Leibniz’s Calculation with Compendia

Procedures of Leibnizian infinitesimal calculus: An account in three modern frameworks

“…30 Arthur's inference of alleged Archimedean nature of Leibnizian dv is dubious. 31 In a letter to Wallis two years earlier, Leibniz already emphasized the distinction between Archimedean and non-Archimedean techniques:…”

“…Effectively, the concept of analysis with the 30 The Leibnizian passage is also quoted by RA [104, p. 437] who similarly fail to account for the fact that talking about a translation into the style of Archimedes entails the existence of a separate method exploiting infinitesimals à la rigueur. 31 Furthermore, Arthur's attempt to account for the Leibnizian derivation of the law d(xy) = xdy + ydx in Archimedean terms, by means of quantified variables representing dx, dy, d(xy) taking assignable values and tending to zero, would run into technical difficulties. Since all three tend to zero, the statement to the effect that "the error that could accrue from this would always be smaller than any given" understood literally is true but vacuous: 0 = 0 + 0.…”

Leibniz�s well-founded fictions and their interpetations

“…(Breger [28], 2008, p. 188) Breger's claim that to understand Leibnizian infinitesimals one should "look at the process of ever-decreasing divisions" may not be easy to reconcile with Leibniz's claim that one does not reach infinitesimals by a process of ever-decreasing divisions; see e.g., Section 2.4. Breger goes on to offer a sharp criticism of Bos' position in [28, pp.…”

“…[7] Et quae tali quantitate non differunt, aequalia esse statuo, quod etiam Archimedes sumsit, aliique post ipsum omnes." (Leibniz [73], 1695, p. 322; numerals [1] through [7] added) 22 In reference to this passage, Breger claims that "the unassignable magnitudes are fictitious, they cannot be determined by any construction" (Breger [28], 2008, p. 196), but fails to deal with Leibniz's very next sentence concerning Euclid V.5 (V.4 in modern editions).…”

“…(Breger [28], 2008, p. 185) (emphases added) We disagree with Breger's claim of alleged incoherence of Leibniz's verbal descriptions, but we agree concerning the need to focus on mathematical usage. [G]râce à ce triangle inassignable, c'est-à-dire à l'intervention de la raison entre quantités inassignables CD et (C)D (que notre calcul différentiel donne au moyen des quantités ordinaires ou assignables), on peut trouver la raison entre les quantités assignables T B et BC, et donc tracer la tangente T C. (Leibniz [89], 2018, p. 155; emphasis added) Leibniz's tangent line T C is thought of as passing through both infinitely close points C and (C).…”