...Our patron, Nicole Oresme, reviewing the last session's
Daniel J. Curtin (Northern Kentucky University) and Daniel E. Otero (Xavier University) have organized the Ohio River Early Sources in Mathematical Exposition (ORESME) Reading Group. The Reading Group has been bringing together scholars interested in the history of mathematics from the Cincinnati tri-state area twice a year since 1998 to read original source materials in mathematics. The organization was named after the French Scholastic philosopher Nicole Oresme (1323-1382) whose "latitude of forms" constituted early graphical representations of mathematical functions, long before Descartes. The Reading Group was inspired by the experience the organizers shared during the summers of 1995-1997 at the Institute on the History of Mathematics and its Use in Teaching held at American University (Washington, DC), and was established under the guidance of Fred Rickey, one of the organizers of the Institute and an erstwhile Ohioan.
Individuals interested in participating in ORESME's meetings are encouraged to contact either of the organizers.
Plans are to extend our discussion on the foundations of analysis by reading from Bernhard Riemann's Habilitationschrift of 1854.
The thirty-first meeting was held on October 25-26, 2013, at Xavier University. We continued the thread begun at the last meeting by reading from the work of Karl Weierstrass (1815-1897). Along with Cauchy, Weierstrass has long been regarded as one of the central figures in the rigorization of analysis in the nineteenth century. We read two pieces "by" Weierstrass (given that the first was actually written by his student, H. A. Schwarz):
This task was made far more challenging by the fact that neither of these pieces are available in their entirety in English translation and facility with German is weak among the members of the seminar. So Wiebke Diestelkamp (U Dayton, and current President of the MAA Ohio Section) agreed to produce a rough translation for us of both pieces. For this we acknowledge her kind assistance.
The thirtieth meeting was held January 25-26, 2013, at Northern Kentucky University (which, incidentally, marked the 15th anniversary of the ORESME Reading Group!). We recognized the recent publication by Rob Bradley and Ed Sandifer, the co-founders of our sister seminar Arithmos, of Cauchy's Cours d'Analyse (Springer, 2009) by reading from their translation of these highly influential lecture notes by Augustin-Louis Cauchy (1789-1857) for a course that he never gave (!) at the École Polytechnique. It is in these lecture notes that Cauchy first introduced the concept of limits into the calculus. We read the following selections from the Cours d'Analyse:
The twenty-ninth meeting was held September 21-22, 2012, at Xavier University. We celebrated the bicentennial anniversary of the publication of Théorie Analytique des Probabilités by Pierre-Simon de Laplace (Courcier, Paris, 1812; 2nd ed., 1814; supplements in 1816, 1818, 1820, contained in 3rd ed., 1820; reprinted with a fourth supplement, 1825) by reading a portion (Chapter IV, Section 18) of this seminal work, the section in which he presents his version of the Central Limit Theorem, a result that lies at the foundation of much of modern statistical methodology. Conveniently, one of our members, Dick Pulskamp, recently wrote a piece for publication on TAP, which was ideal for helping us understand the context and content of Laplace's work. We also consulted these supplementary texts:
• E.C. Molina, "The Theory of Probability: Some Comments on Laplace's Théorie Analytique", Bull. AMS 36 (1930), 369-392.
• Anders Hald, A History of Mathematical Statistics from 1750 to 1930, Wiley, 1988, sections 17.1-17.2.
The twenty-eighth meeting was held January 20-21, 2012, at Northern Kentucky University. We took up two selections of that giant of modern mathematics, David Hilbert (1862-1943):
• his proof of the Hilbert Basis Theorem (as Hilbert understood it, any ideal in the ring of polynomials in n variables with rational coefficients is finitely generated), which appears in the paper Über die Theorie der algebriaschen Formen [On the theory of algebraic forms], original pub. as Math. Annalen, 36 (1890), 473-534; our English translation was by M. Ackerman in Lie Groups: History, Frontiers and Applications, Math Sci Press, 1978;
• one of his most popular works, Grundlagen der Geometrie, Teubner, 1899, the first edition of a work that has seen no fewer than 19 editions (!!); the English translation we used was the "authorized" translation of E.J. Townsend: Foundations of Geometry, Open Court, 1910.
The twenty-seventh meeting was held September 23-24, 2011, at Xavier University. We extended our study of the history of modern algebra by turning to the work that started it all: Evariste Galois' Mémoire sur les conditions de résolubilité des équations par radicaux, written days before his too-early death in 1831 (pub. posthumously by Joseph Liouville in his Journal de mathématiques pure et appliquées, 1846, pp. 417-433). An English translation appears as an appendix in Harold M. Edwards' Galois Theory (Springer, 1984).
The twenty-sixth meeting was held February 25-26, 2011, at Northern Kentucky University. This meeting was a natural follow-up to the previous meeting, continuing a line of reasoning first begun by Ernst Kummer, with his ideal complex numbers. We undertook a a study of the evolution of the concept of an ideal (in a ring) by reading from Richard Dedekind's first exposition in the Supplement to the 2nd (1871) edition of Dirichlet's Vorlesungen über Zahlentheorie (Lectures on Number Theory). The relevant sections have appeared in a recent English translation by Jeremy Avigad (Carnegie Mellon University). Together with the German originals for this text and the revised texts that Dedekind provided for the 3rd (1879) and 4th (1894) editions of Dirichlet's textbook, we will also draw from two papers by Harold M. Edwards, who wrote extensively on this subject in the early 1980s. Here's a full bibliographic list:
• Richard Dedekind. Supplement X von Dirichlets Vorlesungen über Zahlentheorie, 2. Auflage, 1871.
• Jeremy Avigad. Dedekind's 1871 version of the theory of ideals. Carnegie Mellon Technical Report CMU-PHIL-162, 2004.
• Richard Dedekind. Supplement XI von Dirichlets Vorlesungen über Zahlentheorie, 3. Auflage, 1879.
• Richard Dedekind. Supplement XI von Dirichlets Vorlesungen über Zahlentheorie, 4. Auflage, 1894.
• H.M. Edwards. The genesis of ideal theory. Arch. Hist. Exact Sci. 23, 1980, 321-378.
• H.M. Edwards. Dedekind's invention of ideals. Bull. London Math. Soc. 15 (1), 1983, 8-17.
The twenty-fifth meeting was held October 8-9, 2010, at Xavier University. We read Zur Theorie der complexen Zahlen (On the theory of complex numbers), J. Reine Angew. Math. 35 (1847) 319-326 [Collected Papers, A. Weil (ed.), pp. 203-210], in its English translation as found in Smith, A Source Book in Mathematics, 119-126. The paper presents Ernst Kummer's first description of his invention of ideal complex numbers, famously used to illustrate that unique factorization of integers into primes is not always available in certain rings of algebraic integers.
The twenty-fourth meeting was held January 29-30, 2010, at Northern Kentucky University. We read from two papers by the Scottish mathematician Joseph HM Wedderburn (1882-1948): A Theorem on Finite Algebras, Trans. AMS, Vol. 6, No. 3 (Jul., 1905), pp. 349-352; and Non-Desarguesian and Non-Pascalian Geometries, Trans. AMS, Vol. 8, No. 3 (Jul., 1907), pp. 379-388. The first of these papers includes three proofs (!) of the celebrated theorem that carries Wedderburn's name, that every finite division algebra is a field; the second proves the existence of finite geometries in which Desargues's Theorem fails. (As it turned out, we had little time to explore the second of these two papers.)
The twenty-third meeting was held October 2-3 , 2009, at Xavier University: to celebrate the bicentennial of its publication, we read the paper that made Carl Friedrich Gauss (1777-1855) famous as something other than a mathematical prodigy, his work on the computation of the orbit of Ceres (in which he successfully predicted where astronomers would be able to relocate it, within a half a degree of arc, after its passage behind the Sun). Our attention focuses not on the astronomical calculation but on his work in this same paper wherein he lays out his method of least squares for minimizing measurement errors and the function that describes the probability distribution now named after him. We read Section III of Book II of his Theoria motus corporum coelestium in sectionibus conicis solem ambientum (Hamburg, 1809) in the 1857 English translation, Theory of the motion of the heavenly bodies moving about the sun in conic sections, by Charles Henry Davis (available in a 2004 Dover reprint).
The twenty-second meeting was held January 16-17, 2009, at Northern Kentucky University. We read from the work of Bernard Bolzano on the foundations of analysis. In particular, we read an English translation of his seminal 1817 paper Rein analytischer Beweis des Lehrsatzes, das zwichen je zwei Werthen, die ein entgegengesetzes Resultat gewahren, wenigstens eine reele Wurzel der Gleichung liege (A Purely Analytic Proof of the Theorem that between and two Values which give Results of Opposite Sign, there lies at least one real Root of the Equation) from Steve Russ' The Mathematical Works of Bernard Bolzano (Oxford U. Pr., 2004).
The twenty-first meeting was held September 19-20, 2008, at Xavier University. In a departure from tradition, our topic was not an important work of an individual mathematician, but rather a survey of the development of a central idea that required many decades to take shape, the determinant. We followed (essentially) the work of Thomas Muir (1844-1934), the Scottish mathematician famous for a monumental multi-volume work, The Theory of Determinants in the Historical Order of Development (Macmillan, 1890, 1906) which to this day remains the authority on the subject. We also read a single modern paper, by Bruce Hedman, concerning Maclaurin's contributions.
1. Leibniz, Specimen Analyseos novae, qua errores vitantur, quasi manu ducitur, et facile progressiones invenientur [A Model for a new kind of Analysis, by which error is avoided, the mind is led as if by the hand, and patterns are easily discovered], Leibnizens mathematische Schriften, C.I. Gerhardt, ed., Part II, Volume 3, Berlin, 1863, pp. 7-8. Unpub. ms., dated June 1678.
2. Muir, pp. 6-10: a description of the following ms.
3. Leibniz, Brief an de L'Hosptial, VI, Hanover, 28 Avril 1693 [Letter to L'Hôpital, VI, Hannover, 28 April 1693], Leibnizens mathematische Schriften, C.I. Gerhardt, ed., Part I, Volume 2, Berlin, 1850, pp. 238-241.
4. Hedman, An earlier date for "Cramer's rule", Historia Mathematica 26 (1999) 4, 365-368: a relatively new piece of scholarship which argues the advertised claim.
5. Maclaurin, From Treatise of Algebra, 2nd ed., London, 1756, Chap. XII. pp. 81-85.
6. Muir, pp. 11-14: a description of the following two pieces.
7. Cramer, From Introduction a l'Analyse des Lignes Courbes algébriques [An Introduction to the Analysis of algebraic Curved Lines], Genève, 1750, pars. 37-38, pp. 57-60.
8. Cramer, From Introduction a l'Analyse des Lignes Courbes algébriques [An Introduction to the Analysis of algebraic Curved Lines], Genève, 1750, App. No. I, pp. 656-659.
9. Muir, pp. 14-17: a description of the following excerpt.
10. Bézout, Recherches sur le degré des équations résultantes de l'évanouissement des inconnues, et sur les moyens qu'il convient d'employer pour trouver ces équations [Researches on the degree of equations resulting from the vanishing of unknowns, and on the means which are convenient to use in order to solve these equations], Hist. de l'Acad. Roy. des Sciences, Paris, 1764, pp. 288-295.
11. Muir, pp. 17-24: a description of the following excerpt.
12. Vandermonde, From Mémoire sur l'élimination [A memoir on elimination], Hist. de l'Acad. Roy. des Sciences, Paris, 1772, 2° partie, pp. 516-525.
13. Muir, pp. 24-33: a description of the following paper.
14. Laplace, Recherches sur le calcul intégral et sur le système du monde, Sec. IV [Researches on integral calculus and the system of the world, Sec.], Hist. de l'Acad. Roy. des Sciences, Paris, 1772, 2° partie, pp. 294-304.
15. Muir, pp. 63-66: a description of the following excerpt.
16. Gauss, From Disquisitiones Arithmeticae [Investigations in Arithmetic], Leipzig, 1801, Sect. V, Pars. 153-159, 266-270, in the English edition by Arthur A. Clarke, rev. by William C. Waterhouse, Cornelius Greither, and A. W. Grootendorst, Springer, New York, 1986, pp. 108-115, 292-297.
17. Muir, pp. 80-92: a description of the following excerpt.
18. Binet, From Mémoire sur un système de formules analytiques, et leur application à des considérations géométriques [A memoir on a system of analytical formulas and their application to geometric considerations], Journal de l'Ecole Polytechnique, 1812, T. IX, Cah. 16, pp. 280-302.
19. Muir, pp. 92-131: a description of the following excerpt.
20. Cauchy, Mémoire sur les fonctions qui ne peuvent obtenir que deux valuers égales et de signes contraires par suite des transpositions opérées entre les variables qu'elles renferment [A memoir on functions that can have but two equal values, and on the contrary signs they must hold because of transpositions performed between the variables], Oeuvres de Cauchy, Ser. II, T. 1, pp. 91-169.
21. Muir, pp. 176-178: a description of three papers, the third of which follows here.
22. Jacobi, Ueber die Pfaffshce Methode, eine gewöhnliche lineäre Differential-gleichung zwischen 2n Variabeln durch ein System von n Gleichungen zu integriren [On Pfaff's Method, an ordinary linear Differential equation between 2n Variables in terms of a System of n equations to integrate], Werke, IV, pp. 17-29. Special thanks to Danny Otero, Dick Pulskamp and Chuck Holmes for preparing English translations of the materials above from, respectively, Latin, French and German originals.
The twentieth meeting (and our 10th anniversary!) was held January 25-26, 2008, at Xavier University. We read three papers by Ernst Zermelo (1871-1953) in celebration of the 100th anniversary of his publication of the axioms of set theory and a proof of the axiom of choice. The papers were: Beweis, daß jede Menge wohlgeordnet werden kann, Math. Ann. 59 (1904), no. 4, 514--516 (an English translation of the original text of this paper, part of a letter to Hilbert dated 24 Sep 1904, has the title Proof that every set can be well-ordered, in Jean van Heijenoort's From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Harvard, 1967); Neuer Beweis für die Möglichkeit einer Wohlordnung, Math. Ann. 65 (1907), no. 1, 107--128 (an English translation of this later version of the well-ordering property for sets has the title A new proof of the possibility of a well-ordering, in Jean van Heijenoort's From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Harvard, 1967); and Untersuchungen über die Grundlagen der Mengenlehre. I, Math. Ann. 65 (1908), no. 2, 261--281 (an English version of this paper that lays out the axioms for set theory has the title Investigations in the foundations of set theory. I, also in Jean van Heijenoort's From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Harvard, 1967).
The nineteenth meeting was held October 5-6, 2007, at Northern Kentucky University. This was the second meeting in celebration of the Leonhard Euler tricentennial. We read from Euler's Introductio in analysin infinitorum (Introduction to the analysis of the infinite), translated by John D. Blanton (Springer-Verlag, 1988). The selections from Book One: Chapter I. On Functions in General; Chapter VI. On Exponentials and Logarithms; Chapter VII. Exponentials and Logarithms Expressed through Series; Chapter VIII. On Transcendental Quantities Which Arise from the Circle; Chapter XVII. On using recurrent series to find roots of equations; and Chapter XVIII On continued fractions; and from Book Two: Chapters I and II.
The eighteenth meeting was held January 26-27, 2007, at Xavier University. This was the first of two meetings planned to celebrate the Euler tricentennial. We read selections (from v.1, Sections I.xxi - I.xxiii on logarithms; II.v on series; IV.viii on square roots of binomials; IV.x - IV.xv on cubics and quartics; from v.2, Sections IV - VII on solutions to Pell's equation) from Leonhard Euler's Vollständige Anleitung zur Algebra (St. Petersburg Akademie der Wissenschaften, 1770), available in a 1984 Springer reprint of John Hewlett's 1828 English translation as Elements of Algebra.
The seventeenth meeting was held September 15-16, 2006, at Northern Kentucky University. The topic of the meeting was the work and career of Alan Turing. We read two of his most important papers, Computing machinery and intelligence, in Mind (new series), vol. 59, no. 236 (Oct 1950), pp. 433-460, in which Turing presents what is known today as the Turing test for artificial intelligence; and On computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. Soc. (2) 42 (1936), 230-265, in which Turing explores a definition of computability.
The sixteenth meeting was held January 20-21, 2006, at the University of Cincinnati, hosted by Charles Groetsch, one of the founding members of the Reading Group. The topic of the meeting was the famous and important work of Galileo Galilei (1564-1642) on kinematics, as it appeared in Days Three and Four of his Discorsi e dimonstrzioni mathematiche intorno a due nuove scienze (Discourses and Mathematical Demonstrations Concerning Two New Sciences), published posthumously in 1654 and available in English translation by Stillman Drake (U. Wisc. Pr., 1974).
The fifteenth meeting was held October 21-22, 2005, at Northern Kentucky University. The focus of the meeting was the mathematical career of Hermann Weyl (1885-1955). We read his book on Symmetry (Princeton University Press, 1952, reprinted in 1982 and 1989).
The fourteenth meeting was held January 28-29, 2005, at Xavier University. The readings included the paper in which John von Neumann first proved the Minimax Theorem and launched the serious mathematical theory of games: Zur Theorie der Gesellschaftspiele (Math. Annalen 100 (1928), 295-320), translated as On the theory of games of strategy by Sonya Bargmann, in Contributions to the Theory of Games, IV (Annals of Mathematics Studies 40), A. W. Tucker and R. D. Luce, eds., Princeton U. Pr., 1950, pp. 13-42; additionally, the paper that started von Neumann thinking about games of strategy: Emile Borel, La théorie du jeux et les équations intégrales à noyau symétriques (C. R. Math. Acad. Sci. Paris, vol. 173 (1921), 1304-1308) , translated as Theory of games and integral equations with skew symmetric kernels by Leonard J. Savage, Econometrica, vol. 21, no. 1 (Jan 1953), 97-100. For details see Danny Otero's report of the meeting.
The thirteenth meeting was held September 17-18, 2004, at Northern Kentucky University. The readings were two papers by Maj. Percy MacMahon: The design of repeating patterns, Part I ( Proc. Royal Soc. London, Ser. A, vol. 101, no. 708 (Apr 1, 1922), 81-94) , and On the thirty cubes that can be constructed with six differently coloured squares (Proc. London Math. Soc. 24 (1893), 145-155). For details see Danny Otero's report of the meeting.
The twelfth meeting was held January 30-31, 2004, at Xavier University. The readings were: the Preface, and Chapters 1, 6, and 11 from George Polya's Mathematics and Plausible Reasoning (1954); the Preface and Appendix to vol 1, and Chapters 13-14 of vol. 2 from Mathematical Discovery (1961). This was the second meeting on the work of Polya, and we focused deliberately on his work in teaching mathematics. For details see Danny Otero's report of the meeting.
The eleventh meeting was held September 26-27, 2003, at Northern Kentucky University. The reading was the Introduction and first Chapter of George Polya's Combinatorial Enumeration of Groups, Graphs and Chemical Compounds (Springer, 1987), the English version with R. C. Read of a translation of his influential 1937 paper Kombinatorische Anzahlbestimmungen fur Grüppen, Graphen und chemische Verbindungen (Acta Math., 68, 145-254) in which the eponymous Enumeration Theorem first appears. This is the first of two meetings devoted to Polya. For deatils see Danny Otero's report of the meeting.
The tenth meeting was held January 24-25, 2003, at Xavier University. The readings were by E. H. Moore: first, his A doubly infinite system of simple groups (in Mathematical Papers Read at the  International Mathematical Congress, Macmillan,1896), in which he contributed to early work on the classification of finite groups; and also On the foundations of mathematics (Science, vol. XVII, no. 428 (March 13, 1903)), Moore's Presidential address upon retirement from that AMS post. For details see Danny Otero's report of the meeting.
The ninth meeting was held September 20-21, 2002, at the University of Louisville. Michael J. Crowe, Distinguished Scholar in Residence and our special guest, led the discussion of work of Hermann Grassmann and, more generally, on the history of vector analysis. Danny Otero's report of the meeting. It includes links to Crowe's notes "A History of Vector Analysis."
The eighth meeting was held May 10-11, 2002at Xavier University. The reading was L. E. Dickson's (1874-1954), Recent progress on Warings's Theorem and its generalizations (Bull. Amer. Math.Soc., 39, 701-727). For more, see Danny Otero's report of the meeting.
The seventh meeting was held September 14-15, 2001,at NKU. We read three papers and a letter of Georg Cantor dating from the 1880s and 1890s on his transfinite numbers. For more, see Danny Otero's report of the meeting.
The sixth meeting was held January 26-27, 2001,at XU. We read a paper by the British mathematician William Burnside that was influential in the development of group theory at the beginning of the 20th century. The paper, On an unsettled question in thetheory of discontinuous groups (Quart. J. of Pure and Applied Math. 33 (1902) 230-238), introduced what is now known as the Burnside problem: are all finitely generated torsion groups finite? For more, see Danny Otero's reportof the meeting.
There was no meeting in Fall 2000. The members decided to support instead the Midwest History of Mathematics Conference at NKU, October 13-14, 2000.
Our fifth meeting was held March 24-25, 2000, at Miami University, hosted by members David Kullman and Chuck Holmes. John Fauvel of the Open University (UK) was our special guest, and led us in a reading of Isaac Newton's "De Analysi" (1669) [in The Mathematical Papers of Isaac Newton, D.T. Whiteside, ed., Cambridge, 1967-, vol. 1, pp. 206-247]. In lieu of a formal report, we have photos of the meeting taken by our illustrious guest, John Fauvel.
Our fourth meeting was held at NKU on September 17-18,1999. We completed our study of Klein and his Erlangerprogramm. See Danny Otero's report of the meeting.
Our third meeting was again at Xavier, January 29-30,1999. This was the first of two meetings on Felix Klein's Erlangerprogramm. The primary reading was Haskell's translation A Comparative Review of Recent Researches in Geometry, (Bull. NY Math.Soc. 2 (1892-3), 215-249). Dick Davitt prepared a preliminary bibliography of Klein and the Erlangerprogramm for the members. See Danny Otero's report of the meeting.
The second meeting took place on September 18-19, 1998, at Northern Kentucky University. The paper Sur une courbe continue sans tangente obtenue par une construction géométrique élémentaire, (Archiv fur Matematik, Astronomi och Fysik, 1 (1904) 681-702, [trans.by Ilan Vardi in Classics on Fractals, Gerald Edgar, ed. (Addison-Wesley, 1993)]), in which Helge von Koch presented the Koch Snowflake, was the topic of the meeting. This material is reprinted, with some additions, in Une méthode géométrique élémentaire pour l'étude de certaines questiones de la théorie des courbes planes (ActaMathematica, 30 (1906), 145-174). The last four pages are available here in JPEG format. Dick Pulskamp's translation is available in dvi format. Here is a translation of some additional material, not in the 1904 paper, proving the curve is simple.
The inaugural meeting of ORESME was held January 30-31, 1998, at Xavier University. We read William Fogg Osgood's paper, A Jordan Curve of Positive Area (Trans. AMS, 4 (1903) 107-112). See Danny Otero's report of the meeting.