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/