A Centennial Celebration of Two Great Scholars: Heiberg’s Translation of the Lost of —1907 Heath’s Publication on ’s Elements—1908

Shirley B. Gray

he 1998 auction of the “lost” palimp- tains four illuminated sest of Archimedes, followed by col- plates, presumably of laborative work centered at the Walters Matthew, Mark, Luke, Art Museum, the palimpsest’s newest and John. caretaker, remind Notices readers of Heiberg was emi- Tthe herculean contributions of two great classical nently qualified for scholars. Working one century ago, Johan Ludvig support from a foun- Heiberg and Sir Thomas Little Heath were busily dation. His stature as a engaged in virtually “running the table” of great scholar in the interna- bequeathed from antiquity. Only tional community was World War I and a depleted supply of manuscripts such that the University forced them to take a break. In 2008 we as math- of Oxford had awarded ematicians should honor their watershed efforts to him an honorary doc- make the cornerstones of our discipline available Johan Ludvig Heiberg. torate of literature in Photo courtesy of to even mathematically challenged readers. The Danish Royal Society. 1904. His background in languages and his pub- Heiberg lications were impressive. His first language was In 1906 the Carlsberg Foundation awarded 300 Danish but he frequently published in German. kroner to Johan Ludvig Heiberg (1854–1928), a He had publications in Latin as well as Arabic. But classical philologist at the University of Copenha- his true passion was classical Greek. In his first gen, to journey to (present day Is- position as a schoolmaster and principal, Heiberg tanbul) to investigate a palimpsest that previously insisted that his students learn Greek and Greek had been in the library of the Metochion, i.e., the mathematics—in Greek. Indeed, he made his “daughter” or “sharing” house of the Church of the debut into public life at the age of thirty when Holy Sepulcher in Jerusalem. Heiberg’s proposal he led a movement to require the study of Greek to the foundation was to make photographs of a in all Danish schools. He even founded a Greek “damaged and readable, but very hard to decipher” social society that continued until after his Byzantine Greek work on vellum. Over its thousand death. year history, the palimpsest had journeyed be- He had entered Copenhagen University as a tween these two libraries. young man of fifteen and immediately took up Indeed, the overscript is a euchologion most likely mathematics, Greek, and classic philology. He com- written near the turn of the thirteenth century by pleted his doctoral work at the inordinately young a priest of the church. Today the palimpsest con- age of twenty-five with a thorough, very meticulous chronology of the works of Archimedes. He was Shirley B. Gray is professor of mathematics at California especially attracted to The Sand-Reckoner problem State University, Los Angeles. Her email address is sgray@ dealing with the of grains of sand needed calstatela.edu. to fill the . His capstone graduation trip The author thanks Nigel Wilson (Lincoln College, Oxford), was naturally to Italy to investigate more deeply Adrian Ponce (Jet Propulsion Laboratory, Pasadena, CA), the background of his dissertation, Quœstiones Akif Tezcan (University of California, San Diego) and Erik Archinedeœ. Petersen (Det Kongelige Bibliotek).

776 Notices of the AMS Volume 55, Number 7 possibility of finding additional Greek mathemat- ics . Following the award of the Carlsberg grant and subsequent travel, Heiberg arrived in Con- stantinople in 1906 for the first of two trips. He painstakingly transcribed the original Greek under- script using only a magnifying glass and various light sources. Judging by the proposal, one feels that Heiberg probably left Copenhagen thinking he could successfully photograph the palimpsest and then later read the prints at his leisure upon returning home to Denmark. Being a meticulous scholar, Heiberg returned to Constantinople in 1908 to refine and check his earlier work. Then

Photograph courtesy of the author. the of courtesy Photograph satisfied with his notes, he was joined by a German . colleague, H. G. Zeuthen, who assisted him in veri- The Metochion faces the main courtyard and fying a transcription of the text. This watershed entrance to the Church of the Holy Sepulcher in discovery was announced in Eine neue Archimedes- Jerusalem. Handschrift, Hermes XLII (1907), pp. 235–297. This author attended the 1998 auction at Chris- tie’s in New York and has examined the palimpsest Upon returning to Denmark he initiated a ca- both prior to and after its Christie’s sale; also, the reer of translating classical manuscripts. While one stolen page now in the Cambridge University not abandoning Archimedes, his first academic Library, Add. 1879.23. In addition, the 65 photos recognition came through a parallel interest in taken by Heiberg in 1906 were examined both when various editions of Euclid. Dijksterhuis, the Dutch on loan to the Walters Art Museum in Baltimore scholar, would later write that “If a reader was to and in their home in Copenhagen’s Royal Library. become acquainted with the work of a writer of Ar- The condition of the palimpsest is poor. Mildew chimedes’ level, an understanding of the Elements

over the past century has marked some pages with Bibliotek Kongelige Det of courtesy below Photograph of Euclid was—and still is—a prerequi- site, though by no means a sufficient preparation.” Heiberg’s publications underscore his multiple talents. In ad- dition to mathematics, he practiced his textual skill by putting the 1269 Latin works of Flemish Dominican Willem van Moerbeke back into their original Greek. Heiberg’s work was recognized by his appointment (1896) to be profes- sor at the University of Copenhagen. For over two decades and in several publications, Heiberg refined the work on classical Archimedean manuscripts initiated in his doctoral dissertation. When he wrote his proposal to the Carlsberg Foundation, Heiberg was aware of two major unexplored refer- ences. First, there was mention of a palimpsest in a travel book, Reise in der Orient (Leipzig 1846), written by the German biblical scholar Constan- tine Tischendorf and translated into English by W. E. Shuckard (London 1847). A half century later, A. I. Pa- padopoulos-Kerameus cataloged 890 Greek manuscripts at the church in Heiberg's original publication (1907) draws attention to the missing Constantinople. This catalog, which portions of Propositions 14 and 15 in the Method. Even with the most was published in 1899, came to the at- advanced technology, it may not be possible to image these critical tention of H. Schöne in Germany, who sections, thus leaving mathematicians to speculate on Archimedes' then alerted Heiberg in Denmark of the approach.

August 2008 Notices of the AMS 777 a purple stain. Browning around the edges suggests was primarily a source of interest only to the con- that in addition to being quite aged, the palimpsest noiseur of mathematics (Wilson, 1999). There are may narrowly have escaped a fire. Far worse today, numerous manuscripts and editions of Euclid in much of the text that Heiberg tediously translated European libraries, but the textual transmission is no longer visible. Thus the photos taken in Con- of Archimedes across the centuries “hangs by a stantinople one century earlier have become a valu- slender thread.” There are not many occasions able guide for contemporary scholars. The photos, when today’s translators can catch a glimpse as enlargements of the original leaves, are easier of past experts at work on Archimedes. Among to read than the Archimedean underscript visible Heiberg’s papers in Copenhagen is a hand-written in 2008. In translating the manuscript listing where he found references to , Heiberg copies of Archimedes and, all too frequently, their sequenced the pages and later disappearance. Primarily his sources, bridging identified the original book, centuries of time, were found in the Vatican, other i.e., Equilibrium of Planes, Italian libraries, and Basel. For example, Heiberg Floating Bodies, Method of wrote that Leonardo da Vinci had found one copy Mechanical Theorems, Spi- in the library of the Bishop of Padua, and noted rals, On the and Cyl- Renaissance men truly did seek inder, Measurement of the manuscripts to translate mathematics. The Dutch Circle, and Stomachion. Of Archimedean scholar E. J. Dijksterhuis continued these seven texts, four (see Heiberg’s “philological investigations” of manu- later sections, each marked scripts and printed editions. Wilson, Heiberg, and with an asterisk) were al- scholars before them have noted where they were ready known from other unable to trace a missing copy. sources. Also, readers might As mathematicians, readers of the Notices will note that the The Sand- recall classic problems from each of the books. We Reckoner entered our litera- briefly remind readers of their contents. ture from other sources and Equilibrium of Planes* is not in the palimpsest. Archimedes, in writing on rectilinear figures, There is an additional chal- followed the century-earlier investigations of lenge. When the thirteenth , but constructed propositions knowing In the 1450s Pope Nicholas V century scribe set about to the mathematical dictates of Euclid. This treatise commissioned a new translation scrape off the original Ar- uses the to concentrate on of Archimedes. This richly illu- chimedean manuscript to the center of gravity of , trapezoids, and minated manuscript, considered reuse the parchment for a parabolic segments. We thus find a veiled sugges- to be one of the Vatican’s trea- prayer book, he simply se- tion of modern as well as the principle sures, ends with “Finis librorum lected the leaf he felt was of moments, also known as the law of levers. Ar- Archimedis quos transcrivi ius- in the best condition. Thus chimedes has been called the “Father of Classical sit dominus Franciscus Burgen- today sequential pages of ”. sis sempre deo laus”. Basilese, the overscript in the leg- Floating Bodies 1544. (Note the small cone, ible prayer book represent The palimpsest is the only known surviving copy sphere, and cylinder in Archime- randomly selected leaves of of Floating Bodies in the original Greek. In this des’ left hand.) unorganized mathematics text we find the principle of buoyancy. Familiarity in the underscript. One can with these investigations was probably a factor imagine the chaos of mixing in Archimedes’ legendary naked run through the the pages of this issue of the Notices translated streets of Syracuse yelling the famous “Eureka, into a foreign language and then trying to read Eureka!” Similarly, his understanding of hydro- 178 pages of mathematics overwritten by religious statics and levers combined to supposedly move text. The ancient Greek mathematics in the under- a floundered ship built for King Hiero. This led script, if visible, is faded, tedious to translate, and to “Give me a place to stand and I can move the randomly organized when compared to its original .” Travelers to third world countries may source. In brief, this might be called a scholar’s still see the Archimedean water screw being used nightmare. Fortunately, the very best are hard at to irrigate fields, drain marshes, or empty bilge work on the translation. Nigel Wilson at Lincoln water from a boat. College, Oxford, and Reviel Netz, Department of Spirals* , Stanford University, are the principal Unlike many expressions in mathematics, the scholars involved in the task. Spiral of Archimedes r ​ ​a​θ​ seems to be properly Another factor emerges. Nigel Wilson makes = Photograph above courtesy of the Vatican Library. Vatican the of courtesy above Photograph named. He appears to be the very first of a long list the point that while Euclidean was a of distinguished mathematicians, e.g., Descartes, significant portion of the quadrivium and thus Fermat, Bernoulli, and Euler, to investigate its spe- widely studied in the Middle Ages, Archimedes cial properties. In particular the bounded by

778 Notices of the AMS Volume 55, Number 7 the Spiral is one of 28 propositions in this text. He of game or puzzle consisting of fourteen pieces wrote, “The space bounded by the spiral and the of ivory in various triangular shapes cut from a initial line after one complete revolution is equal square. It is assumed that these pieces were to be to one-third of the circle described from the fixed reassembled into other forms, e.g., a ship, sword, extremity as center, with radius that part of the or tree. After much thought, Netz concluded that initial line over which the moving point advances Archimedes asked the modern combinatorial ques- ​1 in one revolution.” [A ​ ​​π​(2π​a​)2 ]. tion, “In how many ways can you put the 14 pieces = 3 On the Sphere and Cylinder* together to form a square?” Answer: 17,152. Netz George F. Simmons calls Archimedes’ discovery headed a team of Persi Diaconis, Susan Holmes, of the formula for the volume of a sphere “one Ronald Graham, Fan Chung, and William Cutler of the greatest mathematical achievements of all who systematically counted all possibilities and time” (Simmons, 1992). There are several levels of then validated their calculation with a computer importance. Here Archimedes described to Eratos- program. thenes, his friend at the Museum in , Method of Mechanical Theorems a “most wonderful demonstration” that the sum The Method holds a very special niche among pure of slices of cross sections of a sphere and a cone mathematicians and classicists. The book opens would counterbalance the sum of slices of a related with a letter sent by Archimedes in to Eratos- cylinder. Then, by his choice of words in Greek, we thenes in Alexandria. For decades mathematicians are certain that Archimedes recognized the differ- have savored its closing: ence between a heuristic investigation, θ​​ωρ​​ιν, and the very cornerstone of mathematical rea- I am convinced that it (the method) soning—proof or α​π​oδ​ικ​ν​υ​ν​α​ι. Moreover, he will prove very useful for mathemati- was applying the principle of the lever balancing cians; in fact, I presume there will be a center of gravity from classical mechanics. By some among the present as well as thinking in terms of solids being dissected into future generations who by means of the slices, he was again on the brink of modern calcu- method here explained, will be enabled lus. Heiberg translated the same passages into Ger- to find other theorems which have not man as “……dass er sie vorher durch die Mechanik yet fallen to our share. gefunden, nachher geometrisch bewiesen habe” (Heiberg, 1908). —(Dijksterhuis, pp. 313–315, et al) The one leaf in the Cambridge Library, now known to be taken from On the Sphere and the This has to be one of the more provocative Cylinder, is thought to have been stolen by Tisch- statements ever written by a great mathematician. endorf, for it was sold by his executors in 1876. Clearly Heiberg, Heath, and dozens of other schol- This leaf is embedded in air-tight plastic and its ars were and are intrigued. wrote, condition is strikingly better than the heavily mil- “It is…extraordinary that fully eight decades later dewed manuscript auctioned at Christie’s. Both there is still no adequate translation of this work the leaf and the manuscript are scorched at the into English; for the version by Heath is based on edges and thus must have been in a fire. We owe Heiberg’s text.” Alas, a scholar cannot translate the identification and transcription of this leaf to that which cannot be seen; thus, the importance Nigel G. Wilson who also prepared much of the text of new imaging techniques for the current quest for the Christie’s catalog. to translate the palimpsest. While presenting Measurement of the Circle * meticulous details about other propositions, Dijk- The most significant contribution of this particular sterhuis basically skips Propositions 14 and 15 (pp. text is Archimedes’ method for approximating a 332–335) due to gaps in the visible evidence. value for π​. (Both he and the early Chinese math- In particular, the interpretation of methods in ematician Liu Hui (ca. 250 AD) used only words, not Proposition 14 and Proposition 15 is at stake. Be a symbol, to express the ratio of the circumference reminded that the shoulders of giants stood on by of a circle to its diameter. The symbol π​ did not Archimedes surely included the following: appear until much later in the eighteenth century.) 1. From Eudoxus (408-355 BC) he knew the Once again, his approach was the method of ex- concept of a “method of exhaustion”. haustion. This time he calculated a difference in 2. From (460-380 BC) he knew the by successively inscribing and circumscrib- special case for finding the of the circle ing regular polyhedra of 6, 12, 24, 48, and 96 sides (the area of a curvilinear lune equals the area of about a circle. one-half of a 45o​ -45o​ -90o​ inscribed in a Stomachion semicircle). Heiberg recovered only the opening sentences 3. From Euclid (323-285 BC) he surely had though other references (Dijksterhuis, 1987, pp. learned that in two similar solids (a) the lateral and 408–412) to this text have become known as the surface areas have the same ratio as the square loculus Archimedius (Archimedes’ box), a kind of any pair of corresponding segments; (b) the

August 2008 Notices of the AMS 779 In Proposition 14 Archimedes deals with a proof introduced earlier regarding the volume of a cylinder “hoof”; thus, he deals with cross sections of a paraboloid to conclude that the volume of the “hoof” is two-thirds of that of the prism and consequently one-sixth of that of the cir- cumscribed about the cylinder. His method now avoids the concept of equilibrium, but continues with visualizing a solid as the sum of its parallel intersections. To quote Dijkersterhuis, “This proof therefore does not yet satisfy the requirement of exactness.” Proposition 15 was “lost” one century ago, but appeared to Heath to be the same posited in Proposition 14, but with a different method to drawing the same conclusion. “As however the two propositions are separately stated, there is Both figures this page used with permission of Det Kongelige Bibliotek. Kongelige Det of permission with used page this figures Both no doubt that Archimedes’ proofs of them were The geometrical illustrations in the original distinct.” (Well, maybe; the gaps are enormous.) palimpsest were drawn with a finer stylus Dijksterhuis asserts “the concept of volume as than the pen used for text. In several instances being the sum of the areas of parallel intersections Heiberg and Heath were forced to guess at is abandoned” and then takes a giant scholarly leap their reconstruction. This illustration of the in publishing “the method of the indirect limiting “Cylinderhüf” or “cylinder hoof” was drawn by process is applied.” He inserts his personal proof Heiberg (Heiberg, 1908, p. 7). by contradiction. Continuing on Proposition 15, he refers readers to Heath’s reconstruction (p. 48) but fails to elaborate. Heath writes, “There is no doubt that Archimedes proceeded to, and completed, the rigorous geometrical proof by the method of exhaustion” (p. 51). Did Archimedes really grasp integration as we think of it today? Several points are to be made. In London, Heath, nearing completion of his monu- mental work on Euclid, perused the investigations of his colleagues’ research from the Middle East to write, “One of the two geometrical proofs is lost, but fragments of the other are contained in the manuscript which are sufficient to show that the method was the orthodox method of exhaustion in the form in which Archimedes applies it elsewhere, and to enable the proof to be reconstructed.” A purist would remark that a reconstruction is not Heiberg’s illustration of Archimedes’ signature investigations a verbatim or literal translation. The missing por- of related volumes of cones, , and cylinders (Heiberg, tions, the lacunae, are important. 1908, p. 5). The four premises known to Archimedes have been refined over their 2,000 year history. To the volumes have the same ratio as the cube of any method of exhaustion have been added modern pair of corresponding segments. concepts of limit, , and convergence. Today 4. Archimedes experimentally determined that we speak of the center of gravity and calculate the the areas of cross sections of two bodies have an moment of a system about an axis. Cross sections of thickness must be parallel to one another and intimate relationship: the length of the lever arm d​x perpendicular to an axis. and notation times the area of one cross section equals a con- are far less cumbersome than doing mathematics stant times the area of a cross section of another expressed in literal sentences. Heath informally ob- solid. served, the “dx” thickness of perpendicular cross In translating the letter from Archimedes to sections cancels on both sides of Archimedes’ , Heiberg used the German verb expression of equilibrium. “knüpfen” or “tying together” for what Archimedes Heiberg summarized the evolution of Archime- hoped to achieve with their collaborative knowl- des with different emphases: “Even stranger is the edge of mathematical truths in the Method. boldness with which the assumption operated that

780 Notices of the AMS Volume 55, Number 7 (1) a plane is filled with lines and that (2) a body con- exploited, but care must be exercised in such sists of or is filled with circles. This is, in fact, our experiments, since UV light has the potential to infinity-based calculus, which Archimedes already bleach the delicate traces of ink in the thousand- used as a method while the meaning of infinity in year-old manuscript. Recall the darkened libraries Greek mathematics was usually strongly rejected and art museums prescribed by conservators. as exactly not comprehensible and therefore The most recent imaging technique has been to dangerous. That is why Archimedes emphasizes scan the palimpsest using rapidly emitted, finely that these ‘Raisonnements’ cannot be proven and focused x-ray beams that excite fluorescence in instead only yield likely suspicions that require the latent inks. Thus the metal atoms in ancient a strict form of exhaustion, the geometric proof” inks come alive. Related spectroscopic methods (Heiberg, 1908). have long been used by physicists and chemists, C. H. Edwards writes “While it is true that especially at the Department of Energy’s Stanford Archimedes’ work ultimately gave birth to the Linear Accelerator Center (SLAC), to determine calculus, three indispensable ingredients of the the precise separations of atoms in biological and calculus are missing in his method” (Edwards, other complex molecular structures. The intensity 1979). Briefly, (1) the explicit introduction of limit of the fluorescence in key regions can be ampli- concepts, (2) a computational for the fied to produce clear images if any trace of the ink calculation of areas and volumes, and (3) a rec- remains in previously hidden writings. (Mathemati- ognition of the inverse relationship between area cians will enjoy discussion with colleagues in the and tangent problems. Any instructor teaching physical and biological sciences concerning the rel- calculus can recognize the symbiotic relationship ative merits of steady- between Archimedes and chapters we routinely state fluorescence and teach today. Yet there is an incompleteness to the the newer time-resolved thoughts of Archimedes that will be debated even absorption-emission after the palimpsest project finishes its imaging methods that employ and translation. femtosecond and pico- second x-ray pulses.) 2008 Technology We wish to see the The state of the art for photography was not text and geometrical sufficiently developed in 1906 to enhance the illustrations from The pale, stone-scrubbed underscripted ink, though Method that may or may Heiberg translated approximately 80 important not validate our fondest mathematical illustrations that could not be seen. wish that Archimedes Of the 1771 leaves noted by Heiberg, Heath later did, in fact, have the reported 29 leaves that were destitute of any trace thoughts required for Thomas Little Heath. of underscript, and 9 pages that had been “hope- Photo courtesy of the Royal modern techniques of lessly washed off; on a few more leaves, only a few Society London. finding volumes by in- words can be made out; and again some 14 leaves tegral calculus. We want have old writing upon them in a different hand and to see and to read all of the world’s oldest copy of with no division into columns.” Archimedes’ method. Both Heiberg and Heath were captives of the technology of their generation. Imaging is in a far Heath more advanced state today. Mathematicians and Heiberg’s announcement of finding the palimpsest paleographers are intrigued by the wish to see the (1907) and his subsequent publications appearing missing late twelfth or early thirteenth century in German must have come as a great surprise to illustrations and proof—not their reconstruction. Heath. He immediately set about to update The We have no need to question the work of the great Works of Archimedes (1897) with a supplement, scholars at the turn of the last century. But today The Method of Archimedes Recently Discovered we have far more imaging options. Powerful opti- by Heiberg (1912). In English, this publication cal spectroscopic methods can be employed along found in many university libraries has become an with digital enhancement techniques to reveal indispensable source for scholars of Propositions features in the palimpsest that hitherto have not 14 and 15. been legible. (The fluorescence and other spectro- Heath was truly in his element when writing scopic properties of pre-Columbian inks used by about Heiberg’s translation. Like Newton, Heath North American Indians are well known.) Strik- was born in Lincolnshire and educated at Trinity ingly characteristic emission spectra of different College, Cambridge. Following his “first class” inks observed under ultraviolet excitation can be on the tripos, he went up to London to become a civil servant in the Treasury. He was quick, ac- 1In examining the palimpsest in 1999 Nigel Wilson counted ​ curate, neat, and thorough in all written work. 178 leaves. In 1906 Heiberg made 65 photos, some of a single page, others of an opening. His honesty was never questioned. However, the

August 2008 Notices of the AMS 781 Bach’s ‘48’. Watching him, it was obvious at what point his difficulty was cleared up, when he would go back to his desk and continue his writing.” He even published Euclid in the original Greek (1920), hoping his notes would fascinate schoolboys into doing more proofs.

Heiberg and Heath after Constantinople Heiberg was awarded two additional honorary doctorates, one from Leipzig (1909) and one in medicine from Berlin (1910). Throughout his career he had been both a prolific author and an editor,

Photograph courtesy of the author. the of courtesy Photograph but largely turned to Greek medicine in his final

Photograph courtesy of the author. The Palimpsest is now under the watchful care years. He also served as rector of the University of the Walters Art Museum in Baltimore. of Copenhagen (1915–1916). As a shy young man, few would have predicted he would become a gre- very qualities that would merge into his becom- garious social leader, much loved by his students ing one of the world’s greatest classical scholars and colleagues. While his friends were strongly would also lead to his demise in the office place. religious, he attempted to remove both Oriental Victorian to the hilt, Heath opposed women in the and mystic influences in his writings. He had office, telephones, and oral briefings; moreover, little interest in Roman contributions. Outside of he cut budgets. Heath was at ease with the stuffy, antiquity, he admired Kierkegård. Travelers to Co- candle-saving formality of the old Treasury prior penhagen today may visit his grave in Holmens and to World War I. But modernization reforms under see his portrait in the restored Renaissance-style Lloyd George resulted in his being pushed aside. Frederiksborg Castle on a lake near Hillerød. Those who cut budgets are seldom popular. Pos- Like Heiberg, honors filled Heath’s career. Un- sibly his pedantic stiffness was unsettling to those doubtedly his thoroughness in translating Euclid less dedicated in their work. led to his being knighted (K.C.B., 1909; K.C.V.O., Heath was one of a small number of British civil 1916) and named a Fellow of the Royal Society servants who managed to use leisure time to make (1912). He joined Heiberg in being awarded an hon- a major impact on scholarship. Thus, he achieved orary from the University of Oxford (1913). in two simultaneous careers, one as a scholar and His college (Trinity) at Cambridge named him an the other as a government employee. While work- honorary fellow (1920). His professional service ing from 1884 until his retirement at the age of included being on the council of the Royal Society 65 he published major treatises on and serving as president of the Mathematical As- (1885), Apollonius (1896), his original Archimedes sociation. American readers will find it unusual (1897), and seven other publications. His style that his wealth at death (£18.427 9s 2d) in 1940 is was to use modern mathematical notation while published information (resworn probate, April 25, faithfully translating the literal expositions from 1940, CGPLA Eng. & Wales). difficult Greek and Medieval Latin. He rendered the text to be readily understood by all contemporary Archimedes and the Palimpsest at the Year mathematicians. His polished, meticulous prose 2008 with modern mathematical notation culminated Archimedes can become addictive. Sherman Stein in the monumental three-volume non pareil edi- (p. ix) wryly notes that each time he teaches the tion of Euclid’s Elements (1908). Though published he puts more time on one century ago, his work on Euclid remains in Archimedes. Finally on his fifth time, seven of his print and is in virtually every university library. In twenty lectures were devoted to Archimedes, while particular, his treatment of rational and irrational Newton and Leibniz received hasty treatment. He magnitudes in Book X is considered a masterpiece. also asserts Heath and Dijksterhuis are hard to Heath became the indispensable gatekeeper for read, while Heiberg is available only in German or opening the works of ancient Greek mathematics translation. Chris Rorres has shared his lifelong to modern students. Readers of the Notices also passion for collecting Archimedean materials and know A History of Greek Mathematics (1921) as the was one of the very first to launch a mathematics definitive two-volume work on the subject. website (1995). See http://www.mcs.drexel. Heath had a passion for trains and rock climb- edu/~crorres/Archimedes/contents.html. ing. He was an unerring sight-reading pianist. His Reviel Netz remarks, “I am always humbled by wife, Lady Ada Heath, wrote “Music served as a Heiberg” but has the reservation that when Heiberg solvent of difficulties which arose in the writing inserted bracketed passages that are not clearly of his books on Greek Mathematics. He would Achimedes’ own, Heiberg might have been too fast wander over to the piano and play one or more of in his judgment. By modern standards, we know

782 Notices of the AMS Volume 55, Number 7 not to second guess an ancient translator or author [10] W. R. Knorr, The Method of Indivisibles in Ancient too quickly. Yet, a scholar’s opinion has value. Geometry, in Vita Mathematica, (R. Calinger, ed.), Following the Christie’s auction, the current Mathematical Association of America, 1996, 71–74. [11] G. Kolata, In Archimedes’ puzzle, a new eureka owner placed the “lost” palimpsest in the hands moment, New York Times, December 14, 2003. of the Walters Art Museum in Baltimore; William [12] D. C. Macgregor II, Mathematics and physical sci- Noel is heading the project of restoration, imag- ence in , in Chapters in the History ing, and translation. Both the seller and the buyer, of Science, (C. Singer, ed.) Oxford University Press, exchanging US$2 million “under the hammer”, 1922. strongly wish to remain anonymous. This is highly [13] R. Netz, K. Saito, and N. Tchernetska, A new read- understandable—indeed, as a priest who leads pil- ing of Method Proposition 14: Preliminary evidence grimages to the in Jerusalem and to Is- from the (Part I), Sources and Commentaries in Exact Sciences 2, April, 2001. tanbul remarked to this author, “The palimpsest of [14] L. D. Reynolds and N. G. Wilson, Scribes and Schol- Archimedes is to the Greek Orthodox Church what ars: A Guide to the Transmission of Greek and Latin the Elgin Marbles are to the Greek government.” Literature, Oxford University Press, 1968. Both the New York Times and the KATHIMERINI of [15] S. K. Ritter, Imaging in 2020: Seeing is believing, report the owner has pledged not to limit Chemical and Engineering News 77 (45), November 8, access to the ancient manuscript. 1999, 30–35. [16] G. F. Simmons, Calculus Gems: Brief Lives and Memo- Concluding Remarks rable Mathematics, McGraw-Hill, 1992, 242–245. [17] S. Stein, Archimedes: What Did He Do Besides Cry Eu- In 2008, one century after publication, Heath’s reka?, Mathematical Association of America, 1999. edition of Euclid’s Elements is still being printed [18] D’Arcy W. Thompson, Obituary notices of fellows of and sold throughout the world. Only a handful of the Royal Society 3 (9), January, 1941, 408–426. mathematics publications can make this claim. A [19] N. G. Wilson, Archimedes: The palimpsest and visitor to the National Portrait Gallery in London the tradition, Byzantinische Zeitschrift 92 (1999), can see Heath’s photograph, along with DeMorgan 89–101. and Boole, but not Hardy and Littlewood. Without [20] ——— , From Byzantium to Italy: Greek Studies in the Italian Renaissance, Duckworth, 1992. Heath’s translation, Heiberg would be largely unknown in a world now dominated by English. Rare sources or sources in languages other than What is more, we should not forget to pay tribute English: to Heiberg. Were it not for his determined efforts [1] I. Boserup, ed., Københavns Universitet 147-9-1979: to find grant money, to make difficult journeys Klassisk filologi efter 1800 8 (1992), 367–374. across , and to publish demanding scholarly [2] J. L. Heiberg, Eine neue Archimedeshandschrift work, we as mathematicians would be much the (Nebst einer Tafel), Hermes: Zeitschrift für Philologie poorer. It is altogether fitting that we pay tribute 42 (1907), 235–303. to both Heiberg and Heath on the centennial of [3] ——— , Mathematici graeci minores, historisk-filolo- their landmark work. giske meddelelser, Det Kgl Danske Videnskabernes Selskab 13, 1926. References [4] ——— , Fra Hellas til Italien II. 1929, 390–419. [5] ——— , Archimedes, seine Entwickelung und die Wirkung seiner Schriften, MS in Det Kongelige Bib- Modern sources in many university libraries: liotek. NKS 3643 4°, 1908. [1] M. W. Browne, Ancient Archimedes text turns up, and [6] T. L. Heath, The Method of Archimedes: Recently it’s for sale, New York Times, October 27, 1998. Discovered by Heiberg: A Supplement to “The Works [2] ——— , Archimedes text sold for $2 million, New York of Archimedes 1897”, Cambridge University Press, Times (national), October 30, 1998. 1912. [3] Christie’s, The Archimedes Palimpsest, Sale catalog, Relevant Websites: October, 29, 1998. http://www.cs.drexel.edu/~crorres/ [4] E. J. Dijksterhuis, Archimedes, Princeton University Archimedes/contents.html Press, 1987, 44–45; 408-412. http://archimedespalimpsest.org/ [5] C. H. Edwards Jr., The Historical Development of the http://www.thewalters.org/ Calculus, Springer-Verlag, 1979, 68–76. http://www.slac.stanford.edu/gen/com/ [6] S. H. Gould, The method of Archimedes, American slac_pr.html Mathematical Monthly 62 (1955), 473–476. [7] E. Spang-Hanssen, Filologen Johan Ludvig Heiberg 1854–1928. 2nd ed. 1969. [Contains a list of Heiberg’s MS. and letters.] [8] M. F. Headlam, Sir Thomas Little Heath, Proceedings of the British Academy, xxvi (1940), 424–438. [Con- tains a complete list of Heath’s publications.] [9] T. L. Heath, The Thirteen Books of Euclid’s Elements: Translated from the Text of Heiberg, 2nd ed., Dover, 1956.

August 2008 Notices of the AMS 783