Reference / Paper · 2022
Mediterranean Archaeology and Archaeometry 22(3), 103–131. Reads fragment D's 63-tooth gear as part of a draconic train.
Mediterranean Archaeology and Archaeometry Vol. 22, No 3, (2022), pp. 103-131 Open Access. Online & Print. www.maajournal.com DOI: 10.5281/zenodo.7268844 ASSEMBLING THE FRAGMENT D ON THE ANTIKYTHERA MECHANISM: ITS ROLE AND OPERATION IN THE DRACONIC GEARING Aristeidis Voulgaris*1, Christophoros Mouratidis2 and Andreas Vossinakis3 1City of Thessaloniki, Directorate Culture and Tourism, Thessaloniki, GR-54625, Greece 2Merchant Marine Academy of Syros, Ermoupoli, GR-84100, Greece 3Thessaloniki Astronomy Club, Thessaloniki, GR-54646, Greece Received: 06/05/2022 Accepted: 01/10/2022 Corresponding author: Aristeidis Voulgaris ([email protected]) ABSTRACT The unplaced Fragment D of the Antikythera Mechanism with an unknown operation was a mystery ever since it was discovered. The gear r1, which was detected in Fragment-D radiographies by C. Karakalos is preserved in excellent condition, but this was not enough to directly relate it to the existing gear trains of the Mechanism. The suggestion that Fragment D could be a part of the hypothetical planet indication gearing is a hypothesis, in contradiction with the AMRP tomographies, and leads to mechanical malfunctions. The analysis of AMRP tomographies of Fragment D and its mechanical characteristics, revealed that it could be part of the Draconic gearing. Currently, the three lunar cycles Synodic, Sidereal and Anomalistic are represented on the Antikythera Mechanism. Although the Draconic cycle was well known during the Mechanism’s era as representing the critical fourth Lunar cycle, it seems that it is missing from the Antikythera Mechanism. The study of Fragment D was supported by our bronze reconstructions of the Fragment D, leading to the reconstruction of the Draconic gearing. These results are very important, because with the use of only one hypothetical/missing gear, the Draconic gearing is revealed on the Antikythera Mechanism. Fragment D(raconic) can be perfectly correlated with the Draconic cycle of the Antikythera Mechanism without mechanical problems and the four lunar cycles appear on the Mechanism. Based on the Draconic pointer, the eclipse events can be calculated and engraved on the Saros spiral cells. KEYWORDS: Fragment D, Lunar cycles, Draconic gearing, Nodes, eclipse prediction, Antikythera Mechanism gearing Copyright: © 2022. This is an open-access article distributed under the terms of the Creative Commons Attribution License. (https://creativecommons.org/licenses/by/4.0/). 104 A.VOULGARIS et al 1. INTRODUCTION The Antikythera Mechanism, a creation of an ingenious manufacturer of the Hellenistic era, was a geared machine capable of performing complex astronomical calculations, mostly based on the lunar periodic cycles Seiradakis and Edmunds (2018). After 2000 years under the Aegean/Ionian sea, Jones (2017); Voulgaris et al., (2019b), the fragments of the Antikythera Mechanism is now a permanent exhibit at the National Archaeological Museum of Athens, Greece. At ancient time, the Synodic lunar cycle was used as the primary time unit of calendars Hannah (2013). Each month of the ancient Greek calendar was based on the New Moon and Full Moon phases Bowen and Goldstein (1988); Hannah (2013). Even the Olympic Games were occurred every 4 years, but started on the 8th (or 9th) Full Moon after the Winter Solstice, every 49 or 50 lunar synodic months Vaughan (2002). Although one tropical year does not include an integer number of lunar Synodic months, the lunar cycle prevailed instead of the tropical year, which was considered less important Theodosiou and Danezis (1995). The Metonic calendar known as Enneadecaeteris or Enneakaidekateris (from Ancient Greek: Εννέακαιδεκαετηρίς: 9+10 έτη-years) correlates the integer number of 235 lunar Synodic cycles (also 254 Sidereal cycles), to the integer number of 19 tropical years. Each Metonic year of 12 or 13 Synodic months deviated from the actual start of the tropical year by several days, in order to keep the lunar Synodic month intact Manitius (1880); van der Waerden (1984b); Theodosiou and Danezis (1995); Spandagos (2002); Freeth et al., (2008); Anastasiou et al., 2016b. The ancient astronomers chose the lunar cycles instead of the solar cycles for several obvious reasons: it is easy to calculate a time span based on the Moon’s every day changing phase, it is visible both at night and at day and when the Moon is on the sky it is easy to observe bright stars. In contrast, the Sun does not exhibit any phases, it is impossible to detect stars when the Sun is above the horizon and the relative position of the Sun on the sky changes by about 1/12th of the lunar angular velocity. Also a particular “solar phase” i.e. a solar eclipse is a rare astronomical event to observe from a specific place. Thus, the ancient astronomers chose to base their measurements and time keeping events on the easier to follow lunar Synodic cycle. By studying the Antikythera Mechanism it is obvious that the operation, design and the gear teeth selection, were mostly based on the Synodic lunar month: the bin axis-Lunar Disc-Input (the fastest of the rotating axes), the small Lunar phases sphere on the Lunar Disc and the cells on the two spirals (Saros and Metonic), were all precisely based on the Synodic lunar month. The selection of the teeth number of most of the gear trains was made in accordance to the Synodic and Sidereal lunar cycles, e.g. the number of the gear teeth e3 (223 Synodic months of Saros) and ((c2/d1)*d2 = (2 X 127) = 254 Sidereal months of the Metonic cycle). The Lunar Disc rotation, which is the proper driving/input for the Antikythera Mechanism gearing trains (see Section 4.1), essentially defines the two lunar cycles - the Sidereal and Synodic month - as the basic time units for the calculations of the Mechanism. Today, only one out of the seven pointers of the Mechanism is clearly related to the tropical year: the Golden Sphere-Sun pointer, Freeth et al., (2006); Voulgaris et al., (2018b); Voulgaris et al., (2019b), i.e. one full rotation equals one tropical year. All of the above lead to the conclusion that the Antikythera Mechanism was a Luni-(solar) time/calendar geared machine computer, based on the lunar Synodic cycle. 2. RESEARCH AIM The authors’ research aims to the detection of a proper position for the unplaced Fragment D/gear r1, on the existing gearing trains of the Mechanism. Today, the preserved/necessary gears of the Mechanism are 35 (+1 unplaced gear/Fragment D). Freeth and Jones (2012) and Freeth et al., (2021), suggested that the Fragment D/gear r1, could be a part of the hypothetical planet indication gearing, representing the planet Venus motion on the Antikythera Mechanism. Freeth et al., (2021) suggested 3D representation design of Fragment D, differs from the AMRP tomographies and presents critical mechanical problems when we constructed it in bronze (see Section 5.2 and in authors’ Supplementary Material at the end). For the hypothetical planet gearing design, a large number of hypothetical gears (>33) and hypothetical additional moving parts (>40), is needed. However, today not one of these is preserved. The suggested planet indication gearing on the Antikythera Mechanism is still a hypothesis, requiring a lot of assumptions and presents critical mechanical problems concerning the functionality of the device (see Section 4.1). In this work we will present that the hypothesis of the planet indication gearing is not the only-one, in order to be justified the gear r1 operation and the Fragment D cannot be considered as a proof for the planet indication gearing (see Section 5.2) [see also, Supplementary Material at the end]. Our present research aim is based on specific data were related to: a) The knowledge of the Hellenistic Astronomy of the Antikythera Mechanism era (around 200BC100BC), Mediterranean Archaeology and Archaeometry, Vol. 22, No 3, (2022), pp. 103-131 ASSEMBLING THE FRAGMENT D ON THE ANTIKYTHERA MECHANISM b) The mechanical characteristics, properties, specifications and mechanical limitations of the Antikythera Mechanism, c) The specific dimension of the Mechanism parts, d) The existence of the real preserved-unplaced parts of the Mechanism. Finally, our results answer to the question “how did the ancient Manufacturer calculate the eclipse events and the hours on which they occurred, presented on the Saros spiral cells of the Antikythera Mechanism?” applying the minimum of possible hypotheses and adapting the minimum of possible hypothetical parts. 3. LUNAR CYCLES AND ANTIKYTHERA MECHANISM 3.1. The lunar motions studied in the Hellenistic era The four well known lunar motions - the Synodic, Sidereal, Anomalistic and Draconic cycles - where extensively observed and studied by the ancient astronomers, with the aim to interrelate them Spandagos (2002); Steele (2000a,b); Hannah (2001); Steele (2002). 105 Ptolemy in Almagest Toomer (1984) extensively referred to the primary lunar cycles, giving a large number of calculations in tables, Pedersen (2011). The attempt of Ptolemy, Hipparchus and the previous era astronomers to (better) correlate - incorporate the four lunar cycles, was obvious. The ancient astronomers realized that a interrelation - phase synchronization (periodicity) of two of the four lunar motions/cycles, revealed a repeatability of the solar/lunar eclipses, while the synchronization of three lunar cycles, presented eclipses with highly similar geometrical characteristics and classification Oppolzer (1962); van den Bergh (1955); Meeus et al., (1966); Neugebauer (1975); Meeus (1998, 2004). The periodicity of the lunar motions, led to the adoption of the “interrelated cycles ratio in integer numbers”, a very useful canon, which was the key for the solar/lunar eclipses prediction. Ptolemy refers to the Saros cycle with the name “Περιοδικός” (Periodic), Voulgaris et al., (2021), since it is the smallest single period which contains an integer number of returns of the various motions. Figure 1. For our research needs, a pulley is adapted on the Lunar Disc of the Antikythera Mechanism functional model (designed/constructed by the FRAMe Project) and an electrical motor rotates the Lunar Disc via a belt (images taken from video frames). The position/phases of the Moon relative to the Golden sphere-Sun and the zodiac sky are presented. A) The red dot that marks the Egyptian month of 1st MECHIR (ΜΕΧΕΙΡ) and also the 18th zodiac day of Gemini (ΔΙΔΥΜΟΙ) is the starting position of the Lunar Disc and the Golden sphere (as the Egyptian and the Zodiac dial ring are freely rotated, we can turn them to any direction, for our research needs, Voulgaris et al., (2018a). B, C, D) During the rotation of the Lunar Disc, the Golden sphere continuously changes its position relative to the zodiac sky (the slight mismatch of the aiming is a cause of the parallax error during the video recording). Each Synodic month on the Antikythera Mechanism, started right after the New Moon phase. We operate our functional models of the Antikythera Mechanism, by rotating the Lunar Disc. The Lunar Disc offers a slow and precise motion of the pointers, and has a high torque in order to overcome the gearing frictions, as opposed to the hypothetical consideration of the crank-a1 being the input of the Mechanism, see Section 4.1. and Voulgaris et al., (2018b). Mediterranean Archaeology and Archaeometry, Vol. 22, No 3, (2022), pp. 103-131 106 A.VOULGARIS et al Certainly, the most easily observed lunar cycle was the Synodic month i.e. the period between two successive New Moons or Full Moons, which was easily measured with sufficient accuracy Fig. 1A,D. For this reason the calculations of the rest three lunar cycles were based on the Synodic month. The Sidereal month is the time duration between two successive transits of the Moon by the same constellation area/star Fig. 1A,C, but the start of each Sidereal cycle presents a different lunar phase. As the Moon is the brighter celestial body in the sky, it far outweighs the bright stars. The Anomalistic month became evident by the variability of the angular velocity of the Moon against the fixed stars. Geminus mentions that at the start of the Anomalistic month (Apogee), the lunar angular velocity is 11° 06’ 35”/1d (minimum), whereas at the middle of the Anomalistic month (Perigee) the angular velocity is 15° 14’ 35”/1d (maximum). As it may be expected, this angular variation is not easily detected without the use of an astronomical instrument (astrolabe) and the recording of a number of observations and measurements, as Ptolemy does in Almagest. Geminus not only describes the mathematical process for calculating the mean Lunar angular velocity, but also refers to the measuring error as a result of the instrument’s limited resolution in practice. Table 1. The time measuring cycles expressed in days and in tropical years, as recorded by the observations of Babylonians and the ancient Greek astronomers. True and rounded values of the Draconic month are presented. The 1/1000th of a Synodic month is about 42.5 minutes. The ratio of Draconic to Synodic month is rounded to 1.08520. Cycle (invented for…) Saros (eclipses) Tropical years Number of Days Days per Year Synodic months 18y 11.3d 6585.322 365.2233 223 Metonic (tropical year) 19y 6940 365.2631 235 Exeligmos (eclipses) 54y 33d 19756 365.2235 669 Callippic (tropical year) 76y 27759 365.25 940 Hipparchic (tropical year) 345y 126010 365.246 4267 Babylonian (eclipses) 441y +106.3d 161188 365.2646 5458 The variable lunar motion is also included on the Antikythera Mechanism, introduced by the operation of the pin&slot design on the gears k1/k2, off-axis on board the gear e3, Freeth et al., (2006); Wright (2006); Gourtsoyannis (2010); Voulgaris et al., (2018b). The centers of gears k1 and k2 are located at an “off-axis” position and the movement transmission from gear k2 (slot) to k1 (pin) presents a variable angular velocity representing the Anomalistic lunar cycle. The fourth, more complex lunar motion is the time period that the Moon crosses the Ecliptic. The elliptical lunar orbit is inclined relative to the Ecliptic by about 5.15° and crosses the Ecliptic plane at two points, named Ascending and Descending Node. Ptolemy in Almagest calls the time span in which the Moon crosses the same Node, as “Ἀποκατάστασις κατά πλάτος” (return to the same latitude), better known to us Draconic or Draconitic month (from Middle Ages Draconic months (rounded) 242 241.999 (true) 255 255.021 (true) 726 725.996 (true) 1020 1020.84 (true) 4630.5 4630.531 (true) 5923 5922.999 (true) Ratio (Draconic/ Synodic) 1.0852017 1.0851973 Draconic month (days) 27.212074 27.212186 1.0851063 1.0851957 27.215686 27.213445 1.0852017 1.0851958 27.212121 27.212271 1.0851063 1.0851957 27.214705 27.192312 1.0851886 1.0851959 27.213043 27.212861 1.0851960 1.0851958 27.213911 27.213916 myth of the Dragon which “eats” the Sun during a solar eclipse, see Kircher 1646, page 548), also known as nodal or nodical month. In Greek language is used the word Draconic (Δρακωνικός, instead of Draconitic, Δρακωνιτικός), not in compliance to the word Anomalistic (Ανωμαλιστικός). The duration of the Draconic month is about 27.2122d, a bit shorter than the Sidereal month (27.3218d) Barbieri (2017). The two Nodes return to the same position relative to the stars after 18.612 tropical years (larger than a Saros cycle, see Espenak and Meeus, 2008-NASA eclipse page). This implies that the Nodes change their projected position on the sky by about 1.564°/1 Synodic month (but not at constant velocity, due to the Anomalistic cycle), transiting each constellation in about 19.2 Synodic months. As the lunar angular velocity varies, the Moon comes to conjunction, opposition and the Nodes at different velocity each time. Mediterranean Archaeology and Archaeometry, Vol. 22, No 3, (2022), pp. 103-131 ASSEMBLING THE FRAGMENT D ON THE ANTIKYTHERA MECHANISM The Metonic, Callippic and Hipparchic cycles were invented in order to better correlate the Synodic month to the tropical year Oppolzer (1962); Meeus et al., (1966); Neugebauer (1975); Meeus (2004). Saros and Exeligmos both fit integer numbers of Draconic and Synodic cycles, but not an integer number of tropical years, and for this reason Saros is the proper choice for the eclipse predictions. 3.2. Requirements for a solar/lunar eclipse - The Draconic cycle One of the most important astronomical events in antiquity as well as today is a solar eclipse (https://sites.williams.edu/eclipse/archive/, Voulgaris et al., 2022). In ancient Mesopotamia, Egypt and Greece, eclipses were correlated to an extended mythology regarding the Gods’ fight, the Kings, their thrones etc. As Herodotus mentions, the war between 107 Medes and Lydians stopped because of the eclipse of 28 May 585 BC, which Thales had predicted, Panchenko (1994); Stephenson and Fatoohi (1997). The prediction of a solar eclipse was a considerable challenge for the astronomers of the ancient world, who used their observations and calculations to improve the accuracy of the eclipse predictions, Steele (2015). A solar/lunar eclipse will occur only if two specific astronomical positional parameters are satisfied: 1) The Moon is at its new phase (or Full Moon) and 2) At the same time, the New Moon (or Full Moon) is located at or close to the Ascending or the Descending Node, i.e. close to the beginning or the middle of the Draconic month. The resonance of these two periods in (about) 0 or π phase, guarantees that a solar (or lunar) eclipse will happen somewhere on the Earth. Of course, in order to predict where on Earth the eclipse will be visible, additional calculations and parameters are needed. Figure 2. Harmonic graph of the Draconic vs Synodic cycle, assuming that the two cycles start simultaneously. On each of the resonance (or close to resonance) points (phase π or 2π) of the two graphs, a solar or lunar eclipse will occur. The units on the x-axis are Synodic lunar cycles. (On the calculations for the graph presentation, the phase variation resulting from the Anomalistic cycle was not included). As the Moon and the Earth have angular dimensions (about 0.5° for the Moon as observed from the Earth and about 1.5° for the Earth as observed from the Moon), their corresponding shadows are extended. Hence, an eclipse can occur even if the Moon is located at some angular distance from the Node. Thus, a zone on the Ecliptic can be defined with angular dimensions of about ±17° 25’ on either side of the Nodes (for all locations on Earth), called the Zone of Eclipses or Ecliptic limits. The ecliptic limits vary depending on the angular dimensions of the Moon, as a result of its varying distance from the Earth (Apogee/Perigee) i.e. by the Anomalistic cycle Fig. 2. The exact phase of the Anomalistic cycle (Apogee/Perigee position) during the resonance of the Synodic and Draconic cycles, defines the type of the eclipse event (total, annular or hybrid, also less affected by the Earth’s Aphelion/Perihelion), Espenak and Meeus (2008) (NASA eclipse page) and the exact time in which the Moon approaches the Nodes or Mediterranean Archaeology and Archaeometry, Vol. 22, No 3, (2022), pp. 103-131 108 A.VOULGARIS et al crosses the ecliptic limits, acting as a variable timer (see Section 5.4). The lower half area of the Back plate of the Antikythera Mechanism was dedicated to the eclipse events information, Freeth et al., (2008); Anastasiou et al., (2016b); Freeth (2019); Iversen and Jones (2019), presented on the Saros four turns spiral, divided into 223 cells/Synodic months. The ancient Manufacturer calculated, designed and constructed the proper gear train for this operation. At the same time, the Manufacturer engraved the specific sequence of eclipse events on the Saros spiral, i) By copying the eclipse information written on a papyrus (?), ii) By using the Babylonian eclipse records (?) Carman and Evans (2014); Iversen and Jones (2019) or iii) The eclipse events were directly (self)calculated making use of his own construction (?) (see Section 7). 3.3. Eclipses information/prediction mechanism on the Antikythera Mechanism The Antikythera Mechanism could predict, the phases of the Moon, the position of the Sun on the sky as it is projected onto a Zodiac constellation, at the corresponding Egyptian and Zodiac month/date, Freeth et al., (2006); Voulgaris et al., (2018a), and also the Metonic month. Additionally it could predict the year and the date in which the athletic games would occur, Freeth et al., (2008). By turning the Lunar Disc, which is the ideal input for handling the Mechanism Voulgaris et al., (2018b), see Section 4.1), the user of the Mechanism could set the lunar pointer aiming directly to the Golden sphere (or to opposite position). At these positions the small half white/half blackened lunar phase sphere of the Lunar Disc shows its black (white) hemisphere, Wright (2006); Carman and DiCocco (2016); Voulgaris et al., (2018b). Geminus in Chap. “VIII-About months“ and “XIAbout the Lunar eclipse” mentions that the Full Moon occurred in mid-month, Διχόμηνις, and in Chap. “IXAbout the light of the Moon”, and “X- About the solar eclipse”, that the New Moon occurred at the last day of the Synodic month, on (29th) or 30th day, named Τριακάς (Triakas), Theodosiou and Danezis (1995); Spandagos (2002); Jones (2017). Therefore, in each Synodic month, the New Moon phase comes after the Full Moon. By each re-aiming of the Lunar Disc pointer to the Golden sphere (or to the opposite position-Full Moon), there was a possibility that a solar (or lunar) eclipse would occur. Just by observing the Front dial plate of the Mechanism it was not possible to know for sure if a solar or a lunar eclipse would occur. Information but, not calculation about upcoming eclipses was only presented on the Saros spiral, on the Back plate of the Mechanism, Freeth et al., (2008); Anastasiou et al., (2016b); Iversen Jones (2019) and New Saros cell numbering by Voulgaris et al., (2021). It seems that the calculation of the Draconic cycle - the fourth lunar cycle (well known in Ancient Greece, Egypt and Babylon), which is quite critical for the eclipses prediction - is not presented or preserved on the Antikythera Mechanism. 3.4. Two speeds on the Antikythera Mechanism machine gearing In the following Section the “nature” of the Antikythera Mechanism as a measuring instrument is to be presented. By rotating the Lunar Disc-Input of the Mechanism about 389°, one Synodic month is completed Fig. 1. In addition, Saros pointer crosses one cell out of 223 (and the Metonic pointer one out of 235 cells), Anastasiou et al., (2014). The pointer’s angular shift is about 6.4°/cell-Synodic month Fig. 3. Thus, the same time unit – a synodic month – is presented by different pointers on the Mechanism. The Front plate pointers move relatively fast, making an extensive route, offering High- Resolution information regarding their geometrical position. In contrast, the Back plate pointers are Low-speed with Low-positional resolution. A large number of compressed information is also engraved on the scales of the Low-Speed pointers: 235 words (months) on 235 Metonic spiral cells and a number of eclipse events on some of the 223 Saros spiral cells. Each Saros (and Metonic) cell corresponds to 29.53d (mean duration). The mean dimensions of these cells are about 7mm4mm. This means that the Saros pointer changes its position about 6.4°/29.53d ~ 0.257°/1d, Fig. 3. It is impossible to detect/measure time with precision of a day directly on each of the Metonic/Saros cells (7mm/29.53 ~ 0.216mm/1d). If the pointer aims at the middle of the cell, this does not mean that the date is precisely at the middle of the month. This calculation “inconsistency” arises from the short dimension of each minimum unit, the engagement of the gears with triangular teeth shape, the gear periodic errors, the small precession of the axes, the (slight) off-center positioning of the central gear holes, the random teeth shape mismatches, the mechanical limits of the specific gear train etc., Edmunds (2011). Therefore, the Saros and Metonic spirals’ minimum measurable time unit is one cell per one Synodic month. Mediterranean Archaeology and Archaeometry, Vol. 22, No 3, (2022), pp. 103-131 ASSEMBLING THE FRAGMENT D ON THE ANTIKYTHERA MECHANISM 109 Figure 3. A) A close up of the Saros spiral with the eclipse information. The angle which the Saros pointer travels in one cell/Synodic month is about 6.4°. Note also the two pairs of successive cells in which the solar eclipse (Η) is mentioned on the preceding cell and the lunar eclipse (Σ), to the following cell. Also, the cells with two eclipse events on the same Synodic month are visible (the lunar eclipse on the first line and the solar on the second line). B) Close up of the micrometer dial of the tool post slide hand-wheel of a conventional lathe. C) Each line of the micrometer dial changes the tool post slide position by 0.025mm. D) The position between two lines, the movement of the tool post slide could not be defined as 0.025mm/2, because of the mechanical limitation of the system, this movement presents an uncertainty. Finally, the speed of a gear train output defines the resolution of the measurement. The Low-Speed pointers manage an extensive information capacity but in a low geometrical/positional resolution. Therefore, the Saros spiral is more of an “eclipse information table”, rather than an eclipse prediction scale. Likewise the Metonic spiral is a month name information table. It seems that the Antikythera Mechanism lacks High-speed gear train/high-resolution calculations for eclipse predictions that are exclusively based on the geometry of the position. The authors believe that the ancient Manufacturer would not rely on a pointer of “eclipse information table” to predict the eclipses - the most important events of the Mechanism - without taking into account geometrical/high-resolution calculations for these events. The existence of two different speeds on the Mechanism leads the authors to consider that the Antikythera Mechanism could have had an additional gear train, with a pointer dedicated to the eclipse prediction calculation. This pointer should be the output of a High-speed gearing train, based on geometry, offering a high-resolution calculation so that the eclipse prediction results would be highly accurate. 3.5. The Fragment D - description and analysis of the gear r1 Fragment D is an enigmatic and unplaced part of the Antikythera Mechanism. It was first noted by I. Svoronos and photographed by A. Rehm in 1905/1906. It was then misplaced and it was re-found in 1973, Price (1974); Lazos (1994); Freeth and Jones (2012). Fragment D is covered by a layer of calcified deposits, visible by the naked eye. Its mechanical design information cannot be clearly detected by the naked eye Fig. 4. Mediterranean Archaeology and Archaeometry, Vol. 22, No 3, (2022), pp. 103-131 110 A.VOULGARIS et al Figure 4. A) The front (original image) and B) back (mirrored image) visual photographs of Fragment D (Credits: National Archaeological Museum, Athens, K. Xenikakis, Copyright Hellenic Ministry of Culture & Sports/Archaeological Receipts Fund). C) Same scale AMRP corresponding radiography (enhanced). Insert: the displacement of the upper part of the shaft-r and its stabilizing pin is evident (combined tomographies in pseudocolors). D) 3D reconstruction in pseudocolors using AMRP tomographies via 3D Slicer software, Fedorov et al., (2012). The shape of the upper plate is visible. Tomographies processed by the authors. AMRP Tomographies revealed that Fragment D consists of three parts, Freeth and Jones (2012). The first part is a partially preserved Circular plate, with a diameter of 43mm (thickness about 1-1.5mm), and a square central hole. On this Circular plate, three per- pendicular pins were placed symmetrically at epicenter angles of 120° and are clearly detected Fig. 5. By observing the preserved circular perimeter shape, one concludes that this plate could not be a gear or a scale, as there is no detection of any tooth or engraved line or subdivision(s) on the plate’s surface. Mediterranean Archaeology and Archaeometry, Vol. 22, No 3, (2022), pp. 103-131 ASSEMBLING THE FRAGMENT D ON THE ANTIKYTHERA MECHANISM 111 Figure 5. Selected AMRP tomographies of a) the Circular plate, b) the gear r1, c) the Additional plate. A) the digital reconstruction of the Circular plate tomography, using well-preserved areas of the plate, B) the digital “cleaning” of r1 gear, from its corrosion and the deposits, in order to make its mechanical characteristics better visible. The stabilizing pin has also been placed on its correct position, C) the digital “cleaning” of the Additional plate from its corrosion and the deposits. Also, the circular hole of the plate and the circular cross section of the gear shaft (probably the end of the shaft’s edge), which is placed on the circular hole of the Additional plate, are presented. AMRP tomographies were processed by the authors. 1), 2), and 3) the corresponding bronze material reconstructions of the three layers of Fragment D, designed/constructed by the first author. In contact with the Circular plate, gear r1 is clearly visible (thickness about 1.5mm) and has been preserved in great condition. Its radius slightly varies as a result of the shrinkage deformation and the random cracks, Voulgaris et al., (2019b), between 16.7mm17.2mm. This gear also has a square central hole. C. Karakalos, Price (1974); Wright (2005); Freeth et al., (2006); Freeth and Jones (2012), measured 63 gear teeth (61 teeth well preserved and two teeth missing). Around the perimeter of the gear stacked deposits of salts, mostly calcites, showing a strong absorption in X-rays, Voulgaris et al., (2018c) and petrified silt are located, following the shape pattern of the teeth. However, a large percentage of these formations have been peeled out of the teeth boundaries. On this gear, the three pins which stabilized it on the Circular plate (Fig. 5) are sharply detected (stabilizing pins have also been detected on the gears c1/c2 and the l1/l2). Thus, gear-r1 and the Circular plate are fixed to each other and rotate with the same angular velocity as one body. It seems that this Circular plate is a base for the r1 gear, increasing its stability. Mediterranean Archaeology and Archaeometry, Vol. 22, No 3, (2022), pp. 103-131 112 A.VOULGARIS et al Figure 6. The four stages for the relocation on the Additional plate’s correct position and the broken r-shaft. A) The present position of the Additional plate and the shaft. B) Digital relocation of these two parts in order to align the broken shaft to the same axial direction with the central part of the shaft (i.e. the remaining shaft on the gear). C) Relocation of the shaft and the stabilizing pin in order for the pin to be level to the gear surface (white line). By the specific position of the hole on the main part of the shaft (i.e. below the gear surface), it is evident that the main part of the shaft (which is preserved inside the gear) is also displaced or shrunk due to deformation, Voulgaris et al., (2019b). D) A digital relocation of the main part of the shaft in perpendicular direction up to the stabilizing pin is presented. The relocation travel is about 0.4mm-0.5mm, (this procedure is also presented in Anastasiou Thesis (2014), without the fourth stage). Note that the perpendicular stabilizing pin is located between the gear-r1 and the Additional plate. The shaft’s hole for the pin adaptation is located just on the upper surface of the gear (white lines). E) By subtracting the two AMRP CT’s of Fragment D of different depth (CT-A surface of gear, CT-B, 1mm deeper) it is revealed that the head of pin-2 has been displaced. F) A “visual tomography of image A”. The bronze Circular plate, the gear and the shaft were cut in half in order to simulate the X-ray CT of A. The upper part of the shaft was displaced as is the original. Bronze parts construction and images by the first author. Close to the gear surface and perpendicular to the gear shaft, a hole can be detected, in which the stabilizing pin of the gear was adapted. In this area, the shaft is broken. Above this hole, the AMRP tomographies reveal the stabilizing pin in contact to the broken shaft and a strangely shaped piece, the Additional plate. These pieces have been displaced from their original position, as is evident by the difference in position of the corresponding formations (gear shaft, hole and pin) Fig. 5,8. The Additional plate (thickness about 1mm-1.5mm) is partially preserved in a particular shape, Fig. 5,8. Visible on the Additional plate is a circular hole, in which the broken and displaced circular axis is adapted (in Supplementary material at the end, a further analysis regarding the position of Fragment D parts is presented and discussed). Mediterranean Archaeology and Archaeometry, Vol. 22, No 3, (2022), pp. 103-131 ASSEMBLING THE FRAGMENT D ON THE ANTIKYTHERA MECHANISM The extended study of the AMRP Tomographies (original and also processed by the authors), did not give any proof for the existence of the three pins on the Additional plate, which could lead someone to consider that the Additional plate was fixed/stabilized on gear r1. The three pin’s edges are clearly detected only between the Circular plate and the gear 113 Fig. 6. Moreover, one of the three pins is totally out of the boundaries of the Additional plate Fig. 7. Additionally, the existence of the perpendicular stabilizing pin of the shaft, between the gear and the Additional plate prevents any contact between the surfaces of gear r1 and the Additional plate Fig. 5B, and 9C,D. Figure 7. A) A 3D representation using the 3D Slicer software, Fedorov et al., (2012), of the digitally reconstructed and “cleaned” AMRP tomographies of Fragment D, processed by the authors. B) The Fragment D is represented in transparent layers. C and D) Two side views in order to make the pin position visible. E) Left, the transparent and Right, the 3D representation of the “artificial radiographies” of the three assembled original bronze reconstructed parts, using the 3D Slicer software. Photos and processed images by the authors. The gap between r1-gear and the Additional plate is the thickness of the stabilizing pin, at least 0.9mm1.0mm (in the bronze original a bit larger, taking into account the material corrosion and shrinkage). Therefore, the Additional plate is not in contact with the r1gear and seems to be utterly independent from it. Inside the circular hole of the Additional plate, the broken part of the gear shaft is also preserved, which at this point has a circular cross section Fig. 5,9. The circular cross section design means that during the rotation of the r1 gear and its shaft/axis the Additional plate does not rotate and is independent of the gear/shaft rotation. The authors cannot find any realistic mechanical operation for this Additional Plate, other that it being a simple oblong plate - curved by one side (of which today only this part remains), acting as one out of two bearing points-plates for r-shaft (Fig. 14,16): most of the Mechanism axes/shafts, are supported on two opposite points-plates, Voulgaris et al., (2021). For example, the preserved shafts f, g, h, i, are supported between two parallel plates Middle and Back (this is also mandatory for the lost shafts m, n and o) and the d- shaft, between the Middle plate and the Ω-shaped Retention bar, Voulgaris et al., (2018c). Mediterranean Archaeology and Archaeometry, Vol. 22, No 3, (2022), pp. 103-131 114 A.VOULGARIS et al In Efstathiou et al., (2011); Anastasiou (2014); Basiakoulis et al., (2017), the Additional Plate is represented as fixed on gear-r1 and rotating with the same angular velocity. It is also suggested that this plate acts as a cam leading to the Equation of Time calculation. This consideration does not agree with the AMRP tomographies and the observations presented in this paper (the Additional plate is not in contact to the gear, it has not fixing pins, and it has circular hole - not square, i.e. it does not rotate with the gear, see also Supplementary material). It is evident that if the ancient Manufacturer wanted to make the Additional plate rotate with the same angular velocity as the r1 gear, he should have at least made the hole of the Additional plate square, as well as the corresponding cross section of the rshaft. Moreover, the stabilizing pin between the r1 gear and the Additional plate prevents the perfect immobilization of the Additional plate on the gear, even if it had a square cross section. The existence and the specific position of the perpendicular stabilizing pin leads to the conclusion that the pin was adapted in order to stabilize the gear on the specific position, as such perpendicular stabilizing pins are preserved in shafts f, g, h and i. Therefore, the Circular plate is the base of gear r1 and from now on we call it Circular base. According to our design suggestion, the Circular base is in contact with the Internal wooden casement of the Mechanism, Voulgaris et al., (2019b). In this way, the diameter of the base is bigger than the gear’s diameter and the stability of the gear is maximum. The engraved letters “ME” are detected on three different places on Fragment D. Engraved letters have also been detected on two other gears of the Mechanism, on m1 (letter H) and b1 (letter N). These letters could be the assembly numbering of the parts (?? but not all gears have engraved numbers on them), or maybe a more imaginative scenario may be in play, e.g. the Manufacturer could have written his name or a phrase spread out on several parts of the Mechanism (?). 3.6. The engaged gears b1-a1, the output on the a shaft The b1 gear is the Mechanism’s larger gear, Freeth et al., (2006). In contrast to the e3 gear (the second larger), which is made by one circular plate, the b1 gear consists of several assembled parts, which form four radial bars and a ring with teeth and probably was constructed by the remained bronze parts-scrap. The rest surface area of the gear does not have any bronze material. The absence of the material resulted in the irregular and relatively strong, three-dimensional shrinkage deformation of the b1 gear, mostly by the long-time stand still of the Mechanism on the sea bottom, the effect of gravity and its abrupt dehydration after its retraction from the sea, Voulgaris et al., (2019b). Figure 8. A) The graph of the b1 gear radius vs angle position. Because the geometrical/mechanical centers of the two axes bin and bout, differ by about 0.4mm (as a result of the contractional deformation), the geometrical center of b1 gear central circular hole was selected as the center for the radius and epicenter angle measurements. B) The graph of the measured gear teeth number of b1 gear vs angle position. This graph more or less has similarities in the monotonicity to the above graph as a result of the radius variation (shorter radius results to smaller number of Mediterranean Archaeology and Archaeometry, Vol. 22, No 3, (2022), pp. 103-131 ASSEMBLING THE FRAGMENT D ON THE ANTIKYTHERA MECHANISM 115 teeth). Measurements by the Authors. Teeth number measurements of gear b1 (in its present condition) were made by C. Karakalos (223-226 teeth), D.S. Price (225 teeth), M. Wright (216-231 teeth) and Freeth et al., (2006) (223-224 teeth). An additional problem for the precise measurement of the teeth number is the difficulty for the detection of the actual teeth boundaries and tip position: many of the preserved gear teeth are filled with depos