notes on the antikythera mechanism

8/2/2019 Notes on the Antikythera Mechanism

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4 Sylvi(l COUdWHd

Dans ce cas, la sornme de routes lcs valeurs est en effet 220 com me lescribe du papyrus BM ]()520 l'annoncc.On sait que la Iormule corrcctc pour calculer la double somme

dune progressi on estk ~ ~ i= C ' : 2 ) C ' ; I I )cc qui corre spond cxac tc rncn t ,j cc que Ie copiste dernotiquc a rnis enpra tique dans sa f ormule . Ccci parait d'autant plus probable que cettefonnule est facile il prouver par une deduction de n. it Il + 1Si lon ucccpte cetre interpretation, du point de vue des mathe-matiques. la situation est des lars completernent changce. Nous pos-

scdcrions iei UI1 cas tou t ~I fait extraordinaire pour l 'Egypte ant ique.Car. si la sornme de la progression anthmenque au dcuxierne degrecrait que lquc chose de bien connu chez les Babylonicns et les Grecs, iln'cn va pas du tout de mernc pour la double somme d 'une progress ionarithmeriquc.Aucun pcuple de l'Antiquirc n'a jusquici laisse trace de la conn a-

.ssance cl-une Iorrnul e de ce typc.F':" Seule l'Egyptc anti que, it traversI'CIlOllCC du prohleme numcro 53 du papyru5 dcmotique 81\-1 10520 et1" solution qui peut lui etre apportcc , portcrai t ternoignage de ceuecormaissance rres avancec puisqu'il nous faut attendrc cn effet lestemps modernes, avec Leibniz et Ie calcul i nfinitesimal, pour retro-uvcr un raisonncrncnt sernblable en mati ere darithmctique.

REFERENCESI. R. A. Parker Ocmottc Mothenunicnt Papyrus ( Brown E gy pt o lo gic al S tu di es . V ii . B rownU ni vc rs i tv P re ss . P ro vi de nc e. 1 97 2) p . (l4.

.., D. Neugebauer MmhcII/alisc/rc Kciischrift icxrc I {Qucllcn und Srudicn zur Ccsclsictnc der, 'H(If!Jc1IwNk.. Abt . i\; BcI ,'., Berlin jI)J5) P: 'N.

3 . N ic om acb c l it : Gcms a lntroductian {II'ArirhmeliqHl' New Yor -e 1926.4. 0, Xcugcbaucr . A. SUL'hs. Mothcmoticul cnncifornrs fCXJ (Amc r. Or ien ta l s er ie s 2 /1 iibrr rii(' (i()_Idlich/(~ dvr antikrn malU l1Iofisc/wtl iVi.\Sf/J .vdlljien.

Lrstcr Band: Vorgriechicltc :\'l({{lIl'IIUllik ( Spri nger Ve rl ag. Ber li n, 196Y) , p . 171 sq.(1. 13. I.. Van dcr wnc rc cn . r.rwachrndc Wis_w~I'.\ch(lJI.Kypliscl!e, /wbyhmi.l"(/lc 1(11(/griec/Jil'C/H!

.14at!w1I!(I.'ik (Birkhtiuser Vcrtag. Ha.;.::1.19(0). r- 12~~l

Notes on the Antikythera Mechanism~by

ALLAN G. BROMLEY

Inhis lengthy pape r "Gear s from the Greeks: the Antikythera Mecha-n ism ~ A Calender Computer from ca. 80 BC" Derek De Solla Pricehas given a detai led description of this earliest known mathemat icalgearwork. Price demonstrates that the Greeks wen: capable of fab-ricati ng gearwork of a complexity not previously known before thegreat astronomical clocks of the middl e ages. Further, the gcarwork isnot just impressive for its quantity but also for thc remarkable sophis-tication of the d iffe rent ia l mechanism i t incorpora te s.An important fruit of Price's many years of careful scholarship is a

proposed reconstruction of the Antik ythera Mechanism in h is F igure33 ( shown in Figure 1 of this paper) and described at length elsewherein his 1J


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6 AI/cUI G. Brorulry

I'll--,1

~II~I ' : ; :I 11

w 12 1w 1V>"" I

I

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moon's nodes and hence with eclipse cycles such as the Saros. An ap-proximation to t~e Saros, employing thc movable dial rings in thesame manner as the front calendar dial, fits the available evidencewell.Final ly, I di sagree wi th Price' s opinion that gear wheels with certainnumbers of teeth would be easier to divide than others and that thegear trains were arranged accordingly. The dividing and cutting ofsuch gear teeth are discussed in a forthcoming paper by M. T. Wright.I sec no inherent difficulties in making gears with any arbitrary num-ber of teeth and hence suppose no restraints on ancient designers fromthis source.

I am attracted to the idea that dividing plates, possibly of timher,might have been divided first and then used to divide ur cut any num-ber of gears with the same tooth count, or related multiples and sub-multiples, irrespective of the modulus. These dividing plates wouldhave formed part of the "capital" of a gear making workshop. I Theiruse might explain the recurrence of certain gear teeth numbers andmultiples in the Antikythera mechanism. Other methods of dividinggears are certainly possible, such as the use of a protractor by Richardof Wallingford, but all lead me to disagree with Price's opinion thatthe dividing would have influenced the design of the gear trains.

I have not had an opportunity to examine the fragments of the Anti-kythera Mechanism at firs t hand. My recons tructions mus t, therefore ,be cons idered as tentat ive only pending such examinat ion by myself orother scholars.

Price')' ReconstructionThe logical structure of Price', reconstruction is shown in Figure 2.The input is the contrate gear A which drives the main drive wheelB1, gears B2 and 83, and a sun position dial through a ]15g ear reduc-tion. From this a moon position dial (sidereal month) is driven by agear ratio of 254/19 (or a step-up of about 13.4:1) through the gearsB2/Cl+C2/Dl+D2/84.' This ratio is based on the Metonic cycle.The sun and moon positions are geared to the inputs Eland E2 ofthe differential. The output, the differential turntable E3, is thusdriven at a rate E3='i>[E2-El] and hence rotates once in two lunar


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Allan G, !Ju;m!cy

Figlll'i.' 2.

synodical months. The dif ferent ial turntable. thus rotates ;H about 6.2times the speed of wheel BJ. A gear train E3/Fl+F2/G2 with a ratioof 211 drives a lunar synodical dial at a rate 12.4 times the speed of BI.The intermediate gears Fl and F2 rotate at a rate 24.7 times the speedof the main drive wheel. The main wheel of the differential turntableE4, is unused in Price's reconstruction. 'A simple gear train GIIHl +H2!I with a ratio of 1211 drives an in-dicator for a lunar year of twelve synodical months.In Price's reconstruction the main drive wheel drives a gear train

R2/1.1 +L2/M I +M2/N with a ratio of 4/1 of which the last step to thegear l\ is uncertain but may be intended to drive a "4 year dial". Thecontinuat ion of this train N / O I +02 is even more uncertain and lacks ac le ar as tronomical funct ion.

Criticisms of Price's ReconstructionFrom a mechanical point of view a great weakness of Price's recon-struction is the substantial step-up ratios required by his gear trains.'The step-up is 6.2: 1 to the large (lifferential turntable with its load ofaddit ional gears J. Kl and K2 and the friction of its peripheral support

The /\:rFik.yrlu:ra M~dJ(/lIi:;1I/

ring and brackets. The step-up is 24.7:1 to the gears Fl and F2 in thesynodic month ;,train which arc, however, only lightly loaded by fur-t i1Cr reduction gearing.To tes t t he fea sibi li ty of this reconstruction 1 have built an approxi-mate model of the main gear train using Meccano components, tire

size of whose gear wheels and shafts arc similar to those of the Anti-kvthera mechanism.-If mounted with the dials vertical it was found essential to coun-terbalance the wheels J, KL K2 and their supports carried with thedifferential turntable. This would involve a counterweight of equalmass to these parts fixed to the turntable in roughly the present posi-tion of the axis I .. This supposition is perfectly feasible in the light ofPrice's descripuon of the remaining fragments. The corrosion of thecDunterwI,; ighL or a stress produced by its initial fixing might have con-tributed to the breaking away of the l ef t hal f of t he d if feren ti al t urn-table.With the addition of the counterweight the mechanism is still diffi-cult to set as the momentum of the gcar train, particularly the differ-ential turntable, makes it difficult to move the main drive wheel, Bl,in such small increments as the one degree required for a day. The SI Ireduction ratio provided by the eontrate A helps, but not a great c1eal.In my model the periphery of the differential turntable is unsup-ported These additions will diminish the momentum effects but in-crease the friction. Backlash is also severe and would contribute muchuncertainty in practice to the Indications of the lunar dials. The divi-sion of the lower back dial tu indicate the age of. t he muon in half dayunits, as Price suggests, cer tainly could not be eff ectively exploited.Granted the relatively inefficient triangular tooth form used in thefragments I have very grave doubts as to whether the mechanismcould be made to work at all in the manner proposed by Price 's recon-struction.Much higher step-up ratios are commonly employed in clock mech-anisms. These, however, use efficient low friction bearings made bvsmall pivots turning loosely in side plates, efficient tooth forms, andhave low inertia at the high speed end of the gear train. Most import-antly, the control by the clock escapement is exercised from the highspeed end of the gear train which acts in the direction of the step-uponly as a mea ns of transmi tting force.


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10 A{{/ l 1 I U. 11/ "011111. :) '

To my mind a major defect of Price's reconstruction is that there isno natural indication of a day, surely the most conspicuous of theastronomical phenomena whieh might he expected to be representedin a calendrical mechanism. Pricc's suggestion thar the crank handlebe divided and marked in 73 days lacks both evidence and conviction.Further, it would provide an approximation to a year of considerablyless accuracy than that of the gear train that embodies the Metonic cy-cle or of the arrangements of the front calendar dial ring.

There remains, of course, the riddles posed by the purpose ofthe main gear E4 of the differential turntable and thc gear trainH2ILl +L2iM I +M2/N/Ol +02 .

Alternative Uses ()f the DifferentialMy starting point for a new reconstruction of the Antikythcra Mecha-nism has been the high step-up ratios required by Price's reconstruc-tion. These step-ups can be avoided most simply if till: gt: s idereal monthas input, the year formcd by a 191254 reduction, and the synodicalmonth formed by subtraction in the differential.

In Figure 3C the synodical month is applied as the only input to thediffereniinl. The two outputs of the differential arc constrained hy the19/254 gear train to then exhibit the sidereal month and year. In prac-ticc the gear train acts to transmit power in whichever direction is ea-sicr and acts. therefore, as a 19/254 r eduction r atio. In ef fect, the gear

II

l-lgurc J.

train connecting the two outputs of the differential acts as a feedbackloop to correctly constrain both outputs to the rates required.

Thi s thi rd inter connection of the differential warrants some f ur therjustification. In essence, a diffe.rential acts only as a kinematic con-straint on the r elative speeds of its connecting shafts, the designationof ill puts and outputs is not inherent in thc diffcrcntial mechanisrn butis characteristic only of each particular instance of our use of it. (Inprinciple a spur gear train can be driven from any point.) Here the ap-pr opr iate constraint is E3= ~;;[E2-E I J which might better be writtenF.1-E2+2E3=O ro not distinguish one gear wheel from the others.The sidereal gear train imposes the further conxtr aint 19E2=254El,the r elationship between solar and lunar sidereal per iods ill the Meton-ic cycle. If the second constraint is substituted into the first we find19(2EJ)=235El which, aside from the factor of 2 f O J " E3 which musthe made up in the gear trains connecting with the differential turn-


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table, establishes the relationship of the Metonic cycle between thesolar sidereal ami lunar synodical periods. These relationships applyequally to all three arrangcrncntx of the mechani sm shown in Figure J.

We remark fur ther rhar the arrangement of Figure JC is that used inthe common automobile; diffcrem ia l . The tail shaft, from the engine,drives the carr ie r of the d iff er en tial as its only input. The road surfaceprovides the constraint on the two half axles, carrying the wheels,which arc the outputs from the differential. On a s traight road the twohalf axles turn at the same rate, but in a corner the road constrainsthem to turn at rates proportional to the radii of the circles aroundwhich they track.

Possible Synodical Gear TrainsOf the three arrangement s shown, Figure 3C seems most appl icable tothe Antikythcra Mechanism as the main gear FA of the differentialturntabl e, which is unused in Price', rcconsuuction , is an obvious endpoint for a gear train providing the synodical month input. The inputto this gear train could be a shaft turned once per day or, as in the alBiruni design and the London Sundial-Calendar Wield and Wright], ashaft turned once per week although the date of the Antikythera Me-chanism seems too carlv for the week to be naturally used.

I commence by designing a gear train for the synodical month withthe differential turntable gear 1:::4 as the final gear. Price ascribes tothis gear 222 teeth, on t he radiographic evidence of Ka rak alos , thougha near approximation should be acceptable. I adopt the further con-straint that the synodical gear train should have an accuracy com-parable with that of the Mctonic cycle, the only other place in whichastronomical constants are embodied in the geared mechanism.

It is evident from the arrangements of the front dials thai the JIl~-ehanism represents sidereal year and month displays (rather thantropical displays). I have therefore taken the following constants after[Pedersen] but reduced to a uniform number of decimal places and theepoch of the Antikythera mechanism with [Allen]. (Secular changesto the present do noi , in fact, alter the calculated gear trains at ali.)

rnn' A/ifil:yrill'1'lI Mcctranism 13

Side real MonthSynod~cal MonthSidereal Year

27.321661 Days29.530581\ Days

365 .256360 Days13.368747 Side real Months12.:168747 Synodical Months

The Metonic cycle y ie lds a side real yea r of 254/19= 13 .368421 side realmonths so the error is about one pan in 40,000. To equal thi s accuracya Dear train for the synodical month from a shaft turned once per daysh7lUid have a ra tio or 29. 530 'i "'x (Ul007()( ). [The sidereal year dif fe;,trom the tropical year by about 1 i n 25,000 and the Metonic cycle givesa tropical year/si dereal month error of about 1 in 70,000. J

The constraints are met by the ratio 945/32=29.531250. As the dif-ferential turntable rotates once in two synodical mouths the ratiowould he used


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14 Altun G. Bnmr/I.'y

-----,,,-~~---------------? BASE PLAT :Back

15

I-ront- - - - - - ;A ~ ~,\- ;; - - - - - - 7----------------_)VRNT/.8LE Back

tor to turn in thc same sense as thc moon position? If so i t i s a complexway of going about a task that is performed far more neatly in thegearing of the differential. Possibly the contrate gear was connected tosome auxiliary display outside the box containing the mechanism, butthe per iod of about 11 5 year has no obvious significance unless a fur-ther reduction of the same amount took place in the auxiliary display.

The alternative possi bili ty is that the synodical gear train is formedby the gears L1, L2, M I and M2 with the contrate gear A serving as aninput shaft turned once per day and the gear Bl introducing the factor225. For this I must suppose that M2 meshes with E4 and that the gearB2 is really two separate gears each of 64 teeth, one of which is joinedto and turns with Bl and drives Ll and hence the d iffe rent ial tur ntab lewhilst the other forms part of the sidereal gear train.

The synodical gear train could thus be:AlB 150/225

L2/MI.54/96

++M2/E416/210

B2a/U64136

++ ++In this gear train M2 turns at the same speed as Bland the interveninggears provide just a 1:1 ratio as in Price's reconstructinn Tn t hat case amuch simpler gear train would have sufficed as discussed later. Also210 Leeth seems unacceptably few for E4. There are alternative geart rains which avoid these de ficiencies :

l-igurc :"l.

50/225501225501224

64/3564/3564/35

+++

17/22116/22416/225

52/9656/9656/96

+++

+++

Any of these seems quite acceptable. A possible reconstruction isshown in Figure 5.This general possi bility, that the synodical gear train i, driven from

the coutratc A, via Bl and the axes L and M, docs nut seem precludedby the evidence in Price's description. However, the complexity of thearrangement leads me to feel that it cannot have been used.The difficulty of the high step-up ratio required by Price's recon-

struction is avoided if a synodical gear train with a reduction of 16/945is used tu drive the turntable, E4, uf the differential. The input to thisgear train is turned once per day and provides a natural indication forthis most important of astronomical phenomena. A simple gear train,such as is shown in Figure 4, in the upper right of the back of the Anti-kythera Mechanism is the most straightforward means of obtainingthe desired ratio. The necessary gearwork is now completely missing.


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t6 AIIIIII G. /J'"(J.IIlcy

The Upper Back DialsThe most natural use of the upper back dials would be: for the indica-tion of eclipses ill conjunction with the synodical month dial below or,possibly. the sidereal dials 01 the front. I can see little purpose in a dis-play of the anomalisuc month of the lunar perigee, or any similarfunctions. which seem of minimal importance compared with eclipses.

There are three forms of indication that might naturally be used:(I) A Nodical (Draconitic) Month, showing the position of the moon

relative to its nodes or, equivalently. of the nodes relative to themoun;

(2) A Nodical (eclipse) Year, showing the position of the sun relat iveto the moon's nodes. or vice versa:(.1) A longer cycle directly rel ated to the eclipses such as the Suros.:'

The only driving func tions convenient ly avai lable 011 the back of themechanism arc the svnodical month from the: differential turntableand the sidereal year-through the train BlUM.

The most obvious mechanism is one based 011 the Saros of 19 eclipseyears ~ 223 synodical months. As Price points out this requires a gearof 223 teeth. which cannot be E4. Such a gear, if made to the st.mdardmodulus , is dif ficult to accommodate within the confines of the me-chanism as indicated by the size of the plates. If mounted ncar thecenter (If the upper dials such a large gear would also obstruct the axis(J of the subsidiary d ial . It seems necessary, therefore, to find a mccha-nisrn that requires only smaller gears in the gear train.

Some good rational approximations to the astronomical ratios arcshown in Table I. These arc derived from the constants

Nodical Month 27.212220 DaysO . 9 2J 4 93 Synodical Months346.62()[)31 Daysl l. 737661 Synodica l MonthsNodical Year

From the table the approximauon 4 eclipse years = = 47 synodicalmonths = = 5l nodical months is appealing. The lower accuracy thanthe Saros may be of no moment since no great accuracy of indicationof thc nodes is required for eclipse purposes. Particularly suggestive(though. as we shall scc , deceptive) is the 47 divisions propoxcd by

Errorrnlr Nodic;,[Years

Synodical NodicnlMourns Months

I~ Ll23 2,)35 3~47 51sz ~l)

1 2 9 1 40[76 11 ) 1123 (SanlsJ 242270 79:14~3 5:15716 777

I: I.(lIX)1/)00

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11 I 5 . 1XI : ) :~~342

1/2~O.: ) IJU 1

17

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IR Allall G. /11(11111


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20 Allan (i. tsro.ntevdial to that suggested in Figure 6. Here it is the position of the sun re-lative to the moon's nodes that is displayed. If driven from the synodi -cal month by the differential turntable using the ratio 4/47 the error inthe gear train is about 1 in 1,000 and the dial would have to be resetevery eight months. This is quite unacceptable. More accurate ratiosdo not lead to likely gear trains.

Alternatively, a nodical year dial might be driven from the siderealyear motion via the gear train B2/L1M/N. There ar e exce llen t approx i-mations to the required ratio obtainable with small integers such as20/19,39/37. and 98/93 - the last having an error of only I in 400,000.However, the small size of \12.14-16 teeth, argues strongly that if thedrive is from the sidereal year the upper back dial turns much moreslowly than the nodical year - indeed, in about the four years sugge-sted by P rice.The final possibility is that the upper back dial indicates a longer pe-

riod such as the Saros of 223 lunations or the 18.61 year period of themoon's nodes. Here the division of the dials described by Price sug-ges ts a pos sible mechanism.

I suppose that the 223 Iunations of the Saros are marked off on thefour rings of the upper back dial by a single pointer on axis N and thatthe subsidiary dial 011 axis ndicates which of these rings is to beread. The divisions on the rings would be marked with an indication ofthe type of eclipse that will occur in that lunation and thus readily ser-ves a pr ed ic tive func tion .With this arrangement each ring would have 223/4 or about 56 divi-sions. Whilst this is nut dose to the 47 or 48 divisions suggested byPrice [p 1 51 his observation is uncertain and further cleaning of thefragment is required to clarify the actual arrangements of the dial. Nouncertainty is suggested by Price [p40] for the division into four of thesubsidiary dial

If the Saros is not to be obtained by a reduction of 223 from the sy-nodical month the alternative is a reduction of about 18.030 from thesidereal year via the train B2/L1M!N. A reduction of 4.500 from B2 to'ican easily be obtained by adjusting Price's train SO that M2 has 14t eeth and 'ihas 63 as shown in Figure 8. These are cxacrlv thc countssuggested by Karakalos [P rice, p~: \6J.The e rror in thi s approx ima tionis a bout 1 i n 6 00 so the error in the indication amounts to about a thirdof a lunation over the 18 year Saros cycle. This is easily corrected by

r 21rrcnt----------------)

B.l.SE P"_ATE ~- - - - - - - - - 1eece

Figure H

resetting the dial rings, the lower back synodical dial enabling the re-setti ng to be done qui te accurately.Aside from the uncertainty of the division of the dial rings this ar-

rangement presents one difficulty when applied to the AntikytheraMechanism. A 1/4 reduction gear train is required from N to 0. Thiscould easily be obtained by a 20 tooth pinion on N and an 1:


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22 AI/(II1 G, Itromli-v

'-igun .. lI.

eclipses. This might he achieved by having the dial indicate the pas-sage of the nodical (draconitic) month. Although a suitable gear traincan he found the arrangement of the main and subsidiary upper hackdials seems quite inappropriate. All III year approximation to the Sa-ros cycle of eclipses fits the available gear trains particularly well andexploits the upper back dials as well as the current evidence of theirdivis ion wil l a llow.

Other PuzzlesA number of other puzzles remain in the reconstruction of the Anti-k ythcru Mechanism. I make brief note of these here, but cannot pre-tend to of fe r more than s imple specu la ti on s.

The Front Dials:P rice notes evidence that some additional, possibly substantial, me-

chanism was car ried by the main drive wheel B I and suggests that itmight have been a similar ge


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24 Alian G. Bromtey 25

i-t",:~:",.25 4Ficurc II.

ConclusionI bel ieve that the h igh step-up r at ios requ ired in Price' s rccons truct ionuf the Antikyt hera mechanisrn , and the consequent uncertainty in theindications given, are an insuperable impediment to his proposals.The mechanical difficulties are entirely removed if the differentialturntable is made the input to the mechanism and is turned at the lu-nar synodical rate. This approach has the considerable advantage thata natural means for indicating the clay as an astronomical unit is alsoprovided.

The upper back dial should most naturally be used to indicate theoccurrence of eclipses. A direct cnurucration of the lunations of theSaros cycle of the eclipses agrees well with the evidence of thc geartrains and the arrangement of the dials.

/\ reconstruction of the mechanism with the I321L/M train used in aSaros cycle drive to the upper hack dials is shown in Figures 10 and 11.This rccousuuction req uircs a complete synodical drive train as wellas another axis in the gear (rain between Nand 0addi tional to thosepreserved in the fragments. However, the trains are quite natural andthe indications of the dials convenient. For these reasons I favour thisreconstruction.


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;\llatJ G, Rromlt.!y

Important difficulties still remain in this reconstruction with respectt o the solar sidereal dial, particularly with regard to the apparent useof the contratc gear A and the large wheel D 1 as a reversing mecha-nism which I consider particularly awkward, and in the purpose of themultiple rings of thc lower back synodical dial. The elucidation ofthese I must leave to others.A mos t s igni fi cant cont ribut ion to c lar ify ing the proposal s d is cussedin this paper could be made by further cleaning of the LIpper back dialsand the axis 0 of fragment B and axis N of fragment D.

AcknowledgementMy interest in mathematical gear work was reawakened by contactwith the I.oncion Sundial-Calendar whilst a Visiting Research Fellowat the Science Museum. London. 1 am ever grateful for the continuinghospitality of the Science Museum and, in this instance, for the pati-ence and enthusiasm of Michael T. Wright to whom 1 owe a particulardebt for drawing my attention to the high step-up ratios in Price's ro;:-construction of the Anrik ytheru mechanism which provided both themotivation and the key to the work presented here. I have also pro-filed by discussions wit h Frank Percival and John Blackler.

REFEHE'iCESAlh-n , I". W _A,\'fm"hysh'.11 (jlumti!;(.',.., Athlonc P re ss . Un iv er si ty o f Lo nd on . Il Jh :1 ,O. omlc y. t\.G _ . "T he De si gn o f A st ro nomi ca l Ge ar T wi ns ", The Noro{ogica/ JOlO(//, Vol . 128rr, 19-2], December 19R5. and PI;. 10-1,. March 19t16,Ficld , 1. V" a nd W ri gh t. M , T, " Ci( !a r~ fr om t he B yz an ti ne s". Annals of Science. Vol . ' :;2. pp.

N7-I:H. IWS,Pedersen. 0, . A Sun'IY ()J the Almt/f,l'J/, Odcnsc Uuivcrstty press. 11.)74,P ri cc , D_ t i l . ' S__"Ge ar s I rom rlu- ( Ir e' '' k. ,, : T he Ant ikyt hc rn Mecha ni sm - ,\ Cal enda r Compu te r1I1I11I1.:U,NI Be" , 'liw,sf/OiOlt.'u)f,h(' Anu-rirun Phi/o,';f}phic-uI5iuo"cty Vol. (11 (lJ7:1; also pub-

le-hcd ;1~a monll!!-rilph, Science Hi-aory l -ubl ir at ious , N ew Yor k. 19/5,Wri~h{, M, T" "E:1r1y Whcclwork ". in rr.:par'ali: 1t)

r 27NOTESl. 1nrc described ill the sequence they arc encountered in tile gcur train

\,'ht'f} [I aced f n uu t he d ri vi ng . cu d. He re ~ l: ar Il 2 d ri ve s C I, ( '2 is f ix ed Ol t he s am e s haf t ; :s (Jan.. drive-r [)2, Etc , If these symbols tksignill..! th.. numbers of teeth on the r cspccrivc gearwheels th. . ..1 111l.'ear Irain prnvidcs 4/02, Sup-up gear(raill:-',:11 which Ilit, OUlpUI (LlIWat a faster rate i hu n t he il1PIII, are ( ks ign; tl cd i n 1111:igul't'}>:'y \hIX":S with ,I bold outline.

.', I n m indebted to Mr. M_ T, Wriehl of the Science 'vtuecum, London, (01 bringmg this hI my.uteution. The poinr nos. I undcrstnnd, been made pubficty hy Professor Zeeman of the Uui-varsity 01 warwick.

d 1 ,1m (\\/:lrl that t l t 11

notes on the antikythera mechanism

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