We draw a radius of a circle. We draw a second radius $23$ degrees clockwise from the first radius. We draw a third radius $23$ degrees clockwise from the second. This continues until we have drawn $40$ radii each $23$ degrees clockwise from the one before it. What is the measure in degrees of the smallest angle between any two of these $40$ radii?
1. **Understanding the Problem:**
   We need to find the smallest angle between any two of the 40 radii drawn at 23-degree intervals on a circle.

2. **Listing the Angles:**
   The angles where the radii are drawn can be listed as follows:
   \[
   0^\circ, 23^\circ, 46^\circ, 69^\circ, 92^\circ, 115^\circ, 138^\circ, 161^\circ, 184^\circ, 207^\circ, 230^\circ, 253^\circ, 276^\circ, 299^\circ, 322^\circ, 345^\circ, 8^\circ, 31^\circ, 54^\circ, 77^\circ, 100^\circ, 123^\circ, 146^\circ, 169^\circ, 192^\circ, 215^\circ, 238^\circ, 261^\circ, 284^\circ, 307^\circ, 330^\circ, 353^\circ, 16^\circ, 39^\circ, 62^\circ, 85^\circ, 108^\circ, 131^\circ, 154^\circ, 177^\circ
   \]

3. **Finding the Smallest Angle:**
   To find the smallest angle between any two radii, we need to consider the differences between consecutive angles in the sorted list. The differences are:
   \[
   23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ, 23^\circ
   \]

4. **Considering the Wrap-Around:**
   The last angle (177°) and the first angle (0°) also need to be considered. The difference between 177° and 0° is:
   \[
   360^\circ - 177^\circ = 183^\circ
   \]

5. **Conclusion:**
   The smallest angle between any two radii is the smallest difference in the list of differences, which is 23°.

The final answer is \(\boxed{23}\).