This is the way I understand the device (I may be off a little, so don't trust me too much
):
The goal is to have an indexing mechanism that is accurate to within 1 minute (exactly 1/60 of a degree). Just to put that in perspective, one entire circle is 360 [degrees] * 60 [minutes/degree] = 21600 minutes in total.
The obvious way to do this is to just have a gear with 21600 teeth on it right? Well... that's a little hard to make. So lets go about this in a more clever way.
Lets say that instead you have two gear meshes, one that offsets every 60 minutes (1 degree) and one that offsets every 61 minutes. so, if you want a total offset of 1 minute, you increment the 61 minute mesh forward one step, and move the 60 minute mesh back one step. Total change = +61 minutes - 60 minutes = 1 minute. Hooray!
The tricky part is building it.... The 60 minute one is pretty easy to do: you have 21600 minutes / 60 minutes = 360 increments. So, you machine a gear mesh that has exactly 360 teeth. Each step is then 60 minutes.
The other mesh isn't so easy... 21600 minutes / 61 minutes = 354.09836 teeth. So you just need a gear that has 354.09836 teeth on it (non-integer! gah!). Good luck.
So what Newbould did is he made a gear that has effectively 354.09836 teeth on it. This gear then has a little half-tooth or space in it to make it the right size. But in order to mesh two of these gears together, you need a gap in one for the little half tooth to rest. Hence the section of no teeth on one of the faces.
At least that's how I understand the concept, someone feel free to correct me if I'm way off.
EDIT: note that for the mesh that increments by 61 minutes, the gap only needs to allow for about 60 increments. For example: if you want to add 74 minutes, you would actually add one degree (notch the 60 minute mesh forward one) and 14 minutes (14 forward on the 61 minute step, and 14 back on the 60 minute step). Because of this, the gap in the 61 minute mesh would only take up about a third of the circle, leaving plenty of teeth engaged for the indexer to work properly.