A New Gearing Scheme

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2 Bulletin of the Scientific Instrument Society No.85 (2005)

M.T.Wright

Introduction

The Antikythera Mechanism, the oldestgeared instrument in the world, becamewidely known through the work ofProfessor Derek de Solla Price;and for any-one seriously interested in this seminal arte-fact his Gears from the Greeks is still essen-tial reading.1 Price was the first to show theremarkable complexity of this unique sur-vivor of what must have been an extensivetradition of instrument making, forcing usto rethink fundamentally our appreciationof the technical achievement of Hellenisticculture in the first century B.C.

Prices monograph stands as a classic forthis reason, but his interpretation of thedevice itself is flawed because his observa-tion of its detail was poor. In giving theSocietys Invitation Lecture in November2003, I spoke about my progress to thatdate in my attempt to understand theMechanism and to prepare a more satisfac-tory reconstruction than Prices, on thebasis of a detailed new examination of theoriginal fragments carried out by the lateAllan Bromley and myself.2

Any such attempt centres around discus-sion of the gear trains.Figure 1 is Prices dia-gram of his gearing scheme.3 This is to becompared with the diagram of my newgearing scheme,Figure 2,which representsmore accurately what is found in the origi-nal and indicates how I think it should becompleted.In further papers I will expandon several novel points that the attentivereader will notice here, supporting myargument by reference to the very consid-erable wealth of detail that can be seen inthe original fragments.

The surviving gearing may be divided intothree groups,corresponding closely to thethree dials with which the instrument wasfurnished:

1.The foundation of the instrument is theprovision of two concentric mobiles behindtheFront Dial, a large wheel and a central

spindle, connected by a reverted train(loop) of gearing so that one revolution ofthe wheel represented one year, and onerevolution of the spindle represented onetropical month. The large wheel wasworked by a gear connected directly to ahand knob,and everything else was workedthrough it in turn.

2.A second train of gearing leads away fromthe one-turn-per-tropical-month motiontowards theLower Back Dial.

3.A third train leads from the one-turn-per-year motion towards the Upper Back Dial.

In a later paper I will discuss a number of

features that point to a strong possibilitythat the instrument may have been radical-ly rebuilt at some time. If so,it was perhaps

The Antikythera Mechanism: a New Gearing Scheme

only at that stage that the present schemeemerged, but it is important to emphasizethe tripartite arrangement of the instrumentas it survives because this validates my

approach in developing a new reconstruc-tion by treating the parts in turn:each partshould make sense within itself.Ultimately,however, one might expect to see the dis-tinct parts as a coherent whole. Furtherpapers, drawing on the material set outhere,will show to what extent this has nowbeen achieved.

I began by developing a reconstruction ofthe front dial,based on the first section ofthe gearing:the initial driving arrangementand the reverted train. I accepted Pricesnumbers for the wheel teeth of that trainprovisionally,but I have now confirmedthem.My demonstration that the front dialmust have been more elaborate than waspreviously supposed offers, for the firsttime,an explanation for several features onand around the Mean Sun Wheel.The pos-sibility that the instrument might plausiblyhave been a complete planetarium is impor-tant because it links this one survivinginstrument more strongly with the traditionof the making of astronomical modelsattested by the literature. Besides, the intri-cacy of what survives seems more in keep-ing with an elaborate function than withthe rather banal display of Prices recon-struction.

The motions of a planetarium call for theintroduction of further gearing: I havedemonstrated the possibility of includingepicyclic trains on the Mean Sun Wheelmodelling the anomaly of the Sun and oneanomaly each for Mercury and Venus, andfurther epicyclic arrangements modellingone anomaly each for the Moon, Mars,Jupiter and Saturn.The arrangements that Ihave built into a working practical recon-struction are described elsewhere.4 I willhowever repeat the point that it is the prin-ciple that is important, not the detail.Although my arrangement is economicaland efficient,embodies very good approxi-mations to the several period relations andis consistent with the evidence of the orig-

inal fragments and with what we know ofthe astronomy of the first century B.C.,there is simply no way of knowing whetheror how closely any such reconstruction rep-resents what has been lost.Therefore nodetail of this conjectural restoration isincluded in the gearing scheme that I pre-sent in here,since my present aim is to elu-cidate what remains within the originalfragments.

New Gearing Scheme: General Features

I retain Prices convenient system ofnomenclature alphabetical letter for axisand number for wheel with some neces-sary modifications.Firstly,not all the wheelsin Prices scheme are actually found in theoriginal.Secondly,in a few cases he was mis-taken in interpreting the evidence as to

where a wheel lay or with which otherwheel it was engaged.Thirdly, in other caseshe argued that he had to restore a wheel tofill out his scheme when,in fact,the schemewas incorrect.Consequently I have movedsome wheels and removed others. I havealso introduced a few new ones. In all casesI have simply abandoned the designationsof moved or removed wheels, and I haveextended Prices system to give a new des-ignation to each wheel that has beenmoved or introduced.Reallocating the aban-doned designations would have given atidier appearance, but comparison of myscheme with Prices and others based on itwould then be very confusing.

Figure 2 shows that the reverted gear train(from axis B through C and D and back toB) now includes axis E.Wheels E6 and E7are those understood by Price as E1,E2i andE2ii,and their changed designations reflecttheir altered functions in my scheme,whichnow accord with actuality. Since they haveequal numbers of teeth,their effect on thereverted train is simply as though an idlewheel were introduced, so that wheel B2(and with it B1, representing the meanmotion of the Sun) and B4 (representingthe mean motion of the Moon) turn in thesame sense.Prices conjectural Sun Positionwheel is therefore redundant and is dis-carded.

Prices wheels B3 and E1 are not present,and so the central arbor on axis E, andwheel E5 on it, remain stationary. Theepicyclic gear based on E4 as a platform istherefore not a differential gear.The wheelsE2ii and K1 within this epicyclic cluster,shown by Price as lying below E4, aremoved to accord with the actuality and arenow called E8 and K3, but the principle oftheir action is not changed.I retain Pricesconjectural wheel J a little tentatively,because it is not quite clear from the phys-ical evidence that it is called for; but I canmake sense of the train of wheels leadingto the Lower Back Dial only with such anidle wheel.5 It remains doubtful whetherwheel J should be placed between E8 andK3 or between E5 and K2,but that detail is

unimportant.Prices single wheel N is replaced by themore extended conjectural restoration ofwheels N1 and N2 and a further axis P withwheels P1 and P2,affording a rational con-nection between axes N and O.Wheels O1and O2 of Prices scheme are rejected infavour of a single wheel O,which is all thatI find.

The provision on axis B of a stationary bosswith a squared upper end indicates that atleast one fixed wheel was planted there. Iindicate this by introducing wheel B5, butsince nothing survives of the epicyclic gear-ing under the front dial to which it gave

motion no attempt is made to show arestoration.


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3Bulletin of the Scientific Instrument Society No.85 (2005)

Fig.1 Gearing scheme according to Price (note 1).Reproduced by kindpermission of the American Philosophical Society.

Fig.2 Gearing Scheme according to M.T.Wright.


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At the top of the diagram I indicate the con-jectural placement a small drumlike com-ponent(Prices description) found in frag-ment C (). The arrangement provides a

rotating Moon ball display, using just twowheels:the contrate wheel Q actually existsin this fragment, but the spur wheel B6fixed to the Sun indicator is a restoration.

Gearwheel Analysis

The tooth counts for the wheels shown inFigure 2,and listed in the Table,are the out-come of my reappraisal of each individualwheel, working from digitized versions ofboth plain radiographs and images takenfrom tomographic sequences.6 Three dif-ferent type faces are used. Tooth-countsprinted in bold type are certain.Those print-ed in ordinary type whether given as sin-gle figures or as ranges are not; furtherinformation on their interpretation is givenbelow.Those that are printed in italic typeare suggested numbers for wheels that areconjectural restorations. In the Table mycounts are compared with those made byKarakalos (as reported by Price) and withthe numbers subsequently adopted byPrice.To the Table I have added a wheel Q,as explained above; wheel 1 from frag-ment D (), which Price restored conjec-turally as wheel N but which is too large tobe placed there; and a further wheel 2which I believe I see within this fragment.I have no clear suggestions as to where thelatter two wheels might have been placed,

but since at the very least there must havebeen an epicyclic train on wheel B1, thereis scope for excluding them from thescheme illustrated in Figure 2 withoutembarrassment.

The following notes refer briefly to thederivation of the tooth-counts given inFigure 2 and the Table.From them the read-er may form an impression of the degree ofconfidence with which the counts aregiven.The assessment is not wholly objec-tive,and I doubt whether, in the more diffi-cult cases, it can ever be so.It depends onjudgment as to how far the pitch of the lostparts of each wheel might have varied,

either by design or by accident, and howgreat a variation in pitch is likely to havebeen tolerable in the immediate circum-stances under which that wheel worked.

A

The limb of this contrate wheel is soencrusted and so damaged that no directtooth-count is possible. It may be estimatedonly roughly,by taking the ratio of its diam-eter to that of B1 which it engaged togeth-er with a tooth-count for the latter which(as noted below) is insecure.This proceduresuggests about 48 teeth. In my reconstruc-tion the gear-pair A : B1 serves only as themeans of giving motion to the mechanism,and the number of teeth is unimportant.

B1

The teeth of B1 are rather poorly preserved.In many places no trace of them is found,while in others we see ghosts, distinct

images of the outline of teeth at severalradii,an appearance that may be an artefactof the corrosion process. Besides all this,the limb does not appear very truly round.It is therefore difficult to give a close tooth-count.I find 223 as a preferredvalue,withprobable low and high limits of 216 and231. Presumably the closer limits given byKarakalos reflect his assumption that thedivision is more uniform than it really is.However, in my reconstruction the numberof teeth in this wheel is unimportant.

B2

Most of the teeth of B2 can be seen. Some

seem rather unevenly cut:one single spacemeasures 1.26 time the mean,but since theadjacent space is small this may be partly aresult of damage.There are two lacunae.Thesmaller must have contained three tooth-spaces.The larger measures 8.2 times themean pitch of the surviving teeth and I sup-pose that it must have contained 8 spaces.Therefore I offer a confident count of 64teeth.

B4

The tomographic sequences show thatthere is no wheel B3 in front of the frameplate,and only this one wheel behind.I see30 teeth with well-constrained geometry.The count of 32 is certain.

B5

B5 appears in the gearing diagram simplybecause we know that there must havebeen such a wheel;but since we have noneof the wheelwork that was driven throughit I cannot offer any meaningful conjectureas to its number of teeth.

B6

If it is correct to place the assembly con-taining wheel Q at the centre of the Moonindicator, then there must have been a

wheel of the same number of teeth fixed tothe Sun pointer. Q appears to have had 24teeth. If so,B6 also had 24 teeth.

C1

Most of the teeth of C1 can be found, andtheir division is fairly uniform, so that I amconfident in taking the two lacunae as 2and 7 spaces and in giving a count of 38teeth.

C2

Nearly half the teeth of C2 are lost.Thosethat remain show some variation in pitch,so that in the middle of a continuous run

of 23 spaces the data points are displacedby nearly 0.4 of a space from points divid-ing the arc equally into the same number.

There is one large lacuna which, with thepreferred count of 48, contains 18 narrowspaces,but we must admit the possibilitythat it contained only 17 wide ones.So,on

the basis of this analysis,a count of 48 is notquite secure;but in reviewing the velocityratio of the whole reverted train of whichthis wheel is a member,we see that it is theonly number that makes sense.

D1

Prices direct countof 24 teeth is puzzling,because the wheel cannot be seen clearlyby direct inspection, and in radiographs Ifind only 19 teeth.The geometry is howev-er well constrained and the two small gapsmust have contained 2 and 3 teeth respec-tively.Therefore the count of 24 teeth is cer-tain.

D2

Many teeth can be seen,and because theyare rather well divided I have some confi-dence in filling the lacunae (with 23,13,10,4 and 3 spaces) to give a count of 127.

E3

The epicyclic assembly is broken almostexactly in half, leading to considerableuncertainty in the tooth count of eachwheel within it.For the platform itself (E4)and the gear ring E3,however,a small frag-ment of the edge from the lost half remainscemented to the frame plate,which reducesthe uncertainty so long as we can be confi-dent that it has not shifted relative to theremaining half during the destruction of theassembly. For E3, the counts for lacunae ofup to about 9 spaces seem safe,but thereare two lacunae of about 23 and about 70spaces.Taking into account the variationsin pitch of the remaining teeth generally,and the pitch of the teeth immediately toeach side of these lacunae,I arrive at a pre-ferred count of 191 within a range of pos-sible values 188 to 192.

E4

The problem of E4 is similar to that of E3,except that I find a more alarming variation

in pitch of the surviving teeth, amountingto over 4%.The two lacunae giving rise touncertainty should contain about 25 andabout 67 spaces.The preferred count is 223,but the range of possible values may be aswide as 218 to 228. It happens that theresult makes no difference to my recon-struction, because this wheel does notengage any other and appears redundant.Its existence is,in fact, one of the planks inmy argument that the instrument has beenaltered.

E5

This wheel, and E8 lying directly under it,

are really difficult.We have only fragmen-tary remains of each.By direct observationE5 is the larger of the two, and analysis


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ken edge of fragment A. I trace runs of 10,4and 2 data points only.The mean of these13 spaces leads to a count of 60 with gapsof 31, 11 and 5 spaces, but the run of 4

points is noticeably misaligned for an equal-ly divided wheel. Perhaps this part of thewheel has been distorted. (Elsewhere I seethat small fragments have dropped out andhave been cemented back slightly out ofalignment, although I do not actually seeclear evidence of that here.) If we take thefour points as reliable,then the pitch variesenough that there could have been 5 smallteeth in the gap of 4.6 mean spaces andthere must be considerable uncertainty indeciding on the number of teeth in thelargest lacuna.The analysis favours a countof 60 (with 61 as a strong second-best) butit is uncertain,with wide limits of 57 to 64.However, I agree with Prices argument that

the number must be 60 so that the velocityratio between axes G and I may be exactly12 :1,because I cannot find any use for thesubsidiary dial if the two periods are notsimply commensurate.

H2

It is common for wheels of few teeth to bepoorly divided.This one is, which may haveled Karakalos astray into counting 16;but Ifind 13 data points and the geometry is suf-ficiently well constrained that there is noroom to doubt that the true count is 15.

I

According to the argument offered above(under H1), wheel I should have 60 teeth,the number found by Karakalos. However,this case demonstrates the difficulty ofobtaining counts from irregularly dividedteeth on fragmentary wheels, because myanalysis leads to a seemingly confidentresult of 59 with lacunae of 13, 8 and 2spaces.Within the range of pitch variationthat I find in some other wheels, however,the largest of these gaps does afford thepossibility of having a run of 14 undersizedteeth, thus permitting the count of 60which is the only number that can makesense.

J

No count can be offered for the conjec-turally-restored idle wheel on axis J,seeingthat we do not even have a position for itsaxis.As I have discussed elsewhere, I havemisgivings about whether the interpositionof an idle wheel,either between E5 and K2or between E8 and K3, is justified by thedirect evidence at all,but I find it necessaryto fit one because only in this way can theoutput period at the Lower Back Dial makeany sense.

K2

As with wheels E5 and E8,it is particularlydifficult to offer secure counts of wheelsK2 and K3.The analyses are very sensitive

to choice of the centre,and neither periph-ery is very truly circular. In one analysis ofK2, taking data points at the roots of theteeth, I find 25 data points for K2,a contin-

uous run of 24 spaces,which looks like partof a well-divided wheel of 49.Repeating theanalysis taking data points from the tips, Iget an equivocal result of 49 or 50.We haveto allow the possibility that the division ofthe lost half of the wheel might have beenless uniform.Counts of 48 or 50 could cer-tainly be achieved without gross variationin pitch,and still wider limits might be set.Price reports that Karakalos saw a doublewheel here and counted 48 and 51.Karakalos was acute to see both sets ofteeth but his numbers carry little weightbecause he was probably misled by sup-posing that the wheels were concentric.

K3Here I find 26 data points, a run of 26spaces with one point missed.Again thepattern is close to that of a well-dividedwheel of 49, but the limits must be set atleast as wide as 48 to 50,perhaps wider.

L1

I find a continuous run of 27 data points,26 spaces.The lacuna equals 11.6 times themean of these single spaces, which mightcontain 12 small spaces or 11 large ones.The former represents a better approach touniformity, leading to a preferred count of38.Although 37 remains a possibility on the

basis of the analysis,a rational explanationfor the use of the Upper Back Dial rein-forces my preference for 38.

L2

I find a continuous run of 48 points, 47spaces.These show rather uneven divisionbut even so the remaining lacuna can onlyhave contained 6 spaces,leading to a securecount of 53.

M1

This wheel lies near the edge of fragmentA so that about one third of its peripheryhas been destroyed,and therefore the count

is sensitive to the placement of the centre,which is in turn made difficult by lack ofroundness. Analysis leads to a preferredcount of 96 with 97 as second best,but theuncertainty due to the poor division of theremaining teeth is expressed by suggestinga range of 95 98.Again, a rational expla-nation for the use of the Upper Back Dialsuggests that the count of 96 is correct.

M2

Only about half the wheel is preserved,showing 8 teeth in a continuous run.Theresult is therefore sensitive to centre place-ment, which probably explains the diver-

gence in results between Karakalos (whocounted 14), Price (who seems to haveexplored how far he could manipulate the

evidence in adopting 16) and me.With sosmall a number of teeth it seems a validapproach to inspect the effect of taking inthe tooth analysis program centre-positions

which give integral results for the numberof tooth-spaces in the lost half.In this way Iobtain 15 as by far the best fit for roundnessand equality of division together, and thecentre position so found appears satisfac-tory when checked back on the image.Therefore I give a count of 15 with confi-dence.

N1

Axis N lies at the break between fragmentsA and B.The stub of the arbor survives inthe latter, but there is no wheel on it.Thesize,and therefore the approximate wheel-count,of the wheel to be led by pinion M2,

in fragment A, can be found only roughly,by estimating the centre-separation of axesM and N.This depends on fitting the twofragments together correctly,which can bedone when we notice that traces of axis Ocan be found in both,and that the line NOis at right angles to the centre-line of theinstrument through G,B and M,which alsocontained N. It is clear that the wheel thatPrice placed here (which I call 1) is fartoo large, but when (as here) the pinionleads the wheel the actual size and numberof teeth in the large wheel are not welldefined.A rough calculation suggests 53,and this number suits my reconstruction.

N2Wheels N2, P1 and P2 are conjecturalrestorations.With wheel O (see below),they form a train connecting axes N and Oand,as with the wheels connecting axes Gand I of the lower dial,I argue that the peri-ods of revolution of the main and subsidiarypointers of the upper dial must have beenin some simple ratio.In a later paper I shallshow that in this case it should be 20 : 1,and since wheel O probably had 60 teethwe require that (N2 x P2) P1 = 3.Thechoice of N2, P1 and P2 having 15, 60 and12 teeth respectively would suit.Wheels ofthese numbers of teeth,and of a pitch with-

in the range found among the other wheels,would fit.The pinion of 12 is smaller thanany found among the surviving wheels,butthere is no reason why the maker shouldhave avoided it where,as here, it leads thewheel. Axis P would have been plantedbeyond the broken edges both of the frameplate and of the back dial plate,and so nofurther constraints are imposed.

O

I find only a single continuous run of teethover about one third of a circle, 21 pointsor 20 spaces.The tooth count is highly sen-sitive to the choice of centre, and on thebasis of the analysis alone it is hard tochoose between preferred values of 59 and60.The very large size of the sector to berestored introduces a further uncertainty,


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7Bulletin of the Scientific Instrument Society No.85 (2005)

widening the limits perhaps as far as 57 and62. However, as with the wheels connect-ing axes G and I in the lower dial, I arguethat the periods of revolution of the main

and subsidiary pointers must have been insome simple ratio.Later I shall show that inthis case it should be 20 : 1, which arguesstrongly for this wheel having 60 teeth.

P1, P2

See under N2.

Q

This small, mutilated contrate wheel wasfound within fragment C by Bromley andme.7 It is seen edge on when one views thefragment full-face,and so a direct count ofits teeth is impossible. By comparing itsdiameter with the size of teeth as seen in a

full-face radiograph from a tomographicsequence, I obtain a rough estimate of 24teeth.In my reconstruction the actual num-ber is unimportant, but B6 must have thesame number of teeth.

1

Price placed this wheel, found within frag-ment D (), on axis N and called it wheelN,but it is too large to be fitted there.I seea complete circle of teeth with just onemissed,and the count is certainly 63.

2

Within fragment D () I see a ghostly trace

of teeth outside the periphery of wheel

1,rather similar to the appearance notedabove in some places around wheel B1. Isee two long continuous arcs of teeth,which I believe to be independent of1because they lie on a well-defined circleeccentric to 1.Analysis offers an insecurecount of 65 teeth.

Conclusion

This dry but necessary paper provides thebasis for some interesting conclusions andfurther developments.

Firstly,the velocity ratio of the reverted geartrain, from axis B through axes C,D,E and

back to B, connecting the two concentricmobiles under the front dial, is confirmedas 19 : 254. In Prices treatment thisappeared to be a leap of faith.

Secondly, the tooth-counts for the train tothe lower back dial, axes E to K, includingthe epicyclic cluster on axis E,provide thedata for an analysis of the intended functionof this train,and of the Lower Back Dial thatit served, which is published elsewhere(note 5).

Thirdly, the counts of the wheels on axes L,M and O contribute to my identification ofthe intended velocity ratio for the train tothe Upper Back Dial and so to the function

of the dial itself,which is reflected in theconjectural restoration of axis P and thenumbers of the wheels on it and on axis N.8

Finally,the discovery of wheel Q within thedrumlike componentin fragment C leadsto the restoration of this assembly as a rotat-ing Moon display of a type not otherwise

attested before the late Middle Ages.Besidesthis, the feature provides evidence that theinstrument was in a slightly disordered statewhen it was lost.These points will be dis-cussed fully at another time.

Notes and References

1. D.J. de S.Price:Gears from the Greeks,Transactions of the AmericanPhilosophical Society, 64 No.7 (1974).Reprinted as an independent monograph,Science History Publications (New York,1975).

2. M.T.Wright,The Scholar, the Mechanicand the Antikythera Mechanism,Bulletin

of the Scientific Instrument Society,no.80(March 2004),pp.4 11.

3. Price (note 1) fig.33, p. 43, reproducedby kind permission of The AmericanPhilosophical Society.

4. M.T.Wright,A Planetarium Display forthe Antikythera Mechanism,HorologicalJournal, 144 No. 5 (May 2002), pp. 169 173,and 144 No.6 (June 2002),p.193;M.T.Wright & A.G. Bromley, Towards a NewReconstruction of the AntikytheraMechanism, S.A. Paipetis, editor,Extraordinary Machines and Structuresin Antiquity (Patrs: Peri Technon, 2003),

pp.81 94;M.T.Wright,In the Steps of theMaster Mechanic,

(Ancient Greece and the Modern World)(University of Patras,2003),pp.86 97.

5. The point is discussed in detail else-where: M.T.Wright,Epicyclic Gearing andthe Antikythera Mechanism,AntiquarianHorology.Part 1 appears in 27 No.3 (March2003), pp.270-279;parts 2 and 3 are forth-coming.

6. The procedure is outlined in Wright(note 2), and is treated in more detail in afurther paper: M.T.Wright & G.J.T.Wright,Computer-Aided Analysis of Radiographic

Images, applied to the AntikytheraMechanism, in preparation.

7. Initially we misread the evidence asshowing a small rack.We took the featureas a case study in:M.T.Wright,A.G.Bromley& H. Magou,Simple X-ray Tomography andthe Antikythera Mechanism, PACT, 45(1995),pp.531 543.

8. M.T.Wright,Counting the Months andYears: The Upper Back Dial of theAntikythera Mechanism,Bulletin of theScientific Instrument Society,forthcomingin the September issue.

Authors address:c/o The Science Museum, London SW7 2DD

e-mail:[email protected].

To mark the Millennium,theWorshipful Company of ScientificInstrument Makers (see page 24),gave the City a speciallycommissioned, stainless steel andglass obelisk: the MillenniumMeasure

The Measure takes the form of a"length standard" divided into 2000mm symbolising 2000 years ofhistory and bearing at appropriatelymarked intervals the dates ofsignificance, from the birth of

Christ to the year 2000,in the fieldsof science and instrumentation, theCity and the Nation,and religioushappenings.There are a number ofhappy coincidences,for example,the letters "mm" signify millimetreswhilst "MM" is the Roman numeralequivalent of 2000 as well as theinitials of the words "MillenniumMeasure".

The designer was WCSIMLiveryman Joanna A. Migdal.The

Measure stands below theMillennium Bridge at the St Paulsside of the river.

The Millennium Measure

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A New Gearing Scheme