The Cosmos in the Antikythera Mechanism

2.5. Main Drive Wheel

The physical evidence from the Main Drive Wheel suggests that a complex system was mounted on this wheel, as Price noted in a previous publication:⁵⁷⬈#p112

“This main drive wheel preserves clear evidence of some sort of superstructure mounted over it. The spoke in the ten o'clock position has a lug mounted on it 8.3 mm long, 3.9 mm wide and standing 6.3 mm above the surface of the wheel. The three other spokes contain holes indicating that they may also have had similar lugs on them and in addition there is a square depression on the spoke in the one o'clock position. Furthermore, on the rim, exactly midway between each of the spoke positions, there are traces of former fixtures. In the eleven o'clock position is a rectangular depression with a rivet hole at the center; in the eight o'clock position just the rivet hole remains, and the other two corresponding places are obscured by debris. The evidence seems to suggest that pillars rising from these four places on the rim and another four on the spokes supported some sort of plate above and parallel with that of the drive wheel, turning with it.”⬈#p113

Following Wright,⁵⁸ we believe that the likely functions for this system were to calculate the variable motions of the Sun and the planets and to display their ecliptic longitudes on the Zodiac Dial. First we examine the physical evidence.⬈#p114

2.6. Fitments and bearings on the spokes of the Main Drive Wheel

In our examination of the evidence, features were identified using the photographs, PTMs and X-ray CT. Measurements were made using reference scales in the photographs and using Volume Graphics VGStudio Max software on the X-ray CT. This software includes very accurate measuring tools. We give our measurements to the nearest tenth of a millimetre. Due to corrosion, it is not always possible to be confident of measurements to this degree of accuracy. We estimate that most of our measurements are probably accurate to a few tenths of a millimetre. The reason that we give this rather imprecise overall estimation of errors for our measurements is that we do not believe that anything more precise is meaningful. The features that we are measuring in the fragments are invariably heavily corroded, they are often affected by heavy calcification and they are sometimes broken (and in places glued back together). So we do not believe that it is helpful to try to give more precise error estimations.⬈#p118

Throughout this discussion, we make what we believe are plausible inferences about the functions of the features described. However, due to lack of evidence, it is not possible to be dogmatic about our reconstruction and we are open to other ideas about interpretation of the evidence. In addition, there are some features, which do not have a function in our model and whose purpose we do not understand. Despite these uncertainties, it is our view that the spokes of b1 are very likely to have carried gearing for the solar anomaly and the inferior planets.⬈#p119

On b1, there are apparent bearings on some of the spokes as well as areas where there appears to have been attachments and a pierced lug. We refer to the spokes and features as being at the 1 o’clock, 4 o’clock, 7 o’clock and 10 o’clock positions.⬈#p120

Fig. 12 Flat area with rivet at 1 o’clock. ©2005 Antikythera Mechanism Research Project. All rights reserved.⬈#p123

There is a depressed flattened area in the 1 o’clock position of dimensions 19.0 mm x 15.5 mm (the full width of the spoke). Its edges are 22.1 mm and 41.1 mm from the central axis. It appears that there was some fitment attached here with a rivet and possibly also solder. In our model, we reconstruct a bearing here for the epicyclic gear of the Venus mechanism. In addition, there is a circular feature outside the flattened area, which might have been for a rivet. It might have provided some additional support for the bearing of the large Venus epicycle and carrier disk, but we have not used it in our proposed model.⬈#p124

Fig. 13 Apparent bearing in the 4 o’clock position. ©2005 Antikythera Mechanism Research Project. All rights reserved.⬈#p127

In the 4 o’clock position, there is a prominent hole, which looks like the remains of a bearing. Its outer diameter is 9.7 mm and its inner diameter is 6.6 mm. It is 27.1 mm from the central axis. In our model, we reconstruct a bearing here for the epicyclic gear of the Mercury mechanism.⬈#p128

Fig. 14 Apparent bearing in the 7 o'clock position. ©2005 Antikythera Mechanism Research Project. All rights reserved.⬈#p139

(A) Photo of bearing and flat area on the 7 o’clock spoke. (B) Close-up photo of bearing. (C) X-ray CT slice of bearing and flat area on the 7 o’clock spoke. (D) Close up of the bearing, showing a ring and a small hole drilled within the body of the spoke; (E) orthogonal section, showing the hole.⬈#p140

In the 7 o’clock position, there is another apparent bearing. This has a central hole of 4.3 mm in diameter, with an outside ring of 18.1 mm diameter. The central hole has a light streak across its diameter. On examination of this feature in the X-ray CT, it does not appear to have any mechanical significance. The outside ring is not visible from the front surface of the spoke, though it goes right through the rest of the spoke from 0.6 mm below the surface to the back of the spoke. It appears to be a bearing set into the spoke. Drilled from the outside of the ring, towards the outside of the wheel, is a small hole of length 4.1 mm and diameter 1.1 mm, which is not visible on the surface. It is possibly part of a lubrication system, though we advance this idea with some diffidence. This small hole is drilled accurately within the body of the spoke, where the only access for drilling would have been through the hole in which the ring is set. This must surely have been a difficult achievement for the technology of the time.⬈#p141

In our model, we reconstruct the main feature on the 7 o’clock spoke as a bearing for the middle epicyclic idler gear of the solar anomaly mechanism.⬈#p142

Fig. 15 Flattened area on the spoke in the 7 o'clock position. ©2005 Antikythera Mechanism Research Project. All rights reserved.⬈#p145

PTM and X-ray CT slice.⬈#p146

There is a raised flattened area on the spoke in the 7 o’clock position, which is 17.8 mm long and 15.0 mm wide (the full width of the spoke). This starts at 38.2 mm from the centre and extends to the circumference ring. The X-ray CT suggests that it had a rivet near its centre. We reconstruct this as the place of attachment of a bearing for the second epicyclic gear in the solar anomaly mechanism. In addition to the rivet, the bearing may have been soldered to the spoke, though we have no direct evidence for this. The way that the spoke is dovetailed to the circumference ring is clear in the X-ray CT.⬈#p147

Fig. 16 Pierced lug in the 10 o'clock position. ©2005 Antikythera Mechanism Research Project. All rights reserved.⬈#p152

In the 10 o’clock position, there is a pierced lug. The dimensions, including the corrosion, are about 7.3 mm long, 2.2 mm wide and 5.7 mm high. Its inner core is a well-defined tapered block, with length 5.6 mm tapering to 5.1 mm; width 1.5 mm; and height 4.9 mm. It is difficult to be precise about its original dimensions because of the corrosion. The diameter of the hole is 1.4 mm. This lug does not feature in our model and we do not understand its function.⬈#p153

2.7 Pillars on the periphery of the Main Drive Wheel

There are three pillars on the Main Drive Wheel (b1). One of these is bracketed to the periphery of the wheel and attached with rivets. The other two are simply attached directly to the wheel, with the bottom part of the pillar shaped to form an oval rivet. The surviving evidence shows one long support pillar and two short pillars. Their function has long been a subject of debate, but no satisfactory and detailed explanation has so far been offered.⁵⁹ Yet they are a very striking feature of the Main Drive Wheel and demand a good explanation. They play a critical role in our proposed model, so we shall examine them in some detail.⬈#p154

Fig. 17 Fragment A-1, showing the support pillars attached to b1. ©2005 Antikythera Mechanism Research Project. All rights reserved.⬈#p157

The two short and one long support pillars, near the input gear a1. From top to bottom, we shall call them pillar 1, 2 and 3.⬈#p158

The pillars are all close to the input crown gear and one of them has in fact merged with this gear as a result of corrosion and calcification. We shall label them from top to bottom as pillar 1, pillar 2 and pillar 3. Pillar 1 is longer than the other two.⬈#p159

2.7.3. Short Pillars

A⬈#p197	B⬈#p198
C⬈#p201	D⬈#p202

Fig. 22 Short pillar attachments on the Main Drive Wheel. ©2005 Antikythera Mechanism Research Project. All rights reserved.⬈#p205

(A) The Main Drive Wheel, showing the pillars on the right-hand side. (B) Front view of the two short pillars. (C) The symmetrically opposite position on the Main Drive Wheel. (D) The X-ray CT shows holes in these positions, which suggest that rivets might originally have been there.⬈#p206

The short pillars are riveted directly to b1, rather than attached with brackets. In the symmetrical position on the opposite side of b1, possible rivet holes can be seen in the X-ray CT. In our model, we reconstruct four short pillars consisting of two pairs on opposite sides of the Main Drive Wheel. Why these pillars are offset from a symmetrical position relative to the spokes is not clear, though it may be designed so that their rivets avoid the dovetail joints, which attach the spokes to the rest of the wheel. Like the long pillars, the short pillars also appear to be designed to carry a plate.⬈#p207

⬈#p208

⬈#p209

2.8. Pointers

Fig. 24 Surviving pointers. ©2005 Antikythera Mechanism Research Project. All rights reserved.⬈#p217

(A) The Metonic pointer.(B) The hub of the broken-off Exeligmos pointer.⬈#p218

The final component of our new model will be the pointer system on the Front Dials, so we here examine the sparse evidence for pointers in the Antikythera Mechanism. There are only two incomplete pointers that survive in the fragments. The first is the Metonic pointer. The surviving part of this pointer is 55.0 mm long, 4.2 mm wide and 2.2 mm thick. The second is part of the Exeligmos pointer, which we identify here for the first time. This is 5.4 mm across and about 1.0 mm thick. Its end is broken off. We have modelled our planetary pointers on the Front Dials with dimensions 68.7 mm long, 2.0 mm wide and 1.2 mm thick.⬈#p219

3.1.Babylonian Astronomy & Period Relations

The discovery that many astronomical phenomena are periodic was one of the foundations of the Babylonian astronomy of the first millennium BC.⁶⁰ Through records of dated observations beginning in the seventh century BC if not earlier, astronomers in Babylonia identified time intervals, generally shorter than a century, which separate very similar occurrences of a single kind of phenomenon, for example the Saros comprising 223 lunar months, which separates lunar eclipses of almost identical magnitude and duration, and the Metonic cycle comprising 19 solar tropical years and 235 lunar months, which separates full or new Moons at which the Moon is at almost identical longitudes. For the planets, the most important set of intervals was the so-called "Goal-Year" periods, which were used to forecast repetitions of phenomena such as first and last visibilities and stationary points; these are sub-century intervals approximately comprising whole numbers of solar years and whole numbers of a planet's synodic cycles, so that after one Goal-Year period a planet will repeat its phenomena at very nearly the same longitudes. Besides the Goal-Year periods, many other approximate periods are attested in Babylonian texts, including longer and more accurate periods (of the order of centuries) that were built up out of the shorter ones to serve as the basis for advanced methods of predicting planetary motion.⁶¹⬈#p220

For the purposes of forecasting future occurrences of phenomena one-to-one from observed past occurrences, it sufficed to know the duration of a suitable period for the planet in question without having to take account of the planet's behaviour during the period. As a basis for mathematical modelling of a planet's apparent motion, a period relation took the form of an equation of a whole number x of synodic cycles, a whole number z of revolutions of the planet around the ecliptic, and a whole number y of solar years; for example, Saturn's Goal-Year period of 59 years becomes:⬈#p221

57 synodic cycles = 2 longitudinal revolutions = 59 years.⬈#p222

Greek astronomers' records of dated planetary observations began only about 300 BC, and were never as systematic as the Babylonian records, so that their knowledge of period relations, beyond the crudest periods, was derived from Babylonian sources; thus we are informed by Ptolemy that Hipparchos, in the mid second century BC, had a set of planetary period relations that we recognize as the Babylonian Goal-Year periods. The Greeks also applied observational evidence and mathematical algorithms to obtain other period relations as modifications of the Babylonian ones. In some cases these were more astronomically accurate, but some were preferred for other reasons, for example so that one could have a simultaneous repetition of the phenomena of all the planets in a single vast "Great Year" period running to tens of thousands of years or more.⁶²⬈#p223

3.2. Geometrical theories of planetary motion

During the second century BC, Greek astronomers seeking geometrical models to describe the motions of the Sun, Moon, and planets employed geocentric models that may be described anachronistically as representing a heavenly body's position relative to the Earth as the sum of two uniformly rotating vectors of constant length.⁶³ When a planetary model of this kind is translated into a heliocentric system, one of the vectors turns out to represent the planet's mean revolution around the Sun (i.e. treating its orbit as perfectly circular), and the other represents the negative of the Earth's mean revolution around the Sun. Since the vectors can be added in either order, each body's motion can be effected by two geometrical models that result in identical paths for the body while suggesting distinct physical interpretations. Taking the longer vector as the primary revolution around the Earth, and the shorter vector as a secondary revolution superimposed on the primary revolution, we obtain the deferent and epicycle model, where the body revolves uniformly along an epicyclic circle whose centre revolves uniformly along a deferent circle concentric with the Earth. Reversing the role of the vectors, we obtain the eccentre model, where the body revolves uniformly along a circular orbit that encloses the Earth but is not concentric with it, while the centre of the orbit revolves uniformly around the Earth. Greek astronomers appear to have preferred the deferent and epicycle model for the two inferior planets, Venus and Mercury (see Fig. 26), probably because it gave an intuitive explanation of the fact that these planets alternately run ahead and behind the Sun without ever surpassing a certain maximum elongation from it.⬈#p224

For the superior planets, the Greeks employed both varieties of model (Fig. 26). Assuming deferent and epicycle models for the superior planets resulted in a system in which all five planets revolve on epicycles, whereas assuming eccentre models resulted in a system in which the mean Sun plays the same role for all five planets. (The eccentre models can be thought of as "extreme" epicyclic models in which the epicycle has become so large that the Earth is inside it.)⬈#p228

Because these models translate into a heliocentric system in which all the planets revolve uniformly on circular orbits concentric with the Sun, they successfully explain the planets' synodic cycles with their alternations of prograde and retrograde motion, but they fail to predict the variations in a planet's successive synodic cycles that result from the fact that the true orbits are not uniform and circular but elliptical and subject to Kepler's Second Law. Passages in Ptolemy's Almagest (12.1 and 9.2) indicate that they were employed by Apollonios of Perga in the early second century BC, whereas during the third quarter of that century Hipparchos showed that invariable synodic cycles were inconsistent with observational evidence.⬈#p232

3.3. Mechanisms for planetary and solar motion

Our focus is on geared mechanisms for the planets that are based on the ancient Greek deferent and epicycle theories that combine two circular motions. Prior to this study, to the best of our knowledge all attempts to build such mechanisms into the Antikythera Mechanism took a direct form. A gear is turned at the rate of the deferent and a second gear mounted epicyclically on the first gear is turned at the rate of the epicycle. A slotted follower, turning on the deferent axis, follows a pin attached to the epicyclic gear. The follower is connected to a tube and a pointer is attached to the tube. This outputs the variable motion. This is the obvious way to model the theory with gears.⬈#p233

3.3.2. Inferior Planet Mechanisms

Simple proposals for inferior planet mechanisms for the Antikythera Mechanism have essentially been of this form (though Wright has chosen to use compound gear trains, which are not discussed here).⁶⁶ For an inferior planet, such as Venus, the central gear X is fixed and the axis of the epicyclic gear rotates at the rate of the mean Sun as it is carried round on an “epicycle carrier” gear, which is not shown here. In the Antikythera Mechanism the epicyclic carrier is b1. The epicyclic gear rotates at the rate determined by the number of teeth in the two gears. The tooth counts exactly reflect the Babylonian period relations. This mechanism uses the 8-year cycle of Venus, with 5 synodic periods, which we shall refer to as (5, 8). This is scaled up to make the tooth counts reasonable—in this case x = 40 and y = 64. Following the example of Wright's model, the bar that carries the pin is replaced with a disk to ensure smooth running of the assembly. These mechanisms track the planets very well in a Simplified Solar System, where the planets all orbit the Sun in circles in the same plane—their accuracy being limited in this context only by the accuracy of the period relation. In fact, the systems are equivalent—though there are considerable errors by comparison with the actual Solar System.⬈#p328

We shall measure all rotations as rotations per year. It is worth pointing out that there are two inputs to this system: the fixed gear, which turns at 0 rotations per year, and the rotation of the epicycle carrier, which turns at the rate of the mean Sun at 1 rotation per year. This will be true for all our planetary mechanisms. We shall refer to these as the 0-input and the 1-input to the planetary mechanisms.⬈#p329

Let r be the rotation of the planet round the Sun in rotations per year. Let p be the distance of the planet from the Sun in Astronomical Units (AU). Let d be the distance of the pin from the centre of Y. Let d(X, Y) be the inter-axial distance between the gears, then it is easy to establish the basic equations of the mechanism:⁶⁷⬈#p330

1/r = 1 + x/y or r = y/(x + y)⬈#p331

d = p * d(X, Y)⬈#p332

For our Venus mechanism, r = 8/(5 + 8) = 0.615385 years. This compares with the actual figure for the mean rotation period of Venus round the Sun of 0.615185 years. Note that the pin is outside the face of the gear. If we make the assumption that the radius of a gear is proportional to its tooth count, it can easily be established from Kepler’s Third Law that this must be true for every inferior planet mechanism.⬈#p333

This is clearly a heliocentric way of viewing these mechanisms. For the ancient Greeks, the period relation would almost certainly have been obtained from Babylonian astronomy or from direct observation of dates of synodic phenomena, and the pin distance from observation of the maximum elongation of the planet from the Sun.⬈#p334

Previous studies have proposed how such inferior planet mechanisms might have been included in the Antikythera Mechanism at the front of b1.⁶⁸ Mechanically, this is comparatively easy—though even here the planetary mechanisms must be “interleaved”, rather than simply stacked adjacent to each other. From back to front, we have Mercury fixed gear, Venus fixed gear, Mercury output, Venus output. So the process is not entirely straightforward. The reason that the solar anomaly and the mechanisms for Mercury and Venus can all be included here is that they “go with the Sun”, so their anomalies are limited to a fixed number of degrees of elongation from the mean Sun. They can therefore be mounted on different spokes of the wheel without their slotted followers interfering with each other, as we shall see later.⬈#p335

Historical parallels for these “pin-and-slot” devices occur about a millennium and a half later with the remarkable Astrarium by Giovanni de Dondi (1348-1364), an astronomical clock, which implemented the full Ptolemaic system for the planets.⁶⁹ In this device, the Sun, Moon and planets each had their own individual dials, so avoiding the complexities of coaxial outputs. Anomalies in the de Dondi instrument are generated by pins and slotted followers and similar clocks flourished in the centuries afterwards, such as the magnificent Dresden Planetenlaufuhr by Eberhard Baldewein (c. 1565).⁷⁰⬈#p336

3.4. Superior Planets

3.5. Wright's Planetarium Display for the Antikythera Mechanism

Before assembling our new model, we examine Wright's model.⁷¹ We cannot emphasize its importance too strongly. Until its appearance, the prevailing assumption was that any planetary display on the Mechanism could have shown only mean motions.⁷² Wright's model gave the first solid grounding to the idea that it might indeed be feasible to make a model of the Antikythera Mechanism that includes pointers displaying appropriate cycles of prograde and retrograde motion for all five planets—both conceptually and mechanically. The author made a working model of this scheme with eight coaxial outputs, “...entirely by simple methods available to the original workman.” This proved in principle that such a scheme could not only work but could have been made in the era of the Antikythera Mechanism. From the inside of the central axis outwards, the eight coaxial outputs are: Moon, Sun, Mercury, Venus, Date, Mars, Jupiter and Saturn—as they are in our model as well.⬈#p340

In our view, the main objection to seeing this model as an actual reconstruction of the lost planetary gearwork is that it does not conform sufficiently to the physical evidence from the fragments and that it is not in harmony with the design simplicity that has been revealed in the surviving gearing. (In fairness we should point out that Wright states that, "This reconstruction is not significantly more complicated than the original fragments; it is simply more extensive."⁷³) Our current study is aimed at tackling both of these problems. Because of the importance of this model, we will examine it in some detail to discuss whether it matches the physical evidence. Images of Wright’s Planetarium model can be seen in Wright 2002, pp. 171, 172.⬈#p341

Wright observed that b1 turns on a bearing attached to a central pipe, fixed to the Main Plate, which ends in a squared boss.⁷⁴ He suggests that fixed gears were attached here—and these might very plausibly have been the fixed gears for inferior planetary mechanisms. These gears provide the 0-input to the inferior planet mechanisms in this model, as they do in other models that include the inferior planets. In most cases, b1 is the epicyclic carrier and single epicyclic gears are attached to b1 for each inferior planet. b1 provides the 1-input to the mechanisms.⬈#p342

The inferior planets are arranged on b1. Some of the arbors turn on bearings on the spokes of the wheel and some do not—meaning that they must be supported between the spokes. With this scheme, it is not easy to understand why the wheel had spokes, rather than being a simple disk.⬈#p343

Once the inferior planets and the solar anomaly have been added, there is a problem with mechanizing the superior planets. Because the mechanisms already attached to b1 get in the way, a fixed gear for the 0-input cannot be added to the squared boss on axis b and a mean solar input for the 1-input apparently cannot be contributed directly by b1. Wright's model solves these problems, by adding a separate module for each superior planet, where the module is fixed to the side of an extended case for the 0-input and an auxiliary axle provides the 1-input. This axle is at the side of b1 and takes its rotation via a small gear that meshes with b1. This rotation is then transmitted to the planet module via another small gear of equal size and a gear the same size as b1 for each module. In our view, it is a cumbersome arrangement. Indeed Wright has considered ways of dispensing with this arrangement.⁷⁵ From the 1-input to each superior planet module, gearing calculates the mean rotation of the planet around the Sun, and an epicycle is attached, which is geared to have the rotation of the mean Sun. Hence the module directly models the deferent and epicycle model. Each module uses seven or eight gears for each superior planet, leading to a system of considerable complexity.⬈#p344

Our study of the pillars on b1 strongly supports the idea that there was a circular plate attached to these pillars (see 3.6). Wright reconstructs these pillars, but an attached plate here appears to be redundant.⬈#p345

In summary, this model was an important evolutionary step in understanding how the Mechanism might have included the planets, but it cannot be considered to be a reconstruction of the Antikythera Mechanism. Our model uses a different design, which circumvents these problems. It is described in 3.6.⬈#p346

3.6. Building the New Model

Fig. 29 Computer reconstruction of b1 with plates carried by the pillars. ©2011 Tony Freeth, Images First Ltd. All rights reserved.⬈#p351

(A) The Date Plate is attached to the short pillars. The input crown gear can be seen on the right. (B) The Superior Planet Plate (SPP) is attached to the long pillars.⬈#p352

Our proposal is that a mechanism for the solar anomaly and mechanisms for all five planets were contained within the space in front of the Main Drive Wheel, b1. In our model, mechanisms for the solar anomaly and the inferior planets are attached to the spokes of b1; and mechanisms for the superior planets are attached to the rear side of the circular plate carried by the support pillars. To enable our model to work this circular plate is annular and we shall refer to it as the superior planet plate (SPP). Between these two distinct assemblies, our model includes a rectangular plate carried by the short support pillars. This plate has three functions: its primary use is to carry the rotation of b1—the mean sun rotation—to the Date Pointer on the front Calendar Dial; in addition, it supports the outputs from the superior planet mechanisms and serves as a bearing for the output tubes from the mechanisms attached directly to b1. We shall refer to it as the date plate (DP). Our new model explains all the observed evidence from b1, including all the bearings, the support pillars and their dimensions—except for the pierced lug on the 10 o’clock spoke of b1, whose function we do not understand.⬈#p353

In our model, the distance from the front of b1 to the back of the Date Plate is 16.2 mm and the space between the front of the Date Plate and the SPP is 9.7 mm. These distances determine the number of layers of gearing that can be accommodated. If the layers of gears of the solar anomaly and inferior planet mechanisms were at a similar spacing of about 1.5 mm as they are in some other parts of the surviving mechanism, then they would take up just over half the space between the front of b1 and the SPP. If this is all that was contained in this space, it is hard to understand the function of the long support pillars and why they are so long. So it is very plausible that there was a plate attached to the long pillars and that the rest of the space was taken up with superior planet mechanisms attached to the back of this plate. In our model, these just fit into the remaining space between the Date Plate and the SPP: there are six layers of gears for the superior planets at 1.5 mm intervals (1.2 mm gears and 0.3 mm gaps), using up 6 x 1.5 mm = 9.0 mm of space out of the 9.7 mm available.⬈#p354

3.6.1. Solar anomaly & inferior planet mechanisms

A⬈#p355	B⬈#p356
C	D

Fig. 30 Computer model of Solar Anomaly & Inferior Planets & Mechanisms on b1. ©2011 Tony Freeth, Images First Ltd. All rights reserved.⬈#p359

(A) b1 with the bearings and arbors for the solar anomaly and inferior planets. (B) The gears for the solar anomaly and inferior planet mechanisms with Venus on the 1 o’clock spoke, Mercury on the 4 o’clock spoke and the Sun on the 7 o’clock spoke. (C) The solar and inferior planet mechanisms. (D) The solar and inferior planet mechanisms. b1 has been rotated so that Mercury is at 1 o’clock; the Sun at 4 o’clock; and Venus at 10 o’clock.⬈#p360

Assembling the inferior planets and the solar anomaly mechanisms on the spokes of b1 is now fairly easy. The mechanisms do not interfere with each other because they all “go with the Sun”. The output levels of the slotted followers are chosen in our model so that their output tubes are in the order, Sun, Mercury and Venus. The Sun tube must be next to the Moon output arbor, since these two outputs are subtracted in the lunar phase mechanism.⁷⁶⬈#p361

It is notable that there is a spoke that is not used in our model, with a fitting that must have had some function. It would certainly put significant stress on the engineering constraints to include another function. It is also not clear what that function might be. It could possibly have been an output that showed both nodes of the Moon on the Zodiac Dial—a double ended pointer or Dragon Hand, as the Chinese called it. This was fairly common on astronomical clocks from mediaeval times onwards.⁷⁷ However, it is very hard to see how this might be arranged in the context of the Antikythera Mechanism, though the gear ratio of 12/223 for the rotation of the nodes (derived from the Saros and Metonic cycles) looks possible to achieve.⬈#p362

3.6.2. The Gear in Fragment D

Fig. 31 Both sides of Fragment D in 2005. ©2005 Antikythera Mechanism Research Project. All rights reserved.⬈#p365

The question arises as to whether the gear in Fragment D, whose function is unknown, might have been part of a planetary mechanism. Fragment D was first recorded in a picture taken in 1902-1903.⁷⁸ Price writes about it as follows:⁷⁹⬈#p366

“This is a detached fragment of mechanism (Svoronos, fragment D) which was seen by Rehm, but then misplaced in the museum and not refound until March, 1973. It is a highly calcified mass about 40 mm in diameter and 5 to 8.5 mm thick which appears in the radiographs to contain a single gear wheel which Karakalos counts at 63 teeth. Just possibly the great thickness and the blur which gives a double row of teeth might indicate that we are dealing here with a pair of identical or nearly similar wheels. The center shows a round hole with a wedge and pin fixing as in the axes of C and D. There are also three fixing (?) rivets or axes arranged in an equilateral triangle on the gear face.⬈#p367

All subsequent tooth counts have asserted a count of 63 teeth.⁸⁰⬈#p368

Fig. 32 X-ray CT of Fragment D. ©2005 Antikythera Mechanism Research Project. All rights reserved.⬈#p374

Three parallel slices through Fragment D. The 1^st and 2^nd slice are 2 mm apart and the 2^nd and 3^rd 1.5 mm apart. Two orthogonal slices through the hub of the gear.⬈#p375

The X-ray CT reveals some interesting features of the gear in Fragment D. Measurements are given to the nearest tenth of a millimetre, but the decimal component cannot be regarded as being reliable. The first X-ray slice shows a disk of radius 21.4 mm, with three rivets and a squared centre; the second shows the gear in Fragment D, with pitch radius 16.0 mm, and the third shows a plate with a semi-circular end. The rivets pass through both disk and gear. The disk is about 1.0 mm thick and the gear 1.5 mm. Because there are two sets of teeth—seen on the right hand side of the second CT slice—it has sometimes been thought that Fragment D might contain two gears.⁸¹ However, looking at the first orthogonal slice, a pin can be seen end-on going through the arbor, just below the level of the gear. The same pin can be seen side-on in the second orthogonal slice. It is clear that this pin has split the hub of the gear, causing one side of the gear to shift laterally, as can be seen by direct observation of the 3D X-ray CT volume. A strip of teeth has come away from the main body of the gear, giving the illusion of two gears. So we believe that there is only one gear in Fragment D. On both the surface of the gear and of the disk the letters “ME” are clearly inscribed. This stands for “45” in the ancient Greek letters-for-numbers system. It may have been a gear number, but we cannot find any other significance in this inscription.⬈#p376

Since (63, 20) is a period relation for Mercury attested in Babylonian astronomy, the first question that arises is whether this gear is the fixed gear on the central axis in a mechanism for Mercury. However, this cannot be the case, since this gear evidently had a pin through its hub and this would exclude the lunar output that comes through the middle of the central axis.⬈#p377

From the X-ray CT slices it can be seen that the gear had a circular disk riveted to one of its faces and the gear ran in a circular bearing in a plate on the other side of the gear. This is exactly the same structure that we are proposing for the epicyclic gears in our inferior planet mechanisms, where a disk is attached to the gear and a pin is attached to the disk. Might this gear be an epicycle for an inferior planet mechanism? The only plausible planet would be Venus with period relation (5, 8), which multiplies to (40, 64) for reasonable tooth counts. Is it possible that this gear had 64 teeth?⬈#p378

Fig. 33 X-ray CT section of gear in Fragment D. ©2005 Antikythera Mechanism Research Project. All rights reserved.⬈#p381

The X-rays have a false-colour scale to enhance the teeth tips for counting. Two alternative counts are shown in red for 63 teeth (on the left) and 64 teeth (on the right).⬈#p382

In these X-Ray slices the tips of the teeth have been identified for counting. These have been marked on the remnants of the teeth, which are still attached to the gear, since the other teeth have broken away and so their original positions are not evident. There are 61 surviving teeth with a small gap of 16.9°. The 61 teeth represent 60 gaps over a total of 360° - 16.9° = 343.1°. This means an average of 5.7° per gap. The 16.9° should therefore represent 16.9/5.7 = 2.96 gaps. It is therefore natural to infer that there are 3 gaps, meaning 2 teeth are missing. This would imply a gear count of 63.⬈#p383

The mean angle between teeth is 5.7° with standard deviation 0.273°. The minimum and maximum angles are 5.1° and 6.6°. Even if all the missing teeth were at the minimum observed angle relative to each other, the number of gaps would only be 16.9/5.1 = 3.3 gaps. In the right-hand image, four gaps have been inserted at regular intervals of 4.23° to show the consequence of a 64-tooth count. For this to be correct, we would have to believe that the gaps between the missing teeth all just happened to be significantly less than the minimum surviving gap. This is not plausible.⬈#p384

It could be argued that the gear was made with 63 teeth by mistake, when it should have had 64 teeth. However, a 64-tooth gear is very easy to lay out by repeated halving of the sectors—much easier than a 63-tooth gear. In addition, it would be expected that the gear would show the symmetry of its layout, which the gear in Fragment D does not display.⬈#p385

There is another argument that supports the idea that the gear might be the epicycle for a Venus mechanism. Measurements of the gear and the disk attached to it suggest that the size of the disk is within the permissible range to include the pin for the epicycle of Venus. However, the tooth count of 63 makes it very difficult to accept this hypothesis. A further question arises. Is it possible that the designer used a 63-year cycle of Venus? The associated period relation would be (39, 63). However, this implies a mean rotation for Venus of 0.617647 years. This compares with the figure implied by the 8-year cycle of Venus of 0.615385 years and the actual figure of 0.615185 years. So the period relation (39, 63) is considerably worse than the usual period relation. In addition, we know of no ancient source that uses a 63-year period for Venus. In conclusion, it is difficult to interpret the gear in Fragment D as being part of an inferior planet mechanism. Its function remains a mystery to add to the mystery of the unused spoke on b1.⬈#p386

3.7. The New Superior Planet Mechanisms

We here describe new mechanisms for the superior planets, which will complete our proposed Cosmos model. Some key questions arise. Is our model consistent with the evidence? Do the dimensions of the gears fit the spaces available? Is the front-to-back spacing of our proposed gears consistent with the surviving layers of gearing? Is the conception and design of the planetary mechanisms in accord with the surviving gear trains? Are the parameters of the planetary mechanisms consistent with the astronomical knowledge of the era? Are the engineering requirements feasible for the era of the Mechanism? No previous model has been able to meet all these challenges. For our model, we give evidence for positive answers to all these questions.⬈#p387

The superior planet mechanisms for our proposed model are unlike any previous mechanisms in that they do not model the deferent and epicycle theories in a direct way. They are based closely on the lunar anomaly mechanism. This mechanism is hard to understand and these new planetary mechanisms are equally difficult and surprising. It is very unexpected that essentially the same design works for both the lunar anomaly and the planets. Though the anomalies of the Moon and planets have distinctly different causes—the elliptical orbit of the Moon and the heliocentric orbits of the planets—the deferent and epicycle theories provide a unified solution (to a first order approximation). This unity is reflected in the forms of these new mechanisms.⬈#p388

Fig. 34 Proposed mechanisms for the superior planets. ©2011 Tony Freeth, Images First Ltd. All rights reserved.⬈#p397

The numbers refer to the tooth counts of the gears. All the mechanisms use period relations from Babylonian astronomy. Taking Mars as an example, each superior planet mechanism contains four gears: the fixed input gear G1, the pin gear G2, the slot gear G3 and the output gear G4. The tooth counts of G1 and G2 directly reflect a Babylonian period relation for the planet. The pin gear G2 is mounted epicyclically on a carrier, turning at the rate of the mean Sun. G2 meshes with G1 and is induced to turn by the rotation of the carrier. G3 is also mounted on the epicyclic carrier on a different axis than G2. The slot on G3 engages with the pin on G2 at distance d from the centre of G2. This induces a variable motion in G3. With an offset of d/p between the axes of G2 and G3 (where p is the mean distance of the planet from the Sun in AU), this models the synodic phases of the planet. G3 engages with the equally-sized output gear G4, which reverses the output and carries it to the planet’s pointer on the Zodiac Dial via a tube. A geometric proof that these mechanisms exactly generate the deferent and epicycle models is given at the end of these notes in 4.1. The lunar anomaly mechanism is shown for comparison. The blue gear on the left turns around axis e at the rate of the sidereal Moon, 254/19 rotations per year. The gears on the right are mounted epicyclically on eccentric axes on e3, which turns at the rate of the Line of Apsides of the Moon—(9 x 53)/(19 x 223) rotations per year.⬈#p398

The tooth counts of the first two gears in the system, G1 and G2, exactly reflect the assumed planetary period relation. The tooth counts of the last two gears have no special meaning: they are simply chosen so that two equal gears with much the same module fill the space between the central axis and the axis of the gear with the slot. First, we look in detail at a practical embodiment of these ideas for the planet Mars, then we build all the superior planets onto the SPP in our new model.⬈#p399

3.7.1. Mechanism for Mars

Fig. 35 Computer Reconstruction of Mars Mechanism. ©2011 Tony Freeth, Images First Ltd. All rights reserved.⬈#p408

(A)On the left is the 37-tooth fixed input gear for Mars and on the right the 79-tooth pin gear. The pin is attached to a ring on the gear, which is designed to support the slot gear. The thickness of the ring is chosen so that the Mars output is at the right level. The gear turns on a stepped arbor that allows the slot gear to rotate on an eccentric axis, while following the pin. The hole in the input gear allows the lunar and planetary outputs to pass through to the Front Dials. (B) On the right is the slot gear and on the left the equally-sized output gear with its tube attached. As the slot gear turns on its eccentric axis it generates the synodic phases of Mars and this is then transmitted to the output gear. (C)A bridge holds the Mars mechanism to the SPP. It is modelled on the bridge for the lunar mechanism, part of which survives in Fragment A. The bridge is held onto the SPP by pierced lugs and pins, so that it can be removed for calibration. (D) On the left is the pin gear, which rests on a disk attached to the SPP. On the right, the output gear with its tube passes through the fixed gear. Attached to the tube is the pointer, where Mars is represented by a red onyx sphere.⬈#p409

Translating the geometric design for these mechanisms into a physical design is fairly straightforward. Care must be taken to make sure that the fixed input and the output are at the right front-to-back levels and that the output can pass back through the input gear for display on the Front Dials. This latter design is exactly analogous to the way that the output of the lunar anomaly mechanism goes back through the epicycle carrier, e3, and the input gear, e5, in order to reach the Front Dials.⁸² Everything about the design closely mirrors existing principles in the surviving mechanism.⬈#p410

The Mars mechanism is made to the largest module (0.58) that will fit onto the SPP. This is because Mars stretches the design parameters in two ways. The input gear must be large enough so that a hole through its centre can accommodate all eight outputs from the system—the lunar output and seven coaxial tubes for the Sun, Date and planetary outputs. In addition, the slot of the Mars mechanism goes very close to the axis of the slot gear, so it is best mechanically to have the gear as large as possible.⬈#p411

The pin gear has a ring on its face to separate the input layer from the output layer, which must be at the correct level for the coaxial tube system. The pin is mounted on this ring. Just as in the lunar anomaly mechanism, there is a stepped arbor that enables the pin gear and the slot gear to turn on eccentric axes relative to each other. The slot gear and the output gear have the same number of teeth, so that—relative to b1—the output gear mirrors the anomaly generated by the slot gear. The bridges that hold the planetary mechanisms to the superior planet plate are modelled on the similar bridge for the lunar mechanism. A broken-off part of this can be seen at the back of Fragment A.⬈#p412

Before putting this mechanism into our proposed model, we discuss why this design works in our virtual model. (A more formal proof can be found at 4.1.) We shall use Rot (X) to mean the absolute rotation of X and Rot (X | Y) to mean the rotation of gear X relative to gear Y. All rotations are measured as number of rotations per year.⬈#p413

Because G1 is fixed and Rot (b1) = 1:⬈#p414

Rot (G1) = 0⬈#p415

Rot (G1 | b1) = - 1⬈#p416

By the basic property of meshing gears:⬈#p417

Rot (G2 | b1) = - 37/79 * Rot (G1 | b1) = 37/79⬈#p418

If we imagine sitting on b1 observing the system, the gears all turn on fixed axes. In this frame of reference, the input gear turns at the rate 37/79 and the slot gear follows the input gear with a cycle of variability determined by its apogee, when the pin is pointing at the central axis. So the cycle of variability of the slot gear is 37 cycles in 79 years, which is what we want for the synodic cycle of Mars. When this variable motion is mirrored back into the “real world” to the output gear, the period of variability is maintained (being invariant under the change of frame of reference) and the mean period of the output becomes 1 – 37/79 = 42/79—again what we want for the mean rotation of Mars about the Sun.⬈#p419

It is tempting to think that these mechanisms are a direct analogue of the eccentric equivalents of the standard epicyclic models. However this is not the case:⬈#p420

Rot (G2) = Rot (G2 | b1) + Rot (b1) = 37/79 + 1 = 116/79⬈#p421

This means that we cannot view the system as simply reflecting the standard eccentre model for a superior planet because the rotation of the pin gear is completely wrong for this idea. However, to an observer sitting on b1 the gear systems and the eccentre models (suitably scaled) will appear as mirror-reflections of each other, so that it is conceivable that the ancient inventor arrived at the pin-and-slot mechanism by way of an eccentre model imagined in the frame of reference of its apsidal line.⬈#p422

Similar mechanisms can also be constructed for Jupiter and Saturn.⬈#p423

Fig. 36 Computer Reconstructions of Jupiter and Saturn mechanisms. ©2011 Tony Freeth, Images First Ltd. All rights reserved.⬈#p428

(A) Jupiter Mechanism based on the period relation (76, -83). (B) Saturn mechanism based on the period relation (57, -59).⬈#p429

3.8. Assembly of Superior Planet Mechanisms

One great advantage of these mechanisms is the simplicity and economy of their design. Another advantage is that they take the anomaly generation away from the central axis and this means that they can work together without interfering with each other—just as the inferior planetary mechanisms do. This in turn means that we can place them all on the same plate and avoid the auxiliary axle of Wright's model, which was necessary to provide the 1-input to each mechanism.⬈#p430

In our model, the fixed inputs to the planetary mechanisms are all attached to a Front Sub-Plate that fixes to the wooden Sub-Frame. This provides the 0-input to the mechanisms. The input gears are stacked in size order, with the biggest closest to the plate, and they all have a hole of 14.4 mm diameter to allow the planetary outputs to pass through on their way to the Front Dials. The rest of the planetary mechanisms are attached with bridges to the back of the SPP.⬈#p431

⬈#p432

Fig. 37 Computer Reconstructions of the Superior Planet Module. ©2011 Tony Freeth, Images First Ltd. All rights reserved.⬈#p442

(A) The fixed gears for the superior planets are attached to the Front Sub-Plate. The fixed gear closest to the plate is for Jupiter, since it is the largest gear. This is followed by Saturn and Mars. The gears must be in size order so that they can be inserted through the hole in the SPP and engage with the pin gears of the planetary mechanisms. The hole through the centre of the fixed gears is for the lunar output and the planetary output tubes. (B) The SPP, showing the hole for the inputs and outputs. The gear on the left is the pin gear for Mars. The gear at the bottom is the pin gear for Saturn and on the right a few teeth of the pin gear for Jupiter are just visible. The gear in the centre is the output gear for Saturn. (C) The superior planet mechanisms are attached to the SPP with bridges. On the right is Mars, on the left Jupiter and at the bottom middle, Saturn. The pins and slotted followers can be seen for all three planets. In the centre is the output gear for Mars. (D) The superior planet mechanisms from the other side, with the SPP removed. On the left, the pin gear for Mars, on the right the pin gear for Jupiter and at the bottom the pin gear for Saturn. In the centre are the output gears and tubes for the superior planet mechanisms.⬈#p443

Once the basic design of the superior planet mechanisms is adopted, their inclusion at the back of the SPP follows a logical framework with few options for the overall design but some options regarding the details. For example, the angles of the planetary mechanisms relative to each other are immaterial, so long as they do not interfere with each other mechanically. The obvious options are to set them either at 120° or at 90°. We have set the mechanisms at 90°, with Saturn between Mars and Jupiter because of a particular mechanical constraint, which is described below.⬈#p444

The order of the outputs at the front of the Mechanism—from front to back—is Moon, Sun, Mercury, Venus, Date, Mars, Jupiter, Saturn. This means that the output tube for Saturn is the outermost of the coaxial tubes. This in turn means that, among the superior planets, its output gear is closest to the front of the Mechanism, followed by Jupiter and then Mars. A consequence is that the bridges for the superior planet mechanisms vary in height with Saturn being the lowest, followed by Jupiter and Mars. The position of the Date Plate on the front-to-back axis is fixed from measurements taken in the X-ray CT of the short pillars on b1. With the dimensions that we have adopted for the gears, the bridges for Jupiter and Mars are too high to fit within the space allowed by the Date Plate. (For Jupiter, the bridge only fails to fit in our model by 0.7 mm.) So we have arranged that Saturn is the mechanism that fits right in front of the Date Plate.⬈#p445

The fixed gears for the superior planet mechanisms are attached to the Front Sub-Plate. After the assembly of the planetary mechanisms on the SPP, this plate is placed in the device and each fixed gear engages with the pin gear for its planet. The fixed gears must be attached to the Sub-Plate in size order, with the largest being closest to the plate—otherwise, the Sub-Plate cannot be inserted into the Mechanism properly. The Sub-Plate needs to be small enough so that it can be placed into the Mechanism when the Dial Plate is removed and it needs to be robust enough to hold the static gears in the superior planetary mechanisms sufficiently firmly. The plate must also be attached to the wooden Sub-Frame, which we infer from observations of the X-ray CT of Fragment A. There do not appear to be any other size constraints on this conjectural plate.⬈#p446

Our proposed superior planet system includes four gears for each planet—making just 12 gears in all. The inferior planets and the solar anomaly use 6 gears, so the total number of gears in the space in front of b1 is 18.⬈#p447

Fig. 38 Computer model of The Planet Module. ©2011 Tony Freeth, Images First Ltd. All rights reserved.⬈#p452

(A) The planet module disassembled from b1. Superior planet mechanisms for Mars, Jupiter and Saturn attached to the SPP with bridge pieces. Attached to b1 are the inferior planet mechanisms and the date plate. The SPP attaches to the pillars on b1 with pins. (B) The planet module, showing the output tubes with pointers attached.⬈#p453

In our model, the fixed gear for Jupiter is the largest, since we have chosen the period relation (76, -83). We could have chosen (65, -71) or (54, -59), but these would have been less accurate. So the fixed inputs, from front of the Mechanism to back, are in the order Jupiter, Saturn, Mars. The output gears—again from front to back—are Saturn, Jupiter, Mars, since we want the planetary order of the pointers to reflect the ordering (from front to back): Moon, Sun, Mercury, Venus, Mars, Jupiter, Saturn. Note that the order of the pointers is the inverse of the order of the output gears because of the way that the co-axial tube system must work. The different orders of the fixed Sub-Plate gears and the output gears is not a problem: it is just a matter of arranging mechanically for the slot gear to be the right distance from the pin gear so that the output gear is at the right level. We achieve this with the help of spacer rings attached to the pin gears.⬈#p454

The coaxial outputs on the Zodiac Dial are enabled by a system of concentric tubes—as will be familiar with the minute and second hands of a modern clock or watch. There is a precedent in the Antikythera Mechanism for a rotation being carried by a tube in the lunar anomaly mechanism, where the input to the system of the mean sidereal month is carried via a tube through gear e3 to the epicyclic system. For our model of the Antikythera Mechanism, we have outputs for the Moon, Sun, Date, Mercury, Venus, Mars, Jupiter and Saturn. In engineering terms, the manufacture of seven coaxial tubes surrounding the lunar output would have been one of the hardest challenges in the whole design. The total width of this tube system is constrained by the fixed input gear for Mars. This is one of the reasons that we have increased the module of the Mars gears, so that the fixed input gear is as large as possible. In our model, its inner radius is 10 mm. We have left a margin of 2.8 mm between the inner radius of the gear and the hole for the output system. Our output tube system has an external radius of 7.0 mm and this passes through a hole in the superior planet input gears of radius 7.2 mm. The output shaft of the lunar anomaly mechanism has a radius of 2.1 mm. So we have 7.0 – 2.1 = 4.9 mm for all seven output tubes. We have made the tubes with thickness 0.5 mm and a clearance between the tubes of 0.2 mm. The lengths of the tubes vary between 19.9 mm for Saturn and 54.4 mm for the Sun. To make an accurate 54.4 mm tube of external diameter 2.8 mm and internal diameter 2.3 mm in the 2^nd Century BC would have been a difficult achievement. Without the discovery of the Antikythera Mechanism, it would have been hard to conceive that this might have been possible. A similar system of tubes is used in Wright's model.⁸³ Wright cites the ancient aulos (flute), with its concentric sliding tubes, as evidence that the ancient Greeks had this capability.⁸⁴ There is no doubt that the Antikythera Mechanism was made in a culture with a very advanced engineering capacity.⬈#p455

3.9. Dismantling & Calibration

As with all the surviving parts of the Mechanism, our proposed model for the planetary mechanisms is designed to be taken apart for calibration and maintenance. To reach the planetary mechanisms, the Moon Phase Mechanism (one pin) and the other pointers are removed. (How these pointers attached to the output tubes is not clear. In our model, they are simply attached to rings, which are a firm push-fit onto the tubes. This enables them to be set at any angle for calibration.) The four sliding catches of the Front Dial Plate are then slid back and the plate is removed. (Evidence for one of these catches is contained in Fragment C,⁸⁵ and there is another almost identical catch for the Back Cover in Fragment F, as is seen in our direct observations of the X-ray CT of Fragment F. Then the Front Sub-Plate with the fixed superior planet gears is taken out. This then gives access to the four pins holding the SPP. Once these are removed, the module containing the superior planet mechanisms is pulled forwards and taken out of the machine.⬈#p456

To calibrate each superior planet, the two pins holding the bridge are removed. The gear with the slot is then taken off. Calibration consists in moving the gear with the pin to the correct phase for the planet for that date. This can only be done to an accuracy of a gear tooth, because the pin gear must be set at an angle where it will mesh with the fixed gear, attached to the Front Sub-Plate. This gear, for example for Mars where it has 79 teeth, can only be set in steps of 360°/79 = 4.6°. (It is just possible that the fixed gear was adjustable, though this would have been difficult.) Given the inherent inaccuracy of these mechanisms, this is not a real problem. The slot gear is then engaged with the pin and the output gear meshed with the slot gear. The pointer would then be set on the output tube at the correct ecliptic longitude for the calibration date for that planet.⬈#p457

3.10. Accuracy of the Planetary Mechanisms

The designer might have hoped that, after going through the immense trouble of building the Mechanism and calibrating it, the outputs would stay accurate for many years. However, neither the science nor the technology of the era of the Antikythera Mechanism could really be described as “exact”—both the science and the mechanical realization of its predictions were very inaccurate.⬈#p458

We compare the positions of Mars, as reconstructed by NASA with the Mechanism’s predictions over the middle seven retrogrades of Mars in the 1^st Century BC—a period of about 13 years.⁸⁶ Serious error spikes can be seen, amounting to nearly 38°—more than a zodiac sign—at the retrogrades. The deferent and epicycle theories, on which the mechanisms depended, might be regarded as an adequate first-order approximation but were completely inadequate for accurate prediction at the retrogrades, particularly for Mars. More accuracy would have to wait for more sophisticated theories such as those employed by Ptolemy in the second century AD. Added to these inherent theoretical errors were significant mechanical inaccuracies because of the way that the rotations were transmitted through the gear trains.⁸⁷⬈#p462

In short, the Antikythera Mechanism was a machine designed to predict celestial phenomena according to the sophisticated astronomical theories current in its day, the the sole witness to a lost history of brilliant engineering, a conception of pure genius, one of the great wonders of the ancient world—but it didn’t really work very well!⬈#p463

3.11. Markers for the planets

The Back Cover Inscription that describes the Front Dials mentions “the little golden sphere”, presumably referring to the Sun. In addition, there was a tradition of using “magic stones” as markers for the planets.⁸⁸ This paper cites the following extract from a 2^nd or 3^rd century AD papyrus (P.Wash.Univ.inv. 181+221) about an “Astrologer’s Board”, where the astrologer lays out particular stones to represent the Sun, Moon and planets:⁸⁹⬈#p464

...a voice comes to you speaking. Let the stars be set upon the board in accordance with [their] nature except for the Sun and Moon. And let the Sun be golden, the Moon silver, Kronos of obsidian, Ares of reddish onyx, Aphrodite lapis lazuli veined with gold, Hermes turquoise; let Zeus be of (whitish?) stone, crystalline (?)... ⬈#p465

Our model includes pointers for all five planets on the Zodiac Dial. In order to be consistent with the “little golden sphere” inscription on the Back Cover, these pointers include conjectural spherical marker beads in different metals and semi-precious stones, which are placed at different distances along the pointers, so that they create a “cosmos” for Sun, Moon and planets in the order: Moon (silver), Mercury (turquoise), Venus (lapis lazuli), Sun (gold), Mars (red onyx), Jupiter (white crystal) and Saturn (obsidian).⬈#p468

3.12. Historical Context and Significance

The last three centuries BC were a period during which mechanical technology and astronomy both developed rapidly in the Greek-speaking world. The surviving Greek technical literature on mechanical devices attests to intense activity especially with respect to military technology (artillery and the like) and wonder-working devices employing pneumatic and hydraulic principles, whereas we possess scarce written evidence for gear-based technology, none of which goes beyond the basic principles of employing toothed gears and worm gears to multiply rates of rotation up and down, as in simple odometers. The Antikythera Mechanism shows that Greek gear technology was far in advance of the level we would infer from the written record, having attained mastery of differential gearing (as in the Moon ball apparatus) as well as the translation of uniform into nonuniform rates of motion by means of epicyclic gearing and pin-and-slot couplings. While the modelling of astronomical phenomena provided an obvious motivation for the development of such contrivances, it is interesting to speculate about other possible applications they might have had in antiquity, for example in purely mathematical calculating machines and in automata.⬈#p469

In the astronomy of this period certain trends stand out: the investigation of kinematic models built up out of nonconcentric uniform circular motions as explanations of the observable behaviour of the heavenly bodies; the integration of Babylonian astronomy, with its emphasis on quantitative prediction, into this geometrical framework, and the public presentation of astronomy as a discipline capable of explaining not only natural phenomena but also social conventions of time-reckoning. The Antikythera Mechanism, if we have correctly reconstructed its front display, turns out to embody all these aspects. Its gearwork invisibly mimicked the kinds of geometrical model whose theoretical validity was among the chief research questions of the time; through its front display it gave a graphic demonstration of how models based on eccentres or epicycles could account for the varying apparent speeds of the Sun and Moon, the limits of Venus' and Mercury's elongation from the Sun, and the retrogradations of all the planets. Babylonian period relations underlay many, if not all, of the gear trains, and the readout of the heavenly bodies' longitudes on a zodiacal ring graduated in degrees was also a fundamentally Babylonian conception. What is especially remarkable, however, is that the front display could combine this function of generating technical data with a didactic function of portraying the standard Greek cosmology in motion in a form that would have been comprehensible to the educated layman. Given the paucity of scientific artefacts surviving from Greco-Roman antiquity, we are extremely fortunate to have the remains of one of such encyclopaedic complexity.⬈#p470