p In this section, as in the preceding one, we shall frequently speak, for clarity’s sake, of the measurement of lengths, but all our reasoning also relates mutatis mutqndi? to the measurement of other quantities.

p One can assume that things that possess length cannot be compared with one another, and therefore be measured, without a standard. In actual fact, however, if the lengths of objects are commensurable, they possess this property independently of whether or not there is a standard of length. It is by virtue of this commensurability that they convert such an object as the standard metre into a measure of length common for them all.

p A measure that has completed its cycle of development returns in the form of a standard of measurement to the form in which it existed in the individual form of measurement, but this is not just a return to the initial form but rather a new step forward (the ’negation of negation’). On the other hand, the metre as the standard of length expresses its length only in the developed form of measurement, i.e. the metre has no standard of length.

p This idea, or rather its essence, can be formulated in another way: the very concept of a standard as a universal measure requires the existence only of one standard. In metrology this requirement is met through a hierarchy of measures of the measured property, the foundation of which is formed by the so-called primary standard.  [266•* 

p What then is the necessary condition for a thing to function as a standard? For this purpose, it must satisfy a certain set of conditions. The first condition is that a thing representing a common property of a set of compared objects should have the ‘property’ of representing purely quantitative differences. And this property implies uniformity, qualitative identity between copies of the standard. It is realised in metrology by the requirement, for instance, that working metres be made of the same material, that measurements by them be made in identical conditions, that the working metres themselves be made and kept in exactly the same conditions. Working metres, however, are not identical as things: they are affected by differences in macroscopic structure, and by the fact that their conditions of use and production are not absolutely 267 identical. Analysis of facts and considerations of this kind confirms that measurements reducible to standards in essence signify measurements by ideal average standards.

p A second condition of converting a thing into a standard follows from a standard’s having to represent purely quantitative differences: namely, that only that thing can be a standard that can be divided into any number of parts and combined with itself without losing its qualitative definiteness. Working measures or gauges are produced in such a way that this requirement is met as far as possible: measure sets, measure shops, gauges. But what does the arbitrary divisibility of a thing mean, or its arbitrary combinability with itself, the more so that a real finite object does not possess these properties to an absolute degree? Points like that will be considered in the sections that follow; let us note here that measurements that can be reduced to standards are ideal standards from the angle of the second condition of measurement.

p The whole content of the first section actually amounted to this, that the property of compared objects is not created in measurement but simply expressed, i.e. that measurement by itself does not in principle alter the objects compared. But since measurement is experimental comparison, in which the compared objects may form physical relations and physical situations may arise that are ‘bizarre’ from the angle of classical physics, possibilities of this kind must necessarily be analysed so as to construct a theory of mensuration in modern physics. Our further exposition is devoted to the relevant issues. At the same time, since a standard represents an objectivised universal measure, i.e. serves for measurement, it should not change either in the process of measurement or outside it, so long as it remains a standard. Its immutability means that the property materialised by a standard is preserved unaltered in the thing that serves as a standard, and that all changes experienced by this thing due to certain conditions (temperature, various fields, etc.) can be allowed for. This is the third condition for an object to function as a universal measure.

p No real object serving as a standard, of course, has the property of immutability in the absolute, but it is chosen or produced in such a way that it has a certain minimum of constancy that is much higher than the constancy of the things measured by it. Measurements that can be reduced 268 to standards are undoubtedly ideal standards from the angle of the third condition of measurement.

p A standard can thus perform’its job in essence as a standard that is ideal in three respects: (1) the quantity is measured essentially by an ideal average standard; (2) the thing is arbitrarily divisible and arbitrarily combinable with itself only in the abstraction, while remaining qualitatively the same thing, and the standard measures only when it is such an ideal object; (3) immutability is an axiomatic property of a standard, and only an ideal thing can be absolutely constant.  [268•* 

p When things are measured, it becomes necessary to relate them to a standard as a materialised unit of measurement. The latter is then expanded into a scale through division into equal parts or through combining with itself. Every object that serves as a standard has such scale even before being converted into a standard, since the thing, according to the second condition of its functioning as a standard, can be divided into any number of parts and combined with itself. Because, for instance, the lengths of things are related to each other as similar quantities measured by the metre, the latter is turned from a measure of length into a scale.

p The measure of a quantity and scale are two different functions of the standard. The standard of length is a measure of length as the materialised common property of things compared for length; it is a scale as a definite thing. As a measure of length, the standard of length provides the material for expressing lengths, in order to convert the lengths of things into a mentally imagined number of metres; as a scale, the standard of length measures this number of metres. The measure of length measures things as possessing length; a scale, on the contrary, measures various imagined numbers of metres by a given metre (which is then the unit). The definition of the unit of measure, and of its subdivisions and multiples, is a purely arbitrary matter; at the same time it must be generally accepted and be obligatory within the limits of mensuration practice.

p Measurement of the lengths of things thus has a dual, inseparably interconnected significance: (1) to measure the 269 length of any thing means to express its length in terms of a specific thing that has length (this specific thing being called a standard); (2) to measure the length of any thing means to compare the magnitude of this length (expressed in the standard) with the magnitude of the length of the standard adopted as the unit. This applies mutatis mutandis to direct measurements of other quantities.

p Let us now consider the views of certain other authors on the standard and unit.

p Wallot stressed that units should be absolutely constant and readily comparable with the measured quantities, and defined the unit (for direct measurement) by means of a ’ primary measure’ (Urmasse)." According to him this ’primary measure’ was, for instance, the line-standard metre; he did not, however, answer such questions as what the ’primary measure’ was, or why such-and-such measure was primary, and so on.

p The neo-Kantian Sigwart said that if it was impossible to find an absolute scale of value it followed in the final analysis that we were faced, following the direct empirical path in mensuration, with the impossibility of attaining objective results.^^17^^ Sigwart obviously could not manage the dialectics of direct measurement. Because the things that figure as standards are variable (since they are real objects), he doubted the possibility of obtaining objective data about a measured quantity in experiment.

p The neo-positivist Reichenbach denied measurement any objective meaning whatsoever. He distinguished statements about facts from so-called real definitions, by which he meant conventions, purely arbitrary agreements about physical objects. The definition of the metre through its prototype kept in Sevres was, in his opinion, a real definition. He thus reduced the functions of a standard to that of a scale, and since the establishment of a unit of measurement is an arbitrary agreement, he concluded from that erroneously that the defining of the measure of any physical quantity is, in principle, conventional.^^18^^

p In conclusion let us consider the limitations of direct measurement.

p 1. Any thing that is measured must, on the plane of direct measurement, be measured by as many standards as it has properties in common with other objects; or, to put it differently, there must be as many standards independent of one 270 another, from the aspect of direct measurement, as there are kinds of quantities in nature, regardless of whether they are connected by regularities.

p (i) It is impossible to have as many standards as there are different properties, the more so that new physical phenomena are being discovered that have to be covered by the theory, (ii) It is wrong in all circumstances to abstract the regular connections between physical phenomena, but direct measurement does not allow for precisely that point.

p Thus, the problem arises of measuring quantities of many kinds by a limited number of standards. Direct measurement neither does nor can solve it.

p 2. The thing measured is not internally connected with the standard, in direct measurement, i.e. the measured quantity and the unit of measurement are external to each other. That is the reason why, in experimental conditions of measuring, the result of the measurement, in certain circumstances, reflects not so much the measured quantity as variations in the thing serving as the standard. This limitation can only be overcome if the measured quantity is internally connected with the unit of measurement (and that is beyond the limits of direct measurement).

p 3. Direct measurement cannot determine the value of quantities characterising, say, celestial bodies and phenomena, or the values of quantities characterising physical bodies not directly perceptible by the sense organs (atoms, electrons), and, in general, the values of quantities not amenable to direct experimental comparison.

In Section 5 it will be shown that the limitations of direct measurement are overcome by indirect measurement.