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Detlef Laugwitz wrote a monumental biography of Riemann. The book was translated into English by Shenitzer. Laugwitz discusses Riemann's fundamental essay

Uber die Hypothesen, welche der Geometrie zu Grunde liegen (On the Hypotheses which lie at the Foundations of Geometry)

of 1854 on the foundations of what has since become Riemannian geometry. Laugwitz writes: "In the lecture, Riemann pushed to extremes his tendency to use as few formulas as possible." Unfortunately Laugwitz does not elaborate, but the unique formula contained in Riemann's essay is the formula $$\frac{1}{ 1+\frac{\alpha}{ 4}\sum x^2}\sqrt{\sum dx^2}$$ expressing the length element of a metric of constant (sectional) curvature $\alpha$.

What is puzzling here is Riemann's notation. This is of course before dual spaces and tensor calculus. What meaning did Riemann attach to $dx$? It is hard to say it was infinitesimal because Riemann is known for giving a rigorous treatment to the, well, Riemann integral.

I see now that there is also a book by Monastyrsky Riemann, topology, and physics that might be relevant.

Does anyone have a reference that would comment on this?

Note 1. Spivak's Differential geometry, third edition, volume 2, chapter 4 contains an English translation of Riemann's essay. Here on page 155 Riemann speaks of the line as being made up of the $dx$, describes $dx$ as "the increments", and speaks of infinitesimal displacements. On page 156 he speaks of infinitely small quantities $x_1dx_2-x_2dx_1$, etc., as well as of infinitely small triangles.

Mikhail Katz
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    Is it possible that the expresion $\sqrt{\Sigma dx^2}$ stood for the integrand of the path length integral, even at that time, and without any independent meaning for $dx$? – Lee Mosher Aug 07 '14 at 15:14
  • @Lee, interesting idea. Is there a source for this? – Mikhail Katz Aug 07 '14 at 15:17
  • Would $dx$ be a differential? I mean a real-valued identity function $dx: h \mapsto h$ on the real field. – Yes Aug 07 '14 at 15:26
  • @Brian I was hoping to find the answer in Laugwitz's book but alas, I don't know. The point of view of functionals, dual spaces, etc. strikes me as decidedly modern, though. – Mikhail Katz Aug 07 '14 at 15:30
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    Michael Spivak has a long chapter on Riemann's paper in Volume 2 of his Comprehensive Introduction to Differential Geometry. – Per Erik Manne Aug 07 '14 at 16:38
  • @PerManne, thanks. I knew that Spivak translated the 1854 paper but I did not realize he discusses this. I will try to look it up. Did you see it? Does Spivak say anything about infinitesimals? – Mikhail Katz Aug 08 '14 at 07:36
  • @PerManne, I looked up the third edition of Spivak, volume 2. Here on page 165 Spivak's discussion of Riemann starts using the notation of $g_{ij}dx^i dx^j$ without any comment, as if assuming that Riemann was using it in the modern sense. I don't have the impression Spivak is particularly interested in the historical question as much as understanding Riemann's mathematics in modern terms. – Mikhail Katz Aug 08 '14 at 09:30

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Note that Riemann's essay originally was a "habilitation lecture", directed at some larger academic audience. Therefore Riemann tended to avoid too much technical machinery. In the formula at hand he just writes $\sqrt{\sum dx^2}$, but at other places in this lecture he talks about the $dx_i$ and about the fact that the fundamental form has ${n(n+1)\over2}$ terms, etc. Therefore it is obvious that Riemann had the interpretation $$\sum dx^2:=\sum_{i=1}^n dx_i^2\>, \quad{\rm resp.}\quad ds=\sqrt{\sum_{i=1}^n dx_i^2}\ ,$$ in mind, which when computing lengths of curves unpacks to $$L(\gamma)=\int_a^b\sqrt{\dot x_1^2(t)+\dot x_2^2(t)+\ldots+\dot x_n^2(t)}\>dt\ .$$

Christian Blatter
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  • In talking about the "next-order term" in the expansion of the line element, Riemann writes (in Spivak's translation, which is occasionally inacurate): "It obviously = zero if the manifold in question is flat, i.e., if the square of the line element is reducible to $\Sigma dx^2$, and can therefore be regarded as the measure of deviation from flatness in this surface direction at this point. When multiplied by -3/4 it becomes equal to the quantity which Privy Councilor Gauss has called the curvature of a surface." It is apparent from this discussion of the Gaussian curvature that... – Mikhail Katz Aug 10 '14 at 11:42
  • ... Riemann has a pointwise discussion in mind, and not merely an integrand. – Mikhail Katz Aug 10 '14 at 11:43
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To Riemann dx was a nilsquare infinitesimal, that is, the most conventional (if controversial) type of infinitesimal. 'The principle of gaining knowledge of the external world from the behaviour of its infinitesimal parts is the mainspring of the theory of knowledge in infinitesimal physics as in Riemann's geometry, and, indeed, the mainspring of all the eminent work of Riemann.' Hermann Weyl, 1950 (quoted in The Continuous and the Infinitesimal, John L Bell). Weyl's quote should however be qualified: the Riemann hypothesis is part of number theory and therefore presumably does not depend on infinitesimals.

  • John Bell is a fan of infinitesimals (and specifically nilsquare ones) but this is a bit short on detail as far as Riemann is concerned. Do Riemann historians deal with this? It is too late to ask Weyl about this unfortunately :-) – Mikhail Katz Aug 07 '14 at 15:52
  • Deleted: Riemann historians probably don't because according to Bell that theory was suppressed; Riemann used nilsquare in preference to invertible infinitesimals. Bell has a three-page section about Riemann in the book and Weyl's claim is based on his own reading of Riemann. –  Aug 07 '14 at 16:11
  • What does Bell say there? Is this in his book or in his essay "continuous and infinitesimal" for SEP? One should be able to establish the nature of Riemann's own position without relying on Bell's opinion. – Mikhail Katz Aug 07 '14 at 16:16
  • It's in the book of that name, I don't have access to copies of Riemann's papers. To me Weyl's claim makes sense - nilsquares were controversial not because they didn't work but because they were (eventually) considered contradictory. Riemann just made use of them. –  Aug 07 '14 at 16:26
  • The problem with this is that Laugwitz writes on page 54 that "Dirichlet, Riemann, and Weierstrass speak of a sequence, or a function, becoming infinitely small. But this is just another expression for convergence to zero, or for the limit zero." This occurs in the section on "infinitesimals" after a long discussion of Cauchy's infinitesimals. I conclude from this that Laugwitz is not of the opinion that Riemann used genuine infinitesimals. – Mikhail Katz Aug 08 '14 at 07:34
  • On page 156 (as translated in Spivak) Riemann speaks of infinitely small quantities of the fourth order, so apparently these are not nilsquare. How does Bell account for this? – Mikhail Katz Aug 08 '14 at 09:54
  • Would have to know what he does with them. –  Aug 08 '14 at 14:55
  • You are apparently referring to Bell's book "The Continuous and the Infinitesimal in Mathematics and Philosophy". Here riemann is discussed on pages 146-148. However, I did not find any claim that Riemann was using nilsquare infinitesimals. Where is this claim coming from? – Mikhail Katz Aug 10 '14 at 10:50
  • Good question - that's mainly what the book is about. However, looking at your equation I see a problem - the numerator collapses. Here's my own take (I've posted more about this) - neglecting 'nilsquares' is optional. We do it because we can. Give dx a decreasing value - the equation will approximate some measurable value (y?) with increasing accuracy. What is that value? –  Aug 11 '14 at 04:35
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There's an interesting take on this in The Continuous and the Infinitesimal by John L Bell, see here. He's actually talking about relativity, but of course Riemann's mathematics is needed for this so the way the mathematics is used indicates what it really means. In summary - the proofs within this version of calculus (smooth infinitesimal analysis SIA) start off from basic principles [f(x + h) = f(x) + hf'(x)] and at some point higher order infinitesimals are 'neglected' (NSA neglects all infinitesimal terms at the end). From that point on we are dealing with the true derivative (not the 'full derivative' or finite difference quotient). During the linked derivation there is a point where Bell changes from using a 'small' but finite variable to a nilsquare infinitesimal. The equation you give from Riemann is meant to apply before this transition occurs, so if it is actually applied you would at some point make the change - assuming that you're studying the continuous case. Bear in mind that for numerical simulations on computer you wouldn't do this, and these were recently used to simulate gravitational waves which was essential for their detection.

NB The book is available as a pdf; and the chain rule is another example of a seamless transition to the continuous case. "We begin with the somewhat simpler case of smooth maps from surfaces into Euclidean spaces. The key to making sense of the differential of such a map is the chain rule." (From a differential geometry student handout.)