Leibniz’s machina arithmetica

The very first mechanical calculating machine for all four operations of arithmetic was invented in 1671 by the philosopher, mathematician und universal scholar Gottfried Wilhelm Leibniz (1646-1716). The only existing original machine is housed in the State Library of Lower Saxony in Hannover. It was constructed in the period 1694 to 1716. In the literature it is often referred to as the “younger“ of the two big machines built by Leibniz from 1694 onwards. It was his idea to make his machine multiply by means of successive additions and also divide by means of successive subtractions. For this purpose he needed a mechanical memory for numbers so that a certain number could be repeatedly entered into the result mechanism. He first invented the so-called pinwheel, a type of cogwheel with nine movable pin-like cogs. Its construction, however, posed so many problems that he invented the simpler so-called stepped drum, a rotatable cylinder with nine longitudinal ribs of different, evenly spaced lengths. By setting the knobs of the setting mechanism in accordance with a particular number and then turning the crank, transfer-cogwheels engage the stepped drums according to their settings and thus enter the number in the result mechanism. For addition, one turns the crank clockwise, and for subtraction anticlockwise. This implied that Leibniz had to invent a fully automatic tens-carry mechanism for each place and for both turning directions. This mechanism works as in Schickard’s machine by means of a wheel with just one cog. When the dial in a particular place turns from 9 to 0, this cog engages the next dial and turns it on, adding 1. However, while this step takes place in one step in Schickard’s machine, Leibniz subdivided it into two, thus delaying it until the relevant rib of the stepped drum is no longer engaged. This can lead to the loss of a tens-carry step. He also had to incorporate a second cog, one for each turning direction of the crank. These were at right angles to one another and formed a piece which Leibniz called the “double horn”. When two or more tens-carry steps occur together, it is necessary to perform further clockwise turns of the crank with “0” as entry in the result mechanism (i.e. add 0) in order to avoid losing tens-carry steps. In connection with their replica, Badur and Rottstedt studied and described this feature in detail. In order to facilitate the multiplication by multi-digit multiplicands, Leibniz incorporated a place-shift mechanism allowing one to move the setting mechanism with respect to the result mechanism by powers of 10. Thus, multiplying by 12 requires not twelve turns of the crank but just three. An additional counter window records how many times the crank has been turned per place. The result of a division is given in the revolution counter. In contrast to the forty years that it took Leibniz to complete his machine, the replica housed in the Arithmeum took a mere ten years. This gives a clear indication as to why it took so long for Leibniz to have his machine constructed by a watchmaker.