sufficient principles of explanation.  For magnitudes alone are knowable; wherever it is impossible to measure and count, to determine force mathematically, there rigorous, exact science ceases.  Nature a system of regularly moved particles of mass; all that takes place mechanical movement, viz., the combination, separation, dislocation, oscillation of bodies and corpuscles; mathematics the organon of natural science!  Into this circle of modern scientific categories are articulated, further, Galileo’s new conception of motion and the conception of atoms, which, previously employed by physicists, as Daniel Sennert (1619) and others, is now brought into general acceptance by Gassendi, while the four elements are definitively discarded (Lasswitz, Geschichte der Atomistik, 1890).  Still another doctrine of Democritus is now revived; an evident symptom of the quantification and mechanical interpretation of natural phenomena being furnished by the doctrine of the subjectivity of sense qualities, in which, although on varying grounds, Kepler, Galileo, Descartes, Gassendi, and Hobbes agree.[1] Descartes and Hobbes will be discussed later.  Here we may give a few notes on their fellow laborers in the service of the mechanical science of nature.

[Footnote 1:  Cf. chapter vi. in Natorp’s work on Descartes’ Erkenntnisstheorie, Marburg, 1882, and the same author’s Analekten zur Geschichte der Philosophie, in the Philosophische Monatshefte, vol. xviii. 1882, p. 572 seq.]

We begin with John Kepler[1] (1571-1630; chief work, The New Astronomy or Celestial Physics, in Commentaries on the Motions of Mars, 1609).  Kepler’s merit as an astronomer has long obscured his philosophical importance, although his discovery of the laws of planetary motion was the outcome of endeavors to secure an exact foundation for his theory of the world.  The latter is aesthetic in character, centers about the idea of a universal world-harmony, and employs mathematics as an instrument of confirmation.  For the fact that this theory satisfies the mind, and, on the whole, corresponds to our empirical impression of the order of nature, is not enough in Kepler’s view to guarantee its truth; by exact methods, by means of induction and experiment, a detailed proof from empirical facts must be found for the existence not only of a general harmony, but of definitely fixed proportions.  Herewith the philosophical application of mathematics loses that obscure mystical character which had clung to it since the time of Pythagoras, and had strongly manifested itself as late as in Nicolas of Cusa.  Mathematical relations constitute the deepest essence of the real and the object of science.  Where matter is, there is geometry; the latter is older than the world and as eternal as the divine Spirit; magnitudes are the source of things.  True knowledge exists only where quanta are known; the presupposition of the capacity for knowledge is the capacity to count;