ntity of thinking and being. But also the contradiction between continuous and discrete matter; see Hegel.)
The differential calculus for the first time makes it possible for natural science to represent mathematically processes and not only states: motion.
Application of mathematics: in the mechanics of solid bodies it is absolute, in that of gases approximate, in that of fluids already more difficult; in physics more tentative and relative; in chemistry, simple equations of the first order and of the simplest nature; in biology=0
213. In the former case, Engels has in mind Hegel’s remark that in arithmetic, thought moves in “thoughtlessness” (Science of Logic, Book I, Section II, Chapter 2, Observation on the employment of numerical determinations to express philosophic concepts); in the latter case, Hegel’s statement that “already the natural numerical system exemplifies a nodal line of qualitative moments, which manifest themselves in the merely external progression,” etc. (ibid., Section III, Chapter 2, Observation on examples of nodal lines of measure-relations; natura non facit saltum).
214. This expression occurs in the book by Bossut, referred to by Engels in the fragment “Straight and Curved.” In the chapter on “Integral Calculation with Finite Differences,” Bossut examines first of all the following problem: “To integrate or sum the whole-number steps of a variable magnitude x.” Bossut assumes that the difference Dx is constant and he denotes it by the Greek letter w. Since the sum of Dx or of w is equal to x, the sum of w×1 or of wx0 is also equal to x. Bossut writes this equation in the form Swx0=x. Bossut then takes out the constant w and puts it before the summation sign, obtaining the expression wSx0 = x, from x/w which he obtains the equation Sx0= w/w. This last equation is then to used by Bossut to find the magnitudes Sx, Sx2, Sx3, etc., for solving other problems. See Bossut, Traités de Calcul differentiel et de Calcul integral, t. 1, Paris, 1798, p. 38.
215. Ch. Bossut, Traités de Calcul differentiel et de Calcul intégral, t. I, Paris, an VI (1798), p. 149.
216. This is how Bossut terms the curves considered in the system of polar co-ordinates.
217. Engels has in mind Fig. 17 and explanation to it on pp. 148-51 of Bossut’s Treatise. This figure has the following form: BMK is the curve. MT is its tangent. P is the pole or origin of the co-ordinates. PZ is the polar axis. PM is the ordinate of the point M (Engels calls it “real abscissa”; nowadays it is called the radius-vector). Pm is the ordinate of point m lying infinitely close to M (Engels calls this radius-vector the “differential imaginary abscissa”). MH, perpendicular to the tangent MT. TPH, perpendicular to the ordinate PM. Mr, the curve described by the radius PM. As MPm – is an infinitesimal angle, PM and Pm are considered parallel. The triangles Mrm and TPM, as also the triangles Mrm and MPH, are. regarded as similar.
218. See Note 95.
219. This note is one of the three larger notes (Noten) which Engels put in the second folder of materials for Dialectics of Nature.
(See Note 204.) It was written originally as the first sketch of a commentary note to pp. 17-18 of the first edition of [Anti]-Dühring.
The heading “On the Prototypes of the Mathematical Infinite in the Real World” was given by Engels in the list of contents of the second folder of Dialectics of Nature. The sub-heading “To pp. 17-18; Concordance of Thought and Being. – The Infinite in Mathematics” stands at the beginning of the note.
220. Nihil est in intellectu, quod non fuerit in sensu (nothing is in the mind which has not been in the senses), the fundamental tenet of sensualism. The content of this formula goes back to Aristotle (see his Posterior Analytics).
221. This figure is given in an article by William Thomson, entitled “The Size of Atoms,” which was first published in the journal Nature No. 22, of March 31, 1870, and afterwards reprinted as an appendix in the second edition of Treatise on Natural Philosophy by Thomson and Tait (Vol. 1, Part II, new ed., Cambridge, 1883