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.
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.