The Encyclopaedia Britannica in its history of Science article states that Newton integrals were made of infinitesimals, whereas Leibniz' were made of sticks and that the former's theory prevailed.

But very often I read, even in some question here, that both theories are concerned with infinitesimals, and yet my trust in EB suggests they cannot be wrong.

I know that Leibniz was not clear in is published works, but can anyone discuss the issue in one sense or another?

  • $\begingroup$ See Isaac Newton, INTRODUCTION TO THE QUADRATURE OF CURVES (Engl.transl.1710) : "I don't here consider Mathematical Quantities as composed of Parts extreamly small, but as generated by a continual motion. […] Fluxions are very nearly as the Augments of the Fluents, generated in equal, but innitely small parts of Time; and to speak exactly, are in the Prime Ratio of the nascent Augments." 1/2 $\endgroup$ Oct 10, 2019 at 12:52
  • $\begingroup$ "to investigate the Prime and Ultimate Ratio's of Nascent or Evanescent Finite Quantities, is agreeable to the Geometry of the Ancients; and I was willing to shew, that in the Method of Fluxions there's no need of introducing Figures innitely small into Geometry." 2/2 $\endgroup$ Oct 10, 2019 at 12:52
  • $\begingroup$ See also Eberhard Knobloch, Leibniz's Rigorous Foundation Of Infinitesimal Geometry (2002) : "Leibniz said in his summary of De quadratura arithmetica that "it is overly carefully demonstrated that the procedure of constructing certain rectilinear step spaces and in equal fashion polygons can be continued to such a degree that they differ from each other or from curves by a quantity which is smaller any given quantity." $\endgroup$ Oct 10, 2019 at 13:00
  • $\begingroup$ @MauroALLEGRANZA, your comments seem to confirm EB: while Newton explicitly refers yo the continuum, Leibniz refers to smaller quantity $\endgroup$
    – user157860
    Oct 10, 2019 at 13:06
  • $\begingroup$ And see l'Hopital basic definitions in the first treatise of Calculus (that was surely Leibnizian) : "Definition I. Those quantities are called variable which increase or decrease continually, as opposed to constant quantities that remain the same while Others change. […] Definition II. The infinitely small portion by which a variable quantity continually increases or decreases is called the Differential." $\endgroup$ Oct 10, 2019 at 13:07


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