If you are humble nothing will touch you, neither praise nor disgrace, because you know what you are.

*Mother Teresa*

Little is known about the humble Danish bank clerk Johannes Vermehren (1858-1928), which is rather strange, because he was an active inventor for more than three decades (from the end of the 1880s until the early 1920s) and has several dozens of patents mainly for calculating devices not only in Denmark, but also in the USA, Austria, Canada, Sweden, France, Germany, Italy, Great Britain, Switzerland, and Finland. Trying to commercialize his calculators, Vermehren even established his own company (*Aktieselskabet Vermehrens Regnemaskiner*, of Copenhagen, Denmark), but obviously without success, because there is no working example, survived to the present.

During his long career as an inventor of calculators, Vermehren tried mainly to facilitate banking-related calculations and patented several basic types of calculating devices, let’s examine some of them: (1) devices with gears-based calculating mechanisms; (2) devices with cone-shaped calculating members; (3) devices with a circular disk, provided with projecting pins; and (4) devices with a pin-wheels mechanism. Let’s examine the first (chronologically) calculating device of Vermehren, using the US patent No. 398360, filed Jan 1888, granted Feb 1889.

This is a calculating machine, suitable for calculations of commission, interest, annuity, and similar proportionate calculations, in which calculations may be made in a moment of time by adjusting the machine to a certain adjustment indicated in a key or index accompanying the machine and is dependent upon the proportions and the character of the calculation, and thereupon setting the hands or pointers upon one set of disks to the number, which is to be treated in the desired manner, when the machine will indicate upon another set of dials the resulting sum.

When the machine is to be operated, for example, the calculation of the interest of a certain sum in a given space of time-the key or index, which must accompany each machine is consulted to determine the adjustment of the carriage, the said key giving a certain adjustment indicated by the scale upon the slot and upon the nut for the amount of interest and length of time. The handle is now turned so as to bring the hands of one set of dials to point to the given number, when the hands of the other set will indicate the desired number, the hands indicating the desired number or the sum having been revolved a proportionate number of revolutions to the adjustment of the horizontal driving-disk.

The entire principle of the machine is to determine by the key or index, which is calculated beforehand to indicate the various proportions between figures in various calculations, the proportion desired between the hands indicating the given number and the hands indicating the desired number or the sum, when the hands will be moved around with a proportionate speed by the friction disks being revolved at the proportionate speed by the horizontal disk, having their points of contact at the proportionate distances from the center of the said disk.

In the late 1890s, Vermehren designed another type of calculating device, which he patented around 1900 in several countries. It was also (like the first) suitable for proportionate calculations, but had cone-shaped and disk-shaped calculating members (see the nearby drawing from the US patent No. 825363).

This calculator consists of two or more pairs of friction or calculating members, one member of each pair being cone-shaped, while the other one is shaped as a disk or a ring whose edge runs on the outer or inner surface of the cone in frictional contact with it so that the disk or ring is rotated when the cone is rotated. The disk or ring may be mounted in such a manner that its contact points with the cone have a constant but adjustable distance from the apex of the cone or so that it during its rotation also has a lengthwise motion in the direction of its axis, whereby its consecutive contact-points with the cone form a curve the projection of which on the base of the cone forms a logarithmic spiral. By the combination of said pairs of calculating members with a number of counting apparatuses, the machine is able to perform multiplications and divisions of whole numbers and fractions and multiplications, divisions, involutions, evolutions, and calculations by means of logarithms.

This mechanism presents great advantages, especially for calculating the exchange value of bonds and the like. If, for example, one German mark is eighty-eight oere, Danish, and one French franc equals 72.5 oere, the disks *a* and *a’* must be adjusted so as to make *r* equal eighty-eight and *r’* equal 72.5, so that the counting apparatus *t* will indicate the number of francs and the other one, *t’*, the corresponding number of marks; but the said arrangement may also be employed for general multiplications, as may be seen from the example quoted. If, for example, any given number is to be multiplied by sixty-seven, it is the same as multiplying with the fraction 670/10. It is obvious that the machine may be used for performing divisions also.

At the beginning of the 20^{th} century, Vermehren devised a calculating device for calculating fractions, based on a mechanism with a circular disk with projecting pins, which he patented around 1910. This machine (see the nearby patent drawing from US patent No. 953160) can be used for calculating fractions, adapted, not only to give a large number of figures in the products, but also to enable fractions to be calculated both the numerators and denominators of, which have several figures. In order that the machine may be able thus to give a product with an increased number of places, the actuating device (which, for example, may consist of five parts on each side) must be able to engage simultaneously a corresponding series of number wheels belonging to the counting mechanism, and further must be movable along the same, unless it is preferred to move the counting mechanisms while the actuating device remains stationary. This latter arrangement is, however, not generally to be recommended, because the counting mechanisms are far larger than the actuating device. In the example illustrated the actuating device is in the form of a circular disk *a* provided with projecting pins *b* which are arranged in nine concentric circles whose radii are in the ratio 1:2:3: . . . 9.

In the late 1910s, Vermehren designed a different type of calculating device, which he patented around 1920. In the contrast with his previous machines, it is based on the pin-wheel mechanism and is more suitable for general addition and multiplication (see the nearby drawing from the Swedish patent No. 51960).

This is an extended addition machine and working multiplication calculator, consisting of two, as a rule, of each-other extension lying drums, composed of against each digit of the multiplier and multiplicand corresponding slices or the like, each of which is provided with a number of organs, by means of which the number of active bodies can be varied in accordance with the numerical value of the figures, and of which drums one is rotated from the other three turns each time the latter was overturned, at the same time the former the drum, in this case, is controlled by the latter in this way, in the case of each of the above-mentioned bodies, one drum on all records on the other the drum of a series of bodies is brought into action on the Gears of a counting device, which is so arranged, that its numbers are shifted one step between the multiplications by the individual numbers. It works by multiplying and the multiplicand is set before the counting operation, and the entire calculation is carried out without other external more than the turnaround itself.

#### Biography of Johannes Vermehren

Johannes Vermehren was born on 3 July 1858 in København (Copenhagen), Christianshavn. He was the firstborn of the famous Danish genre and portrait painter Johan Friderich “Frits” Nicolai Vermehren (1823-1910), and his wife—Thomasine Ludvigne Grüner (1833-1877). After Johannes, the family had nine more children.

Besides his inventions (he had not only numerous patents for calculating devices, but he was also a holder of a patent for *Security systems for Bank premises and the like* (patent DK19549), and another one for *Apparatus for Preventing Collisions on Railways* (pat. GB189730753)), we know almost nothing about the life of Johannes Vermehren. He used to work many years as a clerk in the *Sparekassen for Kjøbenhavn og Omegn* (Savings Bank for Copenhagen and its surroundings), founded in 1820 and one of the oldest banks in Denmark. At the beginning of the 20^{th} century, Vermehren tried to commercialize his inventions and established his own company (*Aktieselskabet Vermehrens Regnemaskiner*, est. 1904), which despite enlisting as board member famous Danish businessmen such as Villads Emanuel Gamborg (1865-1929) and Holger Petersen (1843-1917), didn’t achieve success.

Johannes Vermehren married on 7 May 1886 in Næstved to the local girl Ida Christine Ingeborg Henriette Larsen (1859-1935), the daughter of procurator Rasmus Larsen and Anne Marie Christine Cæcilie Sessiong Christensen. They had five children: Ingeborg Thomasine (1887-1896), Margrethe Cecilie (1888-), Knud Frederik Rasmus (1890-1985), Erik Christian Nicolai (1893-1945), and Else Ludvigne (1896-). Knud Frederik Rasmus Vermehren became an engineer and a gymnast who competed in the 1920 Summer Olympics and won the gold medal in the gymnastics men’s team event.

Johannes Vermehren died on 23 July 1938 in Copenhagen.