The article was written with the expert help of my correspondent Mr. Stephan Weiss, www.mechrech.info

The German *pfarrer* Johann Reichold (1753-1798) was a bright-minded parson, who was not so interested in theological problems but preferred to spend his time studying and lecturing sciences, especially philosophy, mathematics, and engineering. In 1792 he devised a very interesting *arithmetische maschine* (arithmetic machine), rather similar to the device of his compatriot Philipp Matthäus Hahn.

Reichold probably has been acquainted with and inspired by the calculating machine of Hahn, not only because it was a quite popular device, described in several publications, but also because Hahn’s brother-in-law Johann Schuster (who made several machines by Hahn’s design) from the early 1790s ran a workshop in Ansbach, while Reichold worked as a parson in Dottenheim, a village located only 40 km north of Ansbach.

It seems the only reliable information for the calculating machine of Reichold can be found in a book by Johann Paul Bischoff, written in 1804, but published as late as 1990—Versuch einer Geschichte der Rechenmaschine (Attempt at a History of Calculating Machines), publisher: Systhema-Verlag, editor: Stephan Weiss. In the last quarter of the 18^{th} century, Bishoff undertook several long trips to take a look personally at the calculating machines, which he had heard of, in order to describe them in a book. Admittedly this is the case also with the machine of Reichold, otherwise, Bishoff wouldn’t have gained so detailed information about the mechanism, given in the book. Moreover, Ansbach is only some 40 km south of Dottenheim, where Reichold lived.

Reichold obviously had a rich experience in woodwork, making wooden clocks, and other instruments. That’s why the copy of Reichold’s arithmetic machine, seen by Bishoff, was not a metal one but was made entirely of pear wood (pear wood even now is one of the preferred materials in the manufacture of high-quality woodwind instruments and furniture).

Probably due to its wooden construction, the machine was quite big—it has a cylindrical form, 7.5 cm in height, and about 27 cm in diameter (see the nearby image). Let’s mention only, that almost all other calculating machines from that time were big, for example, the first calculating machine by Anton Braun, although made of metal, was even bigger—40 cm diameter, 21 cm height.

The device has two rows with nine graduated number disks in each row. In the center of outer number disks, used during the entering of the numbers into the machine, are mounted axes, on which are fixed cranks (marked with *K* in *Figure 1*) and gear-wheels (marked with *A, A’* and *A”*) (see the *Figure 2* below). In the center of inner (smaller) number disks are mounted axes (*Q*), on which are fixed pointers (marked with *S*), knobs (*m*), ratchet-wheels (*C, C’, C”*), gear-wheels (*B*) and star-wheels (*E*).

The motion (rotation) from the input mechanism (bigger disks) to the resulting mechanism (smaller disks) is transferred by engaging the teeth of gear-wheels (*A*) to the teeth of gear-wheels (*B*) (*Figure 2*). The mechanism of each digital position is connected also by means of the star-wheel (*E*) and teeth *L* (see *Figures 3 and 4*) to the mechanisms of the next position (if any), in order to propagate tens carry operations when needed.

The number disks in the first (right) pair are divided into four parts, numbered 0, 1, 2, and 3. The second pair of disks are decimal (base ten), divided into 10 parts. The third pair of disks are divided into six parts (numbered 0, 1, 2, 3, 4, and 5). The pairs from 4th to 9th are decimal. The graduation of the disks was made according to the monetary system adopted throughout the southern states of the Holy Roman Empire in the 18th century (240 Pfennige = 60 Kreuzer = 1 Gulden). The first pair of number disks is used for presenting Pfennige (4 Pfennige made a Kreuzer), the second and the third pairs of disks are used for presenting Kreuzer (60 Kreuzer made a Gulden), and the remaining 6 are used for Gulden and decimal calculations.

The crank (*K*) in the center of each bigger number disk rotates the axis and the gear wheel (*A*) below the disk. Under each digit on the bigger number disks, there is a small hole, into which can be inserted a pin, thus limiting (stopping) the rotation of the crank. Under the zero digit of each disk, there is also a pin, which abuts the crank, so that it can be turned only up to this pin and not further in circles. So when we want to enter a number in this position, we have to put a pin in the proper hole and rotate the crank from zero position to the pin and back, from the pin to zero.

The smaller (inner) number disks are divided at the same scale, as the larger disks (i.e. first disk in four parts, second in ten parts, etc.), however, they contain a double row of digits—black digits for addition and multiplication, and red digits for the subtraction and division. Black digits are actually placed as a complement to nine of red digits, e.g. black zero is near red 9, black 1 is near red 8, and so on.

In the middle of smaller disks are mounted pointers, which are used for presenting the result of calculations. The pointers also can be moved with the aid of wooden knobs (marked with *m* in *Figure 1*), and thus, can be set to any number.

In the center of the front panel is mounted an immovable ring (marked with *R*) with nine holes, into which can be inserted a small pin (marked with *n*). This ring is used to describe the place where one left off during the operation. On this ring also are marked the values of the adjacent number disks.

Let’s examine the *modus operandi* of the machine, performing the four basic arithmetical operations (addition, subtraction, multiplication, and division).

1. Addition.

Let’s add 365 Gulden, 50 Kreuzer and 3 Pfennige to 219 Gulden, 19 Kreuzer and 1 Pfennig. Firstly we have to enter the first addend (365 G, 50 K, and 3 P) into the smaller number disks, using knobs of the pointers and black digits (black digits are used for addition and multiplication, red digits for subtraction and division). So we set the pointer of the right smaller disk to show 3 (we have 3 Pfennige). The 2nd disk pointer must be set to 0, and the 3rd disk pointer—to 5 (we have 50 Kreuzer). The 4th disk pointer must be set to 5, the 5th—to 6, and the 6th—to 3 (we have 365 Gulden).

Now let’s enter the second addend (219 G, 19 K, and 1 P) using the cranks and the bigger number disks. First, we have to put the pin into the hole below 1 (we have 1 Pfennig) of the first big number disk and rotate the crank until it is stopped by the pin (thus the crank will rotate the axes to 90 degrees), and to return the crank back to zero position. The movement from the bigger disk mechanism will be transferred to the smaller, rotating the pointer from 3 (it was 3 Pfennige) to the next position (0), causing a tens carry to be propagated to the next digital position (small disk), which was set to zero Kreuzer, but now will turn to 1.

Using the same manner, we have to enter Kreuzer (19)—9 into the 2nd big disk (causing the 2nd small disk to rotate from 1 to 0, and a tens carry to be propagated to the third small disk, which was 5, but now will go to zero, and another tens carry to go to 4th small disk, which was 5, but now will show 6), and 1 to the 3rd big disk, causing the 3rd small disk to rotate from 0 to 1. As an intermediate result, before entering the Gulden of the second addend, we have now 366 Gulden (in the 6th, 5th, and 4th small disks), 10 Kreuzer (in the 3rd and 2nd disks), and 0 Pfennige (in the last disk). We have only to enter 219 (Gulden of the 2nd addend) using the cranks, causing 1 tens carry to be propagated from the 4th to 5th small disk, in order to get the final result—585 Gulden, 10 Kreuzer, and 0 Pfennige.

*2. Multiplication.*

The multiplication was done just like in the other calculating machines from the time, i.e. using consecutive additions. If for example, we have to multiply 815 by 9, we have to put the pin under the digit 5 on the 4th big disk and turn the crank to the pin 9 times. A similar operation must be done for tens and hundreds. If we want to multiply multi-digit numbers, this will require writing down the intermediate multiplication results (multiplying the multiplicand to Ones, Tens, Hundreds, Thousands, etc. of multiplier), and adding them one by one, a rather cumbersome operation.

*3. Subtraction.*

The subtraction is similar to addition, but the red digits must be used instead of black. The minuend must be set into the smaller disks mechanism, using knobs of the pointers and red digits, and then the subtrahend must be entered into the bigger disks mechanism, using the cranks.

*4. Division.*

The division operation is even more cumbersome and error-prone operation, than multiplication, because it was done using consecutive subtractions, and some intermediate calculations are needed, in order to get the final result. The dividend is entered using the red numbers in smaller disks. Then the divisor must be placed under the highest digit of the dividend and deducted from this part of the dividend until the same has become smaller than the divisor. Then if the dividend has more digits available, the divisor must be shifted to the right, and the consecutive subtraction to be repeated, and so on, until the entire dividend will become smaller than the divisor.

As you can see, just like the other calculating machines from the time, which are actually simple adding devices, the machine of Reichold is suitable only for addition and subtraction. For multiplication and division, it is much more convenient to use other calculating techniques and methods.

*Internal mechanism of the device.*

In *Figure 2* with letters *A*, *A’* and *A”* are marked gear-wheels, fixed on the same axes as the cranks *K*. Wheels *A* are engaged with gear-wheels with the same diameter and number of teeth *B*, which are fixed firmly to the axes (*Q*) of smaller number disks.

Directly above each of the gear-wheels *B* on the same axis (*Q*) are attached ratchet-wheels (*C*), which have locking pins and springs (marked with *q*), mounted on wheels *B*. These locking pins are fixing the ratchet-wheels to the gear-wheels *B* and limiting their rotation in one direction regarding wheels *B*. Ratchet-wheels *C* are not firmly attached to axes *Q*, but only to the pointers (*S*). Thus, when we are entering a number in smaller disks mechanisms, rotating the pointers in counter-clockwise direction, this will only rotate on the ratchet-wheel, without affecting the gear-wheel *B* and the star-wheel *E*. However, when a number is entered by rotating the cranks forward and backward, then the locking pins and springs, mounted on wheels *B* will engage the ratchet-wheels and will rotate them, together with pointers, thus changing the result.

On the same axes *Q*, but mounted at some distance below wheels *B* (see *Figures 3 and 4*), are mounted star-wheels (marked with *E*). The wheels *E* have teeth *L*, which are used for propagating the tens carry to the next axis, as once in each full revolution of the axes *Q* will engage the next star-wheel and will rotate it. The wheels *E* have also locking pins and springs (marked with *h*), which are fixed to the body of the machine and are fixing the wheels and limiting rotation only to one direction.

The star-wheels *E*, just like the ratchet-wheels *C*, are not fixed firmly to the axes *Q*, so during the rotation of the cranks *K* they can rotate or not, depending on the direction of rotation and the position of the locking pins *h*.

Let’s imagine, that we are going to enter a number in the rightmost wheel *A*, moving crank *A* from position *d* to *f*, and then back from *f* to *d* (see *Figure 2*). During the first half of the rotation, from position *d* to *f*, the teeth of wheel *A* will engage the teeth of wheel *B* and will rotate it. At this moment, however, the star-wheel *E* will be stopped by the fixing pin *h* and will not follow the movement. The ratchet-wheel *C* will not follow the movement of wheel *B* also because the locking pin *p* is not limiting the movement of wheel *C* in the counter-clockwise direction. So only wheel *B*, but neither wheel *C* nor wheel *E* will follow the movement.

During the second half of the rotation of crank *K*, from position *f* to *d*, the teeth of wheel *A* again will engage the teeth of wheel *B* and will rotate it, but in the opposite direction. Now, the wheel *E* will follow the rotation, because in this direction the fixing pin *h* does not limit its rotation. The same is the case with the ratchet-wheel *C*, which will be pushed and rotated by the fixing pin *p*.

In this manner, we will manage to transfer the number (motion) to the calculating mechanism and to turn back the crank into the initial (zero) position.

Obviously, the design of the machine is basically simple but workable, and if the parts had been manufactured precisely and with metal, it could be a much smaller and more reliable device. Nevertheless, Reichold must have been a very good constructor, unfortunately, struck down young and in his prime.

*Literature: Johann Paul Bischoff, Versuch einer Geschichte der Rechenmaschine, publisher: Systhema-Verlag, 1990, editor: Stephan Weiss.*

#### Biography of Johann Reichold

Johann Christoph Reichold was born on 17 January 1753, in Equarhofen (a small village, located some 90 km west of Nürnberg), in Bavaria, Germany, where his father, the parson Johann Georg Reichold was in service at that time.

The father—Johann Georg Reichold was born in Berneck, Switzerland, in 1707, studied theology at Universität Halle from 1731 until 1736, then worked at the orphanage in Bayreuth, and in 1739 accepted a position of a parson in Equarhofen and Frauental, where he served 40 years until his death on 31 October 1779. In Equarhofen Georg Reichold married Johanna Katharina Silchmüller, and they had two children — the son Johann Christoph, and a daughter — Maria Auguste Luise Reichold-Hoechstetter (20 Oct 1748 – 25 Jan 1803).

From November 1763 until April 1771, Johann Reichold studied at the Fürstenschule (Prince School) of Neustadt an der Aisch (a small town, located 40 km NW from Nürnberg), under Christian Theodor Oertel, Friedrich Wilhelm Hagen, and Johann Friedrich Amthor. Then (according to the University registers—from 11 May 1771 to 22 September 1773) he studied theology at the University of Erlangen, but for some reason (most probably financial) didn’t manage to earn a degree.

Reichold lived in Erlangen, studying and working as a private teacher and Hofmeister (butler) until July 1784, when he accepted the position of a parson in Dottenheim, a village near Neustadt an der Aisch (on the nearby image you can see the church St. Markus in Dottenheim, where Johann Reichold was in service).

Just like his more famous compatriot Philipp Matthäus Hahn, Reichold was a bright-minded parson, who was not so interested in theological problems but preferred to spend his time studying and lecturing sciences, especially philosophy, mathematics, and engineering.

Besides his calculating machine, Reichold was also engaged in the manufacture of wooden clocks and in the construction of instruments, for example, a pedometer device.

Johann Christoph Reichold died on 16 February 1798, in Dottenheim. It seems only the early death of Reichold didn’t allow him to make a significant contribution to the development of calculating machines.