What meaning did Riemann assign to $dx$? 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.
 A: 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
$$L(\gamma)=\int_a^b\sqrt{\dot x_1^2(t)+\dot x_2^2(t)+\ldots+\dot x_n^2(t)}\>dt\ .$$
A: 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.
A: 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.)
