* Translation by Michel Bardel, France
I) Complementary digits (1820-1822)
Early arithmometres have no Add/Sub switch to commute from addition to subtraction. So Thomas uses the principle of complementary numbers to perform the subtraction, as invented by Pascal in the 17th century.
The dials and other gears can only turn in one direction.
“A sliding brass mask shows, in the windows, one of two complementary sets of digits written on cylindrical wheels. The result is equivalent to two rows of dials turning in opposite directions.
How to perform a subtraction?
For instance, on a six digit machine, to subtract 839 from 5327, push the mask in its upper position, so that only the complementary digits show. With the stylus, dial until you see the number 005 327 in the complement windows. To do that, simply place the stylus on the left of the end stop, and turn each dial until you see the correct digit in the window. You will have to turn the two windows showing 9 until you get two zeros on the left of the number (in fact you will have dialed 994 672, which is the 9 complement of the original number).
Without moving the mask, normally add 839. To do that, place the stylus in front of the digit you want and turn until the end stop (you now have 994672 + 839 = 995 511, which is the 9 complement of 004 488). Without moving the mask, you can read 004 488 in the complement windows, which is the result of the operation.” / Jean Marguin as amended by M. Bardel.
Evolution gallery
In the 18th century, Hahn & Schuster also used the complement digits,
but under a circular arrangement.
|
Hahn & Schuster machines (Circa 1780)
|
|
Thomas de Colmar adapts it on his machine.
|
Arithmometre 1820 (photomontage) |
|
Arithmometre 1820 (photomontage)
|
|
Outbreak of dual windows in 1822
|
T1822 |
|
| |
T1822 (Smithsonian Institute, Washington) |
|
Complementary windows on T1822
|
 |
II) The dials
Probably for technical reasons, Thomas made the choice to double the teeth number on the total wheels gears n, going from 10 to 20 (cutting gears with less than 12 teeth is more difficult). That had an impact on the dial numbering, as a digit must correspond to a tooth. So there were a double series of digits 0, 1, 2, 3, … , 9, 0, 1, 2, …, 9.
When considering the complementary set of digits, it is 40 digits he had to print on each dial, spread over two concentric circles, one in the ascending order, the other one in the descending order.
III) Position of the windows
The two windows, being side by side, the complementary digits must be on a same chord of the dial, so that moving the mask displays one digit or its complement (General drawing : Fig. 3).
As can be seen on the following drawing, this was made possible by slightly shifting the windows to the left of the dial axis.
|
