On the Fyrst Plaie of Rhythmomachia
-----------------------------------

Good Barones, it has bene some months since laste I wrote to Your
selves about the plaie of the games. I doo beg your humble pardon for
this, that I have bene in Englande, helping the plan and array of their
Revels. But I am now returned to Calais, and tak pen once more to
continue to tell of the play of The Philosopher's Game.

And I shal not repeat all that I have sayed before, save to say again
that eche player shal have .24. men, .8. Round, .8. Triangle, and
.8. Square. And the numbers uppon these men are as I said before.

Now shal I speke of the fyrst plaie of the game. Know that there are
some three schools of Philosophers who play at it today, and these
eche play with some difference. I shal speke of eche in its turne;
yet know that eche of these plaies are much alike, and that whych
is lerned for the one game shal be of muche use in the others.

In thys fyrst plaie, the men are arrayed upon the board muche as
I shewed in the picture I did send to you before, save with one
difference, that the men begin advanced by two ranks. That is,
your men begin on your rows .3.4.5.6., and your enemies men
lykewise. Yet are they else as in the picture.

Eche kynde of men shal move with different draughts, and so I
shal deal of eche in its turne. And fyrst, there are the Roundes.
The Roundes doo remove into the nexte space diagonally, that is
slantwise, and theye make this move at all tymes.

Nexte are the Triangles, whych cannot move diagonally, but only
orthogonally, that is front and back and sidewise. And they do
move into the thyrd space, counting their own; that is, they
leape over one space into the nexte. Yet they cannot leape over
another man, whether your owne or the enemy, but can only leape
over an empty space.

And the Triangles have another move, call'd theyr flying draught, and
this maye be used unless their move captures an enemy, or is moving to
make a Triumph. And the flying draught is as the move of the Knyght in
Chesse. When in flight, the Triangle moves with its usuall draught,
but also makes one move to the syde from that.  That is, into the
thyrd space frontward and then into the nexte space to the syde, or
the thyrd to the syde and into the nexte space backe. Yet in this
move, he maye yet not jumpe over another man, where the Chesse knight
maye.

And the Squares move muche as the Triangles, yet one further, so they
move into the fourthe space (counting his owne) front, backe, or too
the syde, leaping over twoo empty spaces. And lyke the Triangle, he
may not leape over another man. And lyke the Triangle, he maye move
with a flying draught as well, whych is his usuall move and then one
space to the syde: as, into the fourthe space frontward, and then one
to the syde.

The Kyng is made up of Round, Triangle, and Square, and so hee maye
move as any of these men maye. Yet yf he loseth all of a shape within
hys pyramide, then hee maye no longer move that waye: as, yf he loseth
bothe Triangles, hee maye no more move as a Triangle.

And the nexte subject is the capture. Knowe that there are many wayes
to capture in the Rhythmomachia, by Equalitie, Addition, Subtraction,
Multiplication, Division, and Oblivion.

And there are twoo wayes to capture in these. As in the Chesse, you
maye leape upon the ryght enemie man. Yet as unlike Chesse, you maye
also capture hym by placing your men in array against hym: that is,
yf you move your men that bothe of them maye be moved into the
enemies space in their usuall draught (but not their flying draught),
then the enemy is taken straightway, without needing to leape upon hym.
And I shal show this more below.

The fyrst capture is that of Equalitie, and thys is the only waye in
whych you maye capture the enemie with but one of your men. And a man
maye capture an enemie with Equalitie yf the enemy has the same number
as hee hymself. Thus, the Rounde blacke .25. maye take the Triangle
whyte .25. by hymselfe. And thys hee maye do in two wayes. Fyrst, hee
maye on hys move leape upon the enemy, and thus take hym. But more
often, hee will move so that the enemie is within his draughte, and
is thus taken. So yf the Round .25. moveth so that the Triangle .25.
is nexte to hym slantwyse, so is the Triangle taken at that moment.
And Lykewise, yf the Triangle .25. moveth so that the Rounde is in
the thyrd space from hym to the syde, so is the Rounde taken at
that moment.

The second capture is that of Addition, and requires twoo of your men
to capture one of the Enemy. And your twoo men muste have numbers that
summ exactly to the Enemy; as, the Even .16. and .9. maye capture the
Odde .25., as they summ to hym. And again, they maye capture in two
wayes. Fyrst, one of your men maye leape upon the enemy to take hym.
But more often, you shal move your men, so that the enemy lieth within
bothe of their draughts. As (yf you are Even), you do move so that the
Odde .25. is nexte to yr Rounde .16. slantwise, and in the thyrd
space from yr Triangle .9. forwarde, then is the .25. taken at that
moment.

And the captures of Subtraction, Multiplicatyon, and Devision are
likewise to that of Addition: you shal move twoo of your men, which
combine to the enemy man, so that he lieth within bothe of their
draughts. As the .25. and .20. maye capture the .5. by Subtraction,
the .9. and .9. maye capture the .81. by Multiplication, and the
.225. and .9. maye capture the .25. by Division. And as in Addition,
you maye also capture the enemy by leaping upon hym with either man,
so long as the enemy lieth also within the other's draught.

It is rare that you wyll capture by leaping upon the enemy, yet it
will happen in two instances: that you have missed an earlier chance
to take hym by moving your men to endanger him, and yet your Adversary
dyd not move him from the danger, overlooking your opportunity, or yf
the enemy moveth hys man so that he is endangered by bothe of your
men. Yet bothe of these require you or yr Adversary to make an error,
so it happeneth rarely among players of experience.

And some shal playe so that you maye also Capture by Proportion,
either Arithmetic, Geometric, or Musicall, whych are spoken of
below. In this playe, you shal move twoo of your men against an enemye
lieing in their draughts, so that the three men together make a
Proportion, and so the Enemy is taken. Yet this is accompted an
advanced playe, not for beginners, and you should agree wyth yr
Adversary if you wish to so plaie.

And there is yet another Capture, whych is different from the others,
that of Oblivion. And you make thys capture by placing your men
around the Enemy, so that all foure of hys draughts are hindered.
When your men are arrayed so that he maye not move, then is he
taken. And yf he lieth by the wall of the board, then you need but
surround him by the other sydes. Yet hys owne men will save him from
Oblivion, not hinder him, so yf the Enemy is hindered on twoo sydes
by you, and twoo by his owne allies, he is not taken.

The Kyng maye be captured with hys whole number, that is, with the
summ of the pieces he is made up of (thus, as .190. for the Odde
Kynge and .91. for the Even). But hee maye also be taken piecewise.
Yf you arraye your pieces against the Kyng so that you maye take
one of the pieces from oute of him, so that part of the Kyng is
taken. And manie plaiers think it easier to take the Kyng by parts
than as a whole. And the Kyng maye worke to capture men of the
Enemy campe, with his whole number or with his partes. Thus, the
.36. within the Odde Kyng maye capture the Even .36. by Equalitie,
or helpe to capture other men by Addition, Subtraction, or other
means.

In all of these Captures, when you take an Enemy, hee turns coate,
and begins to ply for you instead. You shal take the Enemy from the
Board, and turne hym on his backe, so that he shews your colour. And
then you shal place hym at your end of the board, in the fyrst rank,
in any space of that row. And from then, hee shal be your man, unless
taken again by your Adversary.

And nexte I shal tell you of the Proper Victories. Yet fyrst I must
tell of the Proportions Arithmeticall, Geometrical, and Musicall.
Eche proportion is three numbers, a lowest, a highest, and one between,
suche that these three do share a common relation. And these are the
relations that these numbers maye have. And you shoulde know that you
maye playe without these Proportions yf you playe only wyth the Common
Victories, whych I shal describe later.

The simplest Proportion is Arithmeticall. In thys proportion, a common
number separates the highest from the middle, and the lowest from the
middle. As, the three numbers .5.25.45. make an Arithmeticall Proportion,
for the .5. and .25. are separated by .20., and so are the .25. and the
.45.

The nexte Proportion is the Geometricall. In thys, a common multiplier
separates the highest from the mid, and the middle from the lowest.
As, the three numbers .4.16.64. make a Geometricall Proportion, for
.4. multiplied by .4. maketh .16., and .16. tymes .4. maketh .64. And
note that the multiplier may on some occasion be Fractionall, as in
the relation .4.6.9., for .4. tymes one and one half maketh .6., and
.6. multiplied by one and one half maketh .9., so they are in
Geometricall Proportion.

The laste Proportion is the Musicall. Thys is the Proportion of the
Harmonies, and in it, a common multiplier separates the greatest from
the least, as separates the difference between the greatest and middle,
and the middle and least. Thus, the numbers .5.9.45. maketh a
Musicall Proportion, for the .45. divided by the .5. is .9., the
difference between the .45. and .9. is .36., the difference between
the .9. and .5. is .4., and .36. divided by .4. is again .9. And the
Musicall Proportion is accompted the hardest with whych to playe, yet
the best plaiers do make use of it.

And know that there are Tables of these Proportions, and these is no
shame in using these tables to decide upon your Victories.

The Proper Victories are three in number: the Great Triumph, the Greater
Triumph, and the Greatest Triumph, and they are muche alike.

For the Greate Triumph, you shal first capture thy Enemies Kyng, either
as a whole or by its partes. Afterwards, you may declare that you shall
begin to prepare yr Triumph, and whych men you shal use for this Triumph.
These muste be three men, whych together do forme one of the Proportions.
>From thence, thy Adversary maye not capture these men, yet the men muste
move only by their normal draughtes, not their flying ones. You shal move
these men into the enemies campe, whych is the spaces in whych hys men
did begin, that is, his rearmost six rowes. In that campe, you shal
arrange your three men into a row, lengthwise, sidewise, or diagonall.
When these men are so arrayed, your Triumph is complete. Yet note that
you muste begin your men carefully, in good place, for without the
flying draughte, you may find it impossible to so place your men.

The Greater Triumph is muche alike, save that you muste choose foure
men instead of three, and these foure muste form two of the Proportions,
as Arithmetic and Geometric, Geometric and Musicall, or Arithmeticall and
Musicall. And they maye be arrayed in a line in the Enemie Campe, or maye
be arrayed as a square wyth a space separatyng eche, as the corners of a
square of .3. by .3.

And the Greatest Triumph is as the Greater, save that your foure men muste
betwene them make all three of the Proportions.

Yet, while the Proper Victories are thought the proper waye to playe the
game (for so are they named), men have created also the Common Victories
for lerning the game more easily. And while I doubt not thy wisdome, I
shal recommend unto you that You learn first wyth these Common Victories,
that you not grow frustrated wyth the game before you learn it fully. For
the Common Victories are far easier to achieve than the Proper, and allow
the player to learn the movement and forme of the game without needing
all of hyt at once. And there are five Common Victories: of Bodies, of
Goods, of Quarrel, of Honor, and of Honor and Quarrel.

The easiest of alle is the Victory of Bodies. In thys, the two
gamesters shal agree to playe until one player has captured a number
of men, as the fyrst to capture .6., or .8., or .10. of hys
Adversaries men.

The nexte is the Victory of Goods. In this, they shal agree to playe
until one has captured a set sum of the numbers upon the men, as
the first to capture men totalling .100. or perhaps .200.

Nexte is the Victory of Quarrel, in whych the gamesters agree upon
a sum and also a number of characters upon the men. As, the .8. has
a single character, the .24. has twoo, and the .225. has three. And
so the players maye choose to stake upon a sum of .100. with at
least .8. characters.

The Victory of Honor is that of Bodies combined with that of Goods.
So the gamesters maye choose to laye upon the fyrst to capture .8.
men wyth a total of at least .150.

The Victory of Honor and Quarrel takes all of the above into itself.
So in this, they maye choose a goal of .8. men, .12. characters, and
a sum of .200., or some other as they shal choose.

And thys is the full account of the fyrst playe of The Philosopher's
Game. There are yet two others, yet they are muche alike, and their
telling shall be shorter, for you have here the knowledge of howe
the logic of the game is accomplisht. I shal write you about these
others at a later daye, and commend this fyrst playe to you to begin
with. I remain your servent, Justin duC, this warming Aprill the
Fourthe, in the Yr of Our Lorde 1598.

Endnotes
--------

I assume here that you have read the last chapter, which describes the
basics, and which was published several months ago. If not, it can be
found online at:

	http://www.inmet.com/~justin/ace_rythmo1.txt

This is my personal favorite variation of Rhythmomachy, but I must
caveat that it is the rarest; I have no reason to believe that it
is attested earlier than the 16th century. It is quite different in
its internal logic than the more common medieval forms, which are much
more like Version 3, which I will describe later. This version is more
consistent and balanced, I believe, although it lacks some of the wild
and interesting variation of the more-common form.

Note a possible source of confusion in the descriptions of movement,
due to period terminology. In period, descriptions of movement usually
included the space the man is currently sitting on. Thus, moving to
the "third" space means jumping over just one -- what we would today
usually call moving into the second space. I've tried to be clear in
the above, but watch out for it.

I've tried to make the rules for capture clear above; hopefully I've
succeeded. I will call out one particular point: you must move one of
the men involved in order to capture an enemy. From the original, I'm
pretty sure that an enemy who moves into range is not automatically
captured; you must jump onto him. And I believe that, by the same
token, you can't set up a position just by moving one of your own
blocking pieces out of the way. One of your two pieces that are
involved in the capture must be moved for the capture to take effect.

A number of caveats are in order for this reconstruction; while Fulke
is mostly pretty clear, there are a number of ambiguities that I haven't
yet resolved to my satisfaction. This description is made up of my best
guesses in those cases.

First of all, the term "flying" isn't completely clear. It is clear that
the flying moves can't be used to capture an enemy, or to move towards
the Triumph; however it isn't obvious to me that it can be used in all
other cases. I am assuming that it can.

The exact connotations of the flying move aren't clear either. Fulke
describes it as being like the chess knight's move, but it isn't clear
whether they thought of that as two forward and one to the side, or
one forward and one diagonally. The difference matters, since it decides
whether a man can make his flying move when his normal one is blocked.
I have chosen to believe not, based on the way Capture by Oblivion works.
(It isn't strictly clear that the flying move can't jump over other
men, either, but I believe that to be the case.)

Use of the King is a point of considerable ambiguity. It is not
actually clear from Fulke that the component pieces of the King can be
individually used to capture (although they certainly can be captured
individually); I like to think so, but that raises questions of its
own. For example, can the King, say, move like a Round, but capture
with one of his Squares? I have no pat answer to this, but it seems a
reasonable limitation to say that, if the King is used to capture
with, say, the number on a component Triangle, then it must be a
Triangle's movement from the enemy piece.

Capture by Oblivion is also a bit ambiguous -- Fulke only says that
you must hinder the enemy. It seems unreasonably easy to be captured
this way if your own men hinder you (especially since many pieces
start out entirely surrounded), and unreasonably hard if standing
against a wall entirely saves you from being so captured. The above
description, therefore, seems the fairest way for things to work.

I don't believe that Fulke intended for the Common Victories to be
used with this variant; it looks to me like he really only meant the
Proper Victories to be used here. However, I include the Common
Victories because I do think they are a bit easier and less daunting,
so I commend them to beginners. Note that I suspect that a period
player would not expect captured men to turn coat if you are playing
to a Common Victory -- the main point of turning coats is to enable
more of the Proper Victories to be achievable. However, it makes the
game more interesting and exciting, so I leave both in. I do recommend
trying out the Proper Victories once you feel up to it.

The Proportions, in concise modern notation, run like this. Each
has three numbers, A, B, and C.

Arithmetic: (B - A) = (C - B).

Geometric: (B / A) = (C / B).

Harmonic/Musical: (C / A) = ((C - B) / (B - A)).

Fulke includes enormously comprehensive tables of all of the various
mathematical combinations involved in the game: all of the possible
Proportions, as well as all of the possible captures by Addition,
Subtraction, Multiplication, and Division. I commend them to you;
they can be found in the online transcription of Fulke, at:

	http://www.inmet.com/~justin/fulke.html

Next time: the second version of the game, and maybe the third as well.

				-- Justin