Out of sheer curiosity:
When was the first discovery/estimate of the rotation rate of our Galaxy?

(To my understanding: in astronomy the angular velocity of the Solar System with respect to the Milky Way's center is used to convert to Galactocentric coordinates, hence it is common to have that angular velocity represent the overall galaxy rotation rate.)

It could well be, I don't know, that estimates of rotation rate of other galaxies were obtained first, as other galaxies can be observed as a whole.

I tried search engine search, of course, but this one is hard to search for because the word 'Galaxy' is both used for other galaxies and our Galaxy. "Milky way" is exclusively our Galaxy, but that name is used far less, it appears.

The reason I'm curious:
Milky way rotation rate is too slow to be measured directly.

Once astronomers had inferred from observation that the Milky Way is a spiral galaxy it followed that it must have an overall rotation rate (contrary to, say, a globular galaxy). I'm curious what kind of astronomic observation can be done to infer the Milky Way rotation rate.

Later edit:
I found a review article by Yoshiaki Sofue:
Rotation and mass in the Milky Way and spiral galaxies

The earliest article listed in that review article is from 1954.

Kwee K. K., Muller C. A., Westerhout G. 1954, Bull. Astron. Inst. Ned.
The rotation of the inner parts of the Galactic System

By the looks of it: by that time the astronomers were already in the nitty-gritty of trying to obtain observations for the rotation rate of the inner parts.

Later later edit:
Progress report: I found historical information in a review article by Gianfranco Bertone and Dan Hooper A history of dark matter

In 1884 Lord Kelvin gave a series of lectures, in one of them he presented some theoretical explorations of galaxy dynamics.
"Lord Kelvin was among the first to attempt a dynamical estimate [...] His argument was simple yet powerful: if stars in the Milky Way can be described as a gas of particles, acting under the influence of gravity, then one can establish a relationship between the size of the system and the velocity dispersion of the stars"

(Copyright on the lecture notes is dated to 1884, Date of publication is given as 1904)

This material was known to Henri Poincaré. There is a 1906 article by Poincaré, in which he writes about some further theoretical exploration he had undertaken.

"Here a first remark presents itself; the result of Lord Kelvin which I have just verified by an approximate calculation obviously agrees with the estimates with the observers have been able to make with their telescopes; so that we must concede that we have very nearly pierced the Milky Way."

Henri Poincare, 1906, Popular Astronomy, vol. 14, pp.475-488 The Milky Way and the Theory of Gases

(Poincaré did of course begin with pointing out 'agrees with' should be understood as 'to within a factor of 10')

Lord Kelvin and Poincaré had simplified the model to a spherical galaxy (knowing the Milky Way isn't actually spherical). According to Bertone and Hooper the first to model the Milky Way as a flattened structure was the astronomer Kapteyn, 1922.

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    $\begingroup$ I may be misunderstanding this question, but I believe that the techniques used to measure the rotation rate of galaxies were only very recently developed, and it is the unexpected results of these measurements that led to the need to postulate the existence of dark matter. $\endgroup$ – Nick Jul 30 at 15:43
  • $\begingroup$ @Nick The assessment that you refer to is dependent on assessing how much mass is present in any particular region of a galaxy. Cold interstellar gas has very low luminosity, maybe that's the non-visible stuff. But interstellar gas will give rise to some kind of absorption. So then you have to find some absorption observation to try and put an upper limit to density of cold interstellar gas. Only after all other possibilities were ruled out did astronomers resort to the supposition of totally transparent interstellar matter. The earliest rotation rate estimates may well go back a long time. $\endgroup$ – Cleonis Jul 30 at 18:17

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