Newton on Indivisibles

International audienceThough Wallis's Arithmetica infinitorum was one of Newton's major sources of inspiration during the first years of his mathematical education, indivisibles were not a central feature of his mathematical production. To judge from his reading notes, he firstly studied Wallis's treatise at the beginning of 1664 ([13], I, 1, 3, § § 1-2, pp. 89-95), and came back to it one year later ([13], I, 1, 3, § 3, pp. 96-121). In the former occasion, he confined himself to the first part of the treatise, and possibly accompanied his reading with that of the De sectionibus conicis and the De angulo contactus, also contained in Wallis's Operum Mathematicarum Pars Altera ([16] and [17]). At the beginning, his attention was retained by some general remarks, but very often shifted to the elaboration of an original, algebraic version of the method of indivisibles, which he applied to get both a correct quadrature of the parabola and an incorrect quadrature of the hyperbola. In the latter occasion, he rather focused on Wallis's quadrature of the circle—which he deeply transformed ([15], pp. 152-181)—whiteout any particular attention to indivisibles. Only the notes relative to the former reading deserve to be considered here. Section 1 is devoted to them. Sections 2 and 3 document the important but often implicit role indivisibles and infinitesimals play in Newton's mathematics and his reluctance to accept them. The differences between indivisibles and infinitesimals, and their roles in Newton's mathematics are difficult to grasp because Newton himself equivocated about them. As we shall see in sections 2 and 3, he used both indivisibles and infinitesimals (but infinitesimals, particularly) abundantly, but starting in 1671, he explicitly ascribed to them a subordinated status vis-` a-vis fluxions and " evanescent " quantities and ratios. This did not prevent him from using in the Principia some principles and arguments closely remembering the method of indivisibles. The end of section 3 and section 4 focus on some examples of this use. The latter is, in particular, devoted the proof of two theorems (propositions LXXI and LXXIV of Book I), where this use is quite manifest