Charlotte Wahl raised some critiques of my Historia Mathematica paper on the rectification of quadratures in her review in the Zentralblatt MATH. I shall reply to them here.

At stake is an argument of mine which goes as follows.

Leibniz found a rule for determining the envelope of a family of curves, but at first he did not seem to value it. In 1692 Leibniz alluded to this rule briefly in an inconspicuous little article. Optical problems are mentioned as motivation.

But two years later Leibniz published a much fuller account, complete with details, figures, and worked examples, calling it a “new” application of his calculus “of no small importance for the development of geometry.”

I argued in the paper that Leibniz realised in between these two publications that envelopes were of use for the rectification of quadratures, which he was interested in for foundational reasons. This hypothesis had three sub-claims:

(A) Leibniz’s enthusiasm about the rule in his second paper stems from his realisation that it is relevant for the rectification of quadratures.

(B) The specific problems of Leibniz’s 1694 paper were chosen for their relevance to the rectification of quadratures.

(C) Leibniz had realised these things while reading a letter from Newton on this subject which he received in late 1693.

However, as Wahl notes in her review:

“The draft of the [1694] paper, which is still extant in the Gottfried Wilhelm Leibniz Bibliothek Hannover (shelfmarks LH 35,6,17 and LH 35,13,3 f. 12), shows that it was originally the first part of a larger paper, whose second part was then expanded and published separately in September 1693 under the title Supplementum geometriae dimensoriae.”

This disproves my hypothesis (C). However, it greatly strengthens and confirms my claims (A) and (B), which were in any case the claims that mattered for my overall conclusions. For the Supplementum paper is indeed about the rectification of quadratures, showing that, just as I hypothesised, the 1694 paper was written in the context of this problem.

Therefore I disagree with Wahl who concludes from the existence of this manuscript that:

“The author’s thesis may well turn out right in a more restricted sense: Newton’s letter may have motivated Leibniz to finally publish his paper – perhaps out of fear that Newton had similar methods.”

My hypothesis should not be restricted to this weak and uninteresting claim, but to its core claims (A) and (B), which are strikingly and independently confirmed by Wahl’s own evidence.

So much for this hypothesis. Wahl also makes another unfortunate argument against me, namely that I neglect Leibniz’s 1693 correspondence with l’Hôpital:

“M. Parmentier [La naissance du calcul différentiel. Paris: Vrin (1989), pp. 268–271] pointed out that some of the problems discussed in Leibniz’s envelope paper already came up in Leibniz’s correspondence with L’Hospital in early 1693.”

This is false, plain and simple. None of the problems of the envelope paper are in the l’Hôpital correspondence. In fact, I explicitly cite this very fact in my paper (page 428) in support of my thesis. As for Parmentier, this is indeed not a very reliable source, as I had already noted before my paper appeared.