counting months and years

8/2/2019 Counting Months and Years

1/6

Bulletin o the Scientic Instrument Society No. 7 (2005)

Fig. 1 Antikythera Mechanism, ragment A, ront.Approximately 50% actual size.

Fig. 2 Antikythera Mechanism, ragment A, back.Approximately 50% actual size.

Counting Months and Years: The Upper Back Dial

of the Antikythera Mechanism1

M.T. Wright

Introduction

Toothed gearing has a natural application

to instrument-making in the provision oa counting mechanism. I show here howa counting unction is embedded withinthe design o the oldest surviving geareddevice, and certainly one o the earliestintricate scientic instruments known, theAntikythera Mechanism.

The general arrangement o this instru-ment, with a principal dial system on theront ace and two urther dial systems,one above the other, on the back ace, isillustrated in an earlier paper in this jour-nal.2 In that paper I adumbrated a newgearing scheme or the instrument which

I presented in a urther paper.3 There Ioered, in a somewhat compressed orm,corrections to the previously-acceptedarrangement o the surviving wheels aswell as revised gures or the numberso teeth in each, based on my analysis oradiographs o the original ragments othe Antikythera Mechanism prepared bythe late Allan Bromley and mysel. In theserespects the work o Derek de Solla Price,on which all previous reconstructions othe Antikythera Mechanism are based, issuperseded.4 I also suggested reconstruc-tions o lost parts o the gear trains. In par-ticular, the reader may have noticed that I

oered a completion o the train leadingto the two pointers o the upper back dial,

which Price did not do. My present pur-pose is to expand on this eature.

The essential problem with the upper backdial is that, due to the way in which theinstrument broke up, the gear train leadingto it is incomplete. Part o the train can betraced within Fragment A, rom wheel B2 which rotates under the centre o the rontdial with a period representing one year through axis L to axis M at the edge o theragment, as indicated in Figures 1 and 2.5

The surviving piece o the upper back dialitsel constitutes most o Fragment B, asseen in Figures 3 and 4. At axis N, its centre,

there is the stub o an arbor but no wheel;at axis O, the centre o its subsidiary dial,there is an arbor bearing the wreck o a sin-gle wheel and what is probably a tiny rag-ment o the rame plate on which most othe wheelwork was planted.6 I suggest thatthere must have been a urther intermedi-ate arbor, carrying two wheels to completea compound train transmitting motion roma second wheel on axis N to the wheel onO. By analogy with the arrangement o thetrain behind the lower back dial, we wouldexpect the arbor to have pivots workingin holes both in the rame plate and in thedial plate, but the remains o these parts

do not extend ar enough to provide rmevidence.

A reconstruction o the train as ar as theupper back dial centre, axis N, dependson estimating the numbers o teeth in theremaining wheels, but it also depends cru-cially on working out the correct relation-ship between FragmentsAand B in orderto judge what is likely to have been lostbetween them. The way in which the rag-ments come together leaves little roomor doubt that a wheel on the arbor at Nwas engaged directly by pinion M2, withno intermediate axis. Price got this ar, butthen he umbled: having positioned thecentre o the upper back dial correctly in

relation to FragmentA, he then suggestedrestoring to axis N the detached wheel thathe ound in Fragment D. This wheel is actu-ally ar too large to t, but perhaps Price eltcompelled to nd a place or it somewherein his reconstruction, in which he imag-ined adding very little to what survives.Seeing, however, that the gearing schememust have been more extensive than hesupposed (as I have argued previously:see note 2), we need have no inhibition insuggesting that Fragment D may well havecome rom some other part o the instru-ment. It may indeed not have been a parto this instrument at all.


2/6

Bulletin o the Scientic Instrument Society No. 7 (2005)

Fig. 3 Antikythera Mechanism, ragment B, outside.Approximately 90% actual size.

Fig. 4 Antikythera Mechanism, ragment B, inside.Approximately 90% actual size.

In consequence o his use o a wrongwheel at axis N, and the limitations o hisestimation o the numbers o teeth in thewheels on axes L and M, Prices suggestionsor the period represented by the rotationo the pointer at the upper back dial wereill-ounded. He came tantalisingly close towhat I will show to be the correct result,but with wrong numbers or the wheels.Finally, however, he seems to have decidedthat one turn o the pointer probably rep-resented the passage o our years. In hisattempt to interpret the unction o the dial

as a whole, Price was led urther astray byhis mistaken observation o two wheels ataxis O.

There have been other attempts to makesense o this dial. Since, however, none hasbeen based on any evidence other thanthat ound in Prices published paper, andmost depend on distorting that evidence inways that it will not bear, I pass over themin silence and oer my own account.

Reconstructing the Gear Train toAxis N

The Table lists the numbers o teeth othose wheels that are relevant to thepresent discussion. It is abstracted rom thelarger table given in my last paper (note 3),to which the reader is reerred or an expla-

nation o how the gures were arrived atand or comparison with the gures pre-viously oered by Price. As in that paper,gures that are certain are printed in boldtypeace while the degree o uncertaintyin the counts o other surviving wheels isexpressed by the range in the right-handcolumn, and conjectural gures or wheelsthat are altogether lost are printed in italictypeace. Note, in addition, that 7 ts theobserved data or wheel M1 slightly lesswell than 6, and 5 and are less likelystill.

In determining the period represented byone turn o the main pointer o the upperback dial at axis N, we are concerned onlywith the rst six rows o gures, remember-ing that one turn o wheel B2 representsone year. The adoption o 53 teeth or theconjecturally-restored wheel N1 is, in therst instance, based on an attempt to judgeits size, and or this we need to considerthe correct juxtaposition o Fragments Aand B.

Fragment E, which is roughly lozenge-shaped and measures about 60 by 35 by 14mm., is seen under radiography to containsmall portions o the back dial plate, includ-ing small parts o the two outermost turnso the spiral system o the lower back dial.Its signicance or our present purpose is

that, resting between FragmentsAand B,it conrms the t o one to the other, asshown in Figure 5. The circular lobe thatis preserved almost miraculously in thecorrosion products at the upper edge oFragmentAbeyond the crumbled margino the rame plate (Figures 1 and 2) is seento be a relic o axis O, probably a washersimilar to those ound where other arborstake bearing in the plate. The break thatormed the edge o Fragment B, runningdownward and slightly to the let throughaxis N (as seen rom the outside) is seen to

continue in FragmentAas the break thatdetached the let hal o the epicyclic plat-orm.

The relationship can also be explored ur-ther using radiographs. Axes G (centre othe lower back dial), B (centre o the rontdial), and M in Fragment Aare seen to lieon the vertical midline o the instrument.Axis N in Fragment B is ound to lie onthe same midline. Fragment B ts with theinner end o the spiral system around theupper back dial also lying on this midline,and with the line joining axes N and O atright-angles to the midline. These exercises

yield a centre-distance between axes M andN o about 1.2 mm.

I proceed rom this estimate o the sepa-ration o arbors M and N, and rom the


3/6

10 Bulletin o the Scientic Instrument Society No. 7 (2005)

Fig. 5 Antikythera Mechanism, ragments A, B and E together.Approximately 59% actual size.

measurement o what remains o pinionM2, to calculate the size and number oteeth o the wheel N1 that it drove; but Ido so with some didence. The orm o thesurviving teeth throughout the mechanismis crude, with roughly straight fanks andpointed tips, appearing to be merely theresult o cutting out the spaces betweenthem using a le having an edge with anincluded angle o about 60.7 Moreover,

while much o the observable irregularityin their orm may result rom damage, themarkedly irregular spacing that is oundin many places must be largely a eatureo their original manuacture. Under thesecircumstances, discussions o pitch and thepitch circle, on which such a calculation isbased, become rather vague.

Thereore a airly crude calculation is all thatis appropriate, and it runs as ollows. Thetip and root radii o pinion M2 are about4.5 and 3.5 mm. respectively, so the radiuso its pitch circle is roughly 4.0 mm. Thecorresponding radius o the pitch circle o

N1 would thereore be about (1.2 4.0)= 14.2 mm., the pitch o wheel N1 shouldequal that o pinion M2, and so the numbero teeth in wheel N1 should be the integerclosest to (15 x 14.2/4.0), which suggests a

wheel o 53 teeth. Even with well-dividedgears having teeth o a more sophisticatedorm, the number o teeth would still notbe very closely dened because the smallpinion M2 leads. In such a case, and espe-cially where the load on the train is lightand uniormity o lead does not matter, acertain latitude in both pitch and size othe wheel may be permissible.

However, a count o 53 teeth or wheel N1,taken together with the certain and mostprobable tooth-counts tabulated or theother wheels, gives a velocity ratio or thetrain rom axis B to axis N o 3.: 1. That is,one revolution o the pointer on the upperback dial represents 3. years or, accordingto the 1-year period relation that is alreadyknown to be embodied elsewhere in theinstrument, 47 synodic months. Thisresult ts with, and makes sense o, all theavailable evidence in providing a satisac-tory solution to the question o the unc-tion o the upper back dial.

The Dial PlateI have drawn attention to the spiral designo the back dial systems (note 2). Thearrangement can now be better illustratedby reerence to my reconstruction o the

dial plate, which is shown in Figures 6 and7. The graduations o the upper and lowerback dials orm spiral scales o ve andour turns respectively, accompanied byspiral slots through the plate. The slots aredesigned to hold moveable markers, riv-

eted loosely into them so that they may bemoved along the scales at will. It is not alto-gether certain that the two spirals shouldactually meet in the middle to orm a con-tinuous S-curve as seen here, but analysiso the geometry and dimensions o thesurviving ragments leads to this as a likelyarrangement. The detail is probably unim-portant, but it would have allowed the userto slide markers rom one dial system tothe other. The inside view, Figure 7, showsthe orm o the strips that are rivetedacross the slots to hold the spiral in shape,attested by two surviving examples in theoriginal. These orm little bridges that leavethe edges o the slots clear or the passageo the rivet-heads.

A period o revolution or the main pointeron the upper back dial representing 47synodic months is compatible with Pricesobservation, conrmed by Bromley andme, that the visible graduations on theragmentary upper back dial suggest thatthe ull circle was divided into 47 or 4parts. I accept 47 divisions as the correctnumber. The dial system, with its 5-turn spi-ral, thereby provides a scale o 235 monthdivisions o a reasonable size, accordingwith the makers known interest in the

1-year cycle o 235 synodic months. Thedivision o each turn o the spiral into anexact number o parts also agrees withboth direct and radiographic observationsthat the dial divisions run radially straightacross the several turns o the spiral.

It is evident that characters were engravedin many perhaps all o the spacesbetween the dial divisions. Presumably thelettering comprised numerals or abbrevia-tions, but too little has been read or anyattempt to be made to restore its sense.10

The Subsidiary Dial

It is useul to consider rst what we knowabout the subsidiary dial o the lower backdial system. Here the subsidiary pointer (atI) was worked rom the main pointer (atG) through a two-stage compound trainplanted at axes G, H and I. Parts o all thewheels on these axes survive, and althoughthe tooth-counts o some are uncertainwe nd that the lower subsidiary pointerturned at roughly one-twelth o the speedo the main one. In discussing that train Iargue that a slow-turning subsidiary point-er makes sense only i its period is a simplemultiple o the period o the main pointer,

and so the only possible value o the ratioin that instance is exactly 12: 1.11

I agree with Price in reading the letterH,engraved within the exposed part o thebounding circle o the lower subsidiary


4/6

11Bulletin o the Scientic Instrument Society No. 7 (2005)

dial, to the lower let, although whereas hissketch shows it set horizontally I see it withits upright strokes aligned to the radius othe circle at that point. Looking obliquelyunder the overlying layer, I see parts oother engraved letters that Price missed:

to the right, slightly above the centre, I seean angular corner where two strokes meet,with a seri ; to the upper let I see twostrokes which appear to terminate in boldseris, very like part o another H, alignedroughly to radii o the circle. I read theseincomplete traces as letters D (to the right)and IB (to the upper let) respectively, andI interpret them as numerals: D = 4, H = and IB = 12. No radial divisions are visiblebut it is notable that the dial plate is brokenalong a airly straight line, vertically down-ward rom the centre o the subsidiarycircle, which may indicate racture alonga scribed radial division. Even without divi-

sion lines, the numerals seem to indicate atripartite division o the subsidiary dial, andto denote the cumulative number o draco-nitic months represented as having elapsed(equal to the number o turns made by themain pointer) as the subsidiary pointersweeps out each third o its circle.

Now we turn to the subsidiary dial o theupper back dial system. Analysis o theragmentary wheel on axis O, at the centreo the subsidiary dial, yields an uncertaincount o 60 teeth. This wheel could havebeen driven directly by a second wheelon the arbor at N, but then the subsidi-

ary pointer would have rotated just a littleslower than the main one, one turn repre-senting about 41/2 years, and in the oppo-site sense. This is not a useul unction, andthroughout the rest o the instrument thedesigner seems to show a preerence orpointers rotating clockwise with advancingtime. In any case the subsidiary pointer othe lower back dial certainly turned in thesame sense as the main one.12 Uniormityo the sense o rotation would most simplyhave been preserved by driving the arbor atO rom the arbor o the main pointer at Nthrough a compound train involving a ur-ther axis, in just the same way as the arbor

at I was driven rom the arbor at G throughwheels on axis H. This arrangement oersus scope to reconstruct a useul unctionor the subsidiary dial. None o these pointsalone makes a strong argument, but I willshow that, together with the evidence othe dial markings themselves, they do leadto a compelling solution.

Thereore I introduce the intermediate axisP as a conjectural reconstruction. The highnumber or wheel O dictates that it, andwith it the pointer on the upper subsidi-ary dial, rotated more slowly than the arborat P and so, almost certainly, more slowly

than the corresponding main pointer onaxis N. It remains to determine the appro-priate velocity ratio or the upper back dial,between axes N and O.

I agree with Price that the subsidiary dial

in the upper system was divided into our,as described below. In the lower system, inwhich the velocity ratio between the twopointers was 12: 1, the subsidiary pointermoved on a third o a turn, rom one markto the next, each time the main one swept

out the whole o its our-turn spiral scale.Applying the same principle to the uppersystem, with the subsidiary dial dividedinto our and the main scale o ve turns,the velocity ratio between the two pointerswould have been 20: 1. One revolution othe subsidiary pointer would then repre-sent the passage o 20 x 3. = 76 years.

This output can be achieved using pin-ions o 12 and 15 and two wheels o 60, asshown in the Table. Other numbers couldo course be used, but this extra wheel-work ts neatly, adopting pitches withinthe extremes ound elsewhere in the

ragments and with axis P planted aboveand between axes N and O. Just possiblya notch in the upper edge o Fragment Bmay be interpreted as a broken-out pivothole in the dial plate, but this is uncertain.The corresponding part o the rame plateis broken away.

Price shows the subsidiary dial dividedinto our quadrants, with the letterS in thelower let quadrant. He must, however, havemuddled his notes, because this part o thecircle is overlaid by accretions; only theupper let part o the circle is visible. Tworadii are visible, and although the observa-

tion is dicult I take them to be drawn atright-angles, horizontally and vertically onthe plate. Within this quadrant I do indeedsee traces o a letter, which I initially readas S but which Bromley read as A orD. Allthat it seems sae to say is that we both sawstraight strokes that did not meet at right-angles. In tomographic radiographs whichresolve the plane o the dial plate, in thisregion I see, tentatively, the two letters LH,aligned as beore upright to the radius. Inthe lower let quadrant I see more clearlythe letterI, ollowed by another, indistinctletter. I suggest that the readings, romlower let quadrant and going clockwise,

should be: IQ,LH,NZ and O. (The una-miliar last character, stigma or digamma,is used to denote the number 6. It appearsin an inscription noted below in the ormo a square C with heavy seris.) Theseare again interpreted as numerals: 1, 3,57 and 76 respectively. By analogy withthe lower subsidiary dial, these indicatethe cumulative number o some periodo interest represented as having elapsedas the pointer o the main dial runs overthe whole o its spiral scale and as the sub-sidiary pointer traverses each sector o thecircle in turn; but this time the period ointerest is one year whereas the period rep-

resented by one turn o the main pointeris 3. years.

By direct inspection one may see someslight, rather uneven graduations outsidethe bounding circle o the subsidiary dial;

and in radiographs I see indistinctly whatmay be lettering in this position. The spac-ing o the marks is such that there mightbe 20 around the circle, in which case theywould serve as indicators or individualturns o the main pointer. My impression is

that these are marks o a type that all whoare amiliar with instruments will recog-nize, those added by the user or his ownconvenience. I have made no attempt toreproduce them on my model.

Conclusion

I conclude by showing that this restora-tion o the display o the upper back dialpossesses a coherence and useulness thatlend weight to its plausibility.

Just as ive turns o the main pointer,sweeping out the whole length o thespiral scale, represents the passage o theMetonic period o 1 years, so one turn othe subsidiary pointer represents 76 years,an interval o time known to astronomersin antiquity as the Callippic Period. Thesignicance o this period, our times thelength o the other, is that, while it pre-served the same excellent period relationbetween the synodic month and the year,it was reckoned to contain exactly 2775days so that it embodied good approxima-tions to the lengths o the synodic monthand o the year in days. It seems also thatastronomers ound it a convenient periodor the reckoning o long intervals o time:

thus, in the Almagest, in giving the dateo observations recorded by his predeces-sors, Ptolemy reers to day and month inthe Egyptian calendar (in which, with itsyear o 365 days, it was understood thatdates drited by one day every our yearsin relation to the seasons) and a numberedyear within a given Callippic cycle. On theinstrument, while the main pointer o theupper back dial showed the correspond-ence o synodic months to years, and therotation o the subsidiary pointer indicatedthe passage o Metonic and Callippic cycles,the end o one month, year or cycle and thebeginning o the next would be indicated

with more precision on the ront dial. Thus,the upper back dial system could unctionas a convenient counter o months, yearsand cycles o years, in support o the dis-play on the ront.

Whether or not one accepts my recon-struction o it as a ull planetarium, it iscertain that the ront dial display entailedsome epicyclic modelling and was there-ore relatively elaborate. One may envisagethe user exploring its indications o astro-nomical events or times rather distantlyremoved rom his own. In casting a per-sonal horoscope, or instance, the astrolo-ger required the places o the planets atthe natives moment o birth, and althoughwe have abundant evidence o the use otables or computation,13 a planetariuminstrument could have been used to ndthe data mechanically. Since it takes about


5/6

12 Bulletin o the Scientic Instrument Society No. 7 (2005)

lower back dial, and yet the inter-pretation o the upper dial nowseems, i anything, more secureand more complete than thato the lower. It may be that theargument can now be reversed,

and that our new insight intothe design and unction o theupper back dial may help usto gain a better understandingo how the indications o thelower back dial were meant tobe read and used.

Taking my reconstruction o theback dial as a whole, and com-paring it with that o Price, itwill be seen that the indicationo the synodic month, surely oneo the more widely recognizedand widely used astronomical

periods, is much reduced inprominence. The alert reader omy last paper (note 3) will, how-ever, have noticed that a neweature is added to the rontdial. It appears at the top omy gearing diagram, but it wasunortunately slightly croppedin the inal printing. To makegood the omission, this part othe diagram is reproduced hereas Figure . This element o myreconstruction, which is sound-ly based on evidence ound in

ragment C, supplies what seemed to have

been lost in abandoning Prices: an easily-read indication o the synodic month. Inaddition, it provides a visual display o theMoons phase. The eature orms the topico a urther short paper, now in prepara-tion.

Acknowledgement

I am grateul to Dr Silke Ackermann o TheBritish Museum and Dr J.V. Field o BirkbeckCollege, both o whom have allowed me toengage them in discussion which has beenmost helpul in clariying the ideas that Ihave attempted to express here. Any mis-

takes are, however, entirely my own.

Table

Tooth-counts or the wheels in the trainleading to the upper back dial.

B2 64

L1 3 37 - 3

L2 53

M1 6 5 -

M2 15

N1 53

N2 15

O 60 57 62P1 60

P2 12

Fig. 6 The Antikythera Mechanism, reconstruction by M.T. Wright: the back dial.

ve turns o the driving knob to move thedisplay o this instrument by one year, itwould have been a great convenience tohave had the use o the upper back dialas a year-counter in working it through aninterval corresponding to a persons age.

The upper back dial, used either alone ortogether with the ront dial, might also beused in converting between any o theseveral luni-solar calendars used locallyin the Hellenistic world or civil purposesand the Egyptian solar calendar, avouredby astronomers or its relative stability. The

user might move the marker-beads alongthe slot to show any event o interest, suchas the beginning o a new year or the astro-nomically-determined time o some estival.In any case it is interesting to note that theront dial, with its division into days andmonths according to the Egyptian calendar,and the upper back dial, with its count omonths and years in the 1- and 76-yearcycles, work together in just the samedivided time-reckoning system that we ndused by Ptolemy in theAlmagest.

My conclusion that each turn o the mainpointer o the lower back dial o the instru-

ment probably represented one draconiticmonth, a unction o use in attempting topredict eclipses (note 11), suggests a ur-ther possible use or the upper dial. Eclipseprediction is uncertain according to any

simple astronomical model, but it was rec-ognized that eclipse events kept roughlyto a pattern that repeats ater 223 synodicmonths, the so-called Saros cycle.14The spiral scale o 235 synodic months islong enough to contain one completeSaros cycle and a little more. The moveablemarker beads might be set, according tothe month graduations, to signal the eclipsepossibilities o a whole cycle at a view.

Finally, there is urther evidence rom theinscription on what Price called the backdoor plate, a lea o bronze that may or

may not have been jointed to the casebut which evidently lay over the back dialas the instrument decayed. Its extendedinscription, apparently related to the unc-tion o the instrument, is now reduced totantalising ragments. In one place charac-ters in one line reer to both the 1-yearand the 76-year cycles, while in the nextline a possibly uncertain reading suggests223 conjunctions.15 The rst line seemsto reer to the periods displayed on theupper back dial; and in the reading o thesecond line we have a clear reerence tothe eclipse cycle.

AfterwordThe arrangement and unction o the upperback dial has in part been reconstructedby analogy with what is known about the


6/6

13Bulletin o the Scientic Instrument Society No. 7 (2005)

Fig. Top part o the reconstructed gearing diagram

Fig.7 The Antikythera Mechanism,reconstruction by M.T. Wright. Back dialplate, inside view.

Notes and References

1. This paper was submitted in its originalorm in May 2005. It was to have appearedin issue no. 6 (September 2005) but wasdeerred to make way or material in memoryo the late J.R. Millburn.

2. M.T. Wright, The Scholar, the Mechanic andthe Antikythera Mechanism,Bulletin o theScientifc Instrument Society, no. 0 (March2004), pp. 4 11.

3. M.T. Wright, The Antikythera Mechanism:a New Gearing Scheme, Bulletin o theScientifc Instrument Society, no. 5 (June2005), pp. 2 7.

4. D.J. de S. Price, Gears rom the Greeks,Transactions o the American PhilosophicalSociety, 64 No.7, (174); reprinted as anindependent monograph, Science HistoryPublications, New York 175.

5. The designation o the ragments by alpha-betical letter is already ound in the earliestpublished description and was used by Price(note 4). The designation o wheels by alpha-betical letter or the axis and Arabic numeralor the wheel is due to Price himsel. Both areconvenient, as is continuity within the litera-ture. Thereore, despite a slight risk o conu-sion between the two systems, more apparentthan real, I perpetuate the use o both systems,with some extension.

6. I nd only one wheel on axis O. I cannot

explain either how Price (note 4) can havepersuaded himsel that he saw two wheels(O1 and O2) or how he arrived at the tooth-counts that he oered or them.

7. Their orm and appearance are easily repro-duced by the use o a saw-le that has the sec-tion o an equilateral triangle. I suggest thatthis is probably very close to the method usedby the maker o the instrument.

. It will be recalled that the 1-year periodrelation, according to which 1 years = 235synodic months (sometimes known in mod-ern literature as the Metonic relation ater

the Athenian astronomer who used it as thebasis or calendrical reorm) is implied bythe relation 1 years = 254 tropical monthswhich is built in to the gearing behind theront dial.

. Bromleys unpublished research notes arein my possession.

10. Bromley and I limited our attempts at read-ing inscriptions, but invited an experiencedepigrapher to collaborate with us. His reportremains in preparation.

11. M.T. Wright, Epicyclic Gearing and theAntikythera Mechanism , AntiquarianHorology. Part 1: 27 No. 3 (March 2003), pp.270 27; Part 2: vol. 2 no. 1 (September

2005). pp.51 63.

12. Elsewhere (note 11, part 2) I point outthe kinematic possibility that, i there wereno idle wheel J within the epicyclic cluster

on axis E, the uncertain numbers o teethin the other wheels in the cluster might bechosen so that the pointers o the lower backdial turned anticlockwise; but I give severalreasons or rejecting this possibility.

13. A. Jones, Astronomical Papyri romOxyrhynchus, Philadelphia, AmericanPhilosophical Society (Memoirs vol. 233),1.

14. The modern application o the name Sarosto this period is a mistake due to EdmundHalley.

15. See Price (note 4), Figs. 36 & 3 and p. 50.I bow to the experience o the epigrapherson whose work Price drew in describing thereading as uncertain and conjectural, but Iread (= 223) quite clearly.

Authors address:Honorary Research Associate

Centre or the History o Science,

Technology and MedicineRoom S223BCentral Library, Imperial College

London SW7 2AZ, Englande-mail: [email protected]

counting months and years

Documents