Galina Weinstein has a new paper on Variation of Mass with Velocity: "Kugeltheorie" or "Relativtheorie":

Kaufmann concluded, from 1905 onwards, that the mathematical expression proposed by Alfred Bücherer could also be in accord with his measurements and that one could not definitively decide between that expression and that of Abraham as it was derived from his experiments. In the same paper, Kaufmann noted that the two theories of Lorentz and Einstein yielded the same equations of motion for the electron, and he gave the first clear account of the basic theoretical difference between Lorentz's and Einstein's views.29Lorentz did credit Einstein with having a slightly different approach, but neither expressed any differences in the conclusions, except for minor technical errors. Lorentz said that the chief difference was that Einstein postulated what he had proved. No one saw much difference until several years later when Minkowski's approach became popular, and Einstein adopted it.

In the annual general meeting of the German Society of Scientists and Physicists (Deutsche Gesellschaft der Naturforscher und Ärrzte) in Stuttgart, on the 19th of September 1906, scientists discussed three world pictures, the electromagnetic theories of Abraham, Bücherer, or the other picture based on Lorentz and Einstein's "Principle of Relativity". A discussion revolving around the foundations of physics was held after Planck's lecture. The participants in the discussion were, among others, Kaufmann, Planck, Bücherer, Abraham, Arnold Sommerfeld and others. Scientists did not yet distinguish between Lorentz's theory and Einstein's theory. There were two main theories relating to the electron: Abraham's and Lorentz-Einstein's. An inclination towards Einstein and Lorentz's theories, on the part of scientists such as Planck and Max Laue, was evident.

Walter Kaufmann's 1906 paper says:

Then, a work by H. A. Lorentz[13] appeared in the year 1904, in which the attempt was made to remove the difficulties which sill existed in the optics of moving bodies, by somewhat modified fundamental assumptions on the electron and also on the molecular forces acting in-between the material body-particles. ... Lorentz now showed, that one could arrive at such a result, when it is assumed that the dimensions of all physical bodies, including their individual molecules and electrons, would change their shape in a very specific way with velocity ...Note that Kaufmann is trying to refute Lorentz and Einstein, and considers refuting one the same as refuting the other. Most of the paper just mentions Lorentz without Einstein, such as the section titled, "Comparison with the theories of Abraham, Lorentz and Bucherer."

It is now very remarkable, that, starting from quite different [492] assumptions, Einstein[17] recently arrived at results, which are in agreement with those of Lorentz concerning the consequences accessible to observation, though in which the previously mentioned difficulties of epistemological kind have been avoided. Einstein introduced the principle of relative motion, at least as regards translations, as a postulate. He thus places the theorem at the top, that physical phenomena observable in any rigid system, must be independent from whether the system (together with the observer) is moving relatively to any other system. ... The results accessible to observation are thus the same with respect to both authors; however, while Lorentz only shows that his hypotheses lead to the desired result without excluding that the same can also be achieved in another way, it is shown by Einstein, that when the desired result, namely the principle of relative motion, is placed at the top of the whole of physics, then the kinematics of the rigid [493] body must necessarily be changed in the way stated, and that the equations of electrodynamics[18] must assume the form stated by Lorentz. ...

The measurement results are not compatible with the fundamental assumption of Lorentz-Einstein.

Kaufmann appears to be not aware of Poincare's theory, or how Einstein got the relativity postulate from Poincare.

Kaufmann says Einstein "places the theorem at the top". That is his version of Lorentz's famous statement that Einstein simply postulates what we have deduced. I give a more technical explanation of what this means.

Kaufmann's argument is a little misleading where he says that Einstein showed "that the equations of electrodynamics must assume the form stated by Lorentz." This sounds like a strong statement, but if you read the footnote, Einstein assumes Maxwell's equations in the rest frame, and hence in the moving frame via the relativity principle. So Einstein is really postulating the equations of electrodynamics. It was Lorentz and Poincare who tried to prove the equations for the moving frame, assuming the equations for the rest frame.

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