lld,[=-aa~'jIR'i~l,z:-]ii~lilzr Dirk Huylebrouck, Editor J

f you follow the summer crowd of square used to be an inn. High up on the I tourists biking along the Danube, you front wall, a plaque: on October 30, Kepler in may discover, close to , a short de- 1613, the astronomer tour leading through shady woods to celebrated here his marriage to Susanne Eferding. This is a quiet little Upper Reuttinger, the daughter of a burgher Eferding Austrian town, offering the usual sight- from Eferding. seems' fare: castle, church, and market- By then, Kepler was a widower of Karl Sigmund place. The first house on the main 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? If 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 The Keplerhof Inn (formerly The Lion) on the main square of Eferding and the plaque com- Mathematical Tourist Editor, memorating Kepler's wedding. The house, which is well over five hundred years old, has lately Dirk Huylebrouck, Aartshertogstraat 42, become derelict and will probably be taken over by a bank. Encased in one of its walls is the 8400 9 Belgium tombstone of a Jewish refugee from Regensburg who had found shelter in Eferding in the e-mail: dirk.huylebrouck@ping,be year 1410.

9 2001 SPRINGERWERLAG NEW YORK, VOLUME 23, NUMBER 2, 2001 47 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. Kepler in 1620, age forty-nine. According to his friend, the poet The drawing is from the title page of a treatise by Johann Frey, which Lansius, this is how Kepler did not look. Lansius jocularly put the was published in 1531 in Nuremberg. The formula (in white) is blame on the motion of the Earth (a heresy at that time, as Galileo Kepler's barrel rule. (From the cover of Analysis 1, 126, I.) came to learn): if the Earth had stood still, the artist's hand would have been steadier.

had broken off his theological studies third law on planetary motion, was al- Susanne was seventeen years in Ttibingen to become professor of ready a celebrity in European science, younger than Kepler and seemed mod- at a college in Graz. Later, but this cut no ice with the suspicious est, thrifty, and devoted. He had met her he joined the astronomer Tycho Brahe farmers, who often chased him ignomi- in the household of a friend with a ring- in Prague, as one of the many scien- nously away from their land. ing name--Erasmus von Starhemberg-- tists, astrologers, and alchemists at- Having gone through an unhappy first whose palace dominated Eferding and tracted there by Rudolf II, the oddest marriage, Kepler took great pains to whose family popped up in every century of all Habsburg emperors, a dreamer avoid all mistakes on his second matri- of 's history. Erasmus had stud- suffering from fits of madness, who monial adventure. We know from a long ied in Strasburg, Padua, and Tfibingen, ended as a prisoner in his own castle. and almost comically candid letter where he may well have met young Kepler, who furnished his fair share of (dated from Eferding one week before Kepler for the first time. He was in sym- horoscopes, eventually held the job of his wedding and addressed to a scholarly pathy with Kepler's religious plight--a Imperial Mathematician under three nobleman) that he had wavered for two few years later, at the outbreak of the Habsburg rulers. Each was more nfili- years between no fewer than eleven can- Thirty Years War, he would himself be tantly Catholic than his predecessor, un- didates, among them a widow and her branded as a "main rebel" by the Catholic fortunately, and this made court life dif- daughter. Some were too young; some, establishment--and had arranged for ficult for Kepler, a staunch Protestant. too ugly. Some seemed inconstant, and Kepler's transfer to Linz. (Later, when After Rudolfs death, he took a second others lost their patience with his tem- Starhemberg was imprisoned, Kepler job as " mathematician" in Linz. porising, which became the talk of the wrote to the Jesuit priest Guldin, a pro- This implied cartographical work, town. Eventually, Kepler decided for fessor at the University of Vienna and among other things. Kepler, by then number five, against the advice of all his no mean mathematician himself, to ask somewhere between his second and friends, who deemed her too lowly. him to intervene at the imperial court.)

THE MATHEMATICAL INTELLIGENCER As for Susanne, she was an orphan: she The fact that their content was mea- lish the leaflet--at that time, an oner- had no money, but on the other hand, sured by more complicated means in ous enterprise that required, for no in-laws either. She was a ward of other countries, for instance on the starters, buying the necessary reams of Baron Erasmus's wife Elisabeth, who Rhine, rendered him suspicious. But a paper. Actually, Kepler had even to patronized an institution for the up- few days sufficed to convince him of convince a printer, first, to set up shop bringing of impoverished young ladies. the validity of what he termed the in Linz. The inevitable delays, which After having taken the plunge, Kepler Austrian method. He wrote a short took almost two years, offered him the never mentioned his spouse again in all note, and dedicated it, as a New Year's opportunity to extend his results con- his copious correspondence, except on gift, to Maximilian von Liechtenstein siderably. His Nova stereometria do- the seven occasions when she gave and Helmhard Jhrger, two of his gen- liorum vinariorum grew to a full- birth. Based on this, all biographers erous supporters. He next tried to pub- fledged book. The first part deals with agree that the marriage indeed was a happy one. The Eferding wedding plays a curi- ous role in the prehistory of . In Kepler's words:

After my remarriage in November of last year, at a time when barrels of wine from Lower Austria were stored high on the shores of the Danube near Linz after a copious vintage, on offer for a reasonable price, it was the duty of the new hus- band and devoted family-head to purchase the drink needed for his household. Four days after several barrels had been brought to the cel- lar, the wine-seller came with a rod which he used to measure the con- tent of all barrels, irrespective of their form and without any further reckoning or computation. The metallic end of the gauge-rod was introduced through the bung-hole till it reached the opposite point on the border of the barrels's bottom.

9 I was amazed that the diagonal through the half-barrel could yield a measure for the volume, and I doubted that the method could work, since a much lower barrel with a somewhat broader bottom and hence much less content could have the same rod-length. To me as a newlywed, it did not seem inop- portune to investigate the mathe- matical principle behind the preci- sion of this practical and widespread Title page of the Nova Stereometria. When well-meaning experts told Kepler that a mathe- measurement, and to bring to light matical text, and in Latin at that, would never find buyers, he produced a German version the underlying geometrical laws. (The Art of Measurement of ), which appeared in 1615 and must be one of the first examples of popular science writing: it was considerably shorter than the Stereometria, Posterity did not record what Susanne written in down-to-earth language, and divested of most proofs. Kepler also wrote the first made out of this. science fiction ever, an account of a voyage to the moon. He decided not to publish the in- Kepler, whose father had been an tegral text of his "Dream" during his lifetime, but it raised rumors of black magic which sur- innkeeper when not abroad as a soldier faced during the nearly fatal witchcraft trial that his mother had to undergo in her last years 9 of fortune, must have been on familiar Kepler seems to have been the first to see science as the cumulative effort of successive terms with wine-casks of all shapes. generations.

VOLUME 23, NUMBER 2, 2001 49 cubatures in general, and in particular But the content of his book was not at and the circle's center for a vertex, has with the volumes of solids of revolu- all classical. In a remarkable display of the same area as the circle. The same tion. The second part deals with bar- intuition, he anticipated parts of calcu- works for the full sphere: it is made up rels. For Kepler, these were sometimes lus, arguing about infmities with a non- of infinitely many pyramids whose ver- cylinders, sometimes they consisted of chalance quite foreign to the rigor of the tices meet in the center; their bases re- two trunks of a cone, and sometimes exhaustion method of Archimedes (who duce to points on the surface of the they were what he termed "lemons" is invoked a great deal). For instance, sphere. In another vein, since a torus (obtained by rotating a semicircle's arc Kepler considers the area of the circle is obtained by rotating a circle around around its chord) whose top and bot- as being made up of infinitely many tri- a line that lies in the circle's plane (but tom had been sliced off. The third part angles having one vertex in the center, does not touch the circle), its volume of his book dealt with practical prob- and the opposite base, reduced to a is the product of the area of the circle lems in measuring the content of to- point, on the circumference. If the cir- times the circumference described by tally or partly filled casks. cle rolls along a line for one full turn, rotating its center around that axis. Kepler tried to avoid all algebra, and the baselines of the triangles cover an Indeed, the toms is made up of infi- wrote in the style of Greek geometers. interval. The triangle with this base, 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 error in assuming that it has uniform thickness; two errors, ac- tually, since the outer part of the wedge is thicker, and the inner part thinner. But these errors 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 what Arthur Koestler termed a scientific "sleep-walker." The Nova Stereometria's main re- sult consisted in finding, among all cylinders inscribed in a sphere, those with the maximal volume (today an easy exercise for first-year students). This implied that among all cylinders having the same "measure" given by the rod-length, those have maximal vol- ume whose height is equal to the di- ameter of the bottom multiplied by square root of two. Kepler added judi- ciously that his result was still ap- proximately valid for barrels of close to cylindrical shape: indeed, "whenever there is a transition from smaller to larger and back to smaller again, the dif- ferences are always insensible, to a de- gee." This anticipates an argtnnent explicitly made only decades later by Fermat: close to a maximum, changes are small; i.e., optima are critical points. Kepler's proof that the area of the circle is half the product of the radius times the length of So the rod-measurement works, as long the circumference (from the Art of Measurement of Archimedes). Early in the book, the num- as the barrels have approximately the ber pi is given as 22/7, but Kepler adds that this is not to be understood too narrowly: "even right proportion: height to bottom, like if one divides the diameter in twenty thousand thousand thousand times thousand parts of diagonal to side of the square. equal length, something of the circumference will remain that is smaller than such a small As it turns out, Austrian barrels had part," i.e., pi is irrational. (and still have) a height that is equal to

50 THE MATHEMATICAL INTELLIGENCER drinkable stuff may be around in copi- Max Caspar, Johannes Kepler, Dover, New ous quantities, York, 1993. Arthur Koestler, The Sleepwalkers, Hutchinson, Et cum pocula mille mensi erimus, London, 1959 (many reprintings). Conturbabimus illa, ne sciamus." For Kepler's relation to algebra, see P. Pesic, Kepler's Critique of Algebra, Mathematical Some dedicated teachers tried for five Intelligencer 22 (2000), no. 4, 54-59. years to teach me some Latin, but I cannot help you with the translation. There are several good Kepler sites on It's about wine and science, though. the net, for starters see www. kepler, arc. nasa. gov/johannes, hmtl EDITOR'S NOTE: About wine and igno- www.es.rice.edu/ES/humsoc/Galilieo/ rance, rather! A scholar informs us that Files/kepler.html the two lines, a learned allusion to www.roups.dcs.st-and.ac.uk/ Catullus's poem "Vivamus, mea Lesbia," history/Mathematics/Kepler.html mean, "and if we have measured each other a thousand vessels, we will con- Institut for Mathematik fuse them, in order not to know." Universit~t Wien Strudlhofgasse 4 REFERENCES 1090 Vienna Mechtild Lemcke, Johannes Kepler, Rowohlts Austria The best figure of all. Wine barrels in the run- Monographien, 1995. e-mail: [email protected] 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 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.

three times the radius of their bottom. The fact that 1.41422 is close to 1.50000 sufficed to persuade Kepler that Austrian barrels "had the best figure of all" (figurae omnium optissimae): in fact, he included this proud claim in the full title, which covers half of the fron- tispice of his book. Kepler goes on to ask, "Who will deny that nature can teach to humans through a vague feeling for form, without any recourse to rational arguments?" He toys with the possibil- ity that once upon a time, some pre- eminent geometer could have taught the rule to Austrian barrel-makers; but then he discards it, with the argument that in this case, other wine-growing countries would also have adopted the same rule. Kepler ends his treatise with a hearty prayer that "our spiritual and material goods may be preserved, and

VOLUME 23, NUMBER 2, 2001 51