Richard L. Kremer

“Abbreviating” the Alfonsine Tables in Cracow: The Tabulae Aureae of Petrus Gaszowiec (1448)

Many would agree that the “Alfonsine Tables” were the most widely circulated astronomical texts in Europe during the fourteenth and fifteenth centuries. Created in Toledo around 1272 and reworked in Paris during the 1320s, these tables soon were arranged into several distinct formats, usually in the interest of reducing the computational labour required for their use. To name some of the better-known versions in roughly chronological order, there emerged John of Lignères’s Tabulae magne in Paris, William Batecombe’s Oxford Tables, John of Gmunden’s tables in Vienna, the Tabulae resolutae of Wrocław (then Prague, Cracow and Salamanca), Prosdocimo de’ Beldomandi’s tables in Padua, Giovani Bianchini’s tables in Ferrara, Abraham Zacut’s ha-Hibbur ha-gadol (The great composition) in Salamanca, and Petrus Gaszowiec’s Tabulae aureae of Cracow. Users, scribes and bookbinders often brought together varied groupings of these tables and canons so that, before the printing of the editio princeps in 1483 (roughly the Parisian version), it is impossible to define a set of tables as the canonical “Alfonsine Tables”. Thus, José Chabás and Bernard R. Goldstein have rightly emphasized the need to speak of the “Alfonsine corpus” rather than the “Alfonsine Tables”. With this term, they are casting a wide net that includes all of the material listed above that shares common underlying parameters (such as the treatment of the motion of the eighth sphere by precession and trepidation or the maximum solar correction of 2;10°). Literally hundreds of tables from this corpus are extant in fourteenthand fifteenth-century manuscripts; the Tabulae resolutae, Bianchini’s tables and the Parisian tables would find their way into print. In defining an Alfonsine corpus, Chabás and Goldstein are operationalizing an earlier suggestion by John D. North that medieval astronomical tables be distinguished by their “underlying parameters” and “outward form” (the latter includes both the layout of the tabular text on the pages and the computational algorithms that text prescribes). That is, any given set of tables in the Alfonsine corpus should yield generally identical results (to minutes of time or position) when computing, say, the location of Mars or the time of an eclipse. The tables that I shall here consider do not easily fit this taxonomy. In 1448, a Cracow master and professor of medicine, Petrus Gaszowiec, introduced a set of tables whose results can differ by more than 6° of longitude from those made by other tables in the Alfonsine corpus. These differences do not arise from any attempt by Gaszowiec to modify Alfonsine models or parameters; rather, he so dramatically abbreviated Alfonsine computational algorithms that the results were affected. This radical simplification might explain why Gaszowiec’s tables have been preserved in only six manuscript copies and why they never were printed. On the other hand, one of the manuscripts bearing the tables was copied in 1512; another was purchased in 1530 and had its radices updated to that year by the new owner. JHA, xxxviii (2007)

Thoughts on John of Saxony's method for finding times of true syzygy

Abstract This article examines John of Saxonys iterative method for finding the times from mean to true syzygy (i.e., conjunction or opposition of the Moon and Sun). It argues that the method, composed c. 1330, contains several ambiguities, but is so robust that only one of these ambiguities affects the time correction. Furthermore, the method yields times of true syzygy that correspond, to the nearest minute, to the time when the true elongation, as computed by the planetary equations of the 1483 Alfonsine Tables, makes its closest approach to 0° or 180°. Hence Johns method yields “exact” Alfonsine times, unlike all other known medieval methods or tables that only approximate those results. It will also be shown that John Somer (1380s) and Regiomontanus (1440–1450s) wielded Johns method with considerable computational skill.

Edward Gresham, Copernican Cosmology, and Planetary Occultations in Pre-Telescopic Astronomy

This article introduces an understudied source in the history of astronomy, the Astrostereon or the Discourse of the Falling of the Planet (1603). Written by the English astrologer Edward Gresham, this text presents, among other things, the earliest known set of predicted planetary occultations (for 1603–1604) and the use of these phenomena to defend the Copernican cosmology. We analyse those predictions and then briefly survey all known pre-telescopic observations of reported planetary occulations and the motivations for such observations. These data suggest that for early observers, the greater the difference in apparent brightness between the two occulting bodies, the greater the angular separation could be for an occultation nonetheless to be reported. An appendix seeks to explain this finding by considering several factors known from modern experimental analyses of human visual performance.

Incunable Almanacs and Practica as Practical Knowledge Produced in Trading Zones

This paper traces the development of practical knowledge in early modern astronomy and astrology by considering two literary genres apparently born in the fifteenth-century print shop, the annual broadside almanac and the annual multi-leaved practica. Taken together nearly 1000 editions printed before 1500 are extant, written in Latin or several vernacular languages. These texts combine astronomical, computistic, astrological and medical material. Examination of the term “practica” as an actors’ category, of the social locations of more than one hundred authors named on the imprints, and of the content of these texts suggests that this pragmatic literature arose in sites Pamela O. Long has called trading zones. Exemplified by the early modern arsenal, mine, princely court, or print shop, these zones enabled artisans and scholars to exchange expertise and experience. The resulting almanacs and practica offered readers timely information about the church calendar, astronomical phenomena, and medical and astrological prospects for the coming year.

Playing with Geometrical Tools: Johannes Stabius's Astrolabium imperatorium (1515) and Its Successors

This article suggests that 16th-century sources describing astronomical instruments may be analyzed in terms of ‘geometrical tools’, that is discrete arrangements of lines and curves that solve particular problems. Geometrical tools provided a means for innovation. By playing, literally, with such tools, mathematicians could invent new instruments or add new functions to existing instruments. For a case study of this process, I shall consider the rectangular astrolabe, first proposed in 1515 by Johannes Stabius and reconfigured in several other versions over the course of the 16th century. Geometrical tools, I conclude, are revealed in diagrams found in the sources, not in the accompanying texts.

How do Writings in the Early Astral Sciences Reveal Mathematical Practices and Practitioners

This article introduces a writing format, the “template table” (suanshi, 算式) that was designed to guide the process of calendrical astronomical calculations in early modern China. In conjunction with another kind of text, known as “detailed procedures” (xicao, 細草), users could perform calculations easily by operating the “template table” and extracting data from given numerical tables. This method, that not only normalized the use of numerical tables but also linked instructions with the corresponding tables in computational practices, became widespread from the Ming period (1368-1644) onwards. Wanting to acquire this computational regimen, the Joseon court of Korea (1392-1897) even sent skilled officers to China to learn it secretly. The circulation of the template method beyond China suggests its significance. The article also discusses the advantages and disadvantages of using this method.

Calculating with Andreas Aurifaber: A New Source for Copernican Astronomy in 1540:

In an article recently published in this journal, Jonathan Green announced his discovery of the earliest known printed astrological practica that mentions Copernicus. Apparently extant in a single copy, this Practica auff das Jar M.D.XLI. durch Magistrum Andream Aurifaber gestellet claims to have been computed “from the tables of Nicholas Copernicus” (Figure 1). Although this imprint has been known, bibliographically, since at least 1874, no one before Green, to the best of my knowledge, noticed the remarks about Copernicus in its dedication, although previous historians have documented biographical links between the practica’s author, Andreas Aurifaber (1514–59), and several well-known early Copernicans. Was Aurifaber’s practica indeed “Copernican”? Might it be possible, from quantitative material therein, to reconstruct the “Copernican tables” from which it claims to have been computed? And might such a reconstruction, based on a source printed three years before the publication of Copernicus’s De revolutionibus, shed light on the chronology of the composition of this latter book? Green hinted at these questions but did not pursue them; to do so is the goal of this paper. Consisting of thirteen folios filled with the usual astrological content found in the dozens of German practica editions printed in the 1530s and 1540s, Aurifaber’s little pamphlet perhaps not surprisingly escaped notice before Green’s discovery. But many threads tie the 1541 practica and its author to Copernicus and the early Copernicans. With its dedicatory letter to the Danzig City Council dated 11 November 1540, this practica probably was printed in late November or December. The colophon identifies the printer as Franz Rhode of Danzig, who had issued Georg Joachim Rheticus’s Narratio prima earlier that year. Not a vigorous printing centre, Danzig had only three printshops active in the sixteenth century; Aurifaber’s was the only annual calendar, almanac or practica to have been printed in that city before 1580. Against this paucity of astronomical publishing in Danzig, Rhode’s two 1540 imprints hence stand out sharply. As Green noted, Aurifaber had met Rheticus in the 1530s at the University of Wittenberg, where both men studied under Philipp Melanchthon. In 1539, as Rheticus headed north to Frauenburg, Aurifaber on Melanchthon’s recommendation became first rector of the newly opened St Mary’s Latin School in Danzig (about 60 km west of Frauenburg). At Rheticus’s request, Aurifaber in February 1540 sent Melanchthon a few sheets from the Narratio prima, then being printed. In April 1540, upon receiving his copy of the Narratio, Duke Albrecht von Hohenzollern described the work as “issued by Magister Joachim Rheticus with his assistant Magister Andreas Aurifaber, schoolmaster in Danzig”. Modern editors of the Narratio have suggested JHA, xli (2010)

Tables, Manners, Erudition, and Confession

“Befitting its subject, this is an odd book.” One might expect a review to open with these words; in this case, it is the book under review that so begins (p. 3). A reader familiar with earlier historiography of universities might indeed find this account of the “origins of the research university” to be rather odd. Although taking up the story in the fifteenth century, William Clark focuses his attention primarily on the 1770s through the 1830s and on the German states. Yet the reforms of Wilhelm von Humboldt, the classic Denkschriften about the purpose of the university by the German Idealist philosophers, the exploits of the modern discipline builders (e.g., the philologist F. A. Wolf or the physicist Franz Neumann) or government ministers (e.g., Altenstein in Prussia or Münchhausen in Hanover), and the emergence of a “research ethos” barely appear in this massive tome, which offers nothing new on these points. Clark does not give us a history of the university as idea, program, or institution; rather, he offers an “erudite” account of the changing habitus and manners of the German professors and of the transformation of their world from orality to writing. The well-studied rise of the public sphere, cameralism, the bureaucratic state, and notions of Romantic genius provide not merely the backdrop, but the deus ex machina, in Clark’s story. Richly exploiting visual images of university life in contemporaneous prints and drawings, Clark in the first part of his book asks who sat and who stood where in university classrooms. He examines the shape of the tables (both wooden and paper—i.e., record keeping) found there. He considers academic regalia and the order of marchers in university parades. He discusses changing formats of lecture and library catalogues, the move from disputations to examinations and grades, the rise of the seminar and assigned student writing, the shift in authorship of the doctoral dissertation from professor to student, the form of archival dossiers kept by governmental ministries running the

Text to Trophy: Shifting Representations of Regiomontanus’s Library

A 1797 biography of Johannes Regiomontanus (1436–76), the most competent European astronomer and mathematician of the fifteenth century, decried the loss of the ‘treasure’ that Regiomontanus’s library once had represented, and explained the loss by quoting a seventeenth-century bibliophile: ‘The duration of private libraries is nearly momentary. They … die with their collectors.’1 Recent scholarship on the cultural practices of collecting during the Renaissance offers a more nuanced explanation for the death of private libraries. Since the 1970s, an increasing number of art historians, economic historians and cultural historians have crafted a history of the Italian Renaissance as a flowering of consumerism and acquisitiveness. For scholars like Richard Goldthwaite, Lisa Jardine, Thomas Dacosta Kaufmann, Anthony Grafton and Paula Findlen, to name only a few, the Renaissance represents a new self-consciousness about things as well as words.2