40tH HISTORIA MATHEMATICA FOUR DECADES OF EXCELLENCE IN THE HISTORY OF MATHEMATICS Editors in Chief thomas Archibald Simon Fraser University, British Columbia, Canada Niccolò Guicciardini Università degli Studi di Bergamo, Bargamo, Italy www.elsevier.com/locate/hm Historia Mathematica publishes historical scholarship on mathematics and its development in all cultures and time periods. In particular, the journal encourages informed studies on mathematicians and their work in historical context, on the histories of institutions and organizations supportive of the mathematical endeavor, on historiographical topics in the history of mathematics, and on the interrelations between mathematical ideas, science, and the broader culture. This volume celebrates the 40th anniversary of Historia by providing an anthology of some of the seminal papers published since 1973. CONTENTS: Introduction Doubling the cube: a new interpretation of its Archibald, T., Guicciardini, N. significance for early Greek geometry. Saito, K. (1995) the mathematical studies of G.W. Leibniz. on combinatorics. Knobloch, E. (1974) Branch points of algebraic functions and the beginnings of modern knot theory. Epple, M. (1995) Gauss as a geometer. Coxeter, H. S. M. (1977) L'intégration graphique des equations différentielles ordinaires. Investigations of an early Sumerian division problem, Tournès, D. (2003) c. 2500 B.C. Høyrup, J. (1982) the first Chinese translation of the last nine books of Euclid's elements and its source. On the transmission of geometry from Greek into Arabic. Xu, Y. (2005) Knorr, W. (1983) the Suàn shù shu , “Writings on reckoning": Changing canons of mathematical and physical rewriting the history of early Chinese mathematics intelligibility in the later 17th century. in the light of an excavated manuscript. Mahoney, M. S. (1984) Cullen, C. (2007) the calculus of the trigonometric functions. On mathematical problems as historically Katz, V. J. (1987) determined artifacts: reflections inspired by sources from ancient China. Sharaf al-Dīn al-tūsī on the number of positive roots Chemla, K. (2009) of cubic equations. Hogendijk, J. P. (1989) the correspondence between Moritz Pasch and Felix Klein Hilda Geiringer-von Mises, Charlier series, ideology, Schlimm, Dirk (2013) and the human side of the emancipation of applied mathematics at the University of Berlin during the 1920s. Siegmund-Schultze, R. (1993) How probabilities came to be objective and subjective. Daston, L. (1994) Introduction Historia Mathematica was established by Kenneth O. May 40 years ago, succeeding an earlier newsletter sponsored by the International Commission on the History of Mathematics. May's initial vision of an international journal reflected his ideological views, and remains encoded on the cover. The ideal that the language of production of scholarship would not be an obstacle to its publication has in the meantime given way, for better or worse, to a world in which English is the new Latin. However, May’s ideal that this journal remain open to contributions of scholars from all over the world is intact thanks to its connection with the ICHM, the broad geographical distribution of the members of the Editorial Board, and the vision of our predecessors as editors in chief.1 We feel that we have the responsibility to continue this tradition by ensuring that history of mathematics is not reduced to the chronicle of past events and celebration of the great men of European mathematics that May so powerfully stigmatized. Historia is, and very much seeks to remain, the venue where scholars from all continents and scholarly backgrounds can meet to discuss and exchange their ideas, recent results, and views on the aims of our discipline. One of the most rewarding aspects of our work as editors is the broad correspondence we entertain with colleagues active on all continents. May, we hope, would be happy to see that Historia has met his intentions to promote a journal with an international public. At the time of Historia’s founding, the field of journals accepting papers in history of mathematics included many journals that either are no longer with us or have modified their remit significantly. The Quellen und Studien of Neugebauer, Toeplitz and Stenzel was gone, as was Boncompagni's Bullettino. A fortieth birthday thus remains rare; Archive for History of the Exact Sciences can boast over 50 years, and continues strong. In the time interval between 1973 and today the practices of academic historiography have changed considerably. It's interesting to note that May himself was influenced by one of the most significant trends, quantitative history. Indeed, May showed from the very beginning a strong interest in defining Historia as a venue where historiographical issues were discussed. We believe that this is still a very important feature that is promoted in Historia in a number of ways: by including a section of Abstracts, a book review department that is authoritative in our field and that often includes thoughtful essay reviews, and rather long obituaries of past colleagues. It occurred to us, and our project was enthusiastically backed by Valerie Teng-Broug and Evalyne Salome Wanjiru of Elsevier, that it would be proper to celebrate this anniversary with the present volume, an anthology of papers published in Historia Mathematica during these last 40 years. We had to choose a rather small selection to make a volume of 300 pages out of a list of nearly a thousand papers! Much good material could not be included within the covers of the present volume, but is of course accessible online at Historia’s webpage. The choice has not been easy, yet browsing the many papers that could have been included in this celebration anthology made us justly proud of the journal we are serving as editors. With no length constraint, a “greatest hits” volume would have been three times as long as this. Looking back at the work of our colleagues, from the contributions of masters active in the 1970s such as Joe Dauben, Eberhard Knobloch, Ivor Grattan-Guinness, and Henk Bos, to 1 It is worth naming, with gratitude, all the past editors: Kenneth O. Ma , Joseph W. Dauben, Eberhard Knobloch, David E. Rowe, Karen Hunger Parshall, Jan Hogendijk, Umberto Bottazzini, Craig Fraser, Benno van Dalen, June Barrow-Green. 1 doctoral students who published their first results in our journal, provides both strong impressions of a heritage issuing from the recent past and intimations of developments that are near at hand. Yet, this feeling of continuity notwithstanding, a great deal has changed in our discipline. The “old days” of Kenneth May were characterized by the attempt to rejuvenate and refound a discipline which had come to a kind of maturity at the end of the nineteenth century, one in which activity had somewhat subsided during the two wars (with notable exceptions, of course). In reading May’s editorials one realizes that, in the 1970s, historians of mathematics were trying to define an institutional and cultural space where they could develop their research: the difficulty was to relate this research to that of mathematicians and mathematics teachers; namely, to claim for historical research a status and role within the mathematical community. In a sense this is no longer our main problem: mathematicians constitute our main readership and most of our authors have received their education in mathematics departments. Doctoral students, at least in Europe and North America, have now many possibilities of acquiring a professional training as historians of mathematics, though recent retirements have meant a certain reduction from the peak of such opportunities and on-going employment for even strong recent doctorates can be difficult to obtain. There is no doubt that a dialogue with the mathematical community is vital for our discipline: the connoisseurship of the practicing mathematician helps us greatly, often decisively, in a competent reading of past texts.2 The new frontier nowadays seems to be that of strengthening ties with historians of science, general historians, and philosophers of mathematics as well, scholars who are increasingly associated with and formed in history and philosophy departments. There is a pronounced interest and respect in those circles for the history of mathematics, which increasingly informs research. This is probably the main opportunity for growth for the history of mathematics in the immediate future. One can mention here some recent facts: the success of the history of mathematics sessions organized by Karen Parshall at HSS meetings, the publication in Isis of two Focus sections related to the history of mathematics, the orientation towards a sophisticated methodological approach to history in works by colleagues such as Karine Chemla, Moritz Epple, Catherine Goldstine, Jeremy Gray, David Rowe, and Norbert Schappacher, to give only a few amongst many possible examples.3 Research in the history of mathematics nowadays, as some of the most recent papers published in Historia prove, is not declined along the divide between internalist vs externalist history: such a cleavage belongs to the past. Historians of mathematics are now happily at home with various methodological approaches to history and their tools, coming from fields as diverse as micro-history, intellectual history, the study of historical networks, anthropology and ethnography. The discipline is now sufficiently mature to establish a fruitful cooperation with colleagues working in history departments: while the technical difficulties of the subject remain a challenge for researchers in other fields, it is part of our task to provide the means for a cooperation that can only be mutually enriching. The philosophy of science is no longer regarded as a normative theoretical enterprise detached from scientific practice. It is, instead, pluralistic: we have philosophers of biology, philosophers of physics, philosophers of medicine or engineering, and so on, in addition to the 2 See, Chang, Hasok “The Historian of Science: Painter, Guide, or Connoisseur.” Centaurus 50 (2008), pp. 37-42. 3 Amir R.