"The origins of calculus are clearly empirical. Kepler's first attempts at integration were formulated as "dolichometry"—measurement of kegs—that is, volumetry for bodies with curved surfaces. This is... post-Euclidean geometry, and... nonaxiomatic, empirical geometry. Of this, Kepler was fully aware. The main effort and... discoveries, those of Newton and Leibniz, were of an explicitly physical origin. Newton invented the calculus "of fluxions" essentially for the purpose of mechanics—in fact... calculus and mechanics were developed by him more or less together. The first formulations of the calculus were not even mathematically rigorous. An inexact, semiphysical formulation was the only one available for over a hundred and fifty years after Newton! And yet, some of the most important advances of analysis took place during this period... ! Some of the leading mathematical spirits... were clearly not rigorous, like Euler; but others, in the main, were, like Gauss or Jacobi. The development was as confused and ambiguous as can be, and its relation to empiricism was certainly not according to our present (or Euclid's) ideas of abstraction and rigor. Yet... that period produced mathematics as first class as ever existed! And even after the reign of rigor was... re-established with Cauchy, a... relapse into semiphysical methods took place with Riemann."