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 proven for a long time. It's known that the difficulty exclusively lies in the question after the passage from rest to uniform rotation. I have formulated this difficulty in the following way:

"Let a relative-rigid cylinder of radius $$R$$ and height $$H$$ be given. A rotation about its axis which is finally constant, will gradually be given to it. Let $$R_1$$ be its radius during this motion for a stationary observer. Then $$R_1$$ must satisfy two contradictory conditions:

a) The periphery of the cylinder has to show a contraction compared to its state of rest:

$2\pi R_{1}<2\pi R\,$

.....;

b) ..... the elements of a radius cannot show a contraction compared to the state of rest. It should be:"

$R_{1}=R\,$

Compare with this the specifications given by : "The distance between two points of the cylinder, not lying on the same diameter, and measured when the cylinder was still at rest, won't be equal to the distance between the same points measured synchronously, when the cylinder is rotating (§ 3, end)."

"In my view, the whole thing seems of be based on a misunderstanding. If we measure a line element along the circumference of the" – stationary rotating – "disc in a synchronous way, then we obtain a value which is smaller than $$2\pi R$$, where $$R$$ means" – synchronously measured at the rotating disc – "the radius of the disc. However, in this lies absolutely no contradiction, but everything explains itself from the definition of synchronous measurement .... in general we can define the true shape and dimension of a rigid body by measurement, when (and only when) the body is at rest. The measurements on moving bodies give only apparent values ...." (concluding remark).

§ 3. In the interest of explaining the meaning of the words:


 * "However, in this lies absolutely no contradiction, but everything explains itself from the definition of synchronous measurement"

I allow myself to respectfully request, that he comments on two questions immediately to be formulated: – For the sake of the precise formulation of these questions, I predefine a convention and an assertion:

Let the spherical disc be equipped upon its entire surface with infinitely many marks that are individually recognizable.

While the disc is at rest, the resting observer $$B$$ holds a tracing paper above it, and traces the marks upon the resting paper.

While the disc is rotating stationarily, the resting observer $$B$$ holds a tracing paper $$P_1$$ above it, and in the moment when his clock indicates $$t$$, he traces with one stroke all marks upon the resting paper.

Eventually, the resting observer $$B$$ measures the marking distribution upon the resting tracing-images $$\Pi$$ and $$\Pi_1$$.

I assert: The periphery- and radius length upon $$\Pi_1$$ measured in this way, coincides in the present example exactly with that, what was called by as the disc circumference or disc radius at the stationary rotating disc, "synchronously measured" by the resting observer at moment $$t$$. (See definition of "synchronous measurement" in § 2 of the work of .)

My questions to are :

Question 1: Is the last formulated assertion correct? If not – wherein is then the difference between the result obtained by the resting observer by "synchronous measurement" of the rotating disc, and the result obtained by the measurement of the resting tracing-image $$\Pi_1$$?

Question 2: If my assertion is correct, then the statements put together by about periphery and radius being "synchronously measured", and that are strangely denoted by him as absolutely without contradiction, are transformed into the following statements concerning the tracing-images: The tracing-image $$\Pi_1$$ has the same radius as $$\Pi$$ while its periphery is shorter. How can one imagine tracing-images of such properties without contradiction?

, October 4, 1910.

(Received October 7, 1910.)