## Christopher Clavius

**-**

## Clavius, Christoph

# CLAVIUS, CHRISTOPH

(*b*. Bamberg, Germany, 25 March 1538;

*d*. Rome, Italy, 6 February 1612), *astronomy, cosmology, mathematics, education*. For the original article on Clavius see *DSB*, vol. 3.

Clavius offered the last serious defense of the ancient Ptolemaic cosmology and published one of the earliest critiques of Copernican theory. Along with his students, he authenticated Galileo’s early telescopic discoveries and prominently recognized their epochal significance in his widely used textbook of elementary astronomy. Clavius attained international estee m for his exposition of Euclid'sElements and spent much of his career establishing an important place for mathematical studies in Jesuit schools. He was also a member of the papal commission that planned and executed the Gregorian calendar reform of 1582 and through subsequent publications became the principal expositor and defender of the Gregorian calendar.

**Biographical Background** . Other than his birth date in Bamberg, Clavius’s origins are unknown, including his original family name, which might have been Clau, Schlüssel, or some variant. Any details of his early life in Bamberg are also absent, and he never returned there, although he took an interest for the rest of his life in the city and its fortunes amidst the Counter-Reformation. Horst Enzensberger’s “Società, cultura e religione a Bamberga” (1995) provides a sketch of the intellectual and political context that must have shaped Clavius’s early life. He entered the Society of Jesus on 12 April 1555 and was then sent to study at the University of Coimbra, which he entered in 1556. His first recorded astronomical observation took place at Coimbra: the total solar eclipse of 21 August 1560. By May 1561 he had returned to Rome to begin advanced studies in theology and other subjects at the Jesuit Collegio Romano and was ordained in 1564. Clavius began teaching mathematics at the Collegio Romano, as he would for nearly all his career, as early as 1563. With rare good fortune for a person in that era, he witnessed on 9 April 1567 a second total solar eclipse in Rome. His account of the eclipse, published in his Sphere commentary, attracted attention in its day because of his controversial conclusion that it was an annular eclipse. At the turn of the twenty-first century, F. Richard Stephenson, J. Eric Jones, and Leslie Morrison used his report to investigate long-term variations in the rotation rate of the Earth (1997). During a brief stint in Messina working with Francesco Maurolico, in 1574, he acquired many unpublished mathematical treatises, including Maurolico’s treatise on the nova of 1572 and a manuscript on light, which Clavius would eventually publish. Aside from another sojourn at the Jesuit College in Naples in 1596, Clavius spent the rest of his long career in Rome, where he died on 6 February 1612. A more complete biography
can be found in James Lattis’s *Between Copernicus and Galileo*(1994).

**Mathematics** . Clavius published his edition of Euclid’s *Elements* in 1574. More a commentary enhancing access to the work than a philological edition of the Greek text, it achieved great popularity and influence. Revising and republishing it at least five times, Clavius went beyond the strict bounds of Euclid’s material to introduce new materials, including his own proof of Euclid’s fifth postulate and his solution to the problem of squaring the circle. Vincent Jullien (1997) and Sabine Rommevaux (2005) show the broad significance of Clavius’s *Euclid* for many seventeenth-century mathematicians, not only Jesuits, and Paolo Palmieri (2001) finds connections between Clavius’s theory of proportions and Galileo’s own struggles with the concept.

Clavius’s other original mathematical contributions include a digression on combinatorics in his *Sphere* commentary in 1581, which Eberhard Knobloch (1979) judges a seminal text, and his publication of the *Spherics* of Theodosius (in 1586). He also published a variety of practical textbooks on arithmetic, geometry, gnomonics, and the construction of instruments. Music, one of the four mathematical sciences of the traditional Quadrivium (along with arithmetic, geometry, and astronomy), was another area of interest for Clavius. His surviving works include eleven motets and two songs, none of which have yet received significant study.

**Astronomy** . Clavius authored one of the most influential astronomy textbooks in history, his *Commentary on the Sphere of Sacrobosco*, which remained a standard for astronomy instruction for three-quarters of a century. It was published at least sixteen times between 1570 and 1618 by printers spread across Europe. He revised the text seven times, often expanding it greatly in scope and detail and taking note of new discoveries and controversies.

In lengthy digressions in his *Sphere*, Clavius defended the Ptolemaic cosmology (a blending of Aristotelian physics and mathematical models of Ptolemy’s *Almagest* into a physical cosmos) against a variety of critics. The critics included both skeptics who doubted that knowledge about celestial causes is even possible, as well as those who advocated alternatives to the Ptolemaic cosmos. Clavius’s “realist” views, which hold that it is possible to deduce celestial causes from observations of the motions of celestial bodies, resonated strongly (if only at the epistemological level) with those of Johannes Kepler, as Nicolas Jardine discusses in “The Forging of Modern Realism” (1979). Prominent among the alternative cosmologies criticized by Clavius stands Copernicus’s heliocentric cosmos. Clavius’s criticisms of the Copernican cosmos

included its inconsistencies with common sense, Aristotelian physics, and the testimony of Scripture, as well as a flawed methodology that would, he said, prevent it from providing reliable astronomical knowledge.

Despite his antipathy toward the Copernican cosmos, Clavius’s *Sphere* expressed admiration for Copernicus’s mathematical skill, and he ultimately incorporated several ideas from Copernicus’s work into his own version of the Ptolemaic cosmology, most notably Copernicus’s model for representing what later would be called the precession of the equinoxes, which motion Copernicus attributed to the Earth, but which Clavius located in the outer spheres of the Ptolemaic cosmos. Clavius also confronts and rejects the cosmological theory of homocentric spheres at considerable length and with even greater vigor than he devotes to Copernican theory, and goes on to reject other cosmic concepts as well. His treatment of these rivals to Ptolemaic cosmology shows that the cosmological debates of the late sixteenth and early seventeenth centuries were far more complex than a simple confrontation between Ptolemy and Copernicus.

Clavius also used his *Sphere* as a vehicle for commentary on the remarkable novas of 1572, 1600, and 1604. In the 1585 (and every subsequent) edition, he published his conclusion that the nova of 1572 must have been located in the firmament of the fixed stars—thus demonstrating, contrary to Aristotle, that celestial matter was capable of qualitative change. He based his conclusion firmly on observations reported by correspondents widely placed across Europe showing that all had observed the nova to be in the same location with respect to nearby stars, putting, in effect, an upper limit on the parallax of the nova. Clavius’s measurement of the location of the nova was thus in agreement with but independent of Tycho Brahe’s more famous conclusion. Galileo’s celebrated discoveries of 1609 and 1610 were also reported in the *Sphere*. In April 1611, Cardinal Bellarmine requested of Clavius an opinion concerning Galileo’s sensational telescope discoveries, which the astronomers of the Collegio Romano then confirmed with their own telescopes. In his final version of the *Sphere*, published in 1611, Clavius noted Galileo’s findings, including the phases of Venus and moons of Jupiter, and famously recognized their significance by calling upon astronomers to accommodate them in astronomical theory. A fuller account of Clavius’s astronomical career and significance is found in Lattis’s *Between Copernicus and Galileo*.

Although Clavius’s *Sphere* was the book by which his astronomical teaching reached the world at large, it is not, as Ugo Baldini (2000) points out, an adequate measure of the level of his astronomical research. Clavius never finished his more advanced treatise in theoretical astronomy, but the surviving parts (fragments of his solar and lunar theories) are interesting and perhaps unique examples of how advanced astronomical theory was taught in the late sixteenth century. The surviving solar theory has been published by Baldini in *Legem impone subactis*(1992), and further discussed, along with the lunar theory, in his *Saggi sulla Cultura della Compagnia di Gesù*(2000). Baldini, in his *Saggi*, judges it doubtful that, even if it had been finished, his theoretical work would have resulted in anything other than an ad hoc adjustment to the established Ptolemaic theories. Clavius found even the geocentric system of Tycho to be incomprehensible as a representation of reality and remained committed to the Ptolemaic cosmos. Clavius’s level of expertise was also very high in the area of instrument design as indicated by his several books on the construction and use of astrolabes, sundials, and meridian instruments. Baldini and Juan Casanovas (1996) identify the sole surviving example of one of Clavius’s instruments, namely a celestial globe constructed in 1575, in which he adopted from Copernicus the location of the vernal equinox and updated star positions.

Galileo drew heavily on Jesuit sources during his early academic career, as is documented by William Wallace in *Galileo and His Sources*(1984), and had personally conferred with Clavius. His cordial relationship with Galileo endured through the end of Clavius’s life and generally extended to the other Jesuit astronomers of the Collegio Romano who collectively celebrated Galileo’s telescope discoveries with a ceremony at the Collegio Romano on 18 May 1611. Although Clavius had endorsed and confirmed the observations themselves, he originally expressed reservations about the full meaning of Galileo’s discoveries. Yet the doubts of the senior astronomer seem not to have dampened the enthusiasm of the younger ones, which included Christoph Grienberger, Odo van Maelcote, Paul Guldin, Paolo Lembo, and Gregory of St. Vincent. Relations between Galileo and the Collegio Romano astronomers soured only after Clavius’s death in the wake of Cardinal Bellarmine’s restrictions on the teaching of Copernicanism and the controversies that grew out of Galileo’s feuds with Jesuits Orazio Grassi and Christoph Scheiner.

**Gregorian Calendar** . Sometime between 1572 and 1575, Pope Gregory XIII convened a commission to make recommendations on the reform of the Julian calendar, and the young Clavius was tapped to serve as the commission’s technical expert. As such, he reviewed and explained the various issues and proposed reform schemes and specified the technical terms of the reform that the commission eventually decided on. This, however, was only the beginning of the work, because Clavius went on to write and publish the fundamental works promulgating and explaining the new Gregorian calendar and the transition process from the old calendar to the new. A collection of articles explaining various aspects of the calendar reform appears in *Gregorian Reform of the Calendar* (Coyne, et al, 1983). Many critics, among them Joseph Scaliger and Michael Maestlin, found fault with the calendar reform, and the task fell to Clavius to respond to them in print. An overview of Clavius’s role in the reform and his responses to the critics can be found in Carmelo Oñate Guillen’s “Christopher Clavius y el Calendario Gregoriano” (2000). A proper history of the Gregorian calendar reform has as of 2007 yet to be published.

**Institution Building** . Jesuit scholars achieved great respect for their contributions to mathematical sciences, and Clavius was the architect of the mathematical curriculum in the Jesuit educational establishment. His influence on the *Ratio studiorum*, the plan of studies for Jesuit schools, published in final form in 1599, established mathematics as a vital component in an era when mathematical subjects were rarely or inconsistently taught in many institutions of higher learning. His concerns went beyond curriculum parameters and extended to measures intended to enhance the prestige of mathematical work
and the respect accorded its specialists. Dennis Smolarski surveys Clavius’s pedagogical efforts and his influence on the development of the *Ratio studiorum*. In addition to establishing a curriculum that specified the study of Euclid, arithmetic, astronomy, cosmography, optics, timekeeping, and instrument construction, Clavius’s lifetime of writing provided teachers, Jesuit and otherwise, with textbooks to cover almost the entire mathematical curriculum. By the end of his career, Clavius’s efforts had led to a required rotation of mathematics courses in the hundreds of Jesuit schools and to a growing number of skilled teachers and practitioners of the mathematical sciences. Alistair Crombie, in “Mathematics and Platonism” (1977), largely credits Clavius’s policies and efforts for Jesuit achievements in science during the seventeenth century. Clavius’s impact also went well beyond Europe, carried by mathematically trained Jesuit missionaries such as Matteo Ricci and Johann Adam Schall. Notwithstanding his significance for helping scholars understand the development of early modern science, Clavius’s greatest legacy and impact might be found in his efforts as a teacher and builder of educational institutions.

## SUPPLEMENTARY BIBLIOGRAPHY

### WORKS BY CLAVIUS

*Opera mathematica*, 5 vols. Mainz: Eltz, 1612. Clavius’s collected works.

*Bibliothèque de la Compagnie de Jésus*, compiled by Carlos Sommervogel. Paris: Alphonse Picard, 1891. Reprint, Paris, 1960. Contains a complete listing of Clavius’s publications.

*Christoph Clavius: Corrispondenza*, ed. Ugo Baldini and Pier Daniele Napolitani. Pisa: University of Pisa Press, 1992.

*Theorica solis*. In *Legem impone subactis*, Ugo Baldini. Rome: Bulzoni, 1992. Contains the surviving portion of his solar theory.

### OTHER SOURCES

Baldini, Ugo. “La nova del 1604 e i matematici e filosofi del Collegio Romano: Note su un testo inedito.” *Annali dell’Istituto e Museo di Storia della Scienza di Firenze* 6, fasc. 2 (1981): 63–98.

———. “Christoph Clavius and the Scientific Scene in Rome.” In *Gregorian Reform of the Calendar*, edited by George V. Coyne, Michael A. Hoskin, and Olaf Pederson. Vatican City: Specola Vaticana, 1983.

———. *Legem impone subactis: Studi su filosofia e scienza dei Gesuiti in Italia, 1540–1632*. Rome: Bulzoni, 1992.

———, ed. *Christoph Clavius e l’attività scientifica dei Gesuiti nell’età di Galileo*. Rome: Bulzoni, 1995.

———, and Juan Casanovas. “La sfere celeste di Cristoforo Clavio.” In *Osservatorio Astronomico di Capodimonte, Almanacco 1996*. Napoli: Arte Tipografica, 1996.

———. *Saggi sulla Cultura della Compagnia di Gesù*. Padova CLEUP, 2000.

Casanovas, Juan. “L’astronomia nel Collegio Romano nella prima metà del seicento.” *Giornale di Astronomia* 10 (1984): 149–155.

———. “Il P. C. Clavio professore di matematica del P. M. Ricci nel Collegio Romano.” In *Atti del Convegno Internazionale di Studi Ricciani*, edited by Maria Cigliano. Macerata: Centro Studi Ricciani, 1984.

Coyne, George V., Michael A. Hoskin, and Olaf Pedersen. *Gregorian Reform of the Calendar: Proceedings of the Vatican Conference to Commemorate its 400 ^{th} Anniversary, 1582–1982*. Vatican City: Specola Vaticana, 1983.

Crombie, Alistair C. “Mathematics and Platonism in the Sixteenth-Century Italian Universities and in Jesuit Educational Policy.” In *Prismata*, edited by Yasukatsu Maeyama and Walter G. Saltzer. Wiesbaden: Franz Steiner Verlag, 1977.

Döring, Klaus, and Georg Wöhrle, eds. *Vorträge des ersten Symposions des Bamberger Arbeitskreises “Antike Naturwissenschaft und ihre Rezeption” (AKAN)*. Wiesbaden: Otto Harrassowitz, 1990.

Enzensberger, Horst. “Società, cultura e religione a Bamberga e in Franconia ai tempi di Christoph Clavius.” In *Christoph Clavius e l’attività scientifica dei Gesuiti nell’età di Galileo*, edited by Ugo Baldini. Rome: Bulzoni, 1995.

Flindell, E. Fred. “Christophorus Clavius.” In *The New Grove Dictionary of Music and Musicians*, edited by Stanley Sadie. London: Macmillan, 1980

Garibaldi, Antonio C. “Il Problema della quadratice nella matematica dei Gesuiti da Clavius alla metà del secolo XVII.” In *Christoph Clavius e l’attività scientifica dei Gesuiti nell’età di Galileo*, edited by Ugo Baldini. Rome: Bulzoni, 1995.

Jardine, Nicolas. “The Forging of Modern Realism: Clavius and Kepler against the Skeptics.” *Studies in History and Philosophy of Science* 10 (1979): 141–173.

Jullien, Vincent. “Quelques aspects du caractère incontournable des *Éléments* d’Euclide au XVIIe siècle.” *Science et Techniques en Perspective*, IIe série, 1 (1997): 221–265.

Knobloch, Eberhard. “Musurgia universalis: Unknown Combinatorial Studies in the Age of Baroque Absolutism.”*History of Science* 17 (1979): 258–275.

———. “Sur la vie et l’oeuvre de Christophore Clavius (1538–1612).” *Revue d’Histoire des Sciences* 41 (1988): 331–356.

———. “Christoph Clavius: Ein Astronom zwischen Antike und Kopernicus.” In *Vorträge des ersten Symposions des Bamberger Arbeitskreises “Antike Naturwissenschaft und ihre Rezeption” (AKAN)*, edited by Klaus Döring and Georg Wöhrle. Wiesbaden: Otto Harrassowitz, 1990.

———. “Sur le rôle de Clavius dans l’histoire des mathematiques.” In *Christoph Clavius e l’attività scientifica dei Gesuiti nell’età di Galileo*, edited by Ugo Baldini. Rome: Bulzoni, 1995.

Lattis, James M. “Homocentrics, Eccentrics, and Clavius’s Refutation of Fracastoro.” *Physis* 28 (1991): 699–725.

———. *Between Copernicus and Galileo: Christoph Clavius and the Collapse of Ptolemaic Cosmology*. Chicago: University of Chicago Press, 1994.

Lucchetta, Giulio A. “Componenti platoniche e aristoteliche nella filosofia della matematica di Clavius.” In *Christoph Clavius e l’attività scientifica dei Gesuiti nell’età di Galileo*, edited by Ugo Baldini. Rome: Bulzoni, 1995.

Maeyama, Yasukatsu, and Walter G. Saltzer, eds. *Prismata: Naturwissenschaftsgeschichtliche Studien*. Wiesbaden: Franz Steiner Verlag, 1977.

Oñate Guillen, Carmelo. “Christopher Clavius y el Calendario Gregoriano.” *Letras de Deusto* 30 (2000): 55–70.

Palmieri, Paolo. “The Obscurity of the Equimultiples: Clavius’ and Galileo’s Foundational Studies of Euclid’s Theory of Proportions.” *Archive for History of Exact Sciences* 55 (2001): 555–597.

Remmert, Volker R. “‘Sonne Steh Still uber Gibeon.’ Galileo Galilei, Christoph Clavius, katholische Bibelexegese und die Mahnung der Bilder.” *Zeitschrift für Historische Forschung* 28 (2001): 539–580.

Rommevaux, Sabine. *Clavius. Une clé pour Euclide au XVIe siècle*. Paris: Vrin, 2005.

Smolarski, Dennis C. “The Jesuit *Ratio studiorum*, Christopher Clavius, and the Study of Mathematical Sciences in Universities.” *Science in Context* 15 (2002): 447–457.

Stephenson, F. Richard, J. Eric Jones, and Leslie V. Morrison. “The Solar Eclipse Observed by Clavius in A.D. 1567.” *Astronomy and Astrophysics* 322 (1997): 347–351.

Wallace, William A. *Galileo and His Sources: The Heritage of the Collegio Romano in Galileo’s Science*. Princeton, NJ: Princeton University Press, 1984.

*James M. Lattis*

## Clavius, Christoph

# Clavius, Christoph

(*b*. Bamberg, Germany, 1537; *d*. Rome, Italy, 6 February 1612),

*mathematics, astronomy*.

Clavius entered the Jesuit order at Rome in 1555 and later studied for a time at the University of Coimbra (Portugal), where he observed the eclipse of the sun on 21 August 1560. He began teaching mathematics at the Collegio Romano in Rome in 1565, while still a student in his third year of theology; and for all but two of the next forty-seven years he was a member of the faculty as professor of mathematics or as scriptor. From October 1595 until the end of 1596, he was stationed in Naples.

In 1574 Clavius published his main work, *The Elements of Euclid*. (With the help of scholars, Matteo Ricci, between 1603 and 1607, translated into Chinese the first six books of Clavius’ *Elements*.) His contemporaries called Clavius “the Euclid of the sixteenth century.” The *Elements*, which is not a translation, contains a vast quantity of notes collected from previous commentators and editors, as well as some good criticisms and elucidations of his own. Among other things, Clavius made a new attempt at proving the “postulate of parallels.” In his *Elements* of 1557, the French geometer Peletier held that the “angle of contact” was not an angle at all. Clavius was of a different opinion; but Viète, in his *Variorum de rebus mathematicis responsorum* of 1593, ranged himself on the side of Peletier. In a scholion to the twelfth proposition of the ninth book of Euclid, Clavius objects to Cardanus’ claim to originality in employing a method that derives a proposition by assuming the contraditory of the proposition to be proved. According to Clavius, Cardanus was anticipated in this method by Euclid and by Theodosius of Bithynia in the twelfth proposition of the first book of his *Sphaericorum*.

As an astronomer, Clavius was a supporter of the Ptolemaic system and an opponent of Copernicus. In his *In Sphaeram Ioannis de Sacro Bosco commentarius* (Rome, 1581) he was apparently the first to accuse Copernicus not only of having presented a physically absurd doctrine but also of having contradicted numerous scriptural passages. The friendship between Clavius and Galileo, according to their correspondence, began when Galileo was twenty-three and remained unimpaired throughout Clavius’ life. In a report of April 1611 to Cardinal Bellarmine of the Holy Office, Clavius and his colleagues confirmed Galileo’s discoveries, published in the *Sidereus nuncius* (1610), but they did not confirm Galileo’s theory.

In his *Epitome arithmeticae practicae* (Rome, 1583), Clavius gave a distinct notation for “fractions of fractional numbers,” but he did not use it in the ordinary multiplication of fractions. His means of The distinctive feature of this notation is the omission of the fractional line after the first fraction. The dot cannot be considered as the symbol of multiplication. He offered an explanation for finding the lowest common multiple, which before him only Leonardo Fibonacci in his *Liber abaci* (1202) and Tartaglia in his *General trattato di numeri et misure* (1556) had done. In his *Astrolabium* (Rome, 1593) Clavius gives a “tabula sinuum,” in which the proportional parts are separated from the integers by dots. However, his real grasp of that notation is open to doubt, and the more so because in his *Algebra* (Rome, 1608) he wrote all decimal fractions in the form of common fractions. Apart from that, his *Algebra* marks the appearance in Italy of the German plus (+) and minus (-) signs and of algebraic symbols used by Stifel. He was one of the very first to use parentheses to express aggregation of terms. As symbol of the unknown quantity, he used the German radix For additional unknowns he used 1*A*, 1*B*, etc;, for example, he wrote + 4*A*, 4*B* – 3*A* for 3*x* + 4*y*, 4*z* – 3*y*. In his *Algebra*, Clavius did not take notice of negative roots, but he recognized that the quadratic *x*^{2} + *c* = *bx* may be satisfied by two values of *x*. His geometrical proof for this statement was one of the best and most complete. The appendix of his commentary on the *Sphaericorum* of Theodosius (Rome, 1586)—containing a treatise on the sine, the tangent, and the secant—and the rules for the solutions of both plane and spherical triangles in the *Astrolabium*, the *Geometria practica* (Rome, 1604), and the *Triangula sphaerica* (Mainz, 1611) comprehend nearly all the contemporary knowledge of trigonometry; in the *Astrolabium* for example, is his treatment of the so-called prosthaphaeresis method, by which addition and subtraction were substituted for multiplication, as in

In this he also gives a graphic solution of spherical triangles based on the stereographic projection of the sphere.

Mention must also be made of Clavius’ improvement of the Julian calendar. Pope Gregory XIII brought together a large number of mathematicians, astronomers, and prelates, who decided upon the adoption of the calendar proposed by Clavius, which was based on Reinhold’s *Prussian Tables*. To rectify the errors of the Julian calendar it was agreed to write in the new calendar 15 October immediately after 4 October of the year 1582. The Gregorian calendar met with a great deal of opposition from scientists such as Viète and Scaliger and from the Protestants.

## BIBLIOGRAPHY

Clavius’ collected works, *Opera mathematica*, 5 vols. (Mainz, 1611/1612), contain, in addition to his arithmetic and algebra, his commentaries on Euclid, Theodosius, and Sacrobosco; his contributions to trigonometry and astronomy; and his work on the calendar.

The best account of Clavius’ works and their several editions can be found in C. Sommervogel, *Bibliothèque de la Compagnie de Jésus*, II (Brussels-Paris, 1891). Some information on his life and work can be found in B. Boncompagni, “Lettera di Francesco Barozzi al P. Christoforo Clavio,” in *Bollettino di bibliografia e storia delle scienze matematiche e fisiche*, **17** (1884), 831–837; A von Beraunmühl, *Vorlesungen über Geschichte der Trigonometrie*, I (1900) 189–191; F. Cajori”, “Early ‘Proofs’ of the Impossibility of a Fourth Dimension of Space,” in *Archivio di storia della sciencza*, **7** (1926), 25–28; J Ginsburg, “On the Early History of the Decimal Point,” in *American Mathematical Monthly*, **35** (1928), 347–349; O. Meyer, “Christoph Clavius Bambergensis,” in Fränkisches Land, **9** (1962), 1–8; J. E. Montucla, *Histoire des mathématiques*, 2nd ed., I (Paris, 1799), 682–687; E. C. Philips, “The Correspondence of Father Christopher Clavius S. I.,” in *Archivum historicum Societatis Iesu*, **8** (1939), 193–222; and J. Tropfke, “Zur Geschichte der quadratischen Gleichungen über dreieinhalb Jahrtausend,” in *Jahresbericht der Deutschen Mathematikervereinigung*, **44** (1934), 117–119.

H. L. L. Busard