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LETTERS
Decoding the ancient Greek astronomical calculator
known as the Antikythera Mechanism
T. Freeth
1,2
, Y. Bitsakis
3,5
, X. Moussas
3
, J. H. Seiradakis
4
, A. Tselikas
5
, H. Mangou
6
, M. Zafeiropoulou
6
, R. Hadland
7
,
D. Bate
7
, A. Ramsey
7
, M. Allen
7
, A. Crawley
7
, P. Hockley
7
, T. Malzbender
8
, D. Gelb
8
, W. Ambrisco
9
& M. G. Edmunds
1
The Antikythera Mechanism is a unique Greek geared device, con-
structed around the end of the second century
BC. It is known
1–9
that it calculated and displayed celestial information, particularly
cycles such as the phases of the moon and a luni-solar calendar.
Calendars were important to ancient societies
10
for timing agricul-
tural activity and fixing religious festivals. Eclipses and planetary
motions were often interpreted as omens, while the calm regular-
ity of the astronomical cycles must have been philosophically
attractive in an uncertain and violent world. Named after its place
of discovery in 1901 in a Roman shipwreck, the Antikythera
Mechanism is technically more complex than any known device
for at least a millennium afterwards. Its specific functions have
remained controversial
11–14
because its gears and the inscriptions
upon its faces are only fragmentary. Here we report surface
imaging and high-resolution X-ray tomography of the surviving
fragments, enabling us to reconstruct the gear function and double
the number of deciphered inscriptions. The mechanism predicted
lunar and solar eclipses on the basis of Babylonian arithmetic-
progression cycles. The inscriptions support suggestions of mech-
anical display of planetary positions
9,14,15
, now lost. In the second
century
BC, Hipparchos developed a theory to explain the irregu-
larities of the Moon’s motion across the sky caused by its elliptic
orbit. We find a mechanical realization of this theory in the gear-
ing of the mechanism, revealing an unexpected degree of technical
sophistication for the period.
The bronze mechanism (Fig. 1), probably hand-driven, was ori-
ginally housed in a wooden-framed case
1
of (uncertain) overall size
315 3 190 3 100 mm (Fig. 2). It had front and back doors, with
astronomical inscriptions covering much of the exterior of the mech-
anism. Our new transcriptions and translations of the Greek texts are
given in Supplementary Note 2 (‘glyphs and inscriptions’). The
detailed form of the lettering can be dated to the second half of the
second century
BC, implying that the mechanism was constructed
during the period 150–100
BC, slightly earlier than previously sug-
gested
1
. This is consistent with a date of around 80–60 BC for the
wreck
1,16
from which the mechanism was recovered by some of the
first underwater archaeology. We are able to complete the recon-
struction
1
of the back door inscription with text from fragment E,
and characters from fragments A and F (see Fig. 1 legend for fragment
nomenclature). The front door is mainly from fragment G. The text is
astronomical, with many numbers that could be related to planetary
motions; the word ‘‘sterigmos’’ (STGRICMOS, translated as ‘sta-
tion’ or ‘stationary point’) is found, meaning where a planet’s appar-
ent motion changes direction, and the numbers may relate to
planetary cycles. We note that a major aim of this investigation is
to set up a data archive to allow non-invasive future research, and
access to this will start in 2007. Details will be available on www.an-
tikythera-mechanism.gr.
The back door inscription mixes mechanical terms about con-
struction (‘‘trunnions’’, ‘‘gnomon’’, ‘‘perforations’’) with astronom-
ical periods. Of the periods, 223 is the Saros eclipse cycle (see Box 1
for a brief explanation of astronomical cycles and periods). We
discover the inscription ‘‘spiral divided into 235 sections’’, which is
1
Cardiff University, School of Physics and Astronomy, Queens Buildings, The Parade, Cardiff CF24 3AA, UK.
2
Images First Ltd, 10 Hereford Road, South Ealing, London W5 4SE, UK.
3
National and Kapodistrian University of Athens, Department of Astrophysics, Astronomy and Mechanics, Panepistimiopolis, GR-15783, Zographos, G reece.
4
Aristotle University of
Thessaloniki, Department of Physics, Section of Astrophysics, Astronomy and Mechanics, GR-54124 Thessaloniki, Greece.
5
Centre for History and Palaeography, National Bank of
Greece Cultural Foundation, P. Skouze 3, GR-10560 Athens, Greece.
6
National Archaeological Museum of Athens, 1 Tositsa Str., GR-10682 Athens, Greece.
7
X-Tek Systems Ltd, Tring
Business Centre, Icknield Way, Tring, Hertfordshire HP23 4JX, UK.
8
Hewlett-Packard Laboratories, 1501 Page Mill Road, Palo Alto, California 94304, USA.
9
Foxhollow Technologies
Inc., 740 Bay Road, Redwood City, California 94063, USA.
Figure 1
|
The surviving fragments of the Antikythera Mechanism. The 82
fragments that survive in the National Archaeological Museum in Athens are
shown to scale. A key and dimensions are provided in Supplementary Note 1
(‘fragments’). The major fragments A, B, C, D are across the top, starting at
top left, with E, F, G immediately below them. 27 hand-cut bronze gears are
in fragment A and one gear in each of fragments B, C and D. Segments of
display scales are in fragments B, C, E and F. A schematic reconstruction is
given in Fig. 2. It is not certain that every one of the remaining fragments
(numbered 1–75) belong to the mechanism. The distinctive fragment A,
which contains most of the gears, is approximately 180 3 150 mm in size.
We have used three principal techniques to investigate the structure and
inscriptions of the Antikythera Mechanism. (1) Three-dimensional X-ray
microfocus computed tomography
24
(CT), developed by X-Tek Systems Ltd.
The use of CT has been crucial in making the text legible just beneath the
current surfaces. (2) Digital optical imaging to reveal faint surface detail
using polynomial texture mapping (PTM)
25,26
, developed by Hewlett-
Packard Inc. (3) Digitized high-quality conventional film photography.
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the key to understanding the function
6
of the upper back dial. The
references to ‘‘golden little sphere’’ and ‘‘little sphere’’ probably refer
to the front zodiac display for the Sun and Moon—including phase
for the latter.
The text near the lower back dial includes ‘‘Pharos’’ and ‘‘from
south (about/around)….Spain (ISPANIA) ten’’. These geograph-
ical references, together with previous readings
1
of ‘‘towards the
east’’, ‘‘west-north-west’’ and ‘‘west-south-west’’ suggest an eclipse
function for the dial, as solar eclipses occur only at limited geograph-
ical sites, and winds were often recorded
17–19
in antiquity with eclipse
observations. Possibly this information was added to the mechanism
during use.
Turning to the dials themselves, the front dial displays the position
of the Sun and Moon in the zodiac, and a corresponding calendar
1
of
365 days that could be adjusted for leap years. Previously
1
, it was
suggested that the upper back dial might have five concentric rings
with 47 divisions per turn, showing the 235 months of the 19-year
Metonic cycle. A later proposal
5
augments this with the upper sub-
sidiary dial showing the 76-year Callippic cycle. Our optical and
X-ray microfocus computed tomography (CT) imaging confirms
these proposals, with 34 scale markings discovered on the upper back
dial. On the basis of a statistical analysis analogous to that described
for gear tooth counts below, we confirm the 235 total divisions. We
also find from the CT that the subsidiary dial is indeed divided into
quadrants
1,6
, as required for a Callippic dial. In agreement with the
back door inscription, we also substantiate the perceptive proposal
5,20
that the dial is in fact a spiral, made from semicircular arcs displaced
Figure 2
|
A schematic view of the mechanism to illustrate the position of
major inscriptions and dials.
The front dial has two concentric scales. The
inner scale shows the Greek zodiac with 360 divisions. There are occasional
Greek letters denoting references to the Parapegma inscription, and we add
three further reference letters (Z, H, H) to Price’s description
1
. The
Parapegma is a star almanac showing rising and settings at dawn or evening
of particular stars or constellations, which we will discuss elsewhere. Its form
is consistent with a date of late second century
BC. The outer (originally)
movable scale is a calendar carrying the Egyptian names of the months with
Greek letters. The Egyptian calendar of 365 days, with twelve 30-day months
and 5 extra (epagomenai) days was in standard use in Greek astronomy. The
effect of the extra quarter day in a year could be corrected by turning the
scale one day every four years—and a sequence of holes to take a locking pin
is observed under the scale. We find that spacing of the holes is indeed what
would be expected for a total of 365 days, with a possible range 363–365. The
position of the Sun and Moon would have been indicated by pointers across
the dial scales, and a device
7
showing the phase of the Moon was probably
carried round on the lunar pointer. It is not clear whether the Sun position
pointer would have been separated from a date pointer, or whether any
planetary positions might have been displayed. The spiral upper back dial
displays the luni-solar Metonic sequence of 235 lunar months with a
subsidiary dial showing the Callippic cycle, while the spiral lower back dial
displays the 223-lunar-month Saros eclipse cycle with a subsidiary dial
showing the Exeligmos cycle.
Box 1
|
Astronomical cycles known to the Babylonians
The lunar (or synodic) month is the interval between the Moon being at
the same phase—for example, full moon to full moon. The Metonic
cycle results from the close equality of 19 years to 235 lunar months. It
represents the return to the same phase of the Moon on the same date
in the year. After the cycle, the Sun, Moon and Earth are back in nearly
the same relative orientations. The Moon appears to return to the
same point in the sky relative to the zodiac in a sidereal month, and in
19 years there are 235 1 19 5 254 sidereal months. The 76-year
Callippic cycle is four Metonic cycles minus one day—and improves
the accuracy of reconciling solar years with whole numbers of lunar
months.
The Saros is an eclipse repeat cycle. If either a solar or lunar eclipse
occurs, a very similar eclipse will occur 223 lunar months later
23
.A
record of past eclipses can thus be used to predict future occurrences.
The cycle arises from the coincidence of three orbital periods of the
Moon. These are: (1) same phase to same phase, 223 synodic
months—eclipses will of course only occur at new or full Moon in the
month; (2) the lunar crossing of the Earth
–
Sun orbital plane, 242
draconitic months—eclipses can only occur near these points (nodes)
of co-alignment; (3) similar Earth
–
Moon distances which occur on the
period from apogee to apogee of the Moon’s orbit, 239 anomalistic
months. The distance will determine the magnitude of the eclipse,
ensuring the similarity of eclipses at the period of the cycle. The Saros
cycle is not an integer number of days (6,585M), causing the eclipses
in successive cycles to be displaced by eight hours in time (and solar
eclipses, only visible at limited geographical locations, to be displaced
by 120u in longitude). True repeats come after 3 Saros cycles, the
54-year Exeligmos cycle, but not with identical solar eclipse paths.
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to two centres on the vertical midline. In the CT of fragment B we find
a new feature that explains why the dial is a spiral: a ‘pointer-follower’
device (Fig. 3) travelled around the spiral groove to indicate which
month (across the five turns of the scale) should be read.
From our CT data of the 48 scale divisions observed in fragments
A, E and F, we establish 223 divisions in the four-turn
5,20
spiral on the
lower back dial, the spiral starting at the bottom of the dial. This is the
Saros eclipse cycle, whose number is on the back door inscription.
The 54-year Exeligmos cycle of three Saros cycles is shown on the
lower subsidiary dial.
Between the scale divisions of the Saros dial we have identified 16
blocks of characters, or ‘glyphs’ (see Supplementary Note 2 (‘glyphs
and inscriptions’)) at intervals of one, five and six months. These are
eclipse predictions and contain either S for a lunar eclipse (from
SELGNG, Moon) or G for a solar eclipse (from GLIOS, Sun) or
both. A correlation analysis (analogous to DNA sequence matching)
with historic eclipse data
21
(all modern eclipse data and predictions in
our work are from this reference) indicates that over a period of 400–
1
BC, the sequence of eclipses marked by the identified glyphs would
be exactly matched by 121 possible start dates. The matching only
occurs if the lunar month starts at first crescent, and confirms this
choice of month start in the mechanism. The sequences of eclipses
can then be used to predict the expected position of glyphs on the
whole dial, as seen in Fig. 4. The dial starts and finishes with an
eclipse. Although Ptolemy indicates that the Greeks recorded eclipses
in the second century
BC, the Babylonian Saros canon
17–19
is the only
known source of sufficient data to construct the dial.
The functions of the mechanism are determined by the tooth
counts of the gears. These are based mainly on the CT, using angular
measurement from a nominal centre to the remains of tooth tips. In a
few cases all teeth can be seen, but many gears are incomplete. Counts
are established by fitting models with regularly spaced teeth and
minimising the r.m.s. deviation from the measurements—varying
the centre in software (when unclear) to find the best-fit solution
or solutions (see Supplementary Note 3 (‘gears’)). We have adopted a
systematic nomenclature of lower case letters for the axis of the gear,
with numbering increasing with ordering from the front of the mech-
anism. Hypothetical (lost) gears are denoted by italics.
Several models have been proposed for the gear trains
1,2,4–6,8
.We
agree with the assumption of four missing gears (n1, n2, p1, p2)to
drive the Metonic and Callippic dials
4
. We propose a new reconstruc-
tion for the other trains, which uses all extant gears (except the lone r1
0 1 2 3 4 5 cm
Figure 4
|
Reconstruction of the back dials. A composite of fragments A, B,
E and F. The Metonic calendar is at top, with its subsidiary Callippic dial. The
Saros eclipse cycle is below, with its subsidiary Exeligmos dial. The 16
observed eclipse glyphs are shown in turquoise on the Saros dial, with 35
hypothetical glyphs in violet. The hypothetical glyphs are based on the
criterion that 99% of the 121 sequences exactly matching the observed
glyphs have an eclipse at the month position. Both main dials would have a
‘pointer-follower’ (see Fig. 3) to indicate the relevant lunar month on the
spiral. The monthly divisions on the Metonic upper back dial are not simply
scribed directly across all five turns, as might be expected for simplicity of
construction. There are small misalignments, implying a systematic attempt
at marking full (30-day) and hollow (29-day) months. The incomplete data
do not allow good analysis, other than a hint of bimodality in the interval
distribution. If the marking out of the scale were carried out using the
mechanism’s gearing, then this would greatly pre-date known ‘dividing
engines’
27
by many centuries.
Figure 3
|
The ‘pointer-follower’ lunar month indicator of the upper back
dial.
On the left, false-colour sections through CT images, analysed with
VGStudio Max software by Volume Graphics GmbH. These show two views
at right angles of the pointer-follower in the Metonic dial in fragment B. On
the right, a computer reconstruction of the device from two different angles
(with the Metonic scale omitted for clarity). The pin was constrained to follow
the groove between the spiral scales (the scale is shown in Fig. 4), causing the
device to slide along the month pointer to indicate which ring on the spiral
scale specified the month. A similar pointer-follower would have been present
on the lower back (Saros) dial. The Metonic dial would have required re-
setting every 19 years, the Saros dial after 18 years. The groove-pin may have
been held in place by the small pin through the front of the device, enabling its
removal for re-setting.
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from the separate fragment D). The proposed model is shown in
Fig. 5. We require the assumption of only one further gear (m3),
whose proposed shaft is clearly broken off in the CT. A detailed
description is contained in Supplementary Note 3 (‘gears’).
Of particular note is the dual use of the large gear, e3, at the back of
the mechanism, which has found no use in previous models. In our
model, it is powered by m3 as part of a fixed-axis train that turns the
Saros and Exeligmos dials for eclipse prediction, and also doubles as
the ‘epicyclic table’ for the gears k1, k2. These are part of epicyclic
gearing that calculates the theory of the irregular motion of the
moon, developed by Hipparchos some time between 146 and
128
BC (ref. 22)—the ‘first anomaly’, caused by its elliptical orbit
about the Earth. The period of this anomaly is the period from apogee
to apogee (the anomalistic month). To realize this theory, the mean
sidereal lunar motion is first calculated by gears on axes c, d and e and
this is then fed into the epicyclic system. As explained in Fig. 6, a pin-
and-slot device on the epicyclic gears k1 and k2, clearly seen in the
CT, provides the variation. This was previously identified
4
, but
rejected as a lunar mechanism. The remarkable purpose of mounting
the pin-and-slot mechanism on the gear e3 is to change the period of
variation from sidereal month (that is, the time taken for the Moon to
orbit the Earth relative to the zodiac), which would occur if k1 and k2
were on fixed axes, to anomalistic month—by carrying the gears
epicyclically at a rate that is the difference between the rates of the
Front dials
Lunar phase
Lost epicyclic gearing
Pin and slot
Hipparchos’ lunar mechanism
Possibly
Hipparchos’ solar mechanism
and planetary mechanisms
Hipparchos
sidereal month
Year
Back dials
Luni-solar calendar
Input
Eclipse prediction
Saros × 4Metonic × 5Callippic Exeligmos
Zodiac • Egyptian calendar • Parapegma
Figure 5
|
New reconstruction of the gear trains. A schematic sectional
diagram (not to scale) of the gearing, following the style of Price
1
and
Wright
4
. The viewpoint is looking down from the top right of the
mechanism, and is stretched in the direction of the main axes to show the
structure. Features that are outlined or labelled in red are hypothetical.
Gears are lettered with their shaft, and numbered with increasing distance
from the front dial. The two-or-three digit number on the gear is its actual or
assumed tooth count (see Supplementary Note 3 (‘gears’)). Hypothetical
gears n1, n2, p1, p2 have been proposed previously, the gear m3 on the
broken-off shaft m is our addition. All gears, except the lone one in fragment
D, are now accounted for in the mechanism. The function of the trains is
outlined in the text. We find no evidence in the CT for an idler wheel carried
on e3 and between e5 and k1 or between k2 and e6, as has been previously
proposed
1,2,4
. The CT shows a pin through axis e between gears e1 and e2. We
believe its purpose is to retain the square-bossed e1 on the shaft, but its
passage right through the axis rules out previous reconstructions
1,2,4
where
e1 and e2 were joined by an outer pipe rotating around the shaft e.
Figure 6
|
The ‘Hipparchos’ lunar mechanism mounted on gear e3. The
figure is based on a CT slice of part of fragment A, showing (top) shaft e and
(bottom) shaft k. The complete geometry cannot be seen in a single CT slice.
The two gears on the e axis (e5 and e6) are coaxial, while the two k gears
rotate on slightly displaced axes. k1 has a pin on its face that engages with a
radial slot in k2 (and this was previously reported
5
). In the figure the pitch
circles of e5 and k1 are shown in turquoise and those of e6 and k2 in pink.
The gear e5 drives k1, which drives k2 via the pin-and-slot, introducing a
quasi-sinusoidal variation in the motion, which is then transmitted to e6.
Our estimate of the distance between the arbors on the k gears is about
1.1 mm, with a pin distance of 9.6 mm, giving an angular variation of 6.5u.
According to Ptolemy
28
, Hipparchos made two estimates for a lunar
anomaly parameter, based on eclipse data, which would require angular
variations of 5.9u or 4.5u here—although estimates of the anomaly from
Babylonian astronomy were generally larger. The difference from our
estimated value is probably not significant given the difficulty of precise
measurement of the axes in the CT. The harmonic variation, together with
the effect of carrying the gears on e3 (which rotates at the period of the
Moon’s apogee around the Earth), would simulate the correct variation for
the Moon’s mean (sidereal) rotation rate on the front dial. An (unexplained)
regular pentagon is visible at the centre of gear e5. It is tempting to associate
the conception of the mechanism with Hipparchos himself, but he was not
the first to assume eccentric or epicyclic models.
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sidereal and anomalistic months, that is, at the rate of rotation of
about 9 years of the Moon’s apogee.
Gears with 53 teeth are awkward to divide. So it may seem sur-
prising that the gearing includes two such gears (f1, l2), whose effects
cancel in the train leading to the Saros dial. But the gearing has been
specifically designed so that the ‘epicyclic table’ e3 turns at the rate of
rotation of the Moon’s apogee—the factor 53 being derived from the
calculation of this rotation from the Metonic and Saros cycles, which
are the bases for all the prime factors in the tooth counts of the gears.
The establishment of the 53-tooth count of these gears is powerful
confirmation of our proposed model of Hipparchos’ lunar theory.
The output of this complex system is carried from e6 back through e3
and thence, via e1 and b3, to the zodiac scale on the front dial and the
lunar phase
7
mechanism. Our CT confirms the complex structure of
axis e that this model entails.
The Antikythera Mechanism shows great economy and ingenuity
of design. It stands as a witness to the extraordinary technological
potential of Ancient Greece, apparently lost within the Roman
Empire.
Received 10 August; accepted 17 October 2006.
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Supplementary Information is linked to the online version of the paper at
www.nature.com/nature.
Acknowledgements This work was financed by the Leverhulme Trust, the Walter
Hudson Bequest, the University of Athens Research Committee and the Cultural
Foundation of the National Bank of Greece. For essential support we thank the
Ministry of Culture, Greece (P. Tatoulis), and the National Archaeological Museum
of Athens (N. Kalts as). We acknowledge help and advice from J. Ambers, J. Austin,
G. Dermody, H. Forsyth, I. Freestone, P. Haycock, V. Horie, A. Jones, M. Jones,
P. Kipouros, H. Kritzas, J. Lossl, G. Makris, A. Ray, C. Reinhart, A. Valassopoulos,
R. Westgate, T. Whiteside, S. Wright and C. Xenikakis.
Author Contributions T.F. carried out most of the CT analysis of structure and its
interpretation. Y.B., A.T. and X.M. read, transcribed and translated the inscriptions.
H.M and M.Z. catalogued the fragments, provided guidance on X-ray examination,
and measured the fragments with J.H.S. R.H. led the team (D.B., A.R., M.A., A.C.
and P.H.) that built and operated the Bladerunner CT machine, and provided CT
reconstructions and advice. T.M., D.G. and W.A. built, operated and provided
software for the PTM. M.G.E. was academic lead, and undertook the statistical
analysis. T.F. and Y.B. organised the logistics of the experimental work, with
inter-agency liaison by X.M. and J.H.S. The manuscript was written by T.F. and
M.G.E. including material from Y.B., A.T., X.M., J.H.S., H.M. and M.Z. T.F. designed
the illustrations.
Author Information Reprints and permissions information is available at
www.nature.com/reprints. The authors declare no competing financial interests.
Correspondence and requests for materials should be addressed to M.G.E.
(mge@astro.cf.ac.uk).
NATURE
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Vol 444
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30 November 2006 LETTERS
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Nature
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