Starting from the problem of continuum, discontinuum and indivisibles and its discussions in Antiquity (Zenon of Elea, Xenokrates, Aristoteles) and Middle Ages (Bradwardinus, d’Autrecourt, Gregorio da Rimini, Buridanus, Arabian atomistic theory of Kalâm) up to the seventeenth-century discussions about indivisibles (Galilei, Cavalieri, Roberval, Pascal, Descartes) fundamental for the problem of determination of a tangent at an arbitrary point of a curve (Gregory, Barrow), the author analyses some important texts of Isaac Newton (Philosophiae Naturalis Principia Mathematica, De motu corporum, Liber primus, Sectio prima, 1687) and Gottfried Wilhelm Leibniz (letter to de l’Hospital 14/24 Juin 1695; and the letter to R. J. Tournemine, S.J., 1714 without date) and Definition 4 of the Book V of Euclid’s ‘Elements’, essential for a scientific- historical comprehension and correct interpretation of the foundations of the calculus fluxionum and the calculus differentialis. The author’s analysis (performed in 1942) proves: (1) that the ‘grandeurs incomparables’, ‘quantitates incomparabiles’ as the in actu infinitely small quantities of Leibniz’ calculus represent a non-Archimedean ordered structure, whereas those in Newton’s calculus represent an Archimedean ordered structure; (2) that the Newton’s and the Leibniz’s calculus represent two different mathematical theories; (3) that, consequently the question on the priority of discovery of ‘calculus’ is not justified and does not hold true in the sense as it has been believed by historians of mathematics up to today, but appears in an absolutely new light. The number system \(\Omega_ K\) of D. Laugwitz’ (1959) Non-standard analysis obtained by a modification of the theory of real numbers of Georg Cantor is discussed. It is emphasized that \(\Omega_ K\) contains the in actu infinitely small (and those in actu infinitely great) numbers which coincide with the ‘grandeurs incomparables’ of Leibniz’ calculus and that consequently this calculus is a non-standard differential calculus.