Archimedes' Stomachion.pdf

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SCIAMVS 5 (2004), 67-99
Towards a Reconstruction of Archimedes'
Stomachion
Reviel Netz
Stanford University
Fabio Acerbi
Venzone
(un),
Italy
Nigel Wilson
Lincoln College, Oxford
In memoriam David Fowler
Abstract and Plan of Paper
The
Stomachion
is the least understood of Archimedes' works. This paper provides
a reconstruction of its goal and structure. The nature of the evidence, including
new readings from the Archimedes Palimpsest, is discussed in detail. Based on this
evidence, it is argued that the
Stomachion
was a treatise of geometrical combina-
torics. This new interpretation is made possible thanks to recent studies showing the
existence of sophisticated combinatorial research in antiquity. The key to the new
interpretation, in this case, is the observation that Archimedes might have focussed
not on the possibility of creating many different figures by different arrangements
of the pieces but on the way in which the same overall figure is obtained by many
different arrangements of the pieces.
The plan of the paper is as follows. Section 1 introduces the
Stomachion.
Sec-
tion 2 discusses the ancient testimonies and the Arabic fragment, while Section 3
translates and discusses the
Gr~ek
fragment. Section 4 sums up the mathematical
reconstruction offered in this paper, while Section 5 points at the possible intellec-
tual background to the work. Appendix A contains a transcription of the Greek
fragment, appendix B an English translation with redrawn diagrams, appendix C a
reproduction of the digitally enhanced images of the pages of the palimpsest con-
taining remains of the
Stomachion.
I
The Puzzle
The
Stomachion
is something of a poor relation. Take for instance Dijksterhuis' book
[1987]:1
it goes through all the works extant in the main Greek manuscript tradition,
providing for each a detailed analysis. The only exception is the
Stomachion,
which
Dijksterhuis consigns to the 'Miscellaneous'-together with works known through
testimony or Arabic translation alone. His four pages of commentary are devoted
IThe
Stomachion
is commented on at
pp.
408-412.
68
Reviel Netz,
Fabio Acerbi
and
Nigel Wilson
SCIAMVS 5
more to the nature of the evidence than to the treatise itself (true, the nature
of the evidence is especially complicated in this case, as we shall see below). As
to the contents, Dijksterhuis expresses caution growing into frustration: "it
[scil.
the traditional name
loculus Archimedius]
may indicate that he studied the game
from a mathematical point of view [... ] he calls it necessary to discuss some of
the properties of the so-called Stomachion [... ] In the Greek fragment, however,
we do not find much about this investigation". His conclusion, following upon a
discussion of the Arabic fragment, is that it "can no longer be ascertained whether
this result was the object aimed at or whether it played a part (and if so, what
part) in the investigation as originally announced".
2
In short, Dijksterhuis-in his
typical sobriety of judgment-offers us no indication of what the work, in his view,
was· about.
Knorr's detailed bibliography
[1987]
of studies of Archimedes since Dijksterhuis'
original pub.1ication in
1938,
contains a single entry related to the
Stomachion,
which
has to do with an additional ancient testimony-not to Archimedes himself, but to
the game he was studying [Rose
1956].
Only a handful of studies of the
Stomachion
have been published before or since, none of them going much beyond a summary
of the interpretation in Heiberg.
3
The
Stomachion
was such a poor relation already back in
1907,
when the great
Danish philologist J.L. Heiberg published in
Hermes
his article presenting the sen-
sational find, in Istambul, of a new Archimedes manuscript, the Palimpsest [Heiberg
1907].
Heiberg devoted almost all that article to a preliminary transcription of the
Methodus,
which is of course a work of the greatest importance in the history of math-
ematics (and none of its text known at all prior to the discovery of the Palimpsest).
Heiberg merely mentioned in passing that another text was also read in the same
manuscript for the first time-the
Stomachion.
Heiberg's few comments there were
. dedicated more to the title of the work than to its contents [Ibidem:
240-241].
He
postponed the edition of this small fragment to his major edition from
1915,
and
whatever scholarly interest could have been concentrated on the
Stomachion
was
drowned in the wave of research into the
M ethodus.
Of course, Heiberg's neglect, as that of later scholarship, was the result of there
being so little for us to study. The fragment of the
Stomachion
preserved in the
Palimpsest occupies less than a single page, and contains no more than the in-
troductory passage followed by a little over a single, small, theorem. This was the
Stomachion's
original misfortune already in the thirteenth century: so little of it was
2[Dijksterhuis 1987], quotations from pp. 410 and 412, respectively.
3[Minonzio 2000] is mainly a commented collection of data about the Latin sources mentioning the
game. The paper shows that some traditional ascriptions of the Latin sources concerning the game
are untenable, but the mathematical commentary adds little that is persuasive. The translation of
the Arabic fragment (made by
1.
Garofalo) is valuable and better than Suter's.
SC1AMVS 5
Archimedes'Stomachion
69
used by the maker of the Palimpsest. One would like to know how much longer the
treatise was originally. Now, in the original Archimedes manuscript the
Stomachion
was the last in the sequence of the works (at least as far as extant parts go). The
single extant page of the
Stomachion
constituted, almost certainly, the fifth page
in an original quire, so that the work probably should have had at least three more
pages. More can be said. We may compare this-the end of the original Archimedes
book-to its beginning, where the first in the sequence of the works is
De planorum
aequilibriis.
Remarkably, only the very
end
of
De planorum aequilibriis
is preserved,
in a little over two pages from the beginning of a quire. The work surely began much
earlier, and so it seems that the entire first quire of the original Archimedes book
was discarded by the maker of the Palimpsest. Note, however, that the same maker
has used at least some part of all the following quires as far as the quire containing
the
Stomachion
page. There was a special decision, then, to discard the first quire.
4
Symmetrically, it appears quite possible that the maker has discarded the entire
last
quire of the Archimedes book. Both omissions, of first and final quire, are easy to
understand (we shall see that, for the very same reason, the
Stomachion
is very
difficult to read today). The extremities of books are the first to decay.
It
then
follows that the
Stomachion
could have had as many as nearly twelve pages (though
of course another small work could have intervened to end the original sequence, or
not all of the last quire was used). This may be compared with a little under twenty
six pages used by the First Book of
De sphaera et cylindro
(a long work, some 160
pages of Greek and translation in Heiberg's edition) or the eight pages used by the
Second Book of
De sphaera et cylindro
(a short work, some 60 pages in Heiberg).
In short, it is likely that the
Stomachion
was a respectable-sized work, of as many
as some 90 pages in Heiberg, and that we have less than ten percent of it extant in
Greek.
II
Pieces for the Solution: Before the Palimpsest
Several sources-all later than Archimedes himself-refer to a game called the 'Stom-
achion' ('the Belly-Teaser': attested in Archimedes' Greek fragment and some read-
ings of Magnus Felix Ennodius
5
and Decimus Magnus Ausonius
6 )
or the 'Ostoma-
chion' ('the Bone-Battler', other readings of Ennodius and Ausonius), perhaps even
'Suntemachion' ('the Slice-Fitter', perhaps to be read in the Arabic fragment of
4
And
actually the first quire only, since the dimensions of
De planorum aequilibriis
exactly fit the
missing pages.
5Magnus Felix Ennodius (474-521 A.D.),
Carmina
11.133, title (Hartel).
6Decimus Magnus Ausonius (IV century A.D.),
Cento Nuptialis,
p. 147.39-56 (Green).
70
Reviel Netz,
Fabio Acerbi
and
Nigel Wilson
SCIAMVS5
Archimedes, see below).7 The consistency across time and space is remarkable:
from at least as early as Archimedes himself, through Lucretius,S and down to the
sixth century (date of Ennodius', latest datable testimony), Mediterranean children
played a kind of tangram or 'Chinese Puzzle'. This was rigidly defined by a set of 14
pieces,9 ideally made of ivory (Ennodius, Asmonius, Cresius Bassus), that could be
fitted to form either a square (implied by the Arabic fragment of Archimedes, stated
in Lucretius and Cresius Bassus), or alternatively-and much more prominently in
our literary sources-the figures could be fitted to form many fantastic shapes so
as to suit the player's imagination (Ausonius, Asmonius, Cresius Bassus, who all
repeat what must be a
topos:
the
Stomachion
as a metaphor for the way in which
many prosodic combinations are possible from the same building-blocks). In the
first case of forming a square, this was a game of patience and spatial intuition; in
the second case of forming many fantastic figures, this was a game of creativity. This
distinction would be crucial below, to our understanding of the game as studied by
Archimedes.
The impression made by the ancient testimonies is that the game, in antiquity,
was meant for young children. Perhaps Archimedes' treatise is the work of a young
father.
Moving from the game to the treatise itself, we have two extant fragments. One
is in the Archimedes Palimpsest, and will be discussed in the next section.
The other is preserved inside Arabic mathematical collections where a brief
kitiib,
'treatise', is explicitly said (by a
17th
century scribe?) to be by Archimedes [Suter
1899].
The title provided is 'on the division of the
stmiisyiin
10
figure into fourteen
7Heiberg and Dijksterhuis worry about the variant spelling in various sources, trying to establish
the 'correct' one, but it is in the nature of such objects of popular culture to go through variant
spellings and etymologies. 'Stomachion' is the spelling used in the Greek fragment by Archimedes
himself and shall be used here.
8
De
rerum natura
11.776-787, see [Rose 1956].
9The number 14 is implied by the Arabic fragment of Archimedes, as well as Ausonius and the late
grammarians lElius Festus Asmonius (IV century A.D.-Aftonius was thought to be his name for a
long time until the manuscripts containing his works were better investigated) and Cresius Bassus
(I century A.D.). The last two can be found in
Grammatici Latini
VI, pp. 100-101 and 271-272
(Keil), respectively. These two authors refer to the game as
loculus Archimedius,
'Archimedean
box'. The traditional ascriptions to Marius Victorinus and Atilius Fortunatianus were proved to be
wrong long since; scholarship after Heiberg has nonetheless insisted on sticking to the traditional
names. See the discussion in [Minonzio 2000, part II].
10
Semitic
writings, of course, under-determine many of the vowels. Suter suggested to read this
as Greek
crUVTEf.l<XXWV,
'Piece-Fitter', which Heiberg doubts, preferring to see in the Arabic title a
simple rendition of
crl:O!-l<XXwv:
but notice that Classical Arabic avoided as a rule such consonant
clusters as /st/ at a syllable's onset.
SCIAMVS 5
Archimedes'Stomachion
71
figures in relation to it'. The text is less a treatise and more a single proposition.
It
offers an explicit division of a square into 14 parts.l1 This explicit division is
combined throughout with a calculation of the fraction each of the parts of the
square is of the whole.
12
It
is useful to note immediately the resulting fractions:
1/16, 1/48, 1/6, 1/24, 1/24, 1/12, 1/12, 1/24, 1/48, 1/24, (1/2)(1/6)+(1/2)(1/8),
1/12, 1/12, 1/12. The text explicitly asserts that the goal is to show that all parts
stand in a rational ratio to the square, but one should note immediately how trivial
the result is: clearly, this could not be the original goal of the discussion.
In fact, one is struck by the (i) great ease, (ii) redundancy and (iii) poor expression
of the result ascribed to Archimedes. (i) As for great ease: there is hardly any
geometrical argumentation at all in the treatise, and the reported values follow
directly from the construction, all based on repeated applications of
Elementa
VI. 1.
13
(ii) As for redundancy, it is crucial to see that there is no need to calculate explicitly
the fractions of the parts so as to show that they stand in a rational ratio to the
square. All parts result from the successive bisection or trisection of lines that serve
as bases of parallelograms or triangles; that such bisections or trisection of lines
automatically result in bisections of the respective parallelograms or triangles is
the claim of
Elementa
VLl. But once this is taken for granted, one can show in
general that such a process of division is bound to result in rational ratios (each
division always creates either half or a third of the original), and it would have
been much mOJe elegant simply to prove the general rationality in this direct way.
(iii) As for poor expression, note that the fractions, with one exception, have a
property much stronger than 'rationality': they stand to the square not merely in
the ratio of a number to a number, but in the ratio of
one
to a number-they are all
(with one exception, that of the single pentagon)
unit-fractions.
Had Archimedes
wished to characterize the metrical property of the parts, this would have been a
more informative characterization (in Greek, they would each have been a
tJ-EpO<;
of
the whole). In short, it is implausible that the goal of the proposition-if indeed
one should apply to it the standards of achievement one normally associates with
llNote that Heiberg's figure for the Arabic proposition (bottom of [Archimedes
1910-15, 2:421];
see also p. 80 below) contains an error: the line DC is falsely joined, creating a square with
15
instead of
14
pieces. (This error is not in Suter's original publication, and of course it does clash
with the explicit construction).
It
should be clarified immediately: the identity of the figure as a
square is not certain, nor is Suter's edition in any sense final. This adds a fundamental dimension
of uncertainty to any specific reconstruction of Archimedes' solution: the point will be discussed
again below, following the quotation and discussion of the Greek fragments.
12Speaking of fractions in the context of ancient Greek mathematics is at best misleading, but for
our present purposes the sloppiness will do no harm.
13The content of this basic Euclidean tool is that triangles and parallelograms that are under the
same height are to each other as their bases.

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