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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?

user157860
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  • 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 – Mauro ALLEGRANZA Oct 10 '19 at 12:52
  • "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 – Mauro ALLEGRANZA Oct 10 '19 at 12:52
  • 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." – Mauro ALLEGRANZA Oct 10 '19 at 13:00
  • @MauroALLEGRANZA, your comments seem to confirm EB: while Newton explicitly refers yo the continuum, Leibniz refers to smaller quantity – user157860 Oct 10 '19 at 13:06
  • 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." – Mauro ALLEGRANZA Oct 10 '19 at 13:07
  • EB History of Science does not mention infinitesimals, integrals or Leibniz. Could you quote the passage you refer to, "integrals made of infinitesimals" or "sticks" sound very off? Both Newton and Leibniz used infinitesimals originally, but Newton later shifted towards limits, see What was the notion of limit that Newton used? That is what eventually "prevailed". On the other hand, Leibniz's differentials "prevailed" over Newton's fluxions. – Conifold Oct 10 '19 at 22:56

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