Ancient astronomical calculator under the crosshairs of modern technology

The Antikythera Mechanism is a unique Greek gear-driven device built around the end of the 2nd century BC. It is known to have calculated and displayed astronomical information, particularly cycles such as the phases of the moon and the lunisolar calendar. Calendars were important to ancient societies for timing agricultural activities and establishing religious holidays. Eclipses and planetary movements were often interpreted as omens, while the calm regularity of astronomical cycles must have been philosophically appealing in a turbulent and violent world.

Named for the location of its discovery in 1901 in a Roman shipwreck, the Antikythera Mechanism is more technically complex than any known device for at least a millennium. Its exact functions have remained controversial, as its gears and the inscriptions on its surfaces have survived only in fragments.

This paper reports on surface imaging and high-resolution X-ray tomography of the surviving fragments, which allowed us to reconstruct the gear functions and double the number of deciphered inscriptions. The mechanism predicted lunar and solar eclipses based on Babylonian cycles of arithmetic progression. The inscriptions support theories about mechanical representation of planetary positions, which are now lost. In the 2nd century BC, Hipparchus developed a theory to explain the irregularities in the Moon's motion across the sky caused by its elliptical orbit. The mechanism's gear train reveals a mechanical implementation of this theory, revealing a degree of technical sophistication unexpected for the period.

The bronze mechanism, probably hand-wound, was originally housed in a wooden case measuring 315 x 190 x 100 mm overall. It had a front and back door, with astronomical inscriptions covering most of the mechanism's outer surface. New transcriptions and translations of the Greek texts are provided in the supplementary material. The detailed shape of the letters can be dated to the second half of the second century BC, which means that the mechanism was made in the period 150-100 BC, slightly earlier than previously thought. This is consistent with a date of around 80-60 BC for the shipwreck from which the mechanism was recovered in some of the first underwater archaeological work.

The researchers were able to complete a reconstruction of the inscription on the back door, with text from fragment E and symbols from fragments A and F. The front door is mainly derived from fragment G. The text is astronomical, with many numbers that may be related to the movements of the planets; the word “sterigmos” (STGRICMOS, translated as “standing” or “stationary point”) appears, meaning the place where the apparent motion of a planet changes direction, and the numbers may refer to planetary cycles.

The surviving fragments of the Antikythera Mechanism. Shown are the 82 fragments housed in the National Archaeological Museum, Athens. The main fragments A, B, C, D are located at the top, starting from the upper left corner, with E, F, G immediately below them. The 27 hand-carved bronze gears are in fragment A and one gear in each of fragments B, C and D. The display scale segments are in fragments B, C, E and F. The representative fragment A, which contains the majority of the gears, measures approximately 180 x 150 mm. Three main techniques were used to examine the structure and inscriptions of the Antikythera Mechanism: three-dimensional X-ray microfocus computed tomography (CT) developed by X-Tek Systems Ltd., which was crucial in making the text legible just beneath the flowing surfaces; digital optical imaging to reveal subtle surface details using polynomial texture mapping (PTM) developed by Hewlett-Packard Inc.; and digitized high-quality conventional film photography.

Text near the lower back dial includes “Pharos” and “from the south (near/around)…. Spain (ISPANIA) ten.” These geographical references, along with the previous readings of “to the east,” “west-northwest,” and “west-southwest,” suggest an eclipse function for the dial, since solar eclipses only occur in limited geographic locations, and winds were often recorded in ancient eclipse observations. It is possible that this information was added to the mechanism during use.

Turning to the dials themselves, the front dial displays the positions of the Sun and Moon in the zodiac and a corresponding 365-day calendar, which could be adjusted for leap years. It has previously been suggested that the upper rear dial may have had five concentric rings with 47 divisions per revolution, representing the 235 months of the 19-year Metonic cycle. A more recent proposal supplements this with an upper sub-dial showing the 76-year Callippic cycle. Optical and X-ray microfocus computed tomography (CT) scans support these proposals, with 34 scale divisions detected on the upper rear dial. Based on statistical analysis similar to that described for the gear tooth count, a total of 235 divisions is confirmed.

The CT scan also shows that the sub-dial is indeed divided into quadrants, as required for a Callippus dial. In accordance with the inscription on the back door, it also confirms the insightful suggestion that the dial is actually a spiral, consisting of semi-circular arcs offset to two centers on a vertical centerline. The CT scan of fragment B reveals a new detail that explains why the dial is spiral: a “pointer-follower” device moved along the spiral groove to indicate which month (over five revolutions of the scale) was to be read.

Schematic representation of the mechanism, showing the positions of the principal inscriptions and dials. The front dial has two concentric scales. The inner scale shows the Greek zodiac with 360 divisions. Individual Greek letters appear indicating references to the Parapegma inscription, and three additional reference letters (Z, H, H) are added to Price's description. The Parapegma is a stellar almanac showing the rising and setting at dawn or evening of certain stars or constellations. The outer (originally) movable scale is a calendar with the Egyptian month names written in Greek letters. The Egyptian calendar of 365 days, with twelve 30-day months and 5 extra (epagomenal) days, was standard in Greek astronomy. The effect of the extra quarter day in the year could be corrected by turning the scale by one day every four years - and a series of holes for the locking pin is seen below the scale. It has been found that the spacing between the apertures is indeed as expected for a total of 365 days, with a possible range of 363-365. The positions of the Sun and Moon were probably indicated by hands on the dial scales, and a device showing the phase of the Moon was probably carried on the lunar hand. It is unclear whether the solar position hand was separate from the date hand, or whether any planetary positions were displayed. The spiral upper back dial displays the Metonic lunisolar sequence of 235 lunar months, with a sub-dial showing the Callippic cycle, while the spiral lower back dial displays the 223-month Saros eclipse cycle, with a sub-dial showing the Exeligmos cycle. — Schematic representation of the movement, showing the position of the principal inscriptions and dials. The front dial has two concentric scales. The inner scale shows the Greek zodiac with 360 divisions. There are individual Greek letters indicating references to the Parapegma inscription, and three additional reference letters (Z, H, H) have been added to Price's description. The Parapegma is a star almanac showing the rising and setting at dawn or evening of certain stars or constellations. The outer (originally) movable scale is a calendar with the Egyptian names of the months written in Greek letters.The Egyptian calendar of 365 days, with twelve 30-day months and 5 extra (epagomenal) days, was standard in Greek astronomy. The effect of the extra quarter day in the year could be adjusted by turning the scale by one day every four years – and a series of holes for a locking pin is observed under the scale. It has been found that the spacing of the holes is indeed as expected for a total of 365 days, with a possible range of 363-365. The positions of the Sun and Moon were probably indicated by hands on the dial scales, and a device showing the phase of the Moon was probably carried on the lunar hand. It is unclear whether the solar position hand was separate from the date hand, or whether any planetary positions were displayed. The spiral upper back dial displays the Metonic lunisolar sequence of 235 lunar months with a sub-dial showing the Callippic cycle, while the spiral lower back dial displays the 223-month Saros eclipse cycle with a sub-dial showing the Exeligmos cycle.

According to the CT scan of the 48 scale divisions seen in fragments A, E and F, there are 223 divisions in the four-turn spiral on the lower back dial, the spiral starting at the bottom of the dial. This is the Saros eclipse cycle, the number of which is indicated in the inscription on the back door. The 54-year Exeligmos cycle of three Saros cycles is shown on the lower subdial.

Between the divisions of the Saros dial scale, 16 blocks of symbols, or “glyphs,” have been identified at intervals of one, five, and six months. These are eclipse predictions, containing either an S for a lunar eclipse (from SELGNG, Moon), a G for a solar eclipse (from GLIOS, Sun), or both. Correlation analysis (similar to DNA sequence matching) with historical eclipse data shows that for the period 400-1 BCE, the sequence of eclipses marked by the identified glyphs would exactly match 121 possible starting dates. The match occurs only if the lunar month begins with the first crescent, and confirms this choice of month start in the mechanism. The eclipse sequences can then be used to predict the expected positions of the glyphs on the entire dial. The dial begins and ends with an eclipse. Although Ptolemy indicates that the Greeks recorded eclipses in the 2nd century BC, the Babylonian Saros Canon is the only known source of sufficient data to construct the dial.

Device "tracking pointer" for the lunar month indication of the upper rear dial. On the left are false-colored sections through CT images analyzed with VGStudio Max software from Volume Graphics GmbH. They show two right-angle views of the follower pointer in the Metonic dial in fragment B. On the right is a computer reconstruction of the device from two different angles (the Metonic scale has been omitted for clarity). The pin was forced to follow a groove between the spiral scales, causing the device to slide along the month pointer to indicate which ring on the spiral scale represented the month. A similar follower pointer was probably present on the lower rear (Saros) dial. The Metonic dial would have required resetting every 19 years, the Saros dial every 18 years. The pin in the groove may have been held in place by a small pin through the front of the device, allowing its removal for resetting. — The “tracking pointer” device for the lunar month indication of the upper rear dial. On the left are false-colored sections through CT images analyzed with the VGStudio Max software from Volume Graphics GmbH. They show two right-angle views of the tracking pointer in the Metonic dial in fragment B. On the right is a computer reconstruction of the device from two different angles (the Metonic scale has been omitted for clarity).The pin was forced to follow the groove between the spiral scales, causing the device to slide along the month indicator to show which ring on the spiral scale represented the month. A similar tracking indicator was probably present on the lower rear (Saros) dial. The Metonic dial would have required resetting every 19 years, the Saros dial every 18 years. The pin in the groove may have been held in place by a small pin through the front of the device, allowing its removal for resetting.

The functions of the mechanism are determined by the number of teeth on the gears. These are based primarily on CT, using angular measurements from the nominal centre to the remains of the tooth tips. In some cases all the teeth are visible, but many gears are incomplete. Counts are established by fitting models with regularly spaced teeth and minimising the standard deviation from the measurements – varying the centre in the software (when this is unclear) to find the best solution or solutions.

Reconstruction of the back dials. Composition of fragments A, B, E and F. The Metonic calendar is at the top, with its Callippus subdial. The Saros eclipse cycle is at the bottom, with its Exeligmos subdial. 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 that exactly match the observed glyphs have an eclipse in the position of the month. Both main dials would have "tracking pointer" to indicate the corresponding lunar month on the spiral. The monthly divisions on the upper back dial of the Meton are not simply marked straight through all five turns, as might be expected for the simplicity of the design. There are small discrepancies, implying a systematic attempt to mark full (30-day) and hollow (29-day) months. The incomplete data do not allow a good analysis, other than a hint of bimodality in the distribution of intervals. If the scale markings were made using a gear train, this would be significantly ahead of the known "dividing machines" for many centuries. — Reconstruction of the back dials. Composition of fragments A, B, E and F. The Metonic calendar is at the top, with its Callippus subdial. The Saros eclipse cycle is at the bottom, with its Exeligmos subdial. 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 that exactly match the observed glyphs have an eclipse at the position of the month. Both main dials would have had a “tracking pointer” to indicate the corresponding lunar month on the spiral. The monthly divisions on the upper rear dial of the Meton are not simply marked straight across all five turns, as might be expected for the simplicity of the design. There are slight discrepancies, implying a systematic attempt to mark full (30-day) and hollow (29-day) months. The incomplete data do not allow a good analysis, other than a hint of bimodality in the distribution of intervals. If the scale markings were made using a gear mechanism, this would significantly predate the known “dividing machines” by many centuries.

Several models have been proposed for the gear trains. The authors agree with the assumption of four missing gears (n1, n2, p1, p2) for the drive of the Metonic and Callippic dials. A new reconstruction is proposed for the other gear trains, which uses all the surviving gears (except for the single r1 from the separate fragment D). The proposed model is shown in the following figure. It requires the assumption of only one additional gear (m3), the putative shaft of which is clearly broken in CT.

New reconstruction of gear trains. Schematic sectional diagram (not to scale) of gear train following Price and Wright style. The viewpoint is from top right of mechanism, and it is stretched in direction of principal axes to show structure. Features circled or marked in red are hypothetical. Gears are designated by their shaft letter and numbered with increasing distance from front dial. Two or three digit number on gear is its actual or supposed number of teeth. Hypothetical gears n1, n2, p1, p2 were suggested earlier, gear m3 on broken shaft m is authors' addition. All gears except single one in fragment D are now accounted for in mechanism. — New reconstruction of gear trains. Schematic sectional diagram (not to scale) of a gear train following the style of Price and Wright. The viewpoint is from above, from the right of the mechanism, and it is stretched in the direction of the principal axes to show the structure. Features circled or marked in red are hypothetical.The gears are designated by their shaft letter and numbered with increasing distance from the front dial. The two- or three-digit number on the gear is its actual or assumed number of teeth. The hypothetical gears n1, n2, p1, p2 were proposed earlier, the gear m3 on the broken shaft m is an addition by the authors. All gears except the single one in fragment D are now taken into account in the mechanism.

Of particular note is the dual use of the large pinion e3 at the rear of the mechanism, which has found no use in previous models. In the new model it is driven by m3 as part of a fixed-axis gearing that rotates the Saros and Exeligmos dials to predict eclipses, and also doubles as an “epicyclic table” for pinions k1, k2. These are part of the epicyclic gearing that calculates the theory of the irregular motion of the Moon developed by Hipparchus between 146 and 128 BC – the “first anomaly” caused by its elliptical orbit around the Earth. The period of this anomaly is the period from apogee to apogee (the anomalistic month). To implement this theory, the mean sidereal motion of the Moon is first calculated using pinions on axes c, d and e, and then fed into the epicyclic system.

As explained in the following figure, the pin and slot device on the epicyclic gears k1 and k2, clearly visible in the CT scan, provides the variation. This has previously been identified but rejected as a lunar mechanism. The remarkable purpose of installing the pin and slot mechanism on gear e3 is to change the period of variation from the sidereal month (i.e. the time it takes the Moon to orbit the Earth relative to the zodiac), which would occur if k1 and k2 were on fixed axes, to the anomalistic month – by epicyclically transferring the gears at a rate equal to the difference between the rates of the sidereal and anomalistic months, i.e. the rotation rate of the Moon's apogee of about 9 years.

"Hipparchus's Lunar Mechanism"mounted on gear e3. The figure is based on a CT slice through part of fragment A, showing (top) shaft e and (bottom) shaft k. The full 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 offset axes. k1 has a pin on its surface which fits into a radial slot in k2 (this has been reported previously). In the figure, the pitch circles of e5 and k1 are shown in turquoise, and e6 and k2 in pink. Gear e5 drives k1, which drives k2 through the pin and slot, introducing a quasi-sinusoidal change in the motion, which is then transmitted to e6. The estimated spacing between the axes on the k gears is about 1.1 mm, with a pin spacing of 9.6 mm, giving an angular change of 6.5°. According to Ptolemy, Hipparchus made two estimates of the lunar anomaly parameter based on eclipse data, which would require angular changes of 5.9° or 4.5° here - although anomaly estimates from Babylonian astronomy were usually larger. The difference with the estimated value is probably not significant, given the difficulty of accurately measuring the axes in CT. The harmonic change, together with the effect of the translation of the gears on e3 (which rotates with the period of the Moon's apogee around the Earth), would model the correct change for the mean (sidereal) rotation of the Moon on the front dial. An (unexplained) regular pentagon is visible at the center of gear e5. It is tempting to associate the concept of the mechanism with Hipparchus himself, but he was not the first to suggest eccentric or epicyclic models. — *“Hipparchus Moon Gear” mounted on gear e3. Drawing based on CT slice of part of fragment A showing (top) shaft e and (bottom) shaft k. Full geometry cannot be seen in one CT slice.*The two gears on the e-axis (e5 and e6) are coaxial, while the two k-gears rotate on slightly offset axes. k1 has a pin on its surface that fits into a radial slot in k2 (this was previously reported). In the figure, the pitch circles of e5 and k1 are shown in turquoise, and e6 and k2 in pink. Gear e5 drives k1, which drives k2 through the pin and slot, introducing a quasi-sinusoidal change in the motion that is then transmitted to e6. An estimate of the distance between the axes on the k-gears is about 1.1 mm, with a pin spacing of 9.6 mm, giving an angular change of 6.5°. According to Ptolemy, Hipparchus made two estimates of the lunar anomaly parameter based on eclipse data, which would require angular changes of 5.9° or 4.5° here – although anomaly estimates from Babylonian astronomy were usually larger. The difference with the estimated value is probably not significant, given the difficulty of accurately measuring the axes in CT. The harmonic change, together with the effect of the translation of the gears on e3 (which rotates with the period of the Moon's apogee around the Earth), would model the correct change for the mean (sidereal) rotation of the Moon on the front dial. An (unexplained) regular pentagon is visible at the center of gear e5. It is tempting to associate the concept of the mechanism with Hipparchus himself, but he was not the first to suggest eccentric or epicyclic models.

Gears with 53 teeth are awkward to divide. It may therefore seem surprising that there are two such gears (f1, l2) in the gearbox, the effects of which cancel each other out in the chain leading to the Saros dial. But the gearbox has been specifically designed so that the “epicyclic table” e3 rotates at the speed of the lunar apogee – the factor 53 is derived from the calculation of this rotation from the Metonic and Saros cycles, which are the basis for all the prime factors in the gear tooth numbers. The establishment of a tooth number of 53 for these gears is a powerful confirmation of the proposed model of Hipparchus's lunar theory. The output of this complex system is transmitted from e6 back through e3 and on through e1 and b3 to the zodiac scale on the front dial and the lunar phase mechanism. The CT scan confirms the complex structure of the e-axis that this model suggests.

The Antikythera mechanism demonstrates great economy and ingenuity of design. It stands as a witness to the extraordinary technological potential of ancient Greece, apparently lost in the Roman Empire.

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