Wrestling with Nature From Omens to Science

Mixed Mathematics 167

how natural philosophy began to be implicated with practical know- how.
The accompanying association of mixed mathematics with experimental
procedures rendered natural philosophy itself an increasingly contested
territory. The nineteenth- century German label “applied mathematics,” as
an equivalent to “mixed mathematics,” itself captured an important part
of these developments, therefore: physics could not but be in part directed
to instrumental utility (“application”) in the nineteenth century, because
it had taken mixed mathematics into its heart. There it has remained. But
the compromised category of natural philosophy has not survived the
upheaval.

NOTES

1. Gary I. Brown, “The Evolution of the Term ‘Mixed Mathematics,’” Journal of
   the History of Ideas 52 (1991): 81–102; Richard R. Yeo, Encyclopaedic Visions: Scientifi c
   Dictionaries and Enlightenment Culture (Cambridge: Cambridge University Press, 2001),
   64; on d’Alembert’s usage, Lorraine J. Daston, Classical Probability in the Enlightenment
   (Princeton, NJ: Princeton University Press, 1988), 53–56.


2. W. R. Laird, “The Scope of Renaissance Mechanics,” Osiris 2 (1986): 43–68;
   Laird, The Unfi nished Mechanics of Giuseppe Moletti: An Edition and English Translation
   of His Dialogue on Mechanics, 1576 (Toronto: University of Toronto Press, 2000). These
   sources discuss the promotion in sixteenth- century Italy of mechanics as a mathemati-
   cal science rather than an art.


3. David L. Wagner, ed., The Seven Liberal Arts in the Middle Ages (Bloomington:
   Indiana University Press, 1983).


4. Paul Lawrence Rose, The Italian Renaissance of Mathematics: Studies on Humanists
   and Mathematicians from Petrarch to Galileo (Geneva: Droz, 1975); Michael H. Shank,
   “Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism: Goldstein-
   ian Themes in the ‘Defense of Theon against George of Trebizond,’” Perspectives on
   Science 10 (2002): 179–207.


5. Richard D. McKirahan Jr., “Aristotle’s Subordinate Sciences,” British Journal for
   the History of Science 11 (1978): 197–220; cf. Moletti, in Laird, Unfi nished Mechanics,
   189–191.


6. Tartaglia, discussing Pierre d’Ailly’s commentary on Sacrobosco, in his Euclide
   (1569), f01.6r, quoted in Ann E. Moyer, Musica scientia: Musical Scholarship in the Italian
   Renaissance (Ithaca, NY: Cornell University Press, 1992), 132–33.


7. James A. Weisheipl, “The Interpretation of Aristotle’s Physics and the Science of
   Motion,” in The Cambridge History of Later Medieval Philosophy, ed. Norman Kretzman,
   Anthony Kenny and Jan Pinborg (Cambridge: Cambridge University Press, 1982),
   521–36, on 525; see also W. R. Laird, “The Scientiae mediae in Medieval Commentaries
   on Aristotle’s Posterior Analytics” (PhD diss., University of Toronto, 1983).


8. Christophorus Clavius, “In disciplinas mathematicas prolegomena,” Opera Math-