Reference / Paper · 2024
arXiv:2412.07023. Revised draconic gearing and a classification of the Saros eclipse events.
A Revision for the Draconic Gearing of the Antikythera Mechanism, the Eclipse Events of Saros Spiral and their Classification (V.2) 1 2 Aristeidis Voulgaris *, Christophoros Mouratidis , Andreas Vossinakis 1 2 3 3 City of Thessaloniki, Directorate Culture and Tourism, Thessaloniki, GR-54625, Greece, Merchant Marine Academy of Syros, GR-84100, Greece, Thessaloniki Astronomy Club, Thessaloniki, GR-54646, Greece *Email: [email protected] Keywords: Fragment D, Draconic gearing, Draconic scale, ecliptic limits, eclipse events classification, gears of Antikythera Mechanism, gear errors. Abstract Our research is focused on the missing, but important and necessary Draconic gearing of the Antikythera Mechanism. The three Lunar cycles Sidereal, Synodic and Anomalistic are represented on the Mechanism by correlating the Fragments A and C (part of the Front plate), whereas the fourth Lunar cycle-Draconic results after correlating the unplaced Fragment D with Fragment A. The Draconic scale on the Antikythera Mechanism offers the positional information (ecliptic latitude) of the Moon relative to the Ecliptic. Considering the deformation of the Mechanism’s parts during 2000 years underwater and their shrinkage after their retraction from the sea bottom, we present a revised gearing scheme of the Draconic scale. The existence of the Draconic gearing is crucial, because both the preserved and the missing eclipse events can be pre-calculated by the phase correlation of three pointers: of the Lunar Disc, of the Golden sphere/Sun-ray and the Draconic. This means that the eclipse events are calculated by pure mechanical processing and that they are not documented observed events. The phase coordination of the three lunar cycles can be used as a quality criterion for a functional model of the Mechanism. Eudoxus papyrus was the key for the lost words completion of the Back Plate Inscriptions/eclipse events classification of the Antikythera Mechanism. 1. Introduction The Antikythera Mechanism was a geared device of the Hellenistic era ca. 180 BC. By means of gears, pointers and scales, it displayed the Moon phases, the timed sky path of the Sun and the Moon across the zodiac, it predicted upcoming solar and lunar eclipses with date and hour accuracy and it also showed the starting date of the Athletic Games (Wright 2006; Freeth et al., 2006 and 2008; Anastasiou et al., 2016a; Jones 2017; Seiradakis and Edmunds 2018; Voulgaris et al., 2023b). These calculations are based on the duration (beginning and middle) of the lunar synodic cycle (except the tropical year of the Sun) as it results by the measuring units of the Mechanism’s scales (Voulgaris et al., 2025a). Via its pointers and scales, this complex device provided ready-made information on the timing of astronomical and social events, without additional calculations. The preserved parts of the Antikythera Mechanism incorporate three out of four lunar cycles: Sidereal (Lunar pointer returns to the initial point on the Zodiac Dial ring), Synodic (Lunar pointer aims to Golden sphere-Sun) and Anomalistic (pin&slot configuration, Wright 2005; Freeth et al., 2006; Voulgaris et al., 2018b and 2023a). The fourth lunar cycle, the Draconic cycle, named in antiquity as ΑΠΟΚΑΤΑΣΤΑΣΙΣ ΚΑΤΑ ΠΛΑΤΟΣ or ΠΛΑΤΙΚΗ ΑΠΟΚΑΤΑΣΤΑΣΙΣ (restitution in Lunar latitude, Jones 1990; Pirtea 2019; Voulgaris et al., 2022), was also well known and in use during the Hellenistic era: Chaldean/(Babylonian) astronomers (Toomer 1984, p.175) had found the equation of 5458 Synodic months = 5923 Draconic cycles, which leads to the recalculation of 1 Saros: 223 Syn. ≈ 241.998 Drac. (instead of integer 242 cycles). This means that after one Saros, the Moon does not exactly returns to the Node1 (see Voulgaris et al., 2023b in Appendix A, p.62-64) Eudoxus Papyrus, written on 165-164 BC (Blass 1887, P.Par. 1, col. XVIII, 10-20) refers to the Lunar Node and its relation to a solar and lunar eclipse. Ptolemy in Almagest (Toomer 1984, p.175) mentions that Hipparchus used the equation by Chaldeans (also p.206-209) and in Tetrabiblos he refers to the Moon’s position relative to a Node (Ashmand 1900, III, 17, 18). In Vettius Valens (Brennan 2022, I, 16) Theon of Smyrna (Hiller 1878, p.194) Cleomedes (Ziegler 1891, II.5; see also in Spandagos 2002) Simplicius (Heiberg 1894, 2.12, p.495). The Draconic cycle, a very important and critical cycle, which is directly related to the solar and lunar eclipses, that seems to be missing (lost?) from the Mechanism, but can be represented by correlation of fragments A and D (Voulgaris et al. 2022). There are three specific and critical arguments that lead to the correlation of fragments A and D with an additional gearing with unknown operation (that it could be the Draconic gearing, see Figure 2) presented and discussed in Voulgaris et al., 2018b, p.229-230; 2022, p.116-117 and fig. 10; see also 2023b. If the Input of the Mechanism was from gear-a1 (taken as a common sense assumption since 1974), many mechanical problems regarding the functionality and the handling of the device arise. The Input of the Mechanism from gear-a1 introduces low torque to the following engaged gears (Roumeliotis 2018) and their rotation becomes doubtful and non-seamless. This effect is evident on the Research Quality bronze model reconstructions2. Additionally, by starting the Mechanism from a1-gear, the rotation of the Lunar pointer is fast: one tooth (out of 48) rotation of a1-gear, rotates the Lunar pointer by ≈21.3°, i.e. the Lunar pointer runs through ≈70% of a zodiac sign of 30°. In this state, any attempt to aim the Lunar pointer to a desired position is difficult or impossible. Here is an equivalent example regarding the steering wheel of a (hypothetical) car: when the driver rotates the steering wheel by (360°/48 teeth) = 7.5°, the wheels of the car rotate by 21.3°. So, driving this car becomes extremely difficult and very dangerous. This car could not pass the test of the driving standards (which usually require that by turning the steering wheel at 7.5°, the wheels rotate 0.5°). It is difficult to explain why the ancient Craftsman of the Mechanism would make such doubtful design consideration for its device Input knowing it would create 1 A good repetition of Saros occurs for the very central eclipses around the Earth’s equator and tropics. In every Saros cycle the lunar shadow path gradually moves to northern or southern latitudes and the Saros sequence ends with a partial solar eclipse visible from the North or South Pole, see Figure A1 in Appendix-A. In the same manner, in every Saros cycle, a total lunar eclipse gradually became a partial eclipse and ends as a (small percentage) penumbral eclipse, see Figure A2 in Appendix-A. 2 The Protocol of Mandatory Parameters for a Research Quality Bronze Reconstruction of the Antikythera Mechanism, see Voulgaris et al., 2025a, p.277, Table 6. Reconstructions using modern parts for stabilization, axles/shafts from steel and other modifications change the mechanical status of the device. 2 problematic gear motion, risk component damage, present handling difficulties, and offer poor Lunar pointer control. In such a way that he could not fully control the Lunar Disc and its pointer, since the precise alignment and positioning of the pointer is very critical to the time calculations it performed (Lunar pointer aims to the Golden sphere-Sun = New Moon, right after a new month begins/a solar eclipse probability, or in opposite position = Full Moon, midmonth/a lunar eclipse probability). The Antikythera Mechanism is similar to an analog mechanical computer capable for time and events’ calculations. The interaction with the User follows a three step scheme: - The User submits via the Input specific requests to the Mechanism (Data input), - The Mechanism processes the data via its gears (Processing - the process of transforming input information into and output) and - It produces results (predictions) via its pointers and scales (Output). And finally, the User evaluates and uses the results. In our view, the User and the Mechanism constitute a Human-Machine System (Wieringa and Stassen 1999). Human-Machine System is a system in which the actions of a human-user and a machine are interrelated, and are both necessary in order to achieve goals and objectives. This interrelation/interaction can continue if the operation and the control of the machine are effective, ergonomic and optimized for handling by the User. The assumption that the gear-a1 is the Mechanism’s Input contradicts the Human-Machine System and it is also not compatible to the Human Body Kinesiology and Biomechanics (Carlton and Newer 1993; Lu and Chang 2010; Duncan et al., 2013; van Bolhuis et al. 1998; Hall 2019). On the other hand, setting the Mechanism’s Input to gear-b3, which is directly connected to the Lunar Disc in ratio 1:1, results in high torque (Roumeliotis 2018; Voulgaris et al., 2018b and 2022), ease of use and a perfect control of the Lunar pointer, which is needed for the accurate pointing. Moreover, as the main measuring unit of the Mechanism is the lunar Synodic cycle (each month of the ancient Greek calendar started right after the New Moon), a close relation between the Input of the Mechanism and the Synodic cycle is necessary. Therefore, for the proper operation of the Mechanism, it is very important to be easy and in precision to aim the Lunar Disc pointer to the Golden sphere-Sun or in the opposite position. This analysis creates a number of impacts, consequences and results that can relate the unplaced Fragment D to a new procedure for the Mechanism (Voulgaris et al., 2022 and 2023b). The existence of the Draconic gearing on the Antikythera Mechanism provides several results, gives answers to many questions, justifies the eclipse events’ specific sequence and explains the way of the events’ classification. Correlating the generated results by the Draconic pointer and the eclipse magnitudes’ description in Eudoxus papyrus (Blass 1887), lead to the reconstruction of the lost Back Plate Inscriptions. 2. The deformed and shrunken fragments of the Mechanism During 2000 years under water, copper of the bronze Mechanism’s parts (density 8.8 gr/cm3, alloy of ≈94% Cu and 6%Tin, Price 1974) gradually transformed to a new rocky material named Atacamite [Cu2(OH)3Cl] (Voulgaris et al., 2019b), which has much lower density (3.8 3 gr/cm3) and lower absorption in X-Rays (Voulgaris et al., 2018c). Pressure and gravity further deformed the Mechanism’s (new material) parts. When the Mechanism retracted from the sea bed the abrupt environment change and exposure to the dry air led to its shrinkage, deformation and cracking (Voulgaris et al., 2019b). Today, most of its parts are broken, shrunk, displaced, deformed, worn out and many flattened parts deviate significantly from flatness, Figure 1. The deformation in the fragments’ geometry is stronger to the larger dimension parts and leads to inaccurate dimensional measurements. Dimensional measurement values of a shrunk part are smaller than the original values before the shrinkage: E.g. Budiselic et al., 2020; Woan and Bayley 2024 calculated the total number of holes based on the partially preserved Bearing Base ring on Fragment C (today only an arc is preserved beneath the Egyptian calendar ring), and resulted to the number of 354 holes around the ring, whereas the 365 holes are needed for the functionality of this ring on the Mechanism. The Fragment C presents a strong deformation and shrinkage, evident on its cross section, see fig. 1E in Voulgaris et al., 2025a. Since deformation and shrinkage renders the dimensional measurement values smaller than the original ones, the measurements by Budiselic et al., 2020; Woan and Bayley 2024 can be used to estimate the percentage of linear shrinkage of the Mechanism’s fragments: Percentage of shrinkage= part dimension in current condition/original → 354/365 ≈ 96.7%, and the linear dimension of [shrunk/deformed part] minus [original dimension] ≈ ‒3.3%. The calculated percentage of shrinkage/deformation is not intended as a precise value-rule for the deformation of all fragments. However, we know with certainty that Fragments A and C, are currently deformed and have experienced shrinkage. This value is useful as evidence for the approximate estimation of the original dimension of some partially preserved parts, since today this device does not exist in its original bronze form and dimension. Figure 1: A 3D reconstruction of Fragment A from the AMRP RAW Volume data of X-Ray tomographies. The images are oriented to the b1 gear plane (which is just below the yellow horizontal line). The distorted Back and Middle plates should be parallel to the gear-b1. Panel A: The sagittal profile of Fragment A. The deformation of Fragment A (black arrows) at its left side is evident. Panel B: The coronal profile of the fragment. The crown gear-a1 deviates from the perpendicularity relatively to gear-b1. These geometrical deviations (bending and torsion) are a result of the shrinkage, displacement and deformation of the parts. Processed images by the authors using Real 3D VolViCon software. 4 The gear-b1 is the largest gear of the Mechanism and represents the daily timed travel of the Sun on the Zodiac sky (Voulgaris et al., 2018a). The gear is partially preserved; the radius of the gear is not constant across its perimeter and many teeth are missing (see graph of Figure 8 in Voulgaris et al., 2022). Gear-b1 isn’t a solid disc (as gear-e3 is) but consists of a ring and four arms, and it was probably constructed by scrap bronze parts. Today, gear-b1 isn’t flattened, as this non-solid material construction is more prone to shrinkage and distortion. Taking into account the shrinkage of the Mechanism parts, the total number of teeth for the original bronze gear-b1 should be larger than the current measured values. Adopting a value of about 3% of shrinkage, Table 1 gives the most probable values for the original gear teeth of gear-b1. Table 1: Estimation of the total teeth number of teeth for gear b1 taking into account a value of 97% of deformation/shrinkage. Second column presents the different estimates by several researchers (data taken from Freeth et al., 2006, Supplementary Notes). The third column gives the number of gear-b1 teeth by assuming a correction of 103% on the shrunk/deformed gear-b1 (left value) and the calculated minimum number of teeth resulting from the present condition of gear (right value). Measurements by C. Karakalos M.T. Wright D.S. Price 1974 Freeth et al., 2006 Voulgaris et al. 2022 Estimated number of gear-b1 teeth measured on the current condition of Fragment A 223‒226 216‒231 225 223‒224 219‒225 Probable number of gear-b1 teeth introducing the (estimated) correction of the shrinkage 103% (left value) 233(+) ‒ (223) 238(+) ‒ (216) 232(+) ‒ (225) 231(+) ‒ (223) 232(+) ‒ (219) 3.1 A Revised Draconic gearing for the Antikythera Mechanism The coordination of the solar tropical years, the lunar sidereal and synodic cycles in integer number of 19 years = 254 sidereal = 235 synodic cycles, creates the Metonic cycle, which it was used during antiquity in order to unite the solar tropical year with the Lunar year of 12/13 synodic months. The second important time coordination is that 223 synodic equals 242 draconic lunar cycles and also 239 anomalistic. This is the Saros cycle of 18.029787234 years (value resulting from the Antikythera Mechanism gearing), a duration in which the pattern of eclipse sequence and the geometry repeats with a delay of about 8 hours. This time coordination of the three lunar cycles does not correspond to an integer number of solar tropical years. Moreover, 223 synodic cycles does not correspond to an integer number of sidereal cycles, since 223 synodic cycles equals 241.029787234 sidereal cycles. The sidereal cycle is very important for the Mechanism’s calculations, as one full turn of the Lunar Disc (which is the ideal and proper Input for the Mechanism operation, see Introduction) corresponds into one sidereal cycle. The time difference between the Draconic-Sidereal cycles is very small (≈2.6h) and for this reason there are no characteristic numbers for the gear teeth of the Draconic gearing (applying the usual number of teeth of the Mechanism’s gears, presented below). Taking into account the probable number of the b1-gear teeth, which should be (slightly) higher than the present measured/estimated value, we present a revised gearing scheme for the Draconic gearing of the Antikythera Mechanism in order to improve its precision. We set 229 teeth for gear-b1 (or 228 for the 2nd option), since this is the gear for the Tropical solar year of the Mechanism. Gear-b1 rotates the crown gear-a1 (48 teeth). On shaft-a (partially 5 preserved on gear-a1), the gear-r1 (63 teeth, Freeth et al. 2006, Supplementary Notes) of Fragment D is attached and is engaged to the hypothetical gear-s1 (57 teeth). Then gear-s1 is fixed the gear-s2 (56 teeth) which is engaged with gear-t1 (22 teeth). Finally, the Draconic pointer is attached to the shaft-t (see Figure 2). Figure 2: Top-left panel, the configuration of the revised Draconic gearing. The annual gear-b1 rotates the crown gear-a1, and afterwards via gears r1, s1, s2, the motion is transmitted to gear-t1. The Draconic pointer is attached to shaft-t. The gearing is located at the right side of the Mechanism inside the External Wooden Casement (Voulgaris et al., 2019b). Top-right panel, the three parts of Fragment D (gear-r1 fixed on its Circular base and the thin stabilizing plate) were processed as separate parts using the corresponding tomographies, and were afterwards aligned and stacked (Voulgaris et al., 2022). The thin stabilizing plate is quite worn out and its remains have collapsed on gear-r1 surface, as the X-ray tomography of the fragment reveals. Most of the Mechanism shafts need to be stabilized between two plates for their proper operation. Bottom panel, the parts of the revised Draconic gearing were constructed in bronze by the authors. The configuration of the revised Draconic gearing follows: {223 * (254/235)} * (b3/e1) * (e6/k2) * (k1/e5) * (e2/d2) * (d1/c2) * (c1/b2) * {(b1/a1) * (r1/s1) * (s2/t1)} = 18.029787234 * {(b1/a1) * (r1/s1) * (s2/t1)} = 18.029787234 * {(229/48) * (63/57) * (56/22)} = 242.0002901 Equation (1) or 18.029787234 * {(228/48) * (63/34) * (61/40)} = 242.0001791 Equation (2) turns of Draconic pointer/one Saros. Equation (1) yields an error of +0.0002901 turns of draconic pointer per Saros, corresponds into a pointer’s shift ≈0.064476° per Saros (that is equal to one Draconic cycle per 3447 Saros cycles, practically non-detectable error for several decades of Saros cycles, and beyond the time span of the Mechanism operation). 6 The revised Draconic gearing configuration follows the preserved parts’ position (b1, a1), the specific parts of Fragment D and the hypothetical gears s1, s2, t1 according to the constructional characteristics of the ancient Craftsman. This revised gearing scheme meets the dimensional requirements and is fitted to the right side of the Fragment A, between the Internal and the External wooden casement of the Mechanism (Voulgaris et al., 2019b). The placement of the Draconic scale and pointer at the right side of the Mechanism offers a “3D projection” of the Moon relative to the Ecliptic: the Draconic pointer depicts the current position of the Moon relative to the Ecliptic plane-Zodiac Dial ring, (Moon above, on or below the Ecliptic plane) – ecliptic latitude, while at the same time the Lunar Disc pointer on the Zodiac scale depicts the ecliptic longitude, see Figure 3 and 4. Figure 3: The geometrical relation between the Ecliptic plane – Ecliptic longitude (which is defined by the Zodiac Dial ring of the Mechanism) and the Draconic scale/pointer’s position - Ecliptic latitude, at the right side of the Mechanism. The Lunar Disc pointer aims to the Golden sphere-Sun (Bitsakis and Jones 2016b). When the Draconic pointer aims to the Ascending (ΑΝΑΒΙΒΑΖΩΝ, Panel A) or to Descending (ΚΑΤΑΒΙΒΑΖΩΝ) Node (0, π phase of Draconic cycle), is parallel to the Zodiac Dial calendar ring and then the New Moon is located exactly on the Zodiac ring plane (ecliptic latitude 0). When the Draconic pointer is perpendicular to the Line of Nodes (π/2 οr 3π/2 phase of Draconic cycle), i.e. in greatest Northern (ΒΟΡΕΙΟΝ ΠΕΡΑΣ, Panel B) or in Southern ecliptic latitude (ΝΟΤΙΟΝ ΠΕΡΑΣ), and then the New Moon is 5.15° above or below the Ecliptic plane/Zodiac Dial ring and is located above or below the Sun. In this way, the “3D projection” of the Moon is represented by the Antikythera Mechanism. Bronze parts’ designed/constructed and images by the authors. The existence of the Draconic gearing and scale on the right side of the Mechanism’s box offers the following advantages and indirect information for the User: 1) It shows the Moon’s real position relative to the Ecliptic plane: on the Ecliptic = at the Node, or out of the Node or at the ecliptic limit or away of the Ecliptic - North/South), 2) It gives the User additional information related to the Zodiac month ring: a) when Node-A is at constellation x, then the Node-B is at the diametrically opposite constellation.3 This way, the User can find the current position of the lunar orbital plane in the sky. b) When the Full Moon is at its greatest northern ecliptic latitude (maximum Declination) and the Sun is 3 The astrologer Vettius Valens (born in 120 AD) in Anthologies 1.16 (Ἀναβιβάζοντα ἀπὸ χειρὸς εὑρεῖν, A Handy Method for Finding the Ascending Node) describes the way to find the two Nodes in the sky versus date (Brennan 2022). He also describes the Hipparcheion, a method for calculating lunar positions (1.19). 7 located between Sagittarius ‒ Aries (end of Autumn ‒ begin of Spring), then the Moon reaches its maximum altitude of ≈78°-82° for regions in Greece and it lights up the sky all night. This information could be very useful for military or navigation operations or night transportations or hunting. 3) It informs the User of an impending eclipse: When the Lunar Disc pointer aims to the Golden sphere-Sun (or in the opposite position) and the Draconic pointer aims between the ecliptic limits of the Draconic scale, the User can conclude that a solar (or Lunar) eclipse will occur. This is a sign to the User to look at the Back plate of the Mechanism: by observing the cell in which the Saros pointer aims (Anastasiou et al., 2014), he can read additional information about the eclipse event: Figure 4: The geometrical equivalent in 3D representation of the Moon’s position relative to the Zodiac Dial ring/Ecliptic plane, based on the position of the Draconic pointer. When the Draconic pointer aims to the Node (ΑΝΑΒΙΒΑΖΩΝ ☊ or ΚΑΤΑΒΙΒΑΖΩΝ ☋, see Panel B), the Moon is just on the Zodiac Dial ring/plane. When the Draconic pointer aims to the ΒΟΡΕΙΟΝ/ΝΟΤΙΟΝ ΠΕΡΑΣ (max Northern/max Southern Ecliptic passage of Moon), the Moon is +5.15°/‒5.15° from the Ecliptic plane, see Panels A and C). During the operation of the Mechanism the (imaginary) Line of Nodes (white line) moves westward, i.e. it rotates CCW in the Zodiac Dial ring (white arrow). the hour of the event and the direction of the shadow, since the ancient Craftsman has classified the events according to the position of the Moon relative to a Node, which leads to the direction of the Moon projection on the solar disc and the eclipse magnitude of the event (Freeth 2014 and 2019; Anastasiou et al., 2016; Pakzad 2018; Iversen and Jones 2019), see Section 6. A relative eclipse events classification is in use today (Pogo 1937). 3.2 A Second option for the Draconic gearing and scale If we make an assumption that the ancient Craftsman did not separate between the ascending and descending nodes, but was only interested to the relative lunar position with respect to the nodes and the greatest ecliptic latitude (max Northern/Southern), then a 8 second option for the Draconic scale can be deduced from Equation (2): By changing the last gear t1 into 20 teeth, the Draconic pointer rotates two times faster than the result from Equation (2): 18.029787234 * {(228/48) * (63/34) * (61/20)} = 2 * 242.00017912 = 484.0003582 turns of Draconic pointer per Saros, Equation (3). In this way, the Draconic scale has only one (common) position-point for the two Nodes, one common arc for the ecliptic limits having a double central angle, and one common point for the Maximum/Minimum ecliptic latitude of the Moon (North or South), see Figure 5. Figure 5: The second option for the Antikythera Mechanism Draconic scale. The Draconic pointer rotates two times faster (484 turns per 223 Synodic turns of the Lunar pointer). The position for the two Nodes is common (ΕΙΣ ΣΥΝΔΕΣΜΟΝ ‒ At a Node, Panel A), as it is also for the greatest Northern/Southern ecliptic latitude (ΑΚΡΟΤΑΤΟΝ ΕΓΛΕΙΠΤΙΚΟΝ ΠΛΑΤΟΣ, Panel B). The Ecliptic limits are represented in one common arc having values 2 of the initial ecliptic limits angles and they are divided in six unequal sectors for the easier classification of the eclipse events, see Section 6. The Nodes-A and B come in sequence (Node-A in an odd number and Node-B in an even number of Draconic pointer turns). This design offers a better resolution/double magnification of the Draconic pointer’s position between the ecliptic limits and makes the eclipse events classification easier (but it is more sensitive to the gearing positioning errors). 4. Detecting the mechanical errors on the Antikythera Mechanism functional models ‒ A mechanics quality criterion for their operation The Antikythera Mechanism is a unique geared measuring device. As it is a mechanical instrument, its study is also a subject of the Instrumentation of the geared systems which analyses the parameters, the behavior and the effects of the mechanical components of a geared device (Voulgaris et al., 2023b). All geared devices suffer endogenous mechanical errors, Figure 6. The mechanical errors on the equatorial mounts of telescopes, on theodolites, on clocks, on the wavelength scales of spectrographs affect the operation, the measurements and the efficiency of these 9 instruments. A general rule is that “The final results which are calculated by a measuring instrument are affected by the instrument itself”. Geminus in Introduction to the Phenomena (18.14) describing the calculation of the mean angular velocity of the Moon writes: Λοιπὰ ἄρα ἐστὶ τὰ ἐκφυγόντα τὴν τῶν φαινομένων διὰ τῶν ὀργάνων παρατήρησιν μιᾶς μοίρας πρῶτα ἐξηκοστὰ κα′ καὶ δεύτερα ι′ (the rest fractional part of 0° 21’ 10”, apparently escapes observation by instruments, see Manitius 1898; Spandagos 2002; Bowen and Goldstein 1996). The errors define the final limits on the precision of an instrument. A better quality instrument presents smaller errors, i.e. smaller deviation from the theoretical calculations, resulting in a better approach to Reality. The errors define the final limits on the measurement process of an instrument, because they “blur” the results as a lens with aberrations projects the image of an object with low sharpness and contrast (Hecht 2015). The main mechanical errors of the geared devices are the gears’ and axes’ eccentricities and the gear teeth random non-uniformity (Herrmann 1922; Muffly 1923; Voulgaris et al., 2023b). The eccentricity error happens when the rotation axis (of a gear or a shaft or both) does not coincide with the axis of symmetry, i.e. the rotation axis is not at the center of the gear. Therefore, the gear in point-x has a shorter radius and in the opposite point has a larger radius. The eccentricity affects the position of the pointers and the constant angular velocity, e.g. a clock’s minute indicator rotates periodically faster for half period (shorter gear radius) and slower for second half period (larger gear radius). Figure 6: “Bronze: the Real World vs Red: the digital ideal world”. The broken gear of an old clock (it has four arms like gear-b1). The bottom right quadrant broke many times and was repaired using adhesive alloy of tin/lead. The gear has very low eccentricity, but the image reveals that the cause frequent breaking is the result of a division error during the formation process of the 72 teeth, probably because of eccentricity error of the dividing machine of that era (using Computer-Aided Design-CAD software an ideal “digital” gear, having no errors was sketched and aligned exactly on the bronze gear photo (in red color). Arrows A and B indicate positions where there is a perfect match between the real and the digital teeth (zero phase difference). A phase difference gradually appears in arrows C (phase difference ≈π/3) and D (phase difference ≈π/2-half tooth, see insert). Because of the positioning error, the gear teeth were stressed by the steel teeth of the preceding (smaller, higher torque) engaged gear. The image was captured by an optomechanical system designed/constructed by the first author, having low opto-geometrical distortions and in minimized parallax (Voulgaris et al., 2018a). 10 Endogenous mechanical errors, especially the error of eccentricity or the random nonuniformity on the gear teeth could be present in any of the Mechanism’s gears. Eccentricity errors with several values around to 0.1-0.3mm and various directions are present, e.g. on gear m2, with a probable effect detected (see Fig. 8 in Voulgaris et al. 2023b). However, since the preserved parts are deformed and shrunk and other parts are partially preserved or missing, some additional information cannot be recovered. The tooth by tooth motion transmission between engaged gears having triangular shaped teeth also present intermittent (“broken”/non-continuous) motion (see Fig. 7 in Voulgaris et al., 2023b). The eccentricities and the random non-uniformity of the gear teeth affect the accuracy of the pointers (Edmunds 2011; Voulgaris et al., 2023b and the video Triangular vs involute gear teeth sound recording https://www.youtube.com/watch?v=h-qpXYK3bls), causing deviations from the proper/ideal position of the pointers with errors around ±(1°-3°) or ±4°. Much higher positional errors occur on gears that are close to the end of a gearing (i.e. close to the pointers), as result of non-uniform teeth. Digital 3D simulation can calculate the theoretical perfect/ideal position of the Mechanism’s pointers versus time since in this case there are no mechanical errors, friction or other constructional mismatches, and the digital gears/pointers operate in an ideal and perfect mechanical world. These construction errors of the Antikythera Mechanism affect the accuracy of the predicted eclipse events, especially the events located too close - out or inside - the ecliptic limit boundary are affected by the construction errors of the Mechanism (Voulgaris et al., 2023b). Due to the (hypothetical, but very probable) gears’ errors, the Draconic pointer deviates from the theoretical position. In marginal case when the theoretical position of the Draconic pointer is inside the ecliptic zone (= eclipse event) and very close to an ecliptic limit, due to the errors of the gearing, the pointer may indicate erroneously out of the limits (= no event) and vice versa (see Figure 10 in Voulgaris et al., 2023b). This inaccuracy also explains the absence of some eclipse events that should theoretically be calculated (=missing events) and the existence of some other events that should not exist according to the theoretical calculations (=additional events). The gear(s’) eccentricity can also affect the Draconic pointer position relative to the Node: E.g. The pointer should be theoretically located just on Node-X, whereas its real position is before/after (i.e. north or south) the respective Node. Therefore, the classification of the eclipse events (relative to the Node), can be affected by the endogenous gearing errors of the instrument (see Figures A4-A7 in Appendix-A). Finally, each instrument has its “personal errors”, (see Figure 4 in Voulgaris et al., 203b). The eclipse events’ sequence and classification can (slightly) vary in different instruments, and the results can deviate from the theoretical classification deduced by an imaginary ideal/perfect device, see Table 6. The errors on the pointers’ position (Lunar Disc, Golden sphere and Draconic) of a functional reconstructed model of Antikythera Mechanism can be detected and measured by checking the deviation between the (theoretical) position of the pointers during characteristic phase/antiphase time coordination and the pointers’ current position in a certain time span. We call this arithmetical resonance as “The Key Time Points for Mechanism’s pointers” 11 dedicated for Metonic and Saros cycles. This is a mechanics criterion for the quality evaluation of a functional model of the Mechanism, see Table 2. Table 2: The arithmetical resonance of the lunar and solar-tropical cycles’ in 0, π/2 or π phase, for Metonic and Saros periods, can be used as a quality criterion of a functional model of the Antikythera Mechanism: by checking the pointers’ deviation from the theoretical position derived from the arithmetical resonance cycle phase (theoretical position). The following calculations are based on the starting date of the Antikythera Mechanism pointers on 22/23 December 178 BC: New Moon at Apogee (pin&slot starts at Apogee), Golden sphere-Sun and Lunar Disc pointer at 1st day of Capricorn, Draconic pointer at Node-A (Descending) (Voulgaris et al., 2023a, 2023b and 2023c). In specific cycles, the phase correlation in 0 or π or π/2 (should) appear on the Mechanism pointers. The mechanical errors create small or larger deviations by the cycles’ phase coordination. QUALITY CRITERION I: KEY TIME POINTS FOR METONIC CYCLE Percentage of Metonic cycle Synodic cycle (Lunar Disc pointer) vs Solar Tropical year (Golden sphere) Key Time points Sidereal cycle vs Solar Tropical year Key Time points (New Moon, Sun start from 1st Capricorn) Full Metonic cycle 19y, 235 Synodic, 254 Sidereal After 235 repositions of the Lunar Disc pointer to Golden sphere (starting from 1st Capricorn), the Sun pointer must return to its starting position, 1st Capricorn After 9.5 years, the Sun pointer must rotate 9+0.5 turns (= in opposite position to 1st Capricorn). 117+0.5 repositions of the Lunar Disc pointer (= opposite position to Golden sphere) After 58+0.75 repositions of the Lunar Disc pointer (= Last quarter), the Sun pointer must aim at 90° before 1st Capricorn After 176+0.25 repositions of the Lunar Disc pointer (= First quarter), the Sun pointer must aim 90° after 1st Capricorn After 254 full rotations of the Lunar Disc pointer (starting from 1st Capricorn), the Sun pointer must return to its starting position, 1st Capricorn Half Metonic cycle 9.5y, 117.5 Synodic, 127 Sidereal ¼ Metonic cycle y 4.75 , 58.75 Synodic, 63.5 Sidereal ¾ Metonic cycle 14.25y, 176.25 Synodic, 190.5 Sidereal After 127 full rotations of the Lunar Disc pointer (sidereal cycle stars from 1st Capricorn), the Sun pointer must rotate 9+0.5 turns (= opposite position to 1st Capricorn). Full Moon at 1st Capricorn After 63+0.5 rotations of the Lunar Disc pointer (= opposite position to 1st Capricorn), the Sun pointer must aims at 90° before 1st Capricorn After 190+0.5 rotations of the Lunar Disc pointer (= opposite position to 1st Capricorn), the Sun pointer must aim 90° after 1st Capricorn QUALITY CRITERION II: KEY TIME POINTS FOR SAROS CYCLE Percentage of Saros cycle Repositions of Lunar Disc pointer to Golden sphere, Synodic cycle vs Draconic cycle - Key Time points Repositions of Lunar Disc pointer to Golden sphere, Synodic cycle vs Anomalistic cycle - Key Time points Full Saros 18.02978y 223 Synodic, 242 Draconic, 239 Anomalistic After 223 repositions of the Lunar Disc pointer to Golden sphere, the Draconic pointer (after 242 full rotations) must return to Node-A After 223 repositions of the Lunar Disc pointer to Golden sphere, the Pin (after 239 full rotations of gear-k2) must return at Apogee (pin in its largest distance from gear-k2 center After 111+0.5 repositions of the Lunar Disc pointer to Golden sphere (= opposite position to Golden sphere), the Pin (after 119+0.5 rotations of gear k1) must be in the shortest distance from center of gear-k2 (Perigee) (‒) (the phase of 0.75 Anomalistic cycle it cannot be defined on the Pin&Slot system, see Fig. 16 in Voulgaris et al., 2018b) (‒) (the phase of 0.25 Anomalistic cycle it cannot be defined on the Pin&Slot system see Fig. 16 in Voulgaris et al., 2018b) Half Saros - Sar 111.5 Synodic, 121 Draconic, 119.5 Anomalistic ¼ Saros 55.75 Synodic, 60.5 Draconic 59.75 Anomalistic ¾ Saros 167.25 Synodic, 181.5 Draconic 179.25 Anomalistic After 111+0.5 repositions of the Lunar Disc pointer to Golden sphere (= opposite position to Golden sphere), the Draconic pointer must return to Node-A (after 121 full rotations). Full Moon at Node-A After 55+0.75 repositions of the Lunar Disc pointer to Golden sphere (= last quarter), the Draconic pointer must aim at Node-B (after 60+0.5 rotations) After 167+0.25 repositions of the Lunar Disc pointer to Golden sphere (= first quarter), the Draconic pointer must aim at Node-B (after 181+0.5 rotations) 12 The ancient Craftsman could probably use them to (periodically) check and correct the pointers’ position by a minor rotation in a small angle of their corresponding scales (Zodiac month ring and hypothetical Draconic scale). He could also slightly change the direction of the Golden sphere/Sun and its pointer to a specific subdivision (day) of the Zodiac month ring (by slightly aiming the Sun pointer to the correct subdivision). 5. Revised ecliptic limits and eclipse events For the recalculation of the eclipse events we recalibrated DracoNod(-V2) visualization program (Voulgaris et al., 2023b) by revising the ecliptic limits4 in order to improve the best match to the preserved eclipse events, see Figure 7. Figure 7: The Draconic scale topology. The Line of Nodes divides the scale into the left hemisphere, representing the lunar orbit below/South of Ecliptic and into the right hemisphere - lunar orbit above/North of Ecliptic. The revised ecliptic limits, according to the preserved eclipse events, are also sketched. Each ecliptic zone is divided in six unequal Sectors plus a linear sector(s)-area of Node, related to the eclipse events’ classification (see Section 6, Table 5, Figures 9 and 10, and Figures A4A7 in Appendix-A). (Here the Sectors are presented in a “deformed” position due to the gearings’ errors). This revision sheds light to the controversial index cursive letter Ѡ (Freeth 2019; Iversen and Jones 2019), which is engraved on the Back Plate Inscription (BPI of Fragment F, L.31), see at the end of Table 3. Ecliptic Zone A: 6.5° South (CCW) and 22.5° North (CW) from Node-A, Ecliptic Zone B: 5.7° North (CCW) and 15.3° South (CW) from Node-B, (both include the gearing errors). Mean values: 18.9° before Nodes and 6.1° after Nodes), and they are common for the lunar and solar eclipses. The eclipse events were recalculated via DracoNod-V2 visualization program (Voulgaris et al., 2023b) by applying the revised ecliptic limits and by using the equation 223 synodic months = 242 draconic months (= 239 Anomalistic months). The revised eclipse events are presented on Table 3 and in Figure 11. 4 The ecliptic limits define the boundary of the Ecliptic zone. When the New Moon/Full Moon is located inside the ecliptic zone, a solar or lunar eclipse occurs, see Green 1985; Smart 1949. 13 Table 3: The revised Saros spiral eclipse events recalculated using the authors’ DracoNod-V2 program (Voulgaris et al., 2023b). A total 64 eclipse events are