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Early mechanical calculators - Philipp Matthäus Hahn, 1770-1778.

Hahn's calculating machine, Philipp Mathius Hahn. (1795-1823). Full scale operational model by Michael Leibfritz, 2020.

 

In 1770 the Württemberg pastor, astronomer, engineer, and entrepreneur Philipp Matthäus Hahn turned his attention to the creation of calculating machines, devising a simple calculation device (so-called Rechentrommel – calculation drum), then an adding device, and finally a more elaborate cylindrical calculating machine. The first two devices were not so innovative, but Hahn’s cylindrical calculators were the first fully functional popular four-species (addition, subtraction, multiplication, division) mechanical calculating machines in the world (the earlier machines of Anton Braun and Jacob Leupold remained relatively obscured and unknown to the public).

Philipp Hahn was a gifted mechanic, who was engaged mainly in making complex clocks and planetariums. He needed a calculating device, in order to calculate the parameters of his machines, that’s why sometime in the summer of 1770 he started to design several calculating devices. The first working copy of Hahn’s most advanced calculating machine, his cylindrical calculator, was ready in 1773, but it was demonstrated as late as 1778 because Hahn has difficulties with the reliability of the tens carry mechanism. By 1779 four machines were made, till the end of his life, Hahn manufactured about 5-6 devices, two of which still exist in the Württemberg State Museum in Stuttgart and in the Technoseum in Mannheim. After his death, several calculating machines by his design were created by his apprentices, in the photo below you can see a variety of Hahn’s machine, made by Johann Christoph Schuster, an apprentice, and brother-in-law of Hahn.

Hahn certainly has been acquainted with the calculating machine of Leibniz (not only from Theatrum arithmetico-geometricum of Leupold, but also from other sources), and that’s probably the reason he used the stepped drum of Leibniz in the construction of his device. However, the arrangement of the device has some similarities not with the Stepped Reckoner of Leibniz, but with the calculating machines of Leupold and Braun.

In an article in the magazine Teutschen Merkur from 1779, Hahn mentioned his inspiration:
When my time was occupied with making astronomical clocks, I had to deal with calculations of long fractions, multiplication, and division of large numbers, and I was so overwhelmed, that my primary work was close to being stopped. Then I recalled that some time ago I read a book for Leibniz, which mentioned his calculating machine, for which he spent a lot of money, without satisfactory results. I decided to spare some time in this direction. Certainly, I also wasted much time and money experimenting and troubleshooting the construction of my device. Finally, I managed to construct a rather advanced and reliable machine. Most difficulties I met during the construction of the tens carrying mechanism. The design of making a reliable tens carry mechanism bedeviled many of the early calculator designs. -
comments from owner.

Hahn needed quite some time to solve the problem with the tens-carry mechanism (he complained several times about the poor quality work of his mechanics), but he managed to resolve it, partly by changing the initial rectangular form of the machine with a circular. So the first working copies of the machine had ten digital positions, and the latter had 12 digital positions. The change in design from a rectangular to circular design is significant. This allowed for the powering of the gear work as well as the tens carry mechanisms to be initiated in sequence. The crank which is connected to a semicircular toothed rack engages sequentially each input operand and its associated components individually as opposed to all at once as was needed in a linear design. This lessened greatly the needed input torque and subsequent stress on the machine's components resulting in greater reliabilty. - comments from owner. 

 

The main part of the mechanism of each digital position is a small stepped drum (see the staffelwalze in the drawing), mounted on an axis, which can be moved upwards and downwards.

 

During the rotation of the mechanisms of the machine by means of the handle in the middle of the lid, a stepped drum will be engaged with the wheel of the main counter, which is also attached to vertical axes, and according to the vertical position of the appropriate stepped drum, the wheel will be rotated to 0, 1, …, 9 teeth.

The dials are graduated with two scales. The outer ring of digits is black and is used during adding and multiplication, the inner one is red, and is used during subtraction and division. The digits of the inner scale actually are complementing to 9 of those in the outer scale (i.e. below 0 is 9, below 1 is 8, etc.).

The entered in the input mechanism (stepped drums) number is transferred to the main counter by rotating the crank handle. There is also an additional counter, which counters the revolutions of the handle. The module of the additional and main counter is separated by the calculating module (module with the stepped drums) in a separate ring. This means, that the calculating mechanism is separated by the displaying mechanism. Thus, by rotating the ring of the counters, we actually can move the multiplier (divisor), during the multiplication (division). This moving can be controlled by a special arrow-pointer.

The adding operation can be done as follows:
1. The dials must be set to 0 (if it is necessary). By rotating the axes of the main counter, we set the first addend in the bigger dial (with black digits).
2. Then by pulling the axes of the stepped drums we set another addend.
3. By rotating the handle to 1 revolution, the number is transferred to the main counter and the result can be seen in the windows of the dials.

The subtraction can be performed in a similar way, but the minuend is set according to the red digits, while the subtrahend is set by pulling the axes of the stepped drums. After rotating the handle to one revolution, the result can be seen in the windows of the dials.

The multiplication can be done (by performing successive additions) thus:
1. The dials must be set to 0 (if it is necessary). The multiplicand is set by pulling the digital sticks of the stepped drums.
2. The handle must be rotated to the revolutions, equal to the number of the units of multiplier (the number of revolutions can be seen in the small dials).
3. Then we have to multiply the multiplicand by the tens of the multiplier, so we have to shift the multiplicand one digital position to the left, by rotating the ring with the dials.
4. The handle must be rotated to the revolutions, equal to the number of the tens of multiplier.
5. If it is necessary, the same actions must be repeated for hundreds, thousands… of the multiplier by rotating the inner dial table.

The division is done in a way, similar to multiplication, but in this case, are used the red digits of the dials and it is based on successive subtractions.

The calculating machine of Hahn became popular in Germany at the end of the 18th century and was demonstrated to many dignitaries like Kaiser Joseph II, Johann Wolfgang von Goethe (Goethe visited Hahn in 1779 in his workshop in Kornwestheim), Herzog Carl von Weimar zu Gast, and described in the press.¹

This calculator has an eleven digit capacity, ten thousand million or ten billion.

 

 

 

 

 

  

This video demonstrates the addition and multiplication functions.

 

This video demonstrates the division function.

1. Computer Timeline, Georgi Dalakov,  http://www.computer-timeline.com/timeline/philipp-matthaus-hahn/                                

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