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ISAW Papers 11 (2016)

The moon phase anomaly in the Antikythera Mechanism

Christián C. Carman and Marcelo Di Cocco

Abstract: The Antikythera Mechanism is a mechanical astronomical instrument that was discovered in an ancient shipwreck at the beginning of the twentieth century, made about the second century B.C. It had several pointers showing the positions of the moon and sun in the zodiac, the approximate date according to a lunisolar calendar, several subsidiary dials showing calendrical phenomena, and also predictions of eclipses. The mechanism also had a display of the Moon’s phases: a small ball, half pale and half dark, rotating with the lunar synodic period and so showing the phases of the moon. The remains of the moon phase display include a fragmentary contrate gear. According to the reconstruction offered by Michael Wright, this gear is now pointing unintentionally in the wrong direction. In this paper we offer for the first time a detailed description of the remains of the moon phase mechanism. Based on this evidence, we argue that the extant contrate gear direction is the originally intended one, and we offer a conjectural explanation for its direction as an essential part of a representation of Aristarchus’s hypothesis that half moon phase is observably displaced from exact quadrature.

Library of Congress Subjects: Antikythera mechanism (Ancient calculator); Astronomy, Greek.

Contents

Introduction

The Antikythera Mechanism is a mechanical astronomical instrument that was discovered in an ancient shipwreck at the beginning of the twentieth century. The shipwreck has been dated around 60 B.C.1 A consensus does not exist on the question whether the Mechanism was built around that time or significantly earlier.2 After twenty centuries under water, it is incomplete, and broken into several fragments. The extant fragments are nevertheless enough for reconstructing the main structure and functions of the mechanism: it had several pointers, interconnected and worked by toothed gearing, showing the positions of the moon and sun (and probably also the planets) in the zodiac, the approximate date according to a lunisolar calendar, several subsidiary dials showing calendrical phenomena, and also predictions of eclipses.3

The longitudes of the sun and moon were shown on a single dial with two concentric rings: the inner one was divided into the twelve zodiac signs of 30 degrees each, and the outer one into 12 months of 30 days each plus five epagomenal days that constitute a year according to the Egyptian calendar. One pointer rotated one turn per year, showing the position of the sun in the zodiac; probably the same pointer4 showed also the day in the calendar ring. A second pointer rotated one turn per sidereal month, showing the position of the moon in the zodiac.

The moon pointer included a delightful and prominent detail, partially conserved in the back of Fragment C. See Figure 1. One half of a small ball was seen in a small circular window. The ball was half pale and half dark, and it rotated with the lunar synodic period and so showed the phases of the moon. The device was mounted on a small cap, rotating with the lunar sidereal period, to which the (lost) moon pointer was probably attached. See Figure 2. The moon ball was connected to a contrate gear through an arbor. This contrate was driven, either directly or indirectly, by a gear rotating with the solar period. Consequently the ensemble functioned as a differential gear, so that the contrate–and with it the little ball– rotated at a rate that was the difference between those of the sun and the moon pointers. Its period of rotation was therefore the synodic period, representing the elongation of the moon from the sun.

Figure 1: Back side of Fragment C, in which the moon cap is visible using PTM technology. Copyright of the Antikythera Mechanism Research Project.
Figure 1: Back side of Fragment C, in which the moon cap is visible using PTM technology. Copyright of the Antikythera Mechanism Research Project.
Figure 2: Front and back of the moon phase device built by Michael Wright. Figure taken from Wright 2006: 328.

Not having understood the detail of this assembly, Derek de Solla Price (1974: 20) suggested two possible functions for it: 1) it could be “some part of the dial work for the center of the front dial, possibly a plate indicating the position of the Moon” or 2) “a crank handle ... that drives the contrate wheel A”. (This refers not to the contrate in Fragment C, of which Price was unaware, but another contrate in Fragment A.) Price's first proposal was close to the truth, but he related the device only the moon's position, not with the moon's phases. In any case, Price only mentioned these possibilities, without developing them any further.

Michael Wright presented a reconstruction of the component as the boss of the pointer for the Moon’s position, incorporating a display of the Moon’s phase,5 and since then scholars have universally accepted this reconstruction. It could not have been otherwise, because Wright's proposal makes perfect sense and it fits very well with the physical evidence of the extant cap: even to the naked eye, it is easy to see not only the circular window, but also a smooth hemispherical depression behind the circular hole that strongly suggests that the hole was once filled by the small lunar ball. The arbor connecting the place of the ball with the contrate gear is also there, and even around half of the contrate. There is further evidence confirming Wright's proposal, for in the portion of the Back Cover Inscription preserved on Fragment B of the Mechanism a "small golden sphere" probably refers to the sun, while in a more broken context that was probably describing the lunar display another "small sphere" is mentioned with which the word “black” is most likely associated.6

However, the contrate is found with its teeth pointing away from the centre of the cap, not towards it as Wright’s reconstruction requires (see Figure 3). Wright suggests that this component, as found, had been inserted the wrong way round: “It seems therefore that the Antikythera Mechanism suffered the indignity, depressingly common in the author’s experience as curator of a collection of mechanism, of having been taken apart and reassembled incorrectly” (Wright 2006: 327).

Figure 3: Detail of fragment C showing the moon cap. In the upper left corner a detail of the contrate gear showing the direction of the teeth. Copyright of the Antikythera Mechanism Research Project.

Tony Freeth has proposed an alternative solution according to which the gear, as found, is correctly placed.7 Two more gears are added, connecting the contrate gear (with its teeth pointing away from the axis) to a gear fixed on the solar axis (see Figure 4). Gear u is fixed to the solar axis and therefore rotates with the solar period. Gear u moves x which has the same number of teeth, so that it rotates at the same rate but in the opposite direction. Gear y, which is fixed to x, rotating at the same rate, has the same number of teeth as the contrate (z) with which it is engaged. This device produces exactly the same result as Wright’s proposal, while allowing the teeth of the contrate to be in the extant direction.

Figure 4: Freeth's proposal for connecting the contrate gear to the sun axis.

Freeth's proposal is the simplest solution to the problem if, as we will show, there are compelling reasons to consider that the incorrect reassembly proposal should be discarded. But an obvious question still remains: why would the Antikythera Mechanism's maker prefer this more complicated solution to the simpler one proposed by Wright? We know from many other features of the mechanism that the builder was very smart, and reasonably minimalist, even using in several instances parts of the same gear train for different purposes. Therefore, there must be a reason for using four instead of two gears.

In this paper, we will offer for the first time a detailed description of the remains of the moon ball mechanism as it can be seen in the CT data, trying to make sense of every feature (section 1). This examination will incidentally show the surprising intricacy of the tiny components of this apparatus, some of which must have been custom designed for this application. Then we will consider, in the light of this analysis, whether it is possible that the extant contrate gear direction is not the originally intended one, and we will show that the proposal of incorrect reassembly should be discarded (section 2). Finally we will offer a conjectural explanation for a modified version of Freeth’s configuration (section 3).

Section 1: the moon phase mechanism

In this section we offer a detailed description of the remains of the mechanism and, based on it, a plausible reconstruction of the mechanism that made the moon ball rotate.

Figure 5: Master Diagram: In red, the moon cap plate with its features (the Tau, the rectangular hole, the big window and the footprint of the semicircular rim); in blue, the arbor and all that is or was attached to it (the contrate gear, the square collar, the arbor itself and the remains of the moon ball); in green, the cheeks; and in violet, the arbor keeper pipe. The sizes are expressed in mm.

Description of evidence

We have used photographs, PTMs, and CT to examine the physical remains. Measurements were made using the ruler tool included in 3D Slicer 4.4.0 software on the CT. We give our measurements to the nearest tenth of a millimeter. We estimate that most of our measurements are probably accurate to a few tenths of a millimeter.8 In a mechanism such as the moon phase device with its small pieces it would be desirable to have a greater degree of accuracy but, as Freeth and Jones (2012: #p118) noted, “the features … in the fragments are invariably heavily corroded, they are often affected by heavy calcification and they are sometimes broken”. In order to specify the directions of the pieces of the device, we will always determine them as viewed from the Mechanism's "inside", i.e., as we actually see them in photos of Fragment C (face C-2), schemes and drawings (see, for example, Figure 1.

The moon phase device is mounted on a circular plate (plate thickness: 1.4 mm) with a radius of about 33.8 mm. The plate is colored red in the diagram of Figure 5; see also Figure 6. The plate has a rim in its border 7.1 mm high and with the same width as the plate so that the radius of the plate inside the rim is about 32.5 mm. The rim is not complete and has suffered obvious damage, especially in the zone close to the moon ball. The missing rim extends from around 23° counterclockwise of the arbor to 73.4° clockwise. Notwithstanding this gap, it seems probable that the rim was originally complete in the zone near where the moon ball was, since the missing moon pointer was probably aligned with the arbor, and so should have been attached in some way to the rim in this place. (There are no traces of features to hold the pointer at a level close to the moon plate.)

In the inner part of the plate is the footprint of what seems to have been another rim that intentionally formed an incomplete circular arc. This rim was not concentric with the plate. Its center is in the line of the axis of the arbor, but 4.2 mm from the center of the moon cap in the direction of the moon ball.  The footprint has an inner radius of 14.2 mm and an outer radius of 16.4 mm. The arc is open where it would otherwise pass through the arbor window (described below). The opening of the arc is 41.5° on each side of the arbor's axis. See Figure 7. Similar rims are preserved attached (apparently by solder) to the base plate in Fragment A, where they served to stabilize large gears in planes parallel to the base plate. Following Wright (Wright 2006: 328-329) we refer to them as curbs (or, in Wright's English spelling, kerbs).

In the center of the plate is a squared hole of side around 3 mm (the side closer to the plane of the arbor is 2.9 mm long, while the side perpendicular to that one is 3.2 mm long.) The arbor that turned the moon cap and with it the moon pointer passed through this hole. The hole is not aligned with the arbor of the moon ball; relative to the arbor its sides are rotated 19.5° counterclockwise. (We do not see any particular significance in this non-alignment.) See Figure 8, reference A.

Close to the squared hole and beyond it with respect to the contrate gear, there is a clear Tau inscribed on the inner surface of the plate. It is roughly aligned with the arbor axis but slightly rotated counterclockwise with respect to the plane of the arbor axis, but not as much as the squared hole. Its base is 2.7 mm distant from the border of the hole. It is 2.3 mm high and 2.5 mm wide. The purpose of this inscription is not certain, but was probably a symbol for identifying where the part belonged for purposes of assembly (Price 1974: 20 and Freeth and Jones 2012: #p376). See Figure 8, reference B.

The cap plate is traversed by another large window with a complex shape.  See Figure 8, references 1, 2, 3 and 4. The window's long dimension runs along part of a radius of the plate, starting a couple of millimeters from the squared window and extending right to the rim. The window's shape is symmetrical with respect to this radius. For convenience of description we will treat the window as comprising four distinct sections. From the center to the border, these are as follows:  1) The first section is the smallest; it is just one millimeter long and 2.7 mm wide. It was presumably intended to accommodate the tip of the arbor that extended beyond the contrate gear (see Figure 8, reference 1). The wall of the window that is nearest to the plate's center is not perpendicular to the plane of the cap plate but tilts such that, while the hole is 1 mm long at the inner surface of the plate, it is 1.4 mm long at the outer surface (see Figure 9). 2) The second section was intended to house the contrate gear. It is 5.7 mm long and 11.5 mm wide (see Figure 8, reference 2). 3) The third section housed the arbor; it is 14.9 mm long and 3.3 mm wide (see Figure 8, reference 3). 4) The final section is the circular moon ball window, with a diameter of about 8.1 mm. The border of this part of the window is damaged, especially to the left of the arbor (see Figure 8, reference 4).

The arbor (blue in the diagram of Figure 5) is 22.4 mm long and 1.7 mm wide. It is broken into two pieces (Figure 10, reference 1). The smaller is around 6.4 mm long and is closer to the center of the moon cap; the larger is 16 mm long. In the smaller piece, the arbor has a rectangular cross-section. After the break, the arbor is cylindrical. The smaller part holds the contrate gear; just a small part of the arbor towards the center of the cap plate projects about 0.8 mm beyond the contrate gear (Figure 10, reference 4). This tip is located in the first section of the arbor window. 1.3 mm from this end, the arbor is perforated crossways by a pinhole of 0.8 mm diameter (Figure 10, reference 2). The hole is exactly aligned with a rectangular slot on the back face of the contrate gear (Figure 10, reference 3). The pin that the hole and slot were meant to accommodate is missing. The arbor has just enough space along its length to accommodate the contrate between the pinhole and a squared collar of sides 3.7 mm and 2.5 mm length along the axis of the arbor. This collar might have been a separate component attached in somehow to the arbor, but since close study of the CT does not show any traces of a boundary between the arbor and the collar, we believe that the collar is probably one piece with the arbor (see Figure 11). The break in the arbor comes right after the collar. Inside the collar the CT shows a small radiodense globule, almost spherical, with a diameter of roughly 1 mm, whose significance is not clear to us if it is not simply a bit of free metal that was protected from corrosion by the thickness of the collar (Figure 10, reference 6). After the break, as previously noted, the arbor is cylindrical, and is enclosed in a kind of pipe (the arbor keeper) that will be described later. The arbor, as it is preserved, continues a bit beyond the pipe and into the moon ball window, for less than a millimeter. In the arbor's interior, close to the break, is another radiodense globule with an elliptical shape (major axis 1.38 mm, minor axis 0.9 mm, Figure 10, reference 7).

The contrate (blue in the diagram of Figure 5) is also broken. The extant part is almost exactly half of the complete gear and is located above the plate, i.e., in the face of the plate opposite to that of the rim. The gear has an outer diameter of 11.2 mm, an inner diameter of 7 mm, and a width of 5.2 mm (see Figure 12). As already mentioned, the back face of the gear (opposite to that of the teeth) is crossed diametrically by a rectangular slot 1.3 mm wide and 0.8 mm high, which is aligned with the pinhole of the arbor, and thus would have allowed the gear to be fixed to the arbor by means of a pin (Figure 10, reference 3). Ten teeth of the gear have survived. The distance from tooth valley to tooth valley is approximately 1.7 mm, which is consistent with a complete tooth count of 20. The high of the teeth is also on average 1.7 mm.

Attached to the inner part of the cap plate and along each side of the third section of the big window, i.e. the arbor window, are two pieces of metal (green in the diagram of Figure 5), mirror-images of each other, that extend almost the whole length of the arbor window (leaving just 0.5 mm of the window free, at the end closer to the contrate). See Figure 13. Each of these pieces has a complex shape in which it is possible to distinguish two parts: 1) closer to the contrate is a long rectangle that we will call, following Wright’s terminology, a “cheek” (Wright 2006: 326) (Figure 13, reference 1), and 2) adjoined to it (and probably making just one metal piece with the cheek) a cube with a pinhole that we will call the “holder” (Figure 13, reference 2). The cheeks are 10 mm long, 3.2 mm wide, and 1.2 mm high. They are 2.1 mm apart, so they partially cover the arbor window, which is 3.3 mm wide. Each cheek has a small, clearly intentional indent like a very shallow groove running along the side that is partially closing the window. This depression is 0.6 mm wide and just 0.4 mm deep. The depression is as wide as the part of the cheek that overhangs the window. The initial impression obtained from CT or photographs is that the depression is present along just the first 4.1 mm of the cheek closest to the contrate gear; however, closer inspection of the CT shows that the depression –with the same width and depth– continues all the way to the holders, but is covered along the part towards the moon window by a thin tab of metal that belongs to the arbor keeper pipe—to be described later—and not to the cheeks (see Figure 14). At the extremity of each cheek closer to the moon ball is located the second part of this component, the holder. The cheeks suddenly increase in height in a vertical wall, reaching 2.9 mm height. See Figure 13, reference 2. The holders are 4.4 mm long, and are slightly narrower than the rest of the cheeks by 0.6 mm, so that they do not cover any part of the third section of the window, but exactly follow its borders. The holders are perforated by a pinhole (still containing remains of the pin). See Figure 13, reference 5. The diameter of the pinhole is 0.7 mm and its center is located 1.1 mm from the plate and 2.1 mm from the beginning of the holder (i.e., from the wall of the holder where the cheek changes its height). We note in passing that the removable pins used here and elsewhere in the Mechanism were evidently composed of a different alloy from the components in which they were inserted, so that pinholes are very conspicuous in the CT whether or not the pins themselves survive (Anastasiou et al. 2014).

There is a very clear hemispherical depression where the moon ball ought once to have been (blue in the diagram of Figure 5; see Figure 15). This could be remains of the surface of the moon ball (in which case the ball must either have been hollow or partly covered by some kind of lamination overlaying half of it) or just accretion that assumed this semicircular shape by building up in contact with the missing ball. We find such accretion layers in many places in the Mechanism's fragments; for example, the mirror image inscriptions on fragment A and B are due to accretion against the original inscribed plates (cfr. Price 1974: 47). The diameter of the depression is 6 mm, significantly smaller than the moon ball window. It is roughly centered in the moon ball window. (It is hard to determine the exact center of the moon window because it is damaged.) As we have already said, the arbor protrudes a bit beyond the arbor keeper and projects inside the moon ball window. Nevertheless, the arbor does not quite reach the moon ball depression. But, if we imagine extending the arbor to where the arbor should come into contact for the first time with the moon ball, there is a very small semicircular indentation in the rim of the moon ball depression perfectly lined up with and of about the same size as the arbor, which seems to confirm that the arbor originally went inside the ball (see Figure 16). There is also a slight semicircular depression at the other side that might suggest that the arbor, after going through the entire ball, protruded from it. This depression, however, is less regular and not as well aligned as the previous one, so it may be just a random irregularity.  

Inside the arbor window and between the two cheeks and their holders is another component that kept the arbor in place. We will call it the arbor keeper pipe, or simply, the pipe (violet in the diagram of Figure 5). This pipe has a complex shape (somewhat similar to a tobacco pipe with a bowl) that makes it fit perfectly in the space left by the moon window, the cheeks and holders (see Figure 17). It is 14.9 mm long (hence it goes all along the third section of the big window). It is perforated by a cylindrical hole of the same diameter as the arbor, which is inside it (Figure 17, reference 1). The pipe covers the arbor window and slightly protrudes from the outer surface of the cap plate, by about 0.8 mm. On this face it has a regular semi-cylindrical shape. The “bowl” of the pipe (Figure 17, reference 2), 6.4 mm high and as long as the holders (4.4 mm), is lodged between the two holders and is a bit taller than them (about 1.3 mm). This part covers all the space between the two holders (3.3 mm). It also has a pinhole aligned with the pinhole of the holders and of the same diameter (Figure 17, reference 3). The shape of the sides of the “shank” of the pipe has been executed in such a way that they perfectly fit with the sides of the window and the part of the cheeks that partially covered the window. Thus, the two vertical walls of the shank have a depression that accommodates the protruding parts of the cheeks (see Figure 13). Part of the depression is finished with little “wings” that cover the small depressions of the cheeks, so that the exposed top surface of the pipe constitutes a flat tab flush with the exposed top surfaces of the cheeks. There appears to be a second letter Tau shallowly inscribed on the tab, with its top horizontal stroke running along the tab's right edge (Wright 2006: 326 Figure 8). These wings extend from the bowl of the pipe for 4.1 mm (see Figure 14). We already mentioned them when we observed that, because of them, there is a false appearance that the depression of the cheeks did not extend along the whole length of the cheeks. The complete shape of the arbor keeper pipe can be seen in the following animation.

This completes our description of the evidence that can be obtained by accurate analysis of the CT. Now we will try to make sense of this description.

Animation 1: The complete shape of the arbor keeper pipe.
Animation 2: Comparison between the remains of the moon phase device (animated volume rendering made with Osirix) and our reconstruction.
Figure 6: Moon cap plate. Copyright of the Antikythera Mechanism Research Project.
Figure 7: rim footprint on the inner surface of the moon cap plate. Copyright of the Antikythera Mechanism Research Project.
Figure 8: CT image showing the Tau (A), the squared hole (B), and the big window, divided in four sections (1, 2, 3 and 4). Copyright of the Antikythera Mechanism Research Project.
Figure 9: The tilted wall of the big window, section 1. Copyright of the Antikythera Mechanism Research Project.
Figure 10: Arbor. 1: break, 2: arbor’s pin-hole , 3: contrate gear’s rectangular slot, 4) arbor’s tip at the first section of the big window, 5: arbor’s tip at the moon ball window, 6: spherical radiodense globule, 7: elliptical radiodense globule. Copyright of the Antikythera Mechanism Research Project.
Figure 11: Squared collar (reference: 1). Copyright of the Antikythera Mechanism Research Project.
Figure 12: Contrate. Copyright of the Antikythera Mechanism Research Project.
Figure 13: Cheeks and holders. 1: Cheek, 2: holder, 3: cap plate, 4: arbor keeper pipe, 5: pinhole of the holder. Copyright of the Antikythera Mechanism Research Project.
Figure 14: cheeks and holders. 1: cheek, 2: holder, 3: arbor keeper pipe. Copyright of the Antikythera Mechanism Research Project.
Figure 15: moon ball depression. Copyright of the Antikythera Mechanism Research Project.
Figure 16: Moon ball semicircular indentation. Copyright of the Antikythera Mechanism Research Project.
Figure 17: arbor keeper pipe. 1: cylindrical hole for the arbor, 2: bowl of the pipe, 3: pinhole. Copyright of the Antikythera Mechanism Research Project.

Reconstruction of the Mechanism

The general way that the moon phase mechanism works is clear and was first correctly described by Michael Wright (2006: 327-329). As we already mentioned, the contrate gear must rotate with the lunar synodic period. The gear is attached to the arbor through a pin and so the arbor also rotates at the same rate. The arbor is held by pipe and transmits its rotation to the moon ball.

We will propose a way in which all the parts of the mechanism can be assembled and disassembled that is consistent with the description of the parts and makes sense of some apparently unnecessary complexities in certain pieces (such as the particular shape of the cheeks or of the pipe). See animation 2. In order to disassemble the mechanism:

  1. You first have to detach the moon ball. The moon ball was presumably attached to the the part of the arbor inserted in it just by friction or possibly a screw fitting. The small semicircular slot at the moon ball depression where the arbor should go inside the ball (Figure 16) shows that the arbor was still cylindrical when went inside the ball (this would have allowed one to twist the ball by hand when the apparatus was assembled and all the gears connected, in order to set up the correct phase). Because the moon ball is smaller than the moon ball window, the ball could be taken off by drawing it in the opposite direction of the arbor.
  2. Then you have to remove the pin that goes through the holders and the pipe. Steps 1 and 2 could be reversed, but both must be completed before step 3.
  3. The third step consists in removing the entire apparatus: i.e., the pipe with the arbor and the contrate. The walls of the pipe are as wide as the space left by the third section of the moon window, but the window is partially closed by the cheeks, which thus prevent the pipe from being pushed through to the inner side of the plate. On the other hand, the wings of the pipe prevent it from being pushed through towards the outer side of the plate. Consequently, even if the pin is removed, the pipe cannot be moved up or down. But, now that the pin and the moon ball have been removed, the pipe can be slid in the direction of the moon window using the protruded part of the cheeks as guides. If the pipe is moved in this way until it abuts the rim of the cap, the part of the shank of the pipe that has wings would be now partially in the moon ball window and partially in the part of the third section window where the holders are, and so the cheeks would not stand in the way of moving the pipe to the outer side of the plate. It is important to realize that even now it can only be removed in the direction of the outer part and not towards the inner part of the plate, because the shank still is wider than the space left by the cheeks.
  4. Once the pipe, together with the arbor and the gear, has been removed from the moon cap, you can extract the arbor by drawing it in the direction of the gear (you cannot remove it in the other direction because the squared collar obstructs it).
  5. Finally you can remove the pin that fixes the gear to the arbor and take off the gear (the order of steps 4 and 5 can be reversed). The whole mechanism is now disassembled. You can reassemble the mechanism following the steps in the reverse order.
Animation 3: The step by step disassembly of the Moon phase mechanism.

Section 2: the direction of the contrate gear

In this section we will analyse in detail whether, as Wright suggested, the contrate gear was reassembled incorrectly so that the actual direction of the contrate gear is not the originally intended one, or whether the extant direction was the intended one so that we would have to postulate (at a minimum) two extra gears to connect the contrate gear to the Sun's arbor, as Freeth suggested.

We note in passing that that the supposed incorrect assembly must have taken place before the discovery of the shipwreck and not, say, during handling or conservation of the fragment at the Museum. The earliest known photograph of the face of Fragment C bearing the Moon phase apparatus, published by Svoronos in 1903, shows it before any conservation work had been done on it, and the contrate gear is clearly visible pointing in the extant direction (see Figure 18). It also seems very unlikely that an inversion of the gear could have happened during the shipwreck or during the time the Mechanism was under the sea. The perfect alignment of the pinhole of the arbor and the pin slot of the contrate on one side, and the almost perfect alignment of the two pieces of the arbor on each side of the break make the notion that the contrate gear was somehow flipped around just by chance highly implausible. The incorrect assembly must have taken place before the shipwreck, when the Mechanism was essentially intact.

Figure 18: Oldest extant picture of Fragment C (Svoronos 1903). The contrate's orientation is the same as today.

In Figure 19 we schematically represent the parts that would play a role in this analysis in the extant position, i.e., with the contrate gear teeth facing the moon ball.

Figure 19: Schematic reconstruction of the moon phase device as extant in fragment C. Note the three points labeled a, b and c. A break in the arbor is clearly visible between b and c.

If one wants to assert that the gear has been reversed before the shipwreck there are three options. The first two can be easily dismissed by consideration of the evidence, but the third one will require a more detailed analysis. The first option consists of reconstructing the intended arrangement of the moon phase mechanism by leaving all as in Figure 19 except the gear, which would face the other way around. But, as is evident from Figure 20, if you keep the pinhole of the arbor aligned with the slot of the gear so that the pin can be inserted, the gear would not fit inside the gear-window of the moon cap. An additional problem is that the gear would be too close to the sun axis to leave room for a gear with the same number of teeth that would move the contrate gear.

Figure 20: Option 1: the contrate is facing the other direction, keeping all the other elements in their places. The gear does not fit in the gear-window.

The second option assumes that the collar was movable. According to this possibility, the collar would have originally been between the contrate and edge a of the window. This option would make all the components fit within the cap window. The collar would have had to have a hole for a pin (d in Figure 21) that, passing through the collar and the extant pinhole of the arbor, would fix the collar. Also, the arbor would have to have had another pinhole (e in Figure 21) for a second pin that, passing through it and the rectangular slot in the gear, would prevent the gear from moving to the right. The absence of any traces of a boundary between the collar and the arbor, and above all of a second pinhole in the arbor at e or of a pinhole through the collar rules out this option, since, as previously noted, the pinholes in the Mechanism are conspicuous in the CT whether their pins are still present or not.

Figure 21: Option 2: the contrate is facing the other direction and the collar is located between border a and the gear.

The third option takes advantage of the break between surfaces b and c. According to this proposal, in its original position the gear was reversed together with the broken part of the arbor between a and b and the collar attached to it (see Figure 22). Through some accident the arbor was broken, and after this happened, someone put the broken piece with the contrate back in its part of the window pointing the wrong way.

Figure 22: the broken part of the arbor, together with the collar and the contrate gear, has been reversed.

The main problems that we find with this proposal are the following. First, a close inspection of the CT tends to confirm that there is a reasonably good matching break between b and c. See Figure 23.

Figure 23: the matching between border b and c. Copyright of the Antikythera Mechanism Research Project.

Second, in the incorrect assembly hypothesis the most reasonable reconstruction suggests that the squared collar would have been at the left extremity of the arbor. Because the collar was one piece with the arbor, it doesn’t make sense for the arbor to protrude beyond the collar. Hence the little window that we identified as the first section of the big window would be pointless, as shown in Figure 24:

Figure 24: the arbor shouldn’t protrude beyond the collar.

On the other hand, according to the hypothesis that the contrate is preserved in its intended orientation, it is perfectly reasonable for a bit of the arbor to protrude beyond the gear window so that the pinhole in the arbor does not have to be too close to its end and so more liable to damage. The fact that the gear, instead of having a hole for the pin, has a slot could indicate that a hole so close to the edge would be very fragile. The same would be the case with an arbor that was as long as the gear, but not more.

Third, if the pinhole of the arbor was so close to the break, one would have expected the arbor have broken through the pinhole, since this was the most fragile part the arbor.

Fourth, there are distinct remains of the cylindrical arbor at the collar. In the following CT image the circle left by the joining of the squared collar and the cylindrical arbor is perfectly identifiable. It is clear from the CT that we are seeing surface b and not surface c.

Figure 25: remains of the cylindrical arbor at the collar, before the break. Copyright of the Antikythera Mechanism Research Project.

Fifth, a close inspection of the break shows that it probably includes some little parts of the pipe itself, and not just the arbor. It would be very hard to explain on the incorrect reassembly hypothesis why the little broken pieces of the pipe are still in place. These broken parts of the pipe can be observed in two places. See Figure 26 and Figure 27.

Figure 26: possible break of the pipe that remains attached to the collar. The break is below and to the right of the arbor, so it seems to be a piece of the pipe and not of the arbor. Copyright of the Antikythera Mechanism Research Project.
Figure 27: possible break of the pipe that remains attached to the collar. The break is below and to the left of the arbor, so it seems to be a piece of the pipe and not of the arbor. Copyright of the Antikythera Mechanism Research Project.

Sixth, if the contrate originally faced the center of the moon cap and then you attempted to reverse its orientation, the teeth of the gear attached to the solar arbor that originally moved the contrate would obstruct replacing the moon cap in its original place, because the back of the contrate would collide with its teeth.9 So, there are three possibilities. One is that the moon cap was never reinstalled on the Mechanism after the contrate gear was unintentionally reversed. But this is implausible, first because the moon cap is extant in fragment C, close to the zodiac and calendar rings, and second because the cap faces in the correct direction with respect to the outer part of the mechanism and the inscribed part of the rings face, which seems to indicate that it was in place at the moment of the shipwreck. Secondly, someone might have tried to force the cap into place, causing still more damage (as suggested by Wright 2006: 327), but this requires that the person responsible, feeling the resistance, was too careless to look for what was wrong and notice the obvious fact that the contrate was facing the wrong way to engage with the other gear. The third possibility is that the moon cap was indeed reassembled, but the gear attached to the solar arbor was for some unexplained reason also missing, but this is clearly an ad hoc hypothesis. (This objection would also apply to the second hypothesis involving a removable collar and second pin.)

Seventh, we should ask how it is that the broken part is in its place, so very well aligned with the rest of the arbor. We know that the part of the arbor closer to the moon ball was kept in place by the pipe, but the broken part of the arbor, the collar and the contrate gear itself would have had nothing to keep them in place. They ought to have fallen out.

These are the main difficulties of the incorrect assembly hypothesis. But our proposal also presents at least one difficulty with respect to the physical remains. If the extant position is the intended one, we have to postulate some additional lost gearwork to connect the solar arbor's motion to the contrate, most plausibly two extra gears, as Freeth suggested. But it seems that there is no trace of the loss of a part on which the arbor for the two extra gears could have been planted, and it is reasonable to think that it should have left a detectable trace. Based on his own study of the original fragment C, Wright has made the same point in connection with both Freeth’s reconstruction and the one discussed here (private communication). However, we see two possible ways that the extra gears could have been mounted, either by a platform soldered to the top faces of the cheeks (see Figure 29) or by an attachment to the curb whose footprint remains on the plate (see Figure 28).

Figure 28: Reconstruction of the rim attached to the moon cap. The extremities intersect the point at which the arbor of the two extra gears must be located and the opening is just the required to accommodate the gears. If a support of the arbor was necessary at the top of the gears for mechanical reasons, it could be easily added.
Figure 29: Another way to fix the gears. The bridge holding the arbor of the two extra gears could be attached to the cheeks. If a support of the arbor was necessary at the top of the gears for mechanical reasons, it could be easily added.

A further reason for suspecting the presence of the two gears (or at least of something where the two extra gears should be) is the way in which the pipe is assembled and disassembled. We have already explained that, while the pin of the pipe must be removed from the inner part of the cap, the pipe has been designed in such a way that it could only be removed towards the outer part of the cap. In principle it would have been easier to remove both pieces from the same side. So, the fact that the pipe was designed to be removed to the outer face of the cap strongly suggests the presence of something in the inner part, blocking the passage of the pipe. See again animation 3 which is essentially similar to animation 2, but with the proposed gears added.

Animation 4: The step by step disassembly of the Moon phase mechanism with the two extra gears.

In conclusion, all the evidence seems to indicate that the arbor was broken after the shipwreck and that the extant position is the original and intended position. In what follows I will show that the Antikythera mechanism's maker could have had a good reason for putting the gear in this position.

Section 3: non-uniform motion of the moon phase ball.

Freeth’s proposal consists of two pairs of gears having an equal number of teeth. Fortunately, we have another example of this in the mechanism: the pin and slot device for producing the lunar anomalistic motion. The pin and slot device does not change the period of rotation, but produces a non-uniform motion in the output gear (see Figure 30). Gears z and y have the same number of teeth, as do gears u and x. u moves x by engagement, and x moves y by means of a pin inserted in a slot of y. Therefore, x and y rotate at the same rate but, because y is eccentric with respect to x, its motion is not uniform. This non-uniform motion is transmitted by engagement from y to z. Consequently z rotates at the same rate as u, but with a non-uniform motion. The amplitude of the non-uniformity depends on the eccentricity and its period is the period of rotation of x and y.

Figure 30: A schematic representation of the pin-and-slot device for the lunar anomaly.

Proposals have recently been published for extending the pin and slot device to produce the retrograde motion of outer and inner planets.10 It is thus possible that the pin and slot device was not exclusively employed for the lunar anomaly. If the designer intended to model some anomalistic period in the rotation of the moon ball, through the use of a pin and slot device, we would have an explanation for the presence of the two extra gears of Freeth’s arrangement (see Figure 31).

Figure 31: modification of Freeth's proposal (Figure 4) introducing the pin-and-slot device.

This conjectural pin-and-slot device would produce an anomaly with period equal to the synodic month in the rotation of the moon phase ball, without affecting the motion of the moon pointer (i.e., the lunar sidereal motion). So, we should ask: do we know of any ancient report of some sort of anomaly in the lunar phases with a period of a synodic month, independent of the lunar sidereal motion?

The answer is affirmative. We know from Aristarchus of Samos’s On Sizes and Distances of the Sun and Moon (Heath 1913) that, if we accept that the ratio between the Earth-Sun distance and the Earth-Moon distance is not too great, there would be a perceptible inequality in the moon’s phases. Aristarchus distinguished the dichotomy (when the boundary between the moon's dark and illuminated parts appears to be a straight line) from the quadrature (when the moon is exactly 90° elongation from the sun). According to Aristarchus, the dichotomy is not produced at quadrature, but 3° earlier at first quarter and 3° later at last quarter (see Figure 32).

Figure 32: Difference between dichotomy and quadrature due to the small ratio of the earth-sun/earth moon distance.

If this appearance were to be modelled in the mechanism, the motion of the moon ball would have to be non-uniform: it should take more time to go from the first dichotomy to second dichotomy than from second to first. The difference between these two intervals, according to Aristarchus’s proposal, would be around 1 day.

Aristarchus’s ratio between the Earth-Sun distance and the Earth-Moon distance is not the only one attested in ancient texts, and in fact, together with that of Ptolemy (for whom it was almost the same), it is one of the greatest. Of course, the smaller the ratio, the bigger the effect. So, for example, for Eudoxus the ratio was 9:1, implying a difference between quadrature and dichotomy of more than 6°; for Phidias, Archimedes’s father, 12:1, implying a difference of almost 5°; for Eratosthenes the ratio is a bit bigger than 5:1, implying a difference of almost 11°; and for Hipparchus, it is approximately 7:1, implying a difference of almost 8°.11 See the following animation based on Phidias’s proportion.

Animation 5: The moon phase device and its relation with the Sun-Earth / Moon-Earth distance proportion. The animation assumes the ratio proposed by Phidias, 12:1.

A one-day difference could be observed on the Mechanism. One would move the moon pointer until the moon ball shows the dichotomy, note which day is indicated on the Egyptian calendar scale (or on the day-of-the-month display, if such display existed, as has been proposed)12, then crank the mechanism forward to the next dichotomy and check the date. Of course, a crucial question is whether it could be possible to distinguish the dichotomy on the moon ball within one single day. Two independent observations have been made with different reconstructions of the mechanism, and in both cases the answer was positive.13 Moreover, the observations have been made assuming the maximum attested ratio of the distances (that of Aristarchus), but if the ratio between the distances was smaller, then the effect would be even greater and, therefore, easier to be detected.

We must stress that if the Mechanism's designer adopted Freeth's extra pair of gears, or indeed any means of transmitting motion from the solar arbor to the moon ball's arbor that was more complicated than the obvious direct engagement of the contrate with the fixed gear as proposed by Wright, it must have been intended somehow to modify the motion that direct engagement would have produced. The only possibilities are a modification of the period, which is out of the question for a display of the lunar phases, or a periodic modification of the moon ball's rate of rotation—in other words, a modification of the symmetrical periodic function that would otherwise correlate the appearance of the lunar phase to the moon's elongation from the sun. (Strictly, we should not speak without qualification of a change from uniform to non-uniform motion, since the elongation displayed on the Mechanism was already affected by the lunar anomaly, though likely not by the solar anomaly14.) And the only historically plausible anomalistic effect would have been the one produced by the ratio of the distances of the sun and moon. Incorporating this effect would have provided a way of including one of the main topics of ancient astronomy at least since Aristotle’s time, the distances of the luminaries, which would not otherwise have been present in the mechanism. Their relative distances would have been displayed qualitatively by the fact that the moon ball was closer to the dial's center (representing the earth) than the "golden little sphere" that apparently was mounted on the sun's pointer, but it would have been impracticable to set these spheres at the assumed ratio of their distances. If the Antikythera mechanism was intended as a teaching device (Jones 2012), a moon phase anomalistic device would thus have been a suitable springboard for explaining the luminaries' distances.

Previously, the pin and slot device has only been suggested as a mechanical expression of an epicyclic or eccentric geometrical system, which have no obvious relevance for lunar phases. There is, however, an isomorphism between the mechanical and the astronomical configuration. Figure 16 represents the hypothetical pin and slot device for producing the moon phase anomaly, considered in a frame of reference such that the axes E of the driving gear and M of the driven gear are stationary. The pin, S, thus revolves around axis E uniformly (relative to the motion that directly drives it) with the synodic period. The slot (in dashed lines) revolves around the other axis, M, with the same period, but with a non-uniform motion. The non-uniform motion of the slot represents the moon phase anomaly. This mechanical configuration has an astronomical interpretation: E represents the Earth; M, the Moon, and S, the Sun, in a frame of reference such that the Earth and Moon are both stationary, so that the Sun revolves around the Earth at a rate equal to the difference between its own sidereal speed and the moon's sidereal speed, i.e. with the lunar synodic period. The angle that represents the moon's phases is angle SME, i.e., the angle at the Moon between the Earth and the Sun, and the rate of increase of this angle incorporates a non-uniformity because the Sun revolves around the Earth and not around the Moon. In order to produce the correct anomaly, the proportion between the interaxial distance ME and the distance between the pin, S and its axis, E, should be the same as the proportion between the Earth-Moon distance and the Earth-Sun distance. The isomorphism is perfect.

Figure 33: Isomorphism between the pin and slot device and the astronomical configuration of the moon phases.

If this proposal is correct, then it shows the pin and slot device in a new and interesting light. Recently James Evans and one of us suggested that the epicycle and deferent system could have been originated using the pin and slot as an inspiration and not the other way around (Evans and Carman 2013). According to that proposal, pin and slot devices were conceived as a mechanical solution for producing anomalistic motions in geared mechanisms and then, looking at it, some geometer proposed the epicycle and deferent system. The use of the pin and slot for the moon phase would show some independence of the pin and slot device from the epicycle and deferent system: in some sense, the pin and slot device is even more versatile than the epicycle and deferent system, for it could be used for producing anomalies that could not be represented by epicycles.

Acknowledgements

We are very grateful to the Antikythera Mechanism Research Project for preparing the x-ray slices of Fragment C that we used in this study and for the permission (via Professor Mike Edmunds) to reproduce the images here. Thanks are due to the National Archaeological Museum of Athens. We would also like to thank Tony Freeth, Michael Wright and David Teubel for their suggestions. Lina Anastasiou helped us with CT analysis and many interesting suggestions.  James Evans helped us from the very beginning to develop this proposal and generously discussed with us every detail. Alexander Jones encouraged us to go forward in the detailed analysis of the mechanism and most of the discoveries were made in the context of our discussion. Ignacio Silva, Michael Wright and Alexander Jones helped us to improve the English of different versions of this paper.

References

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Notes

1 Weinberg et al. 1965, Price 1974, Cristopoulou et al. 2012, Kaltsas et al. 2012. This paper is partly based on data processed from the archive of experimental investigations by the Antikythera Mechanism Research Project in collaboration with the National Archeological Museum of Athens (See Freeth et al. 2006).

2 Price 1974; Freeth et al. 2006; Jones 2012; Carman & Evans 2014; Freeth 2014.

3 Price 1974, Wright et al. 1995, Wright 2002, 2003a, 2003b, 2005a, 2005b, 2005c, 2006, Freeth et al. 2006, Freeth et al. 2008, Evans et al. 2010, Carman et al. 2012, Freeth & Jones 2012, Wright 2012, Anastasiou et al. 2013, Anastasiou et al. 2014, Carman & Evans 2014, Freeth 2014.

4 Evans et al. 2010.

5 Wright 2006 is the main source, but he anticipated the discovery in Wright 2005c: 4 and Figure 2 at p. 3.

6 Bitsakis and Jones, Forthcoming; Freeth et al., 2006: 588 with Supplementary Information 9 and 13.

7 Supplementary material of Freeth et al. 2008: 22, figure 14 and Freeth and Jones 2012: #p. 84, figure 6. At present Freeth considers Wright’s proposal to be the most plausible (private communication).

8 All measurements are proportionally subject to a possible calibration error in the CT; comparisons with photographic evidence suggest a systematic correction of about 90%.

9 I thank Tony Freeth for this argument (personal communication).

10 For the outer planets, see Carman, Thorndike and Evans 2012 and Freeth and Jones 2012, for the inner planets, see Wright 2012: 290 and Evans and Carman 2013: 164-166.

11 Both Eudoxus's and Phidias's values are mentioned by Archimedes in the Sand-Reckoner (Heath 1893: 223); Eratosthenes's values are attested in doxographical writers (Diels 1878: 362-363); Hipparchus's values are mentioned by Pappus (Rome 1931: 256-257).

12Wright 2006: 329.

13 One has been done by Michael and Anne Wright checking the date on the day-of-the-month scale, on October 15, 2012. He described the experiment in the following words: “Our procedure was as follows. I first set the dial, stopping a little short of the perceived moment of dichotomy. The backlash in the train allowed the observer to move the pointer on in the same direction until, as closely as he/she could judge, the display showed dichotomy. He/she then moved the pointer, first in one direction and then in the other, just until he/she considered that the display showed a departure from dichotomy, and at each such point I made a fine pencil-mark alongside the Moon pointer on the day-of-the-month scale. Each of us repeated the observation both at first-quarter and at last-quarter. At each trial the marks were separated by rather less than one day. That is, it appeared that each of us could detect the moment of dichotomy to within half a day” (personal communication). The other was performed by James Evans using the calendar ring for checking the date, during the same day and his answer was also positive. I thanks all three for their kind help.

14 Cfr. Evans et al. 2010.