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[A (XXV.45)] [B (XXV.46)] [C (XXVI.5)] |
Mathematics
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 Kant  Herder
Notes from Kant’s
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[ Introduction ] [ Metaphysics ] [ Moral Phil. ] [ Physical Geogr. ] [ Logic ] [ Physics ] [ Mathematics ] [ Varia ] How to Use these Pages ➝ to the transcriptsManuscripts: The manuscripts are arranged into groups: two sets each of notes on physics and math, and then additional notes from three of Herder’s notebooks. These groups are accessed by clicking on the red-links in the light yellow window at the top of the page (e.g., [XX.188]). Explanatory Notes / Textual Notes: There are two windows with notes: explanatory and textual. The text in those windows on this page explains what you will find in them. Introduction to the: [ Mathematics Notes ] [ List of Manuscripts ] Introduction to the Mathematics NotesHerder’s notes on mathematics consist of seventeen 4° pages of text – fourteen pages from two signatures and three pages from Herder’s “brown” study book (although these latter three pages are perhaps more likely Herder’s notes for his own teaching efforts, as discussed in the General Introduction). If Herder’s notes come from Kant’s lectures, they would stem from either winter 1762-63 or summer 1763. They would also be the only notes we have from Kant’s mathematics lectures.[1] The text at A1-A3 almost certainly is based on A. G. Kästner’s 20 pp. “Vorerinnerung” that begins his Anfangsgründe der Arithmetik (1758). The text at A4-A7 is based on Christian Wolff’s Auszug aus den Anfangsgründe aller mathematischen Wissenschaften, zu bequemerem Gebrauche der Anfänger, auf Begehren verfertiget (1717, etc.). A1 and B1 are roughly the same (thus: based on Kästner). A3 and B2 are also roughly the same, tracking Wolff’s Auszug, paragraphs 1-12 of the “Rechen-Kunst” section (up through subtraction), although the B-signature does not give paragraph numbers. While A then proceeds with the various axioms presented in Wolff, B discusses fractions (bottom of B2-B3), square and cube numbers (bottom B3-B4), proportions of magnitudes (B5), and then preliminary remarks on geometry (B6-B7). The content of the A- and B-signatures clearly comes from the very beginning of the semester. There is enough overlap between them to suggest that they came from two separate semesters and too many differences for them to be considered as separate drafts of the same set of notes. Nor should either signature be viewed as anything more than a fragment; they are both far too short and incomplete to be representative of an entire semester of lectures. Finally, it is quite possible that one or both are Herder’s preparatory notes for his own teaching The records indicate that Kant lectured on mathematics at least fifteen of his first seventeen semesters at the university, and he appears to have always lectured from one of Christian Wolff’s two widely-used mathematics textbooks: Anfangsgründe aller mathematischen Wissenschaften, 4 parts (Frankfurt/Leipzig, 1710)[2] and its summary version: Auszug aus den Anfangsgründe aller mathematischen Wissenschaften, zu bequemerem Gebrauche der Anfänger, auf Begehren verfertiget (Halle: 1717). Both texts enjoyed many editions.[3] It is remarkable that Kant taught a course on mathematics nearly every semester for the first eight years of his teaching career, and then (as far as we know) never so much as announced another course, much less taught one. His last recorded offering of this course was as a privatissima during winter 1763-64, when Herder was still a student.[4] Irmscher (1964) was the first to transcribe and discuss these notes and expressed some reservations that the mathematics notes might instead come from F. J. Buck’s lectures, also noting that several passages from the XXV.46 manuscript have the appearance of being later additions, and which Irmscher thought might have been Herder’s prepatory notes for giving his own lectures on mathematics; while these marginal notes do indeed appear to be later additions, there is nothing that suggests they are preparatory notes for his own lectures. So there are three possible sources of these notes to be considered: (1) Herder developed them on his own from various texts (e.g., Wolff and Kästner), or Herder wrote them down in the context of a course of lectures by either (2) Kant, or (3) someone else. As with all of Herder’s notes that bear the appearance of lecture notes, there is a default assumption that they stem from Kant who was, by all accounts, the professor whom Herder valued most. We know that Kant lectured on mathematics during Herder’s student years and that Herder was attending Kant’s courses free of charge – presumably all of them – and some of the testimony is especially supportive of Herder having attended Kant’s mathematics lectures. There are four relevant testimonials (the first three were already mentioned above in the discussion of the physics lectures): (1) Herder (Kalligone, FHA 8: 651-52): Herder “heard all of the lectures, some more than once.” (2) Herder (Letters on the Advancement of Humanity, FHA 7: 424-25): “The wellspring of [Kant’s] lectures was the history of men, of nations, and of nature, as well as natural science, mathematics, and his own experience.” (3) Caroline Herder (1830, 68-69): “He most preferred hearing Kant talk about astronomy, physical geography, and in general about the great laws of nature.” (4) Karl Gottlieb Bock, in recounting Kant’s lectures that Herder attended, lists “logic, metaphysics, moral philosophy, mathematics, physical geography” (Herder 1846, 133-3). The specific mention of mathematics and astronomy in these lists of subjects is striking (astronomy was a subject discussed in both the physics and the mathematics textbooks used by Kant). We also know that Herder studied with other professors, including possibly Langhansen but certainly Buck and specifically his mathematics lectures. Herder attended the lectures of Friedrich Johann Buck (1722-1786), the Professor of Metaphysics and Logic from 1759 to 1769, “with great diligence” (Böttiger 1998, 125), and Buck routinely offered private lectures on mathematics.[5] Herder, who had very little money, would also have had the opportunity to attend without cost the public mathematics lectures of Christoph Langhansen (1691-1770) who, apart from being a Professor of Theology since 1725, was also the Professor of Mathematics (1719-65). Langhansen’s public lectures alternated each semester between arithmetic/geometry and trigonometry/astronomy. He had also served as Rector when Herder matriculated at the university in August 1762. The two 7 pp. group of notes make clear references to Wolff, as well as to Kästner’s Anfangsgründe der Arithmetik (1758),[6] although neither of these texts help establish the source lecture since no one is shown in the Lecture Catalog as using Kästner at the time.[7] We do know, however, that Kant was using Wolff in his lectures, and we also know that he was already engaged with Kästner’s work (see, for instance, his 1763 essay on “Negative Magnitudes” where he praises Kästner).[8] Martin (1967) has argued that Kant taught this course as a two-semester sequence with arithmetic, geometry, and trigonometry (the first three parts of Wolff’s Auszug 1717) during the winter semester, and mechanics, hydrostatics, aerometry, and hydraulics (the next four parts of Wolff’s textbook) during the summer semester. Outline of the Textbook Christian Wolff, Auszug aus den Anfangsgründe aller mathematischen Wissenschaften, Zu bequemerem Gebrauche der Anfänger, Auf Begehren verfertiget (Halle: 1717). The Auszug consists of nineteen parts; page numbers are from the 6th edition (1737):
List of Manuscripts [top]At the Staatsbibliothek zu Berlin – Preußischer Kulturbesitz, Nachlass Johann Gottfried Herder: XXV.45 (4°, 17.5 x 20.5 cm): 7 pp. Text in ink. [45(A)] XXV.46 (4°, 17.5 x 20.5 cm): 7 pp. Text in ink. [46(B)] XXVI.5 (4° notebook, 17.5 x 20 cm): 3 p. Text in ink. [XXVI.5(C)] |