Claims I'm evaluating

I've read in multiple sources that Leibniz formulated separation of variables for ODEs in 1691. A couple example sources are below.

My Question

Does anyone know the original source for this claim?

Note 1: If it's a handwritten document (e.g. a letter), then I'd be delighted to see a digital copy of the original.

Note 2: If there are English (or Spanish) translations available, that would be helpful.

Note 3: If the original source is not available online, then any information about the specific equation (or equations) that Leibniz solved, or the context in which the equation arose, would be interesting.

Note 4: If anyone happens to know where I could find John (Johann) Bernoulli's Acta Eruditorum article of 1694, in which he apparently elaborates on the method of separation of variables, then that would be helpful as well.

What I've found so far

The closest source I've found is a letter from Leibniz to Huygens, written on the 29th of December, 1691. The letter is in French, and I read French only a little (just what I can pick up from knowing another romance language). Nonetheless, Leibniz clearly applies separation of variables in this letter to the equation dy = y/a dx.

In Kline (see link above), Leibniz's introduction of separation of variables is said to occur in a letter to Huygens, but I don't see any references to a specific letter. Since Leibniz and Huygens corresponded quite a bit in 1691, it's possible that another letter is being referenced.

Thank you for any assistance you can provide!


1 Answer 1


The reference is probably to a treatise sent to Huygens on 5 October 1691, where Leibniz says (and illustrates with several examples) that "Whenever the subtangent [$=y/y'$, but it would also work for just the tangent $y'$] is a product of two quantities or formulas, of which one is given purely in terms of the abscissa $x$, and the other in terms of the ordinate $y$, then the problem reduces to quadratures [i.e., to integration]." (page 186 of http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/III5A.pdf)

However, that was hardly a new discovery at that time.

In Leibniz's first paper on the calculus, in 1684, he determines a curve from its tangent property, which arguably amounts to a separation of variables. See page 8 of http://www.17centurymaths.com/contents/Leibniz/nova1.pdf.

In his first published paper on the integral calculus, in 1686, Leibniz again effectively uses separation of variables. See page 297 and the note below the image at https://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-integral-calculus.

In 1687, Leibniz separates variables again to solve the differential equation of a descent problem. See page 4 of http://www.17centurymaths.com/contents/Leibniz/ae3a.pdf.

In his lectures on the calculus of 1691-92, Johann Bernoulli uses separation of variables throughout and treats it as one of the most basic ideas of the calculus. See https://archive.org/details/dieersteintegra00kowagoog/page/n19.

Because of examples such as these I do not think it makes sense to view separation of variables as a discovery separate from the Leibnizian calculus, made only in 1691. Separation of variables seems to have been taken for granted as a natural and self-evident part of calculus itself. Separation of variables does not stand out as a distinctive technique in these works because Leibniz et al. worked with relations between differentials in a flexible way rather than expressing all such relations in terms of the derivative $dy/dx$ as we are inclined to do today.

  • $\begingroup$ Wow, thank you so much!! Assuming the technique had not been observed as a systematic method before the letter on October 5, 1691, I see how one might conclude that Leibniz pioneered it there as a general method, since he took the time to describe it in fairly general terms and to give a range of examples. However, your last point is illuminating: if I understand correctly, even if Leibniz laid out the approach explicitly in 1691, it would be misleading to call that a discovery of the method as we know it today, considering differentials in Leibnizian calculus were separate to begin with. $\endgroup$ Jul 9, 2019 at 22:08
  • $\begingroup$ 1. Out of curiosity, do you know the meaning of the overline notation in your first reference, as in $d\bar{x}$? I've done some digging but I haven't had any luck, and I've found copies of these letters that don't have them at all. $\endgroup$ Jul 10, 2019 at 4:42
  • 1
    $\begingroup$ @GregStanton $\overline{abc}$ basically means $(abc)$; the bar indicates the scope of the operator $d$ or $\int$, just as it indicates the scope of which terms $\sqrt{}$ is applied to in $\sqrt{abc}$. $\endgroup$ Jul 10, 2019 at 7:01
  • $\begingroup$ Thanks! That makes sense. I didn't realize the modern radical symbol is a vestigial usage of the overline as a notation for grouping! I also didn't know that (evidently) Descartes was the first to combine the German radical sign with the vinculum. Today I learned :) $\endgroup$ Jul 11, 2019 at 9:25

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