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Train Counting

First, some terminology...

The train of a watch is made up of wheels and pinions, each on an axis, or staff. The pinion the the small part on the inside, and the wheel generally is the large part. Wheels have teeth. Pinions have leaves.  The train starts with the mainspring barrel, then the center wheel (or great wheel), then the third wheel, the fourth wheel, and finally the escape wheel.  In this image, the center wheel is the large wheel at the left.  The train goes clockwise up and to the right from there, in the picture.
Looking at this next view, we see the mainspring barrel at the left, the center wheel in the center, and so on, pinion to wheel, to pinion, to wheel, etc.  

The third wheel is a bit tucked away and hard to see in this photo.

As an aside, this watch has a "motor barrel."

Next, this is a different movement, with a different layout.  The design is the same.

This watch has a "going barrel."

Counting the Train

Here's a commonly used equation to get started.

n1 is the number of revolutions of the center wheel per hour. This is almost always one, this is the minute hand.
n4 is the number of revolutions of the 4th wheel per hour. This is almost always 60, this is the seconds hand.
z1 is the number of teeth on the center wheel.
z2 is the number of leaves on the third wheel pinion.
z3 is the number of teeth on the third wheel.
z4 is the number of leaves on the fourth wheel pinion.

n4    z1 * z3
-- = ---------
n1    z2 * z4

This equation gets to a watch that reads correctly, that is the minute and hour hands are in sync with the seconds hand.  For this we don't care about the number of teeth on the barrel, nor the number of leaves on the center pinion.  This is because time reading begins at the center wheel; it is the minute hand.  

Considering the rate take a little more work.  

"Fast-beat" vintage American watches beat at 18,000 beats per hour, or 2.5 Hz. One way to observe this is the fact that for each second, the second hand jumps five times.

 z6 * 18000
------------- = 1
 z5 * ze * 2

z5 is the number of teeth on the forth wheel.
z6 is the number of leaves on the escape wheel pinion.
xe if the number of "teeth" on the escape wheel.

This equals one because one is the number of revolutions per hour of the center wheel (n1 above). Because of the action of the pallet stones the number of foot-shaped teeth on the escape is multiplied by two.

There's another way to look at this, by fixing the number of revolutions per hour of the 4th wheel (the seconds hand) to 60.

 60 * z2 * z4
-------------- = 1
   z1 * z3

And there you have it.  Pinion leaves are on the top, wheel teeth are on the bottom.

Why worry about this?  Simple, every now and then an obscure watch will be missing a part.  Using the above, and some algebra, it is possible to figure out what the part should be like.  Sometimes, if the balance is missing, we won't know what the beat rate should be.  If the train is there it can be calculated.

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