In the present SI system, the units of length and time are related by defining the speed of light as exactly 299,792,458 m/s. In 1857, Wilhelm Weber and Rudolf Kohlrausch proposed a system of units in which the units of length and time were also related by defining a speed, although not the speed of light. The system was extended to include a unit of mass. The Weber-Kohlrausch system was impractical and was apparently never adopted. Nevertheless, it remains of historical interest as an early attempt at a systematic treatment of physical units.

## I. INTRODUCTION

At least since the development of the metric system in the late eighteenth century there has been a desire to base the units of physical measurement on fundamental properties of nature. Thus, the meter was defined as 1/10,000,000 of the distance between the North Pole and the Equator along the meridian through Paris, the kilogram as the mass of one cubic decimeter (1 liter) of water, and the second as 1/86,400 of the mean solar day.^{1} The meter and the kilogram fit the definitions only approximately, so in 1875, the Convention of the Meter in Paris established the International Bureau of Weights and Measures (Bureau International des Poids et Mesures, BIPM) to develop new units. As a result, for many years, the meter was the distance between two engraved lines on a platinum-iridium bar kept at the BIPM in Sèvres, a suburb of Paris. Similarly the kilogram was, and still is, the mass of a platinum-iridium cylinder kept at the BIPM.^{1–3}

During the twentieth century, the definitions of the meter and second were changed in two ways. First, they were defined in terms of atomic properties rather than properties of the earth. Although the kilogram is still defined in terms of the standard cylinder, there have recently been proposals to define the kilogram in terms of atomic properties.^{4,5} The second change in the definition of the meter and second is that they are related to each other through the speed of light rather than being defined independently. In the present definition,^{1} the second is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom. The speed of light is presently defined as exactly 299,792,458 m/s. Thus, the meter is the distance that light travels in 1/299,792,458 of a second. A brief description of the development of the modern units is given in the appendix.

It is little known that in 1857, Wilhelm Weber and Rudolf Kohlrausch^{6} suggested a system in which the units of length and time are related by a defined velocity. Their system also related the unit of mass to the units of length and time. Although their system was probably impractical and apparently has never been adopted, it is of historical interest. I will describe the system in this paper.

## II. THE SYSTEM

The system of units of Weber and Kohlrausch is based on an equation, developed by Weber,^{7} for the interaction of two moving charges. The equation is

where **F**_{12} is the force exerted by charge *q*_{2}, located at position **r**_{2}, on charge *q*_{1}, located at position **r**_{1}, while **r** = **r**_{1} – **r**_{2} and *c _{W}* is a constant. This and the following equations are written in modern vector notation rather than in the notation of the nineteenth century. Weber denoted his constant by

*c*rather than

*c*; however, it is not the speed of light (but in fact is larger by a factor of $2$). Therefore, I follow Rosenfeld

_{W}^{8}and Assis

^{9}and denote Weber's constant by

*c*. Weber recognized that

_{W}*c*has dimensions of a velocity. If two charges are moving relative to each other with constant velocity such that

_{W}*dr*/

*dt*=

*c*they exert no force on each other. If the two charges are at rest this equation reduces to Coulomb's law and the coefficient of 1 before

_{W}*q*

_{1}

*q*

_{2}implies that charge is measured in esu (statcoulombs).

In 1856, Weber and Kohlrausch^{6} measured *c _{W}* and found it to be 439450 × 10

^{6 }mm/s. This gives $cW/2=3.10738\xd7108\u2009m/s$, which is close to the speed of light measured by Fizeau

^{10}(

*c*= 3.14858 × 10

^{8 }m/s). Weber noticed the similarity of the numbers but apparently attached no significance to it. During the 1850 s and 1860 s, Weber, Kirchhoff, and Ludwig Lorenz used the constant

*c*and found situations in which electromagnetic effects propagate with the speed $cW/2$. In 1864, Maxwell predicted the existence of electromagnetic waves traveling with the speed $cW/2$, and in 1867 Lorenz predicted

_{W}^{11}the existence of electromagnetic waves traveling with speed $cW/2$ (although published after Maxwell's work, Lorenz's paper appears to have been independent). Based on the similarity of this speed to the speed of light measured by Fizeau, Maxwell concluded that light is an electromagnetic wave. Recall that in the mid-nineteenth century, the letter

*c*denoted what I am calling

*c*and was not used for the speed of light (see Ref. 12 for details and a discussion of the various symbols used in the nineteenth century to denote the speed of electromagnetic waves and of light).

_{W}The value of *c _{W}* obtained experimentally by Weber and Kohlrausch became the basis for their system of units. They first defined the unit of length to be the millimeter. In 1857 this was 1/1000 of the length of the then-standard meter bar kept in the French archives in Paris.

^{13}They then defined a unit of time, which I will denote by sw, by defining Weber's constant to be

*c*= 1 mm/sw. It then follows that 1 s = 439450 × 10

_{W}^{6}sw or 1 sw = 2.27557 × 10

^{−12}s.

Weber and Kohlrausch based their unit of mass on the gravitational field of the earth. If earth were a point mass, another point mass at a distance equal to earth's radius would experience an acceleration

where *M _{E}* is the mass of earth and

*R*= 6370 × 10

_{E}^{6 }mm its radius. This same mass at a distance of 1 mm from the earth point mass would experience an acceleration of

The Weber-Kohlrausch unit of mass, which I will denote by gw, was defined by stating that a point mass at a distance of 1 mm from a point unit mass will experience an acceleration of 1 mm/sw^{2}, i.e.

Weber and Kohlrausch did not give a value for the mass of the earth. The modern value of *M _{E}* = 5.97219 × 10

^{24 }kg gives 1 gw = 2.89705 × 10

^{24 }kg.

In the Weber-Kohlrausch (WK) units, *G* = 1 mm^{3}/(gw sw^{2}) and *c _{W}* = 1 mm/sw, so Newton's law of gravity is

and Weber's force [Eq. (1)] is

As noted above, this equation implies that charge is measured in esu.

## III. DISCUSSION

As mentioned in the introduction, apparently no one adopted the WK system of units—not even Weber himself (Kohlrausch died in 1858 shortly after this work was completed). In the reprint of the paper in *Ostwald's Klassiker*^{6} there are a few footnotes on the system but no indication that it was ever adopted. And there is no mention of it in a recent history of the BIPM.^{3} Nevertheless, it is interesting to consider how this early system of units both resembles and differs from the contemporary SI system.

The most obvious similarity is that the system is set up in part by defining a velocity. Of course Weber's constant is not the speed of light and the relation between *c _{W}* and the speed of light only became apparent with the work of Maxwell.

The main difference between the WK system and SI is that the WK units are based on properties of the earth, whereas modern units, with the exception of the kilogram, are based on atomic properties.

Probably, the most serious problem with the WK system is that in the nineteenth century the measurements of Weber's constant, the mass and radius of earth, and the acceleration due to gravity were not as accurate as the measurements of the standard meter, kilogram, and second. Indeed, earth's radius and gravitational field strength vary depending on your location, and the values used by Weber and Kohlrausch are only averages. Thus, the standard units were preferable to the WK units. It was increasing accuracy of measurement, as well as the development of atomic physics, that made the modern SI system possible. Still, the WK system was a clever idea that is worth remembering.

### APPENDIX: DEVELOPEMENT OF THE SI UNITS

Toward the latter part of the nineteenth century, the idea of basing units on atomic properties was taking hold. Already in 1870, in his presidential address to the British Association for the Advancement of Science, Maxwell suggested the use of atomic properties for the definition of units.^{14} At the time this was not possible, so Maxwell's suggestion was really a hope for the future. The first attempts to measure the meter in terms of optical wavelengths were made by Charles S. Pierce and Albert Michelson in the late 1870s and 1880s.^{15} Pierce used a diffraction grating and Michelson his interferometer to measure wavelengths. In 1892, Michelson used his interferometer to measure the length of the standard meter at the BIPM in terms of the wavelength of the red spectral line of cadmium.^{16} In the following decades, the measure of the standard meter became increasingly accurate. Interferometry improved, especially with the introduction of the Fabry-Pérot interferometer, and the use of single isotopes of elements produced sharper spectral lines. After an extensive study of spectral lines of different elements, the BIPM, in 1960, defined the meter in terms of a spectral line of krypton-86.^{1}

As noted above, the second was originally defined as 1/86,400 of the mean solar day. In 1960, the BIPM defined the second as 1/31,556,925.9747 of the tropical year 1900. Meanwhile, atomic clocks were being developed at the National Bureau of Standards (NBS) in the US and the National Physical Laboratory (NPL) in the UK. The first successful atomic clock was developed at the NPL in 1955 and this led to the current definition of the second in 1967.^{17} Finally, in 1983, the speed of light was defined as 299,792,458 m/s, leading to the current definition of the meter.^{1}

The kilogram is still defined in terms of the standard cylinder. However, there have recently been proposals to define the kilogram in terms of atomic properties, by defining the value of Plank's constant *h*. Two experimental approaches to doing this are now being investigated, but so far neither is completely satisfactory.^{4,5}

## References

The standard meter developed by the BIPM after 1875 was made as close as possible to the meter in the French archives.

Reference 5, Chap. 9.

Reference 5, pp. 212–213.

Reference 5, pp. 224–225.