I) The duplicating apparatus

"The main apparatus of the Arithmometer, the one allowing it to reproduce the number entered on the machine as many times as one turns the crank handle, is made of parallel cylinders, in equal number to the order of the numbers to be reproduced, these are connected by bevel gears to a horizontal shaft controlled by a crank handle.  Each cylinder completes a full turn with each crank handle rotation.  
Each cylinder has nine teeth of increasing size, covering half of its surface, the other half being bare.  (More precisely the teeth occupy 9/20 of the cylinder circumference).

Leibniz and his cylinder

Next to each cylinder and slightly above, resides a horizontal, square, steel shaft upon which a small ten tooth gear can slide and which can be engaged by the cylinder’s teeth.
The position of each wheel is determined by a digit pointer connected to it by a hanging blade that fits in a groove created on the wheel itself.”

Cylindres et roues dentées

When the digit pointers are set to zero, the teeth of the number gear are on the outside of the cylinder’s teeth and therefore will not be moved by their rotation.
When the digit pointers are set to one, the number gear can only be engaged by the longest tooth of the corresponding cylinder and therefore will only be moved by one.
If the digit pointers are set to two, the number gear will be engaged twice by the two longest teeth and so on and so forth until they reach the number nine when the gear will be engaged nine times for a full cylinder revolution.
We understand how for each turn, the number set by the digit pointers is reproduced as the shafts are moved by the number gears, which are connected to the counting wheels.” / Sébert, BSEIN 1879, page 395-396

II) Evolution of the cylinder in the Thomas de Colmar Arithmometer

The Thomas cylinder was modified quite substantially between 1820 and 1850.  Contrary to logic, the number of teeth hasn’t always been nine.

A) 1820

In the 1820 patent, the cylinder has ten teeth that cover 10/14 of the circumference.  This extra tooth has a totally different function from the other nine.  It is part of the carry-over mechanism and allows the transmission, when going from nine to zero, of one unit to the next higher digit, that is to the next counter to the left.  In fact it is called a carry-over tooth !

T1820

B) 1822

This carry-over tooth is moved, as soon as 1822, to the outside of the cylinder (tooth Z).  But instead of going back to nine teeth, as one could logically imagine, it now has eighteen teeth on the cylinder!  They have doubled in number!

T1822

Les dents doublées du cylindre T1822 (=9x2)

Potential explanation:

On the 1822 machine, one can notice that the carriage is very wide. 
Thomas is still using the principle of complementary notation because the operation inverter hasn’t been invented yet.  One can see that each digit has a two counter display! 
One to display the result and one to display its nine’s complement.  The display counter is bigger because two numbers have to be displayed for each digit.  Most likely, this requirement forced Thomas to build larger display counters, increasing the number of  teeth on the display (40 instead of the 20 on the 1820 patent).

T1822 (Smitsonian Institute, Washington)

Cadran T1822

C) 1844-1849

After a hiatus of more than 20 years, the Arithmometer resurfaces at the national exposition of 1844.  Many models seem to have been on display ….
In 1848 a new machine comes out of the workshop.  Built by Piolaine, it has a very unique cylinder.   The cylinder is made of ten teeth, one of them being retractable within the body of the cylinder.  Its use is to transmit one increment to the next counter up during a carry-over operation while the display counter goes from 9 to 0.
What is strange is the abandonment of the carry-over wheel which is present in all the other machines from 1820 to 1880.
Consequently, in order to perform a carry-over, Thomas has to lengthen the carry-over tooth so that it can engage the detachable wheel A, whether it is in position 0,1,2,3,4,5,6,7,8 or 9.

We find ourselves with a tooth longer than the cylinder itself!  In order for it not to engage all the time, Thomas develops a retractable tooth, encased by the cylinder.
To execute a carry-over, the tooth comes out and aligns itself next to the nine other teeth!
A spring, located inside the cylinder pushes a cylindrical piece X located on the same axis.
This piece, attached to the tooth by a small perpendicular lever, forces the tooth to come out using a “windshield-wiper” effect.

Cylindre T1849

D) 1850-1907

It is from 1850 that the cylinder takes its final familiar form, with nine teeth.  One can find a similar description in all the patents:
“Inside the main compartment, one can see cylinders with 20 teeth.  Eleven of theses teeth have been completely removed.  The other nine are cut in a staircase fashion, in order to represent every number for the multiplicand, from one to nine.  A tenth mobile tooth is located right after the other nine”.

Cylindre T1865	Cylindre T1865 (photo)

E) Tableau récapitulatif

* Dents, nervures, cannelures