Dimensionality of units

As we have remarked, Station 4 is both cutting-edge research and a textbook, an introduction to the study of electricity and magnetism.⁷² This conception of the Treatise can immediately be seen in the opening sections. Maxwell began with basic remarks on the measurement of quantities. He defined the “Unit” as the standard quantity and the “numerical value” as the number of times the standard quantity is to be taken, and then made the claim that:

There must be as many different units as there are different kinds of quantities to be measured, but in all dynamical sciences it is possible to define these units in terms of the three fundamental units of Length [£]. Time [7], and Mass [Af].⁷³

It is immediately clear that this claim, namely, that there is a distinction between the three fundamental units and other units which are derivative, has far-reaching consequences. In framing a mathematical system of a certain physical phenomenon, these fundamental units are presupposed and all derivative units are deduced from them by a set of definitions and nomological relations. This requirement underlies Maxwell’s incisive remark:

In all scientific studies it is of the greatest importance to employ units belonging to a properly defined system, and to know the relations of these units to the fundamental units, so that we may be able at once to transform our results from one system to another.⁷⁴

This mathematical transformation is most relevant, as we will see, to the ratio of electrostatic to electromagnetic units. In fact, as Maxwell explained,

A knowledge of the dimensions of units furnishes a test which ought to be applied to the equations resulting from any lengthened investigation. The dimensions of every term of such an equation, with respect to each of the three fundamental units, must be the same. If not, the equation is absurd, and contains some error, as its interpretation would be different according to the arbitrary system of units which we adopt.⁷⁵

Dimensional analysis thus offers a procedure for checking the physical consistency and coherence of the equations representing the phenomenon in question. For example, in a section titled, "Derived units”, Maxwell defined the unit of force as the force which produces a unit of momentum in a unit of time, so that its dimensions are [MLT²]. According to Maxwell, this is the absolute unit of force; its dimensional definition is implied in every equation on dynamics. Clearly, the strength of this approach lies in its universality.⁷⁶

Against this mechanical background, Maxwell developed a discussion of force in electromagnetism and demonstrated how the units in this domain could be transformed into the fundamental (mechanical) units. In particular, he developed the dimensions of electric units. We note that in Station 4 Maxwell first developed a dimensional analysis of units in mechanics, and subsequently extended this analysis to units of electricity and magnetism. He was, in effect, going from the familiar appeal to length [L], time [7], and mass [Af] in mechanics to the unfamiliar appeal to these units in electromagnetism.

Following the presentation of the general equations of the electromagnetic field. Maxwell dedicated an entire chapter (ch. 10) to the dimensions of electric units. Based on what is taken as the unit of electricity, one obtained, according to Maxwell, the electrostatic system. If, however, one began with the unit strength of a magnetic pole, a different system is obtained, namely, the electromagnetic system.⁷⁷ Maxwell drew a table of the dimensions of the units according to each system. True to the claim that he had made at the outset of his discussion of measurements of quantities. Maxwell specified the different units for the different kinds of quantities to be measured in the domain of electromagnetism. For each unit Maxwell assigned a symbol and the corresponding dimensions in the electrostatic system and the electromagnetic system. Fixing the dimensionality of each unit in terms of powers of [L], [A/], and [7]. facilitated the dimensional calculation for the ratios of units which are, as Maxwell remarked, “in certain cases of scientific importance.”⁷⁸ He then made the significant claim that reflected his successful discovery in Station 2:

If the units of length [L], mass [A/], and time [7] are the same in the two systems, the number of electrostatic units of electricity contained in one electromagnetic unit is numerically equal to a certain velocity, the absolute value of which does not depend on the magnitude of the fundamental units employed. This velocity is an important physical quantity, which we shall denote by the symbol v.⁷⁹

Maxwell thus established a relation between the two systems, namely, the number of electrostatic units in one electromagnetic unit is either proportional or inversely proportional to a velocity, depending on the unit (with some cases where the square of the velocity is required). In a subsequent chapter (ch. 19), Maxwell compared directly the electrostatic units with the electromagnetic units, and remarked that

If, therefore, we determine a velocity which is represented numerically by this number, then, even if we adopt new units of length and of time, the number representing this velocity will still be the number of electrostatic units of electricity in one electromagnetic unit, according to the new system of measurement.

This velocity, therefore, which indicates the relation between electrostatic and electromagnetic phenomena, is a natural quantity of definite magnitude, and the measurement of this quantity is one of the most important researches in electricity.⁸⁰

Maxwell showed how to obtain a physical conception of this velocity which is, in effect, the velocity of light, denoted v by Maxwell. According to Maxwell, "the first numerical determination of this velocity was made by Weber and Kohlrausch.”⁸¹

The path was then paved for the transition from this abstract analysis of the dimensions of electric units to a practical system. Maxwell remarked:

The electrical units derived from these fundamental units have been named after eminent electrical discoverers. Thus the practical unit of resistance is called the Ohm, and is represented by the resistance-coil issued by the British Association ... It is expressed in the electromagnetic system by a velocity of 10,000,000 metres per second.⁸²

Maxwell went on to refer to the Farad and the Volt. In this way Maxwell put the practice of measurements in electromagnetism and its nomenclature on solid theoretical ground. Most importantly, he demonstrated the importance of velocity as the dimension that relates the two systems, the electrostatic and the electromagnetic.

This analysis is entirely novel in Maxwell's corpus on electromagnetism. There is no such discussion in any of the preceding stations. Surprisingly, the origin of this analysis can be traced back to 1863, when Maxwell was engaged with the issue of standards of measurement, but these results showed up in the study of electromagnetism only in 1873, in Station 4.

In 1863, as part of Maxwell’s contribution—coauthored with the engineer Fleeming Jenkin (1833—1885)—to the report by the committee appointed by the British Association on standards of electrical resistance, the claim was made that given the two systems, electrostatic and electromagnetic, the relations between the different units remain unchanged when passing from the one system to the other.⁸³ The constant ratio between units in the two systems, designated v, has the dimensions [LIT], which is a velocity. Maxwell and Jenkin then made a historical claim and acknowledged the findings of Weber and Kohlrausch:

The first estimate of the relation between quantity of electricity measured statically and the quantity transferred by a current in a given time was made by Faraday*. A careful experimental investigation by MM. Weber and Kohlrauschf not only confirms the conclusion that the two kinds of measurement are consistent, but shows that the velocity v = qlQ is 310,740,000 meters per second—a velocity not differing from the estimated velocity of light more than the different determinations of the latter quantity differ from each other, v must always be a constant, real velocity in nature, and should be measured in terms of the system of fundamental units adopted in electrical measurements ... A redetermination of v ... will form part of the present Committee’s business in 1863-1864. It will be seen that, by definition, the quantity transmitted by an electromagnetic unit current in the unit time is equal to v electrostatic units of 84 quantity.

[Footnotes in Maxwell and Jenkin, 1864, 149; and 1865 509.]

* Experimental Researches, series iii. § 361, &c. t Abhandlungen der König. Sächsischen Ges. vol. iii. (1857) p. 260; or, Poggendorfifs Annalen, vol. xcix. p. 10 (Aug. 1856).

In 1864, Maxwell and Jenkin simply reported the findings of Faraday as well as those of Weber and Kohlrausch, but did not elaborate on the physical consequences; they also stressed the constancy of this ratio with the dimensionality of velocity but, again, without elaborating further. They did not then reflect on the significance of this constant ratio which is of the dimension of velocity and is strikingly similar numerically to the velocity of light, measured independently of this ratio.

To understand in what way the dimensionality of units is a methodology and not just a method, we turn to a brief examination of the report. We seek to show that dimensionality of units generates new knowledge, a feature that renders this analysis a methodology. Maxwell and Jenkin made claims independent of any physical theory which typically comes with an accompanied ontology. In acknowledging the achievements of Weber and Kohlrausch, Maxwell and Jenkin concluded that it could be seen that “by definition, the quantity transmitted by an electromagnetic unit current in the unit time is equal to v electrostatic units of quantity.”⁸⁵ We draw attention to the claim, namely, that this fundamental result is obtained “by definition” and is not theory dependent. Maxwell and Jenkin specify the macro-level phenomena on which they developed their analysis of dimensionality of units:

All our knowledge of electricity is derived from the mechanical, chemical, and thermal effects which it produces, and these effects cannot be ignored in a true absolute system. Chemical and thermal effects are, however, now all measured by reference to the mechanical unit of work; and therefore, in forming a coherent electrical system, the chemical and thermal effects may be neglected, and it is only necessary to attend to the connexion between electrical magnitudes and the mechanical units .... The four equations now given are sufficient to measure all electrical phenomena by reference to time, mass, and space only, or, in other words, to determine the four electrical units by reference to mechanical units.⁸⁶

The key point for our argument is that measurement relates to effects not to causes; hence, the reality at the micro-level (whatever it may be) is irrelevant in this context. It was critical that the analysis should not fall prey to some controversy and that it would simply be accepted by the community of practitioners of electricity and magnetism as independent of any idiosyncratic commitment. In short, the claim is that the analysis is natural, and neutral among theories:

In the opinion of the most practical and the most scientific men, a system in which every unit is derived from the primary units with decimal subdivisions is the best whenever it can be introduced. It is easily learnt; it renders calculations of all kinds simpler; it is more readily

Station 4 (1873) 175 accepted by the world at large; and it bears the stamp of the authority, not of this or that legislator or man of science, but of nature.⁸⁷

In Station 4 Maxwell offered a systematic treatment of units both in mechanics and in electromagnetism in terms of L, T, and M. This systematic discussion of the dimensionality of units displays relationships between two different systems that may not be obvious otherwise. In fact, Maxwell went beyond the practical needs for measurement as he sought connections inherent in nature. This is a contribution to knowledge and thus the underlying procedure is methodological.

Consider the following contrast: in 1873 in the Treatise Maxwell ordered all the various units in terms of L, T, and M, whereas in 1867 in their Treatise, Thomson and Tait did not consider any relation among units of different kinds.⁸⁸ We consider Thomson and Tait’s practical account a method, unlike the contribution of Maxwell which we consider a methodology.

In Station 2 Maxwell depended on a mechanical hypothesis to come to his conclusion concerning the ratio between the two systems of units as equal to the velocity of light. But in Station 4 the argument that the ratio is a velocity is independent of any hypothesis. This is the result of the methodology of dimensionality of units. And then Maxwell argued for the physical significance of this ratio.⁸⁹ The lesson is implicit: Dimensional analysis supports the claim for the ratio as a velocity independent of any hypothesis or theory. This is an important aspect of Maxwell's overall methodology. As Chrystal remarked in his review of the second edition of the Treatise'.

It begins by divesting the facts of all hypothetical raiment, and expressing them in language appropriate to themselves, suggesting nothing but what Nature has indicated, indicating nothing that Nature has denied, supposing as little as may be where nothing has been revealed. Above all banishing from the catalogue of physical conceptions the imponderable electrical fluids that have worked such mischief in indolent minds, and poisoned electrical literature so long.⁹⁰

In sum, the dimensionality of units helped to liberate Maxwell from dependence on the hypothesis.