
Consider the problem of determining the number of revolutions an astronaut performs relative to the ground while being fastened to a circle \( C_3 \) of radius \( R \). This circle rolls inside another circle \( C_2 \) of radius \( 2R \), which in turn rolls along the inside of a fixed circle \( C_1 \) of radius \( nR \), with \( n > 2 \).

The task is to find the number of revolutions of \( C_3 \) when \( C_2 \) completes one full revolution inside \( C_1 \).

### Analysis

1. **Path of Circle \( C_2 \):**
   
   - Circle \( C_2 \) rolls internally along the circumference of circle \( C_1 \). Since the radius of \( C_1 \) is \( nR \) and the radius of \( C_2 \) is \( 2R \), the remaining space for \( C_2 \) is a circle path that effectively spans a circle of radius \((n-2)R\).

2. **Revolutions of Circle \( C_2 \):**
   
   - When \( C_2 \) makes one complete revolution around \( C_1 \), it travels a distance equal to the circumference of the circle path, which is \( 2\pi(n-2)R \).

3. **Path and Revolutions of Circle \( C_3 \):**

   - Circle \( C_3 \) is rolling inside circle \( C_2 \). For each full revolution \( C_2 \) performs internally, circle \( C_3 \), due to rolling without slipping, covers its own circumference \( 2\pi R \) multiple times.
   
   - When circle \( C_2 \) makes one complete revolution, considering that \( C_3 \) is rolling inside it, there are additional relative rotations due to its motion around the center of \( C_2 \). 

4. **Determination of Revolutions:**
  
   - The key is the net additional rotation induced on \( C_3 \). As circle \( C_2 \) rolls along \( C_1 \) once, \( C_3 \) effectively rotates one additional time relative to the center of \( C_2 \).
   
   - Suppose circle \( C_3 \) initially performs \( X \) revolutions due to the rolling of circle \( C_2 \), there is one additional revolution of \( C_3 \) around the center of \( C_2 \).

5. **Final Calculation:**
   
   - Total revolutions performed by \( C_3 \) as \( C_2 \) completes one lap are given by subtracting the revolution induced by rolling: \( n \) revolutions total due to its position relative to ground minus the extra revolution categorized above. 

\[
\text{Revolutions of } C_3 = n - 1
\]

Hence, the number of revolutions that the astronaut performs relative to the ground is:
\[
\boxed{n - 1}
\]
The following construction is used for training astronauts:
A circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls along the inside of circle $C_2$ in such a way that the touching point of the circles $C_2$ and $C_3$ remains at maximum distance from the touching point of the circles $C_1$ and $C_2$ at all times. How many revolutions (relative to the ground) does the astronaut perform together with the circle $C_3$ while the circle $C_2$ completes one full lap around the inside of circle $C_1$?