on the pin-and-slot device of the antikythera

8/2/2019 On the Pin-And-slot Device of the Antikythera

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ON THE PIN-AND-SLOT DEVICE OF THE ANTIKYTHERAMECHANISM, WITH A NEW APPLICATION TO THE SUPERIOR

PLANETS *

CHRISTIN C. CARMAN, Universidad Nacional de Quilmes/CONICET,ALAN THORNDIKE, University o Puget Sound, and

JAMES EVANS, University o Puget Sound

Perhaps the most striking and surprising eature o the Antikythera mechaniuncovered by recent research is the pin-and-slot device or producing the lu

inequality.1

This clever device, completely unattested in the ancient astronomicliterature, produces a back-and- orth oscillation that is superimposed on a steprogress in longitude nonuni orm circular motion.2 Remarkably, the resultingmotion is equivalentin angle (but not in spatial motion in depth) to the standardde erent-plus-epicycle lunar theory. Freethet al. gave a proo o this equivalence,which is, however, a very complicated proo .3 One goal o the present paper is too er a simpler proo that would have been well within the methods o the anciastronomers and that, moreover, makes clearer the precise relation o the pin-a

slot model to the standard epicycle-plus-concentric and eccentric-circle theorieOn the surviving portions o the Antikythera mechanism, only two devices are uto account or inequalities o motion. The frst is the pin and slot, used or the luninequality. And the second is the nonuni orm division o the zodiac scale, which,have argued, was used to model the solar inequality.4 The second would obviouslybe o no use in representing planetary inequalities; so a natural question is to whether the pin-and-slot mechanism could be modifed and extended to the planespecially to the superior planets, or which the likely mechanical representationsless obvious than they are or the in erior planets. The second goal o this paper, this to examine the relation o the pin-and-slot model to the standard concentric-pepicycle theory or the retrograde motion o the planets. We shall see that, indethe pin and slot can be applied to the planets and that this device again gives an erepresentation o the motion in angle, though not in depth.

Scholars have pondered the various holes and structures on the main solar gb1, and wondered whether these might represent the remnants o a lost part o mechanism designed or reproducing planetary motion.5 The evidence is rustrat-ingly sparse. An inscription on the ront cover o the mechanism appears to descplanetary phenomena, including stations, and another inscription on the back cmentions Venus by name,6 but the only hardware surviving is a single wheel o 63teeth (r1) which has no unction in the published reconstruction o the solar a

* This paper is partly based on data processed, with permission, rom the archive o experimeninvestigations by the Antikythera Mechanism Research Project (Freethet al. (re . 1)) incollaboration with the National Archaeological Museum o Athens.

JHA , xliii (2012)


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94 Christin C. Carman, Alan Thorndike and James Evans

lunar portions o the mechanism. It is not much to go on. That a planetary dispcould be possible has been shown by Michael Wright, who, in a eat o mechanivirtuosity, constructed one that works and that is based on a ull representationde erent-and-epicycle theory.7 We ourselves published a more modest proposal, inwhich the mechanism would have o ered a display o the key events in the planesynodic cycles, but not a ull-on kinematic display showing the progress o the plaaround the zodiac with retrograde motion.8 However, in our view, the realization thatthe pin-and-slot mechanism could be applied to the planets (when we know this in act used or the Moon) suggests the possibility o a simpler kinematic modWe remain agnostic about whether the Antikythera mechanism o ered a ull-on kematic display, or a more modest display o in ormation about planetary phases

perhaps a display o the mean motions. However, like Hipparchos, we eel it is wothe attention o geometers to investigate the explanation o the same phenomenameans o hypotheses that are so di erent.9

So the third goal o this paper is to present a new approach or the planetary dispin the Antikythera mechanism. We will ollow the idea proposed by Michael Wrand others, according to which the Antikythera mechanism displayed planetlongitudes on the ront dial, using pointers concentric with those o the Moon aSun and sharing the same zodiac scale. But we will use a pin-and-slot mechan

to produce the inequality with respect to the Sun (retrograde motion). A remarksimple reconstruction o the planetary display becomes possible, which also com ortably onto the our-spoke main solar gear.

The Main Characteristics of the Device for the Lunar Inequality

The device or producing the lunar inequality consists o our gears called e5, k1, and k2, illustrated in Figure 1 (le t).10 On the Antikythera mechanism these are

all o equal tooth number 50 but the key requirement is that they be equapairs: k1 and e5 are equal; and k2 and e6 are equal. The input motion is rom an a(actually, a hollow pipe) at E that turns e5 at the rate o the Moons mean siderrequency; ollowing Freethet al . we designate this requency si. Concentric with e5,but turning reely rom it, is a large wheel e3, which turns at the rate o the Mooline o apsides. From a modern point o view, the orientation o the Moons majaxis does not stay invariable. Rather, the Moons elliptical orbit itsel turns in its plane, so that the perigee advances in the same direction as the Moon moves, takabout 9 years to go all the way around the zodiac. The ancient astronomers waware that the position o astest speed in the Moons orbit itsel advances arouthe zodiac, the Greeks modelling the motion geometrically and the Babylonianmeans o arithmetical period relations. In the Antikythera mechanism the advancthe Moons perigee is modelled by letting e3 turn with a requency n (again usingthe notation o Freethet al. ).

Riding on e3 at centre C1 is gear k1, which is driven by e5 and there ore also turnwith respect to absolute space at requency si. Gear k2 turns about an axis, C2, also


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95On the Pin-and-Slot Device of the Antikythera Mechanism

attached to e3 but slightly o set rom C1. The o set is achieved by using a steppedstud, with its larger diameter centred at C1 and its smaller diameter centred at C2, asshown in the right side o Figure 1 and in the inset perspective view. Thus k1 tuabout C

1and k2 turns about C

2.Wheel k1 has a small pin, which engages a radial slo

in k2. Thus k1, turning at a uni orm speed, drives k2, producing a quasi-sinusooscillation in the speed o k2 a little aster when the pin is closer to C2 anlittle slower when arther rom C2 which is superimposed on the mean motiThe motion o k2 is trans erred to e6, which rotates independently o both e3 ae5 about axis E, and communicates the nonuni orm motion o the Moon to the otparts o the mechanism. Uni orm motion in (at e5) is trans ormed into non-uni omotion out (at e6) around the same axis.11

Simple Proof for the Function of the Lunar Pin-and-Slot Mechanism

We shall now o er a simple demonstration o the e fcacy o this device, and walso make clear its relation to the ordinary tools o Greek theoretical astronomnamely epicycles and eccentrics. Let us or the moment imagine that the Mooline o apsides does not revolve, but remains in the same spatial orientation.

F ig . 1. The device or producing the inequality in the Moons motion around the zodiac, as reconstrucin re . 1. The edge-on view (at right) shows how the stepped stud achieves di erent axes o ro

tion or k1 and k2.


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shall construct a representation o the standard Hipparchian epicyclic lunar theoand superimpose on it a diagram o the pin-and-slot device rom the Antikythmechanism.

Consider frst the epicycle theory. Re er to Figure 2. Circle k1 now represents Moons de erent circle, with centre at the Earth C1. The epicycles centre D is carriedaround clockwise (eastward) at the rate o the Moons mean motion. Thus angl increases uni ormly with time. Simultaneously, the Moon M moves counterclockwon its epicycle, but at exactly the same rate, so angle remains equal to. Angle , the Moons epicyclic anomaly, increases uni ormly at the angular requency a. The direction o the Moon as seen rom the Earth is line C1M. Thus, at the momentshown in the diagram, the angular position o the Moon beyond its point o ast

motion is + q (whereq is called the lunar equation).Now suppose that Figure 2 represents the model realized in the Antikythmechanism. In Figure 2, we are in the rame o re erence o wheel e3. Alternativeone can think o the motion o the line o apsides as being suppressed or the timbeing. Wheel k1 turns uni ormly about centre C1; mounted on k1 is a pin at D. (Wemay think o k1 as being somewhat larger than shown, so that the pin need notexactly at the perimeter o the wheel.) Wheel k2 rotates about a distinct centre2 (slightly o -centre rom C1). k2 has a slot (represented by the heavy dashed line)

which points in the direction o C2.Imagine that k1 was originally oriented so that pin D lay in the direction o asmotion (in the direction o line C1C2). But k1 has now turned ar enough that D liesat angle as viewed rom C1. D will be seen rom C2 at a di erent and larger angle . The question is: just what is the size o this angle ? Because o the equality o and , it ollows that DM remains parallel to C2C1. Then, i we chose the eccentric-ity C1C2 in the pin-and-slot mechanism to be equal to the radius DM in the epicytheory, C2DMC1 must orm a changing parallelogram with always equal to + q.Thus, the motionin angle in the pin and slot mechanism agrees precisely with thmotion in angle o the epicycle theory. That is, a point, such as Z, on the rim omoves about C2 in a per ect circle, but at variable angular speed. The angular postion o Z observed rom C2 is the same as the angular position o M as seen rom C1.Now, we may also allow wheel e3 to move, carrying line C1C2 around with it, andthe demonstration remains intact. Moreover, the rotating ensemble now modelslunar inequality in detail, with its rotating line o apsides.

Relation to the Eccentric-Circle Theory

As is well known (and was equally well known in Antiquity) the epicycle-plconcentric model is geometrically equivalent to an eccentric-circle theory. SFigure 3. Since DM remains parallel to C2C1, it ollows that the physical motion o M through space, resulting rom the dual motion o D around the de erent k1 ao M around the epicycle, is uni orm motion on a per ect circle, shown in dashline, which is centred at C3, one epicycle radius below C1. In the typical proo o the


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equivalence o eccentric and epicycle, one ocuses on parallelogram C1C3MD. (Butit is parallelogram C2C1MD that embodies the quasi-equivalence o the pin-and-slomechanism to the epicycle theory.)

Suppose C1 represents the Earth. Then, as the ancient astronomers knew, thewere two equivalent theories: the epicycle-plus-concentric (epicycle DM plus cik1) and the eccentric-circle theory (circle k3 with no epicycle). The proo o tequivalence was given by Ptolemy in the Almagest , and earlier by Theon o Smyrna,who cited Adrastos as his authority. Traditionally, based on some remarks o Ptolethe proo o the equivalence o these two theories is ascribed to Apollonios o PerBut no one really knows when it was frst proven.12

As ar as is known to us, there is no extant ancient mention o the quasi-equivaleo the pin-and-slot mechanism to the epicycle theory. This is a quasi-equivalebecause the pin and slot model produces the same motion in angle, but not same physical motion in space as the epicycle model. The output o the pin and

F ig . 2. Pin-and-slot mechanism or the lunar inequality in the Antikythera mechanism, compared witepicycle-plus-concentric theory in the standard astronomy o Hipparchos or Ptolemy.


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model is a point moving at non-uni orm speed on a circle k2. But the output o epicycle-plus-concentric model (or o the eccentric-circle model) is a point movuni ormly around circle k3.

One must suppose that an ancient Greek astronomer trained in the philosophigeometrical tradition o Hipparchos and Ptolemy would not have regarded pin-and-slot mechanism as a realistic representation o the lunar theory, or pin-and-slot mechanism suppresses the motion in depth, but nevertheless give

account o the motion in longitude that agrees with what the epicycle theory woprescribe. Aristotle, too, would have considered it not an appropriate solution, ainsists that each simple body (e.g., a celestial orb) should be animated by a sinsimple motion. And here, the fnal output motion is the rotation o e6, which consin a steady rotation with a superimposed oscillation.

Did the ancient mechanic who designed the Antikythera mechanism realize equivalencein angle o the pin and slot mechanism to the epicycle theory? I so, howis it that no proo o the equivalence survives? Or was this mechanism considere

F ig . 3. Three models compared: pin-and-slot (wheels k1 and k2), epicycle-plus-concentric (epicycle and circle k1), and eccentric circle (circle k3).


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rough-and-ready approximation to epicycle motion good or giving the fnal outangle, but not necessarily considered exact? In a way, the striking contrast betwapplied mechanics and accepted celestial physics should not surprise us. Therewell-documented example o a similar contrast. Greek astronomers grounded inphilosophical-geometrical tradition ( or example, Theon o Smyrna, early secocenturya .d .) wrote treatises on de erent and epicycle theory while their contempraries were busy mastering and adapting the non-geometrical planetary theorythe Babylonians.13 The philosophically-based astronomy o the high road explicitendorsed uni orm circular motion as the only motion proper to celestial bodies wthe numerically-minded astronomers (who needed quick and reasonably reliaresults or the purposes o astrology) made ree and easy with nonuni ormity

motion. In a similar way, it is possible that mechanical tricks o the trade such aspin-and-slot mechanism were used in a cra t tradition o model-building, quite arom the practices o the serious (i.e., geometrically-minded) astronomers. Inmedieval tradition o equatoria, one also fnds representations o planetary theothat are not realistic.14 On the other hand, Figures 2 and 3 show that a proo othe equivalence in angle would have been well within the reaches o contemporGreek geometry.

Choice of Active and Passive ElementsAn interesting eature o the pin-and-slot geometry is that the equation o cenunction changes its orm i the roles o the pin and slot are reversed. See Figur(le t), which shows the pin and slot as ound on the mechanism. k1 turns about C1 withthe uni orm, mean motion. It is easy to show that the equationq o centre is given by

wheree = b / r . Hereb = C1C2 is the distance between the axes andr = C1D is the fxeddistance o the pin rom the centre C1 o wheel k1. This is the ordinary expressionor the equation o centre in standard Ptolemaic solar theory.

But i (right side o Figure 4) the pin is instead placed in k2, and the slot is into k1 and directed toward C1, then we fnd instead

sinq = e sin,

where againe = b / r , but nowr is the fxed distance o the pin rom C2. The morecomplex expression has become a simple sinusoidal oscillation.15So we are again le tto wonder: did the mechanic have a deep understanding o the equivalence theoror is it just chance that the pin is placed in k1 rather than in k2? Perhaps, as k1 drk2, he would have thought o the pin as the active element and there ore chosenplace it on k1, the active wheel.


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Transfer of the Lunar Pin-and-Slot Mechanism to the Case of the Outer Planets

In their discussion o the lunar pin-and-slot mechanism, Freethet al . say:As discussed above, in absolute terms, e5 rotates at the rate o siand e3 rotatesat the rate o n. So, relative to e3, e5 rotates at the rate o si n = a. Sincee5 and k1 both have 50 teeth, relative to e3, k1 rotates at the rate o a. This isthe critical actor that ensures that the anomaly introduced by the pin-and-smechanism has the period o the anomalistic month, as required by Hipparchlunar theory.16

Now, we want to trans er the basic idea o the lunar inequality to the planets. Tlunar inequality is an inequality with respect to the zodiac or, i we want tomore exact and take into account the act that the line o apsides slowly advancan inequality with respect to wheel e3. This is why the wheels that produce inequality (k1 and k2) must ride on wheel e3. But retrograde motion is an inequawith respect to the Sun.17 So, in order to represent an epicycle and de erent modeo the planets we need:

(a) a gear rotating with the sidereal period, i.e., the period o the mean longitudethe Moon model, e5).(b) a big carrier gear rotating at n = si a, i.e., at the di erence between therates o mean longitude and o the epicyclic anomaly. In the case o the Moon, tepicyclic anomaly increases in the opposite direction to the sidereal motion, sothe case o a planet, where both the epicycle and the de erent rotate in the same dition, the big gear should rotate at si + a. (It is not clear that the outset whether

Fig . 4. The e ect o reversing the roles o the pin and slot.


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the di erence in the directions o motion on the epicycle will cause problems or nWe shall see that all will be well.)(c) the set o other gears, with the correct eccentricity, or the pin and slot.

Now, or an outer planet, the sum o the planets sidereal requency and its anomlistic requency is equal to the requency o the Suns mean motion: si + a = .(This amiliar result ollows rom the construction in Figure 5.18) This means that,or all the outer planets, the big gear (corresponding to e3 in the lunar mechanishould rotate at the same period, and the period is that o the mean motion o Sun. Fortunately, we do have a big gear rotating at this period, the solar gear, And, ortunately also, this gear is rotating about the correct axis, i.e., the axis opointers. There ore, we need only place on each o three spokes o b1 the pin-aslot mechanism o a superior planet. This coincidence seems to increase signifcanthe plausibility o this proposal or, besides the theoretical reason (to have the sastyle o mechanism or the planets and the Moon), we have an empirical reason, we actually have a big gear rotating at the rate we need.

So, or each outer planet, we only need to have a gear, concentric with b1, rotawith the sidereal requency si or this planet. Then, the addition o a pin-and-slotmechanism rotating at requency (so, riding on b1), with the correct eccentricity

would produce the apparent motion in angle o the planet.Pin-and-Slot Compared to Apollonioss Planetary Epicycle

In the preceding section we gave a plausibility argument that the pin-and-slot menism can e ectively mimic planetary motion. We shall show now that a pin-and-mechanism will account in detail or the motion in angle o the planet. Considermotion in longitude o a superior planet according to standard epicycle theory, as shin Figure 5. C

1is the Earth.

Imagine that at one moment the centre D o the epicycl

and the planet P both lay on the vertical straight line in the direction o astest motBut at the moment represented in the fgure, a certain amount o timet has elapsed,and the planets mean longitude has increased by = si t . The epicyclic anomaly hasincreased by = at . The longitude o the Sun has increased by= t . Because o the standard constraint linking the motion o the Sun and the superior planet we h

= + , and = si + a. At this moment, the direction rom the Earth to theplanet is in the line C1P. The planets angular distance rom the direction o astes

motion is + q, whereq is the equation o the epicycle. increases steadily withtime, whileq undergoes an oscillation.Now let us consider matters as seen in the rame o re erence o the revolvi

solar gear b1. Here the situation is shown in Figure 6. The line rom C1 to the Sunpoints in a fxed direction (vertical in the fgure). Thus, we have rotated Figurecounterclockwise by the angle (compared with Figure 5). The angular positiono the planet with respect to the vertical re erence line (i.e., the elongation o planet rom the Sun) is


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= + q .Note that increases to the le t as time goes by. That is, in the rame o gear b1,

moves counterclockwise. This is simply because si < . Note that Figure 6 is asort o mirror image o Figure 2. In Figure 2, we had D moving clockwise aroundde erent, while M went counterclockwise on the epicycle. Here, with the planettheory, as viewed in the rame o re erence o b1, we have just the opposite: D mocounterclockwise, while P moves clockwise on the epicycle. Thus, in spite o the the planets move in the opposite direction on their epicycles than does the Moonconditions necessary or the theorem do apply.

Because the direction DP o the planet in the epicycle remains parallel to the dition rom the Earth to the Sun, we have DP always parallel to C2C1. I we choose acircle k2, with centre C2 located below C1 by a distance equal to DP, it ollows thatthe quadrilateral C1PDC2 is a changing parallelogram; thus C2D is always parallel toC1P. Now suppose that D represents a pin in a wheel k1, driving wheel k2 by meo a radial slot. (In Figure 6, the slot is suggested by the heavy dashed line, and should imagine k2 as larger.) Then a fxed point Z on k2 in the direction o C2D willmove so that C2Z is always parallel to the direction C1P. Again, this is an equivalencein angle only. To get back into the rame o the fxed stars, we just rotate Figure

F ig . 5. Epicycle theory or the motion o a superior planet. The rotating line rom the epicycles centrto the planet P must stay parallel to the rotating line rom the Earth C1 to the Sun. So + = .


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clockwise by angle ; but the parallelness o C2D and C1P will not be disturbed.This works because o a remarkable circumstance. In the case o the lunar theothe theorem will only hold i the Moon moves backwards on the epicycle. (A thewith both D and M moving clockwise, even i M completes its motion on the epicin one anomalistic period, will not satis y the conditions o Figure 2. In this caspin and slot might still give a reasonable approximation to the epicycle theory, it would not be an exact angular match.)

In the case o the planets, the direction o P on the epicycleis clockwise in thesame direction as D. So it might be thought that the theorem could not possibly apBut the situation is saved by the act that the angular requency o the Sun is greathan that o the superior planets. So, when we rotate Figure 5 counterclockwiseget into the rame o re erence o the solar wheel, we end up with Figure 6 planet moving clockwise on the epicycle, but the epicycle moving counterclockwon the de erent. The opposite directions rescue the theorem.

A date or the design o the Antikythera mechanism can be placed between

late third centuryb . c

. and the early frst centuryb . c

. It seems a very open question

Fig . 6. The epicyclic theory o a superior planet (viewed in the rame o re erence o the Sun wheesuperimposed on a pin-and-slot mechanism.


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whether the mechanic understood the deep theoretical complexities o the pin-aslot mechanism; i he did, we might reasonably suspect that the mechanism comrom rather later in the possible range, a ter a thorough understanding o planettheories had been arrived at. But i the mechanic used the pin and slot as a sorad-hoc way o imposing an oscillation on a steady orward motion, he need not hbeen aware o the existence o an equivalence proo . In this case, he was perhalucky though his mechanical instincts were still remarkable.

The Antikythera mechanism contributes signifcant new evidence or the debover realism and instrumentalism in ancient planetary theory. To put it crudely, mechanic was a happy instrumentalist. The lunar pin-and-slot mechanism represthe motion o the Moon satis actorily in angle, but renounces a description o

motion in space. The planetary pin-and-slot mechanism (i it existed!) represeanother such goal-oriented bit o modelling. And o course, the use o a non-uni ormdivided zodiac to model the solar anomaly fts right into this pattern. There is an irhere, in that Ptolemy, when he gives directions or obtaining the longitude o a plarom his theories, essentially gives a procedure or eliminating any considerationthe depth in space. And he was hardly an instrumentalist. But the fnal output orealistic theory may look like the output o an instrumentalists black box.

Obtaining the Mean Sidereal Motion of an Outer Planet We have a working pin and slot that e ectively model the motion in angle in Aplonian planetary theory. The next requirement or a display o the zodiacal motioa superior planet is a gear turning about the b axis at the rate o the planets siderequency si. One simple way to achieve the sidereal rotation is to use a stationagear, concentric with the b axis, fxed to a plate behind the dial plate. Re er to Fig7A. Gear u withu teeth is astened to the back o the plate. Gears x and y are carrie

on an axle that rides around on b1 in a period equal to the year. x and y are rigfxed to one another (as suggested by the dark line with diamonds that connects thin the fgure). Finally, y engages with a gear z that is concentric with the b axis, is ree to turn independently. The requency o gear b1 with respect to fxed space = 1 revolution/year. b1 rotates clockwise as viewed rom above the plate in diagram, and is considered positive. Presently, we shall work out the requenc z o gear z with respect to absolute space and show that it can be made equal to si.

An alternative way to achieve the same result, shown in Figure 7B, is to makeo the stationary squared-o boss, visible in the centre o Figure 8. It is possible tthis boss was fxed with respect to the plate behind b1, and its squared-o portimay have served as a gear seat.19 Figures 7A and 7B are entirely equivalent in termso generating the output requency z. In each case we are working with a stationarygear u and the rotating solar gear b1 that turns with a period o a year. The arranment o 7B inverts the vertical space order o gears x and y.

Let us now work out the rates o rotation. Re er to Figure 9. (Figure 9 and argument that ollows apply equally well to the models o Figure 7A and Figure


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it does not matter whether u and x are above or below y and z.) The top part o Fig9 shows the positions o wheels u and x be ore (le t) and a ter (right) a rotation o main Sun wheel b1 has taken place. Be ore the motion begins, point p on wheeloriented as shown. The main solar wheel, carrying the axle o x, rotates clockwthrough; the situation is now as in the upper right. Point p has been carried to a nposition. The numbern o teeth that have been engaged on u is given by

n = u( /360).The same number o teeth must have been engaged on wheel x, so it ollows tha

= 360(n / x)= u/x. (1)

The angle in absolute space through which wheel x has rotated is + = (1 +u / x).

Now the lower part o Figure 9 examines the ates o wheels y and z. Be ore trotation (le t) the initial position o point q on wheel z is shown. A ter the rotat(right side o fgure), z will have rotated through an angle in absolute space, so qwill be in a new position. Depending on the tooth numbersu, x, y, z, it can be the casethat q has rotated either clockwise or counterclockwise. We want q to have moclockwise, corresponding to the sidereal motion o the planet, so in the diagramhave placed q below the horizontal. Because x and y are rigidly astened togeththe angles marked in the upper right and lower right are the same. The numberm

F ig . 7A. One way to get wheel z rotating at the mean sidereal requency o a superior planet. The examis or Mars, withu, x, y, z = 37, 79, 58 and 58 teeth.

F ig . 7B. A second way to get wheel z rotating at the mean sidereal requency o a superior planet, usa stationary boss.


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o teeth on wheel y that have been engaged is given bym = y /360

= ( /360)(uy / x).The total space rotation o wheel z is (assumed clockwise). The number o teethengaged on z is z( )/360, and this must be equal tom. So we obtain

= (1 uy / xz).Thus, since turns at the requency o the main solar gear, z = (1 uy / xz). (2)I the tooth numbers are chosen appropriately,

zcan be made equal to

si.

F ig . 8. Composite x-ray o the main solar gear b1. Copyright o the Antikythera Mechanism ReseaProject.


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Here is an alternative, and shorter, argument. Let u and z rotate independenabout a common axis, and let x and y rotate at the same rate as one another abtheir common axis. Suppose x and y rotate clockwise by an angle . Then x drivesu in counterclockwise rotation by an angle u / x = say. Similarly y drives z incounterclockwise rotation by an angle y / z = say. To get to a fnal confgura-tion where u has remained fxed, we rotate the entire assembly as a rigid body

an angle. This returns u to its initial orientation. The fnal orientation o z is + , which is = y /z +u / x. Since = x / u, we get = (1 uy / xz), as be ore.Note that z can be o either sign, depending on the tooth numbers. We want z

positive, i.e., with z revolving in the same direction as the Sun, and equal to si, thesidereal requency o the superior planet. So we obtain

uy/xz = 1 si / ,or, since or a superior planet si+ a = ,

uy/xz = a / . (3)An appropriate choice o gears can be determined rom the relation between the yand the anomalistic period o the planet, which can be drawn rom a Babylonperiod relation. It is easiest to work with the Babylonian goal-year periods (rathan the ACT periods), as they involve generally small integers.20 For example, oneperiod relation attested or Mars is

79 years = 42 sidereal periods = 37 anomalistic (or synodic) periods [Mars]

Fig . 9. Obtaining the mean sidereal motion o an outer planet rom the mean motion o the Sun.


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So, to have gear z turn at the sidereal requency o Mars, we need (u / x) ( y / z) = 37/79.A gear scheme that will work nicely is shown in Figure 7A or 7B. For here we h

(u / x)( y / z) = (37/79) (58/58) = 37/79,

as required, while we also haveu + x = y + z = 116, which means the gears will meshas required. Here, or simplicity, we have taken all the gears in the our-gear traibe o the same module. But wheels y and z could be o any convenient tooth numor o any convenient module, as long as they have the same tooth count and modas one another. As we shall see, we will need this exibility later on.

Similarly, using the Babylonian goal-year periods or Jupiter and Saturn,71 years = 6 sidereal periods = 65 anomalistic periods [Jupiter]

59 years = 2 sidereal periods = 57 anomalistic periods [Saturn],we can construct the solutions

or Jupiter: u, x, y, z = 65, 71, 68, 68, so (u / x)( y / z) = 65/71,whereu + x = 136 and y + z = 136;

and or Saturn:u, x, y, z = 57, 59, 58, 58, so (u / x)( y / z) = 57/59,whereu + x = 116 and y + z = 116.

We now have all the elements we need to construct a workable solution or the supeplanets: the big carrier gear b1, turning at the rate o the Sun; a demonstration tthe point-and-slot mechanism can mimic the Apollonian theory in angle; and thgears turning at the sidereal rates o Mars, Jupiter, and Saturn.

Putting It All Together

Figure 10 shows an obvious concatenation o the results above: we append the pand-slot construction to the output o the mean sidereal motion o Mars. Gears uy, and z result in z rotating with respect to absolute space at si, as shown above. (Theblack diamonds on the axis remind us that gears x and y are used together.) Thgears z, a, b, and c, with the pin-and-slot mechanism linking a and b, produce modulated motion, with synodic oscillation superimposed. The Mars axis, conneto c, has the correct mean motion around the zodiac, and an anomalistic motion ocorrect period. A Mars pointer could be connected to this axis. It is noteworthy all the gear axes are either concentric with the main solar axis, or else ride aroon the main solar gear. The size o the retrogradations may be adjusted by makuse o the parameters o the pin and slot: the key ratio is the eccentricity o whb to the distance o the pin rom the centre o wheel a. This solution requires sevgears. Similar solutions can be ormed or Jupiter and Saturn.

Glancing at the reconstruction in Figure 10, one might be struck by its wasteness: look at all those 58-tooth wheels. And note how the our-gear train madethe mechanism or the sidereal motion seems to be echoed by the our-gear train

the anomaly. Wouldnt it be possible to do the whole thing (mean sidereal mot


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plus anomaly), using only our gears? In Figure 10, we see that y is driving z, that a is concentric with y and is driven by z. Thus y and a rotate at the same (they are both meshed with the same gear and are on the same axis). So y and abe used into one, as in Figure 11. But, now we can take another step and eliminz in Figure 11, or y/a is moved directly by x, and z is there ore useless. But ththe tooth count o y/a becomes irrelevant, and y/a may be considered simply to bpart o x. Thus we arrive at Figure 12A, where the z o Figure 12A is the c o Fig11, and the y o Figure 12A is the b o Figure 11. (An alternative, but equivalesolution is shown in Figure 12B, in which the stationary gear is mounted to the bbelow b1 instead o to the plate above it.)

We o er now an analytical demonstration, showing that the our-gear solutionsFigure 12A or Figure 12B indeed have all the right properties the right sideperiod and the right anomalistic period. Equation 2 requires

si = (1 uy / xz).Thus the sidereal period will be fne as long as we do not disturb the ratio (uy)/( xz).Now, Equation 3 requires

a = (uy / xz).But, in the right sides o Figure 9, we need to be equal to , the increase in epicyclicanomaly. This is because indicates how ar point p has moved around the anomalistic cycle, carrying the pin that is a part o the pin-and-slot mechanism. Imposthis condition on Equation 1 requires

a = (u / x).Comparing the last two equations, we see that we must have y = z, which is in actthe way we have drawn Figure 12A (and 12B). With this choice, the our-gear sotion o Figure 12A (or 12B) builds in not only the correct sidereal motion, but a pin-and-slot device producing the right anomalistic motion.

Fig . 10. Complete solution or Mars, embodying the correct sidereal period and the correct anomaliperiod. Seven gears in all are used.


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A Proposal

In Figure 8, the holes at A and B and the lug at C are placed more or less at eqdistances rom the centre o the wheel. The x-ray shown is a composite: one x-slice shows most o the sur ace o the wheel, including holes A and B; but a porto a second x-ray slice taken several millimetres higher has been added to showlug. This lug stands about 6 mm high. We imagine that apparatus was dropped oit and secured to it. Perhaps another lug was originally astened at A.21

F ig . 11. A step on the way to a simpler solution.

F ig . 12A. Four-gear solution equivalent to Fig. 10. Stationary gear mounted to a plate nearer the ro mechanism.

Fig . 12B. Alternative our-gear solution, with stationary gear mounted on a boss fxed to the plate behind


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The construction described above, using pin-and-slot mechanisms or the ouplanets, lends itsel well to this geometry. See Figure 13. The solutions or eachthe superior planets may be mounted on b1, taking advantage o what appear tomounting holes on two o the spokes, and the lug on another spoke, using gear mules (millimetres o gear diameter per tooth) that are all in the vicinity o the gmodules typical o the Antikythera mechanism (on the order o 0.5 mm per tooThe lug perhaps served to raise the gears vertically. Indeed, it would be necessor the y and z gears o the three planets to be placed at di erent heights. In Fig13, the nomenclature xj 71 means gear x or Jupiter with 71 teeth. The gearx, y and z or each planet are named as in Figure 12A. For each planet, a stepstud is astened to wheel b1. The wheel carrying the pin rotates about centre C1 and

the wheel with the slot rotates about centre C2. The centre o the larger-diameterpart o the stud is marked + and the smaller-diameter part o the stud is indicated.The simplifcation that we can simultaneously requireu + x = y + z and maintain

a constant gear module throughout the our-gear train, which was fne or the cao a zero-eccentricity diagram such as Figure 6, cannot be maintained once ecctricity is introduced. The construction o Figure 13 is based on always puttinggear modules near to 0.5 mm o diameter per tooth (where the diameter is the pdiameter), in the range typical o the Antikythera mechanism. But the module

a given planets y and z is not always exactly the same as the module or u and Mars provides a minor extra challenge because o the large eccentricity C1C2(corresponding to a large epicycle in the Apollonian theory). The slot in wheeldirected toward C2; but becauseb / r > 0.5, the slot would need to be so long that itwould overlap the part o the stud that serves as the C2 axis, i we use the arrange-ment shown in Figure 12A. There are any number o ways to deal with this. An esolution is simply to reverse the direction o the stepped stud shown in Figure 1so that the small end o the stud goes into b1. A metal strap would go over the bo the gears and securely hold the wide end o the stud above wheel x. (It shobe noted that above and below are terms that apply only to the diagrams: innormal operation o the Antikythera mechanism, all the axles shown in Figure 1would be horizontal.) A strap might be in any case a requirement or stabilitythe gear trains; and such a strap was a part o the extant lunar anomaly mechaniThis is the arrangement on which Figure 13 is based. An alternative choice wobe a di erent arrangement o the gears, so that wheel x could be closer to b1 this wheel y; this would make the C

2part o the stud narrower than the C

1part o the

stud (as in Figure 12B).None o the details o Figure 13 should be given much weight there is sim

too little material evidence and there would be many di erent ways o mounting gears. We intend the fgure only as an illustration o the principle that a pin-and-solution or the superior planets is workable.


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Calibration Issues for the Superior Planets

In Apollonian planetary theory (Figure 5), let Re denote the radius o the epicycleand Rd the radius o the de erent circle. In the corresponding pin-and-slot represe

tation (Figure 6), letr (= C1D) represent the distance o the pin rom C1; and letb (= C1C2) represent the o -centredness o circle k2 with respect to circle k1.Ththe pin-and-slot mechanism will give behaviour corresponding to the de erent-aepicycle theory i

b / r = Re / Rd.One might worry that the act that gears y and z in Figure 12A have di erent numb

Fig. 13. A possible kinematic solution or the superior planets, based on the pin-and-slot constructsuperimposed on the main solar wheel b1. Underlying x-ray copyright o the Antikythera Mecnism Research Project.


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o teeth than gear x might require some sort o adjustment to this relationship. no adjustment is required. In Figure 12A, the angular oscillation (with respecthe mean motion) o gear y is determined by the ratiob / r , and the tooth count o ydoesnt enter into it. But then the angular oscillation o z will be the same as thay, since z and y do have the same number o teeth.

In Figure 13, we have or the sake o simplicity shown the C1-C2 lines or Jupiterand Saturn as parallel to one another. This might acilitate setting initial conditibut it is not a necessary mechanical arrangement, which we emphasize by setthe C1-C2 direction or Mars at right angles to those or the other planets. The C1-C2 directions can be chosen at will, and separately or each planet. Suppose that actual planet is in the middle o retrograde motion. Then we want to set the gears

its pin-and-slot mechanism so that the pin is closest to C2. This guarantees that thephasing o the pin-and-slot mechanism matches the retrograde cycle o the actplanet. Then, the pointer that indicates the planets longitude should be set so thpoints diametrically opposite the Sun.

Inferior Planets

The ourth spoke is ree or the in erior planets, which can be accommodated imore ordinary way, using a simple epicycle construction, as shown in Figure 1construction resembling Figure 14 was described by Edmunds and Morgan.22 AndWright used a similar construction or the inner planets, with, however, compogear trains rather than simple gear pairs.23As the sidereal requency o both Mercuryand Venus is exactly , both planets apparatus may be mounted directly on thourth spoke o b1. I the fxed wheel hasu teeth and the epicycle wheel has x, thenthe anomalistic requency is given by a = (u / x). I Re and Rd represent the radiio the epicycle and o the de erent in standard theory, we should then require D

CD = Re / Rd.Trial by Fire

We actually built a Mars model in metal, based on the construction o Figure 1and Figure 13, using the pin-and-slot mechanism to model the inequality. We confrm that it works smoothly, even or a planet with such a large Apollonian epicyOne point that modellers should be aware o is that the roles o the pin and the

cannot be reversed. For Mars, having the slot lead the pin (as in Figure 4, right) wresult in a model in which retrograde motion does not actually occur. This results the two di erent mathematical orms or the equationq in these two cases, combinedwith the act that or Mars si is slightly larger than a. For a planet such as Saturn,or which a >> si and the epicycle is small, one could reverse the roles o the pand slot without dire consequences; the pin and slot would no longer per ectly mithe epicycle model in angle, but the results would still be tolerably good.

With the correct version o the model (i.e., with the pin leading the slot),


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pin-and-slot mechanism matches the Apollonian theory in angle. But Apollonplanetary theory itsel gives but a poor approximation to the actual motion o Mar24 The combination o large epicycle and a large eccentricity required something mcomplex than the simple theory o Apollonios or Greek planetary theory to yielsatis actory representation o the motion o Mars.

Concluding Remarks

Unless new material is discovered, it is conceivable that we will never know or just how the planets were represented on the Antikythera mechanism. But the soluor the superior planets described here has great attraction. It uses or these planinequality with respect to the Sun precisely the mechanism that we know was usedproducing the lunar inequality with respect to the zodiac. And, as we have shown,solution is an exact representationin angle o Apollonian epicycle theory. Moreover,this solution requires but our gears or each o the superior planets, it explains wgear b1 needed to be large, and it can be ftted well to the construction holes tremain on b1. O course, one could object that a complete model would require mconcentric axles and so ones receptiveness to the proposal may in part depenwhat sort o mechanical engineering one considers likely around the second centb . c .25 The chie disadvantage o the present proposal is that it o ers no explanatio the tooth count o b1. It also does not explain the need or the large crossed-o

Fig . 14. A possible epicyclic construction or an inner planet.


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regions o b1; perhaps these were only access ports. Or perhaps they were connewith a display o the our seasons, as we have suggested elsewhere.26 In any case, wehope that this paper has o ered some insights into the richness o the pin-and-smechanism and that it will serve to ocus new attention on the issues raised byintersection o ancient Greek theoretical astronomy and ancient Greek mechani

ACKNOWLEDGEMENT

We are grate ul to Mike Edmunds and Tony Freeth o the Antikythera MechaniResearch Project or their generosity in making data available to us.

REFERENCES

1. T. Freeth, Y. Bitsakis, X. Moussas, J. H. Seiradakis, A. Tselikas, H. Mangou, M. Za eiropolouHadland, D. Bate, A. Ramsey, M. Allen, A. Crawley, P. Hockley, T. Malzbender, D. Gelb, WAmbrisco, and M. G. Edmunds, Decoding the ancient Greek astronomical calculator knowthe Antikythera mechanism, Nature , cdxliv (2006), 58791. There is substantial Supplementaryin ormation linked to the online version o the paper atwww.nature.com/nature.

2. Derek de Solla Price drew attention to this slot, but conjectured that it might be the result oattempt to repair a broken gear.Gears from the Greeks: The Antikythera mechanism A calendar computer from ca. 80 B.C . (Transactions of the American Philosophical Society , n.s., lxiv/7(1974)), 35. M. T. Wright described the pin-and-slot device and its workings and observed thwould suitable or modelling an anomaly, in Epicyclic gearing and the Antikythera mechanipart 2, Antiquarian horology , xxix/1 (September 2005), 5163.

3. Freethet al ., Decoding (re . 1), Supplementary in ormation, 257.4. J. Evans, C. C. Carman and A. S. Thorndike, Solar anomaly and planetary displays in the Antiky

mechanism, Journal for the history of astronomy , xli (2010), 139.The evidence is very simpleand comes rom matching degree marks on the zodiac scale with day marks on the Egyptcalendar scale. Over a preserved stretch o 69 o the calendar scale, an apparent equation o cenrises steadily to a 2.1 e ect, o the right sign and amplitude to represent the Suns inequaliWe shall shortly publish a second study o the zodiac and Egyptian calendar scales, examin

this issue in greater detail.5. Price,Gears from the Greeks (re . 2), 21. M. T. Wright, A planetarium display or the Antikytheramechanism, Horological journal , cxliv (2002), 16973 and 193; and The Antikytheramechanism: A new gearing scheme, Bulletin of the Scienti c Instrument Society , lxxxv, issueo June 2005, 27. Freethet al ., Decoding (re . 1), 590.

6. Freethet al ., Decoding (re . 1), Supplementary in ormation, 89, with translation on pp. 10147. Wright, A planetarium display (re . 5). A flm o Wrights reconstruction in motion can be seen

www.youtube.com/watch?v=Zr MFhrgOFcThis flm was made in 2008 by the science writerJo Marchant, who was then working or New scientist .

8. Evans, Carman and Thorndike, Solar anomaly (re . 4).9. Theon o Smyrna, Mathematical knowledge useful for reading Plato , iii, 26; J. Dupuis, transl.,Thonde Smyrne, philosophe platonicien: Exposition des connaissances mathmatiques utiles pour lalecture de Platon (Paris, 1892), 269.

10. We ollow the gear names used by Freethet al . in Decoding (re . 1), which in some cases representdepartures rom Prices nomenclature inGears from the Greeks (re . 2).

11. The output is by means o a central sha t attached to e6; and this sha t runs inside the hollow pto which e5 is attached.

12. Ptolemys proo o the equivalence or the Sun is in Almagest iii, 3, and or the Moon at iv, 5. G. J.Toomer,Ptolemys Almagest (London, 1985), 14153 and 18090. Ptolemys remarks about
http://www.nature.com/naturehttp://www.ingentaconnect.com/content/external-references?article=0021-8286(2010)41L.1[aid=9568283]http://www.ingentaconnect.com/content/external-references?article=0021-8286(2010)41L.1[aid=9568283]http://www.youtube.com/watch?v=ZrfMFhrgOFchttp://www.youtube.com/watch?v=ZrfMFhrgOFchttp://www.nature.com/naturehttp://www.ingentaconnect.com/content/external-references?article=0021-8286(2010)41L.1[aid=9568283]


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Apollonios are in Almagest xii, 1 (Toomer, p. 555). The classic argument that Apolloniosunderstood the equivalence o epicycles and eccentrics is Otto Neugebauer, The equivaleno eccentric and epicycle motion according to Apollonios,Scripta mathematica , xxiv (1959), 521,recapitulated in A history of ancient mathematical astronomy (Berlin and New York, 1975),14950, 26770. But scholars have sometimes questioned the correctness o attributing thunderstanding to Apollonios. See, most recently, Bernard R. Goldstein, Apollonios o Pergcontributions to astronomy reconsidered,Physis , xlvi (2009), 114. For Theon o Smyrnasproo s see Dupuis,Thon de Smyrne (re . 9), 2719.

13. Alexander Jones, Astronomical papyri from Oxyrhynchus ( Memoirs of the American PhilosophicalSociety , ccxxxiii; Philadelphia, 1999).

14. See, e.g., B. R. Goldstein,Descriptions o astronomical instruments in Hebrew, in D. A. KingG. Saliba (eds), Essays in honor of E. S. Kennedy ( Annals of the New York Academy of Sciences ,d (1987)), 10541, pp. 106 .

15. The expressions or sinq are the same pair that we get i , in the ordinary eccentric-circle theory, wexpressq frst as a unction o the mean anomaly and then as a unction o the true anomaly. SeJames Evans,The history and practice of ancient astronomy (New York, 1998), 2334.

16. Freethet al ., Decoding (re . 1), Supplementary in ormation, 24. We have modifed this quotatioby multiplying all the requencies by a minus sign, in order to make the discussion more eascomparable with our diagrams.

17. This is a distinction in language used by Ptolemy himsel (that one sort o anomaly is related toSun, and is one related to position in the zodiac), e.g., in Almagest ix, 2; Toomer,Ptolemys

Almagest (re . 12), 420.18. See, or example, Evans, History and practice ( re . 15), 338.

19. We thank Michael Wright or this suggestion (private communication), as well as or several ocomments that helped improve the paper.20. Listed in Evans, History and practice (re . 15), 314. The term goal-year texts was introduced by

Abraham Sachs, A classifcation o the Babylonian astronomical tablets o the Seleucid perio Journal of cuneiform studies , ii (1970), 27190.

21. Price,Gears from the Greeks (re . 2), 28, speculated that there might have been other lugs astened to b122. Mike Edmunds and Philip Morgan, The Antikythera mechanism: Still a mystery o Gre

astronomy?, Astronomy & geophysics , xli, issue o December 2000, 6.106.17.23. Wright, A planetarium display (re . 5).

24. See Evans,History and practice ( re . 15), 341.25. In arguing that multiple concentric pipes would not be a problem, Wright makes an analogysurviving ragments o the musical wind instrument called theaulos , in which rotatable sleeveso bronze tube are o ten seen ftted over a structural tube. M. T. Wright, In the steps o the mastmechanic, in Ancient Greece and the modern world (Patras, 2003), 8697.

26. Evans, Carman and Thorndike, Solar anomaly (re . 4), 323.
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on the pin-and-slot device of the antikythera

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