l l d , [= -aa~ ' j IR ' i~ l , z : - ] i i~ l i l z r D i r k H u y l e b r o u c k , E d i t o r J

Kepler in Eferding Karl Sigmund

I f you follow the summer crowd of tourists biking along the Danube, you

may discover, close to Linz, a short de- tour leading through shady woods to Eferding. This is a quiet little Upper Austrian town, offering the usual sight- seems' fare: castle, church, and market- place. The first house on the main

square used to be an inn. High up on the front wall, a plaque: on October 30, 1613, the astronomer Johannes Kepler celebrated here his marriage to Susanne Reuttinger, the daughter of a burgher from Eferding.

By then, Kepler was a widower of 42. Born and raised in Wiirttemberg, he

Does your hometown have any

mathematical tourist attractions such

as statues, plaques, graves, the cafd

where the famous conjecture was made,

the desk where the famous initials

are scratched, birthplaces, houses, or

memorials? Have you encountered

a mathematical sight on your travels?

I f so, we invite you to submit to this

column a picture, a description of its

mathematical significance, and either

a map or directions so that others

may follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 �9 Belgium e-mail: dirk.huylebrouck@ping,be

The Keplerhof Inn (formerly The Lion) on the main square of Eferding and the plaque com-

memorating Kepler's wedding. The house, which is well over five hundred years old, has lately

become derelict and will probably be taken over by a bank. Encased in one of its walls is the

tombstone of a Jewish refugee from Regensburg who had found shelter in Eferding in the

year 1410.

�9 2001 SPRINGERWERLAG NEW YORK, VOLUME 23, NUMBER 2, 2001 4 7

Kepler in 1620, age forty-nine. According to his friend, the poet

Lansius, this is how Kepler did not look. Lansius jocularly put the

blame on the motion of the Earth (a heresy at that time, as Galileo

came to learn): if the Earth had stood still, the artist's hand would

have been steadier.

Measuring barrel contents. The left barrel is measured in the Austrian

way, by help of a gauging rod. The other barrel's volume is deter-

mined by pouring its content into vessels of specific volume. This is

from the title page of a book on analysis, which appeared in 1980.

The drawing is from the title page of a treatise by Johann Frey, which

was published in 1531 in Nuremberg. The formula (in white) is

Kepler's barrel rule. (From the cover of Analysis 1, 126, I.)

had broken off his theological studies in Ttibingen to become professor of mathematics at a college in Graz. Later, he joined the as t ronomer Tycho Brahe in Prague, as one of the many scien- tists, astrologers, and alchemists at- t racted there by Rudolf II, the oddest of all Habsburg emperors, a dreamer suffering from fits of madness, who ended as a prisoner in his own castle. Kepler, who furnished his fair share of horoscopes, eventually held the job of Imperial Mathematician under three Habsburg rulers. Each was more nfili- tantly Catholic than his predecessor, un- fortunately, and this made court life dif- ficult for Kepler, a staunch Protestant. After Rudolfs death, he took a second job as "district mathematician" in Linz. This implied cartographical work, among other things. Kepler, by then somewhere between his second and

third law on planetary motion, was al- ready a celebrity in European science, but this cut no ice with the suspicious farmers, who often chased him ignomi- nously away from their land.

Having gone through an unhappy first marriage, Kepler took great pains to avoid all mistakes on his second matri- monial adventure. We know from a long and almost comically candid letter (dated from Eferding one week before his wedding and addressed to a scholarly nobleman) that he had wavered for two years between no fewer than eleven can- didates, among them a widow and her daughter. Some were too young; some, too ugly. Some seemed inconstant, and others lost their patience with his tem- porising, which became the talk of the town. Eventually, Kepler decided for number five, against the advice of all his friends, who deemed her too lowly.

Susanne was seventeen years younger than Kepler and seemed mod- est, thrifty, and devoted. He had met her in the household of a friend with a ring- ing name--Erasmus von Starhemberg-- whose palace dominated Eferding and whose family popped up in every century of Austria's history. Erasmus had stud- ied in Strasburg, Padua, and Tfibingen, where he may well have met young Kepler for the first time. He was in sym- pathy with Kepler's religious plight--a few years later, at the outbreak of the Thirty Years War, he would himself be branded as a "main rebel" by the Catholic establishment--and had arranged for Kepler's transfer to Linz. (Later, when Starhemberg was imprisoned, Kepler wrote to the Jesuit priest Guldin, a pro- fessor at the University of Vienna and no mean mathematician himself, to ask him to intervene at the imperial court.)

THE MATHEMATICAL INTELLIGENCER

As for Susanne, she was an orphan: she had no money, but on the o ther hand,

no in-laws either. She was a ward of Baron Erasmus ' s wife Elisabeth, who

pa t ronized an inst i tut ion for the up- bringing of impover i shed young ladies. After having taken the plunge, Kepler

never ment ioned his spouse again in all his copious cor respondence , excep t on

the seven occas ions when she gave birth. Based on this, all b iographers agree that the marr iage indeed was a

happy one. The Eferding wedding p lays a curi-

ous role in the preh is tory of calculus.

In Kepler ' s words:

After my remarr iage in November of

last year, at a t ime when bar re ls of wine f rom Lower Austr ia were

s to red high on the shores of the Danube near Linz after a copious vintage, on offer for a r easonab le

price, it was the duty of the new hus- band and devoted family-head to pu rchase the dr ink needed for his

household . Four days af ter severa l bar re ls had been brought to the cel- lar, the wine-sel ler came with a rod

which he used to measure the con- tent of all barrels , i r respect ive of thei r form and wi thout any fur ther

reckoning or computat ion. The metal l ic end of the gauge-rod was in t roduced through the bung-hole

till it r eached the oppos i te poin t on the b o r d e r of the bar re ls ' s bot tom. � 9 I was amazed that the diagonal

through the half-barrel could yield a measure for the volume, and I doub ted that the me thod could work, s ince a much lower bar re l with a somewha t b roader bo t tom

and hence much less content could have the same rod-length. To me as a newlywed, it did not seem inop- por tune to investigate the mathe- matical principle behind the preci-

s ion of this pract ical and widespread measurement , and to bring to light

the underlying geometrical laws.

Pos ter i ty did not r ecord wha t Susanne

made out of this. Kepler, whose father had been an

innkeepe r when not abroad as a so ld ier of fortune, mus t have been on famil iar te rms with wine-casks of all shapes.

The fact that thei r content was mea-

sured by more compl ica ted means in o ther countries, for ins tance on the

Rhine, r endered him suspicious. But a few days suff iced to convince him of the validi ty of wha t he t e rmed the Austr ian method. He wrote a shor t

note, and ded ica ted it, as a New Year 's gift, to Maximil ian von Liechtenstein

and Helmhard Jhrger, two of his gen- erous suppor ters . He nex t t r ied to pub-

lish the l ea f l e t - - a t tha t t ime, an oner- ous enterpr ise that required, for

s tar ters , buying the neces sa ry reams of paper . Actually, Kepler had even to convince a printer, first, to set up shop

in Linz. The inevi table delays, which t ook a lmost two years, offered him the

oppor tun i ty to ex tend his resul ts con- s iderably. His Nova stereometria do- l iorum vinar iorum grew to a full-

f ledged book. The first par t dea ls with

Title page of the Nova Stereometria. When well-meaning experts told Kepler that a mathe-

matical text, and in Latin at that, would never find buyers, he produced a German version

(The Art of Measurement of Archimedes), which appeared in 1615 and must be one of the

first examples of popular science writing: it was considerably shorter than the Stereometria,

written in down-to-earth language, and divested of most proofs. Kepler also wrote the first

science fiction ever, an account of a voyage to the moon. He decided not to publish the in-

tegral text of his "Dream" during his lifetime, but it raised rumors of black magic which sur-

faced during the nearly fatal witchcraft trial that his mother had to undergo in her last years�9

Kepler seems to have been the first to see science as the cumulative effort of successive

generations.

VOLUME 23, NUMBER 2, 2001 4 9

cuba tu res in general, and in par t icu lar wi th the volumes of sol ids of revolu-

tion. The second par t dea ls wi th bar- rels. F o r Kepler, these were somet imes cylinders, somet imes they cons i s ted of

two t runks of a cone, and somet imes they were what he t e rmed "lemons" (ob ta ined by rotat ing a semic i rc le ' s arc

a round its chord) whose top and bot- tom had been sl iced off. The third par t

of his book dealt with prac t ica l prob- lems in measur ing the con ten t of to- ta l ly or par t ly filled casks.

Kepler tried to avoid all algebra, and wrote in the style of Greek geometers.

But the content of his book was not at all classical. In a remarkable display of

intuition, he anticipated parts of calcu- lus, arguing about infmities with a non- chalance quite foreign to the rigor of the

exhaust ion method of Archimedes (who

is invoked a great deal). For instance, Kepler cons iders the a rea of the circle as being made up of infinitely many tri- angles having one ver tex in the center ,

and the oppos i te base, r educed to a

point, on the circumference. If the cir- cle rol ls a long a line for one full turn, the base l ines of the tr iangles cover an interval. The triangle with this base,

Kepler's proof that the area of the circle is half the product of the radius times the length of

the circumference (from the Art of Measurement of Archimedes). Early in the book, the num-

ber pi is given as 22/7, but Kepler adds that this is not to be understood too narrowly: "even

if one divides the diameter in twenty thousand thousand thousand times thousand parts of

equal length, something of the circumference will remain that is smaller than such a small

part," i.e., pi is irrational.

and the circle 's cen te r for a vertex, has

the same a rea as the circle. The same works for the full sphere: it is made up

of infinitely many pyramids whose ver- t ices meet in the center; their bases re- duce to poin ts on the surface of the

sphere. In ano the r vein, s ince a torus is obta ined by rota t ing a circle a round a line that l ies in the circle 's plane (but

does not touch the circle), its volume is the p roduc t of the a rea of the circle

t imes the c i rcumference descr ibed by rotat ing its cen te r a round that axis.

Indeed, the t o m s is made up of infi- nitely many thin discs, whose volumes have to be added up. Kepler admits that

since such a disc is more like a wedge, he makes an er ror in assuming that it has uniform thickness; two errors, ac- tually, s ince the ou te r par t of the wedge

is thicker, and the inner par t thinner. But these er rors cancel each other. The arguments run fast and loose, and a few

of the results are wrong. But they came decades before Bonaventura Cavalieri, Rend Descartes, and Pierre de Fermat ,

and they display in their blind groping toward calculus an uncanny sense of di-

rection. Kepler may well be the fore- most example of wha t Arthur Koest ler t e rmed a scientif ic "sleep-walker."

The Nova Stereometria's main re-

sult cons is ted in finding, among all cyl inders inscr ibed in a sphere, those with the maximal volume ( today an

easy exerc ise for f i rs t-year s tudents) . This implied that among all cyl inders having the same "measure" given by the rod-length, those have maximal vol-

ume whose height is equal to the di- ameter of the bo t t om mult ipl ied by square root of two. Kepler added judi- c iously that his resul t was still ap-

p rox imate ly valid for barre ls of c lose to cylindrical shape: indeed, "whenever there is a transit ion from smaller to larger and back to smaller again, the dif-

ferences are always insensible, to a de- g e e . " This anticipates an argtnnent explicit ly made only decades la ter by Fermat: close to a maximum, changes are small; i.e., opt ima are critical points.

So the rod-measurement works, as long as the barrels have approximately the right proportion: height to bottom, like diagonal to side of the square.

As it turns out, Aust r ian barrels had

(and still have) a height that is equal to

5 0 THE MATHEMATICAL INTELLIGENCER

The best figure of all. Wine barrels in the run-

down entrance of the Keplerhof Inn. In

German textbooks, Kepler's name is associ-

ated with the so-called barrel rule of numer-

ical integration (Simpson's 1/3 rule). In spite

of Kepler's praise of Austrian barrels, the dis-

trict deputies of Upper Austria decided in

1616, after a formal scrutiny of all his publi-

cations, to dispense with his services. How-

ever, influential friends made sure that this

decision was never put into effect.

drinkable stuff m a y be around in copi-

ous quantities,

Et cum pocu la mil le mensi erimus,

Conturbabimus illa, ne sciamus."

Some ded ica ted t eache r s tr ied for five

years to teach me some Latin, but I cannot help you wi th the translat ion. It 's about wine and science, though.

EDITOR'S NOTE: About wine and igno- rance, rather! A scholar informs us that

the two lines, a learned allusion to Catullus's poem "Vivamus, mea Lesbia,"

mean, "and if we have measured each other a thousand vessels, we will con-

fuse them, in o rder not to know."

REFERENCES

Mechtild Lemcke, Johannes Kepler, Rowohlts

Monographien, 1995.

Max Caspar, Johannes Kepler, Dover, New

York, 1993.

Arthur Koestler, The Sleepwalkers, Hutchinson,

London, 1959 (many reprintings).

For Kepler's relation to algebra, see P. Pesic,

Kepler's Critique of Algebra, Mathematical

Intelligencer 22 (2000), no. 4, 54-59.

There are several good Kepler s i tes on

the net, for s tar ters see www. kepler, arc. nasa. gov/j ohannes, hmtl www.es . r ice .edu/ES/humsoc/Gal i l ieo/

Fi les /kepler .html www.roups .dcs .s t -and.ac .uk/

h is tory/Mathemat ics /Kepler .h tml

Institut for Mathematik

Universit~t Wien

Strudlhofgasse 4

1090 Vienna

Austria

e-mail: karl.sigmund@univie.ac.at

three t imes the radius of thei r bot tom. The fact that 1.41422 is c lose to 1.50000 suff iced to pe r suade Kepler tha t Austr ian bar re l s "had the bes t figure of

all" (figurae omnium optissimae): in fact, he inc luded this p roud claim in the full title, which covers half of the fron-

t ispice of his book. Kepler goes on to ask, "Who will

deny tha t na ture can teach geomet ry to humans th rough a vague feeling for form, wi thout any recourse to ra t ional arguments?" He toys with the possibi l -

ity that once upon a time, some pre- eminent geomete r could have taught the rule to Aust r ian barrel -makers ; bu t

then he d i scards it, with the a rgument that in this case, o ther wine-growing countr ies would also have adop ted the same rule. Kepler ends his t rea t i se wi th a hear ty p raye r that "our spir i tual and

mater ia l goods may be preserved, and

VOLUME 23, NUMBER 2, 2001 51