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.
READINGS
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.
