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Home , Astronomical clock, Common year, Epact

THE INFORMATION

BRIAN H AY E S

Clock ofAges

When one '5 bright ideas become irlScrutable legaciesfor the next

5 T HE WO I,LD SPIftALS ON twenty- four feet wid e, with three orna- towJrd 0 1-0 1-00, sm vivalists mented.spires ilnd a gigantic instru ment :I re ho,1rd i'lgosh, c:lnned goods panel ofdi

N(!vc m/ocrlDuc ,"bn 1999 ' THE 'J tical . Calculating the dates of statues are mainly on religious themes- 4.0905533 . The error is less than those holidays requires feats ofmechan- deat.h and resurrection, ell! and salva- a second a century. ical trickery. tion-but they also include portraits of The most intricate calculations are the For the Y2K police, the crucid] CO\11- Urania (tbe muse of ) and ones for leap years and the movable feasts ponent of tbe clock is, of course, the Coperniclls. Another painting portrays of the church. The rule for leap years counter of years. It is an inconspicuous Jean-Baptiste Schwilgue, whose part in states that a year lV has an extra day if N four-digit register that anyone from our this story I shall return to presently. is divisible by 4, unle<;s N is divisi- age ofautomobiles will instantly recog- ble by 100, in which case the year is J nize as 811 odometer. On December 31, r's ALL DONE \\11TH GEAR.s. ALSO PIN- C01111110n year, with only the usu;d 365 at 111idnight mean solar time-and thus IiODS, W0D11S, snails, arbors; pawls and days-but if N happens also to be divis- half an late by French offICial ratchets; cams and cam followers; cables, ible by 400, the year becomes a time-the digits will roll over from one- levers, beU cranks and pivots. again. Thus 1700, 1800 and 1900 were triple-nine to two-triple-zero. The actual timekeeping mechanism- all common years, but 2000 will have a Wait! There's even more! The clock a and much like Februaty 29. How can you encode such is inhabited by enough animated figures the ones present in other -drives Cl nest of if-t.ben-else rules in Cl geClr train? to open a small theme park. The day of the gear train for mean solar time. All the The clock has a wheel \.vith t\venty- other astronomical and calendrical func- four t.eet.h and for an ornitted tions are derived from that basic, steady twenty-fifth. That wheel is driven at a motion. for example, local solar time is rate ofone turn per century, and 50 evelY calculated by applying wo cOlTections to four years a tooth comes into position meJn solar time. The first cOlTection to actuate the leap-year mechanism. compensates for seasonal changes in the The gap where the twenty-flfth tooth length of the day, the second for varia- would be takes care ofthe divisible-by- tions in the earth's orbital velocity as it 100 exception. For the divisible-by-400

PARTS OF THE STRASBOURG CLOCK look deep into the

follows its slightly elliptical path around exception, a second wheel turns once the SUD. The corrections are computed every 400 yeJrs. It carries the missing by a pa.ir of "profile wheels" whose rims twenty-fifth tooth and slides it into are machined to trace out a graph of the place on every fourth revolution of the appropriate mathematical function. A centulY wheel,just in time to trigger the roller, following the profile as the wheel quadricentennialleap year. turns, adjusts the speed oft.he local-solar- The display of leap years calls for as Death and his chimes time pointer accordingly. The computa- much ingenuity as their calculation. On tion of lunar motion reguires five cor- t.he large calendar ring, an open space the is marked by a slow procession rection terms and five proftle wheels. between December 31 and JanualY 1 of seven Greco-Roman gods in ch8ri- They all have names: anomaly, evection, bears the legend COl"lImencemcnt de {'an- ots. Each day at noon (that's mean solar variation, annual equation, reduction. nee corm-nutlc ("start o'f ") [see noon) the twelve apostles appe::n, SJlut- The overall accuracy ofthe clock can photograph 0/1 page 'f 2]. Shortly before ing a figure of Christ, who blesses each be no better than the adjustment of the m.idnight on the December 3 J before a in turn. Every hour a Pl1tto overturns a pendulum, which requires continual in- leap year, a sliding flange that can-ies the sandglass. At VJriOllS other times figures tervention, but for the subsidiary time- first sixty days of the year ratchets back- representing the four ages of man and a keeping functions there is another kind ward by the space of one day, covering skeletal Death emerge to strike their ofetTor to be considered as \vell. Even if up the word (011111nmf Jt one end of the chimes [see photograph above]. the mean solar time is exact, will all the flange and at the same time exposing All of that. apparatus is hOllsed in a solar, lunal· and planetalY indicators keep February 29 at the other end. The Range structure of unembarrassed eclecticism, pace correctly? The answer depends on remains in that throughout the botb stylistic and inteUectual. Tbe cen- how well celestial motions can be approx- yelr, then shifts forward again to cover tral tower of the clock is topped with a imated by rational alitlunetic-specifi- up the twenty-ninth and reveal comlIwne froth of German-baroque frost.ing, caUy, by gear ratios. The Strasbourg clock just as the foUowing year begins. whereas the smaller turret on the left comes impressively close. For example, (\.vhich houses the weights that drive the the true sidereal day is twenty-tbree hOllrs, H E RULES FOR FINDING THE DATE clockwork) has been given a more fifty-six , 4.0905324 seconds, of Easter are even more intricate Frenchifled treatment. The third tower, whereas the mean solar day is, by defin- than the leap-yearruk. Donald E. Knuth, on the right, is a stone spiral staircase that ition, exactly twenty-four hours. The in his Art if Computer might have been salvaged from an Ital- ratio ofthe two intervals is 78,892,313 to remarks: "There are many indications ian Renaiss;mce belvedere. In the base 79,108,313, but grinding gears \vith near- that the sole important application of of the cabinet, two glass panels allowing ly 80 million teeth is out ofthe question. arithmetic in Europe during the Middle a view of brass gear trains are a distinc- The clock approximates the r8tio as the Ages was the calculation of [the] EJster tively nineteenth-century element; they reciprocal of 1 + (450/611 X 11269), date." Knut.h's version of a sixteenth- look like the store windows of an which works Ollt to a sidereal day of century 8lgOli.thm for t.he calculation has apothecalY's shop. The paintings and nventy-three hours, fifty-six minutes, eight steps, and some of the steps

10 TI-IE SCIENCES· NOlJClllilcrIOcU'IH/!CI 1999 are t8irly compl.icated. Here (to paraphrase being completed. The origin,] clock h,d ta to fit the old design. The new mech- the mathematics slightly) is step fiv e: three mechanical Magi thac bowed anism began ticking on October 2,1842. down before the Vi rgi n and child every Sch wilgue was clearly thin king long- Calculate the 11 G + 20 + Z- X, w here hour o n the ho m . when he undertook the project, the ll umbers C, Z and X com e [rom ea rlier steps jn the aJgorithm. Nov..· reduce By the middle of the sixteenth cen- As I noted earlier, the leap-year mech- that sum modulo 30--that is, divide by 30 tUlY, the C lock ofthe Three Kings was anism include ::. P::lrts th::lt engage only and keep only the remainder. Label the no longer running and no longer at the once every 400 ye::lfs-parts that w ill result E, fo r the so- ca lJed epact, the " age" leading edge of ho rologi,,1 techno l- soon be tested for the flfSt time, and then of (h e moon at the start of rhe yea r. ogy. T o supervise an upgrade, the Stras- lie dormant again until 2400. Such very Finall y, if E is equal (0 25 and C is greater bo urgeois hired Conrad Dasypodiu s, rare events nught ha ve been left for (han 11, o r ifE is 24 , then increase Eby 1. the professor of mathemat ics ;u Stras- manual correction: it would have been Prograrruning a modern computer ro bo urg, as well as the cl ock maker Isaac only a slllall imposition o n the clock's perform the Easter calculatio n requires Habrecht and the artist T obias Stimmer. m aimainers co ask [hat [he hands be reset some care; programming a box of br.lSS Those three laid o ut the basic plan of eveL), [our . But Schwilgue evi- gears [Q do rhe arithmetic is truly a tour the instrument still seen today. includ- de ntly took pride and pleasure in get- de force. L have stared at diagrams ofthe ing the three- turreted case and most of ting the deta ils ri ght. H e couldn't know gears and linkages, and tried to trace out the p

' : one wheel turns once a century, another turns only once every 2,500 years.

twenty steps more in the same direction, to look too closely jnto its authenti- computer programs, even ifthey survive then another Z steps; fimlly, it would city. Early in the 1800s, the story goes, the Y2K scare, are explicitly limited to turn X steps counterclochvise. The a beadle was giving a tour of the cathe- dates between 1901 and 2099, The rea- "modulo 30" part ofthe program would dral, and mentioned that th e clock had son for choosing thar parricular span is th3t be taken care ofautomatically ifthe arith- been stopped for twenty yea rs. No one it makes the leap-year rule so simple: it's meti c were done on a circle with thiTty knevv how to fix it. A sm:lil voice piped just a of di visibility by 4. Under the divisions. So [.lr, so good. The thirty- up: " I w ill make it go!" The bo y w ho circ umstances, that design choice seems [Ooth w heel does exist in the Strasbourg prettywirnpy. IfSchw ilgue could take rhe cl oc k, and it is even helpfully labeled (rouble to fabricate wheels that make one " Epacfe . ." W here I get lost is in trying to revolution every 100 years and every 400 understand the va rious lever anns and years, surely a programIller could w rite rack-and-pinion assemblies that drive the the extr:lline o f code needed to check (or epacc wheel , and the cam foUo wers that the cenhuy exceptions, The tin e might communicate irs state [Q the rest of the never be needed, bm there's the s.1(isfac- system . There appear [Q be a number of tion ofk.nowing it's there. optirnizatio nsinthe works, w hich doubt- Other po res ofSchwilgue's clock look less save a li ttle b"J SS bur make the oper- even [lIl1her into the fi.tllre. There is a ation mo re obscure. Perhaps if l had a gear deep in the works ofthe ecclesiasti- model ofthe cl ock that ( could cake apatt cal computer that turns once every 2,500 and pur back together. years. And the celestial sphere out in fi'-Ollt But never mind my £ailures of spa- of (he clock has a s(ill-slower motion. 111 ti otempo ral reasoning. The mechanism addition to the sphere's daily rotation, it does work. Each New Ye.ar's Eve a met- pirouettes slowly on ;morher axis to reflect al ta g that mark::. the date ofEaster slides the precession of the of the along the circumference of the calendar Mechanism for computing church holidays ea rth 's o rbit through the of ri ng and takes up a position over the cor- the . In the rea l solar sys tem, that rect Sunday for the coming year. All the made the declaration was Jea n- B

NMJelllbe,/Dcrcm/le r 1999 • TH E SC IENC ES I I ius and no doubt paisley bdJ-bot- dar '-vas abandoned after 1,500 years; the been redoubled by a recent pro posa l [Q tolllS will be back in fashion. M ayan o ne may have lasted as long as build another IO ,OOO-year dock. The odo meter of as I mentioned 2,000 years, [he: Egyptian, possibly The new plan comes from Danny earlier, runs to 9999. According to so me 3,000. The Chinese h

BUT IF LOOKING AHEAD two or three generations is good, does that mean looking ahead doesn't actually work for his o wn clock, have come and gone. IfSchwil gue had Now, a book by Stewart Brand, the instj- at least no t w.itho ut some further tinker- rebui lt the Strasbourg clock just a few gator oethe JVholc EtfrtlJ Calfl!og. in g. T he pattern of leap years would earlier, it w ould hove listed dates T echnica l details of the Long Now continue correctly (o ssllming there is no ill Brumaire, Thermidor, Fructidor and clock remain to be worked alit, but the change to the Gregori,m calendar), but rhe o cher months decreed by the French provisional design that Brand descrjbes tl1e does not repeat on a Revolution ill September 1792, and the ha s a torsion pendululll (one that twists I O,OOO-yea r cycle. If the Stras- ra ther than swings) and a digital counter bourg clock cre

12 T HE SCIEN CES · Norlf lllb nIO("(Cf!1 /'l1f 119 9 be coming fi'om [h e acceleration of[ech- ma y be evanescent 011 a geologica.l or nology, [he sho rr- horizon perspecrive of astronomical timesc ale. Consicl er th e market-driven economics, the n ex[- plan (Q put one clock in a city (New election perspecrive of democracies, or , say) and another in a dt!sert (N eva- the di strac ri ons of pt!rsonal Illulritask- da). This makes se nse 110\\1, bu t will New in g." The big, slovv clock would offcr a York remain urban and Nevada sparse- coumerpoise to those fl' enetic tenden- ly popltiated for the next 10,000 years? cies: it would "embody . " Manya desolate spot in tlte desert today \vas once a city, and vice versa. H E OF PLANNING AI'IEAD, N eedless to say, the difficulty o f pre- Thusbanding resource-s, saving some- dieting the future is 11 0 W:lrrant to ignore thing for rhose who will come after, leav- it. The curre nt Y2K predica ment is clear evidence th:lt :J. horizon of two digits is roo short. But 200 or 300 generations IS better? fo ur di gits is pl enty. If we take up the habir of building machines meant to la st ing th e world a bener place- it's hard 10000, or ifwe write our computer pro- to quibble with al! that. Concern for the grams with r00111 fo r five-digi t years, we welfare ofone's children and grandchil- are not doing th e future (.1V01'. We're dren is surely a virtue-or at IC:I.'\t a Dar- merely nourishing OU I" OWI1 delusions. winian irnperarive-:md more general In the 15005. D:l sypoclius and his col- benevolence toward future inhabitants leagues could ha ve chosen to restO re (h e ofthe is also widely esteemed. BlIt 200-year-0Id C lock ofthe Three Kings if looki ng ah ea d twO or three genera- in , bur in steacl tions is good, does that lIl e;m looking they ripped om alt rr:l ces of it and buile -ahead twenty or thirty generati ons is bet- a new and better clock. T wo hundred ter? What about 200 or 300 generations? years later, Sc hwilglle was asked (0 Perhaps the answer depend.:; on how [11" the Oasypodiu s clock, bu t ins tead ahead you ca n aC[l]ally see. he evisceraced it :md instal1ed his own The Lo ng Now b'TOUP urges us to act mechanism in th e hollowed-out carcass. in the best interests of posterity. but Today, ;]fcer :l llOChl.!r two centuries, the bcyond :l ct: mury or two I have.:: no idea Long Now group is n O[ threatening to w lut those interests Inight be. To assu me destroy (he Schwil gut: clock. but nei- th:u the va lu es of our own age embody th er are [h ey worki ng to ensure etern:ll ve riti es and virtues is fo oli sh and longevity. They ignore it. They want to arrogant. For aU I know, so me future build a newer, better. difFerent clock. generation will thank LI S for burning lip good for 10,000 years. all th at noxious pt;'croleul11 and curse liS I begin to deteer a p:mern . The f.1C[ for exterminating the smallpox virus. is, winding and dusting and fLxing some- From a reading of Brand's book, I body else's old cl ock is boring. Building don't sense that the Long Now orga- a brand-new clock ofyour own is much ni ze rs ca n sec any further ahead th:1I1 the mo re fUll, particularly if you ca n pretend rest of us; as a matter of f.1ct, they seem that it's going to insp iTe awe ;}nd won- to be li ving in quite a shorr Now. AU der for ages to come. So why not havc th ose affli ctions li sted in theil" pream- the fun n ow and let [h e next 300 gen- ble-the foc us all gl1J.rtt'rl y ea rnings, erations do the borin g p3.rtl) ? qU:ldrcnni:l i electi ons and so forrh- an; bugaboos of recent years and decad es. F I THOUGHT THAT H ILLIS AND HIS They would have been incomprehensi- Iassociates might possibly slIcceed in ble a few centuries ago, and th ere's nor this act ofchronocolonialislll, enslaving 111l1 ch reao:;on to suppose they wi ll make hlture gener:l ti o l1 s to maintain Ollr lega- anybody's li st ofpressing concerns a few cy systeJ1ls, I would consider it Ill y duty centuries hence. to posterity to oppose the project, even The cllIplusis on the superiority of to sabotage it. BlIt in [1Ct I don't wony. binmy digital computing is something else I have f.1ith in th e future. Sometime in that puts a la te-twentierh-centll1Y date the 21 OOs 3 small child touring the ruins stamp on the project. A time may come ofthe C lo ck ofthe Long Now wi ll pro- when Hilli s'o:; binary counters will look claim: "Iwilt make it go!" And (h ac child just as quaint as Schwilgw!'s brass gears. will surely sc rap the whole mess and Long-te1111 thinking is rea ll y hard. Of build a new and berter clock, good fo r course.::, that's the point ofth e Long Now 10.000 years . • proj ect, but it's also a poim ofweakn css. It's hard to keep in nlind that what seems BRIA N HA YES is dJi'celaJI((, writer mid afor· Illost stea df.1st over th e human life span Ill er edirvr q{ AMERICAN SCIENTIST.

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