could capture Circle(1000).

Using the NAND Equation, they could capture Square(1011).

Using the NOR Equation, they could capture Triangle(0011).

The Pyramid is also used to preform such captures. If its moves involves the displacement of a piece, it still may preform the appropriate captures moves. The displaced piece can become either a Compiler, Filter or even a Target.

All captures are mandatory, it is the obligation of either player to assist the other in determining any possible captures which may have resulted from a particular move. Neither player is obliged to suggest any particular move to an opponent.

WINNING THE GAME

When an opponent's last Pyramid is captured, or there are no legal moves or potential captures, victory is determined thus:

Pyramid Victory:

The player who has the most number of Pyramids remaining on the playing field.

Piece Victory:

The player who has the largest number of pieces, including Pyramids, on the playing field.

Value Victory:

The player whose value of pieces, including the Pyramids, on the playing field is the highest total.

The Simple Victory consists of one of the above victories.

The Compound Victory consists of two of the above victories.

The Perfect Victory consists of all three victories.

It is possible that not all of the victories will be available. Example: Both players may have the same value of pieces on the field but not the same number, so neither would be awarded the Value Victory.

Addendum 01.03.2002

It is not necessary to use all formulae in every game. The players may choose to play a NOT-NAND game, a game using the NOT and NAND equations. Or they may play an OR-XOR-NOR game. The desired formulae to be used should be clearly stated before the start of each game.

Addendum 07.11.2002

Allowing the Pyramid to "push" another piece can be optional. Denial of the "push" can increase the opportunity for the game ending with a player having Pyramids but no legal moves. There is no direct penalty for a game ending in such a manner.

--------------

Addendum 02.16.2003

Here is an interesting way to select the equations to be used during the game: Each player secretly writes down two choices, then simultaneously reveals them. This allows for the possibility of either a two-function, three-function or four-function game.

** END **

Subscribe to:
Posts (Atom)