On the Nature of Dimensions

On the Nature of DimensionsAuthor(s): Brian EllisSource: Philosophy of Science, Vol. 31, No. 4 (Oct., 1964), pp. 357-380Published by: The University of Chicago Press on behalf of the Philosophy of Science AssociationStable URL: http://www.jstor.org/stable/186265 .

Accessed: 29/08/2013 12:58

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The University of Chicago Press and Philosophy of Science Association are collaborating with JSTOR todigitize, preserve and extend access to Philosophy of Science.

http://www.jstor.org

This content downloaded from 129.68.65.223 on Thu, 29 Aug 2013 12:58:52 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=ucpress

http://www.jstor.org/action/showPublisher?publisherCode=psa

http://www.jstor.org/stable/186265?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


ON THE NATURE OF DIMENSIONS*

BRIAN ELLIS University of Melbourne

In the first part of this paper it is shown that unit names, whether simple or complex, whether of fundamental, associative or derivative measurement, may always be regarded as the names of scales. In the second it is shown that dimension names, whether simple, like "[M]", "[LI" and "[T]", or complex dimensional formulae, may always be regarded as the names of classes of similar scales. Thus, a new foundation for the theory of dimensional analysis is provided, and in the light of this, its nature and scope are examined. Dimensional analysis is shown to depend upon certain conventions for expressing numerical laws.

There are few concepts used in science which have caused more controversy over the last half-century than that of a physical dimension. In this paper I wish to present a positive thesis concerning the nature of dimensions and to show how dimensional analysis depends upon certain tacit conventions governing the expression of our numerical laws. In order to present this thesis, however, I must first say something about scales and units, and the first section of this paper is devoted to a discussion of these.

Part I: SCALES AND UTNITS

There are two chief systems of classification for our scales of measurement. The first of these, due to N. R. Campbell [2], is based upon an analysis of our measuring operations. Our scales are classified according to the kinds of operations by which they may be set up. The second system is due to S. S. Stevens [7] and, roughly speaking, it is based upon an analysis of the mathematical properties of our scales. In the system here presented an attempt is made to classify our scales according to the kinds of transformations to which they may be subjected, and to the kinds of statistics appropriate to them. For my purposes, the Campbellian system is more useful, although some refinement of it is necessary. Where Campbell distinguishes only two classes of scales, fundamental and derived, I shall distinguish three classes, fundamental, associative and derivative.

In my discussion of these classes of scales, I shall be mainly concerned with the significance of the various kinds of unit names. I shall argue that unit names, whether they be of fundamental, associative or derivative measurement, may consistently be regarded as the names of scales since, as I shall show, the criteria for the identity and difference of units are simply the criteria for the identity and difference of scales.

1. The Concept of a Scale. Let us first consider what is meant by 'a scale of measurement.' I suppose it will be agreed that every measurement must be made on some scale or other. What is it, then, to make a measurement? According to Stevens, it is simply to make an 'assignment of numerals to things according to a rule-any rule.' ([8], p. 19) And the 'is' which occurs in this sentence I take to be

* Received March, 1962.

357



358 BRIAN ELLIS

the 'is' of identity. Hence, if Stevens' criterion is adopted, we must say that we have a scale of measurement if and only if we have a rule for making numerical assignments.

However, this criterion is not entirely satisfactory. Certainly, it provides a necessary condition for saying that we have a scale of measurement, but it does not also provide a sufficient condition. If, for example, we havTe the rule that for any two substances A and B, the numerals 'a' and 'b' are to be assigned to A and B respectively in such a way that:

( scratches a R b according as A does not scratch and is not scratched by B,

a is scratched by

we do not yet have a scale of measurement. Nor, it seems to me, do we make a measurement of anything if we merely assign a pair of numerals to a pair of given substances according to this rule. For we cannot yet say on what scale they are assigned. The scale has not yet been defined. A set of such assignments may indeed be used in subsequently defining a scale (as in Mohs' hardness scale). But I think we must distinguish between those numerical assignments we make in setting up a scale, and those that we make on a scale which had already been set up. Only the latter may be termed measurements.

Hence, before Stevens' criterion can be accepted, some restriction must be placed on the nature of the rule. Measurement cannot be just the assignment of numerals to things according to any rule. For, this admits too much. The restriction I propose is that the rule must be determinative. That is, it must be such that:

a) Anyone who follows the rule with sufflicient care would be led to make the same numerical assignments to the same things under the same conditions, and

b) the results thus obtained are independent of any previous numerical assignments (if any) that may have been made to these things.

The condition a) ensures that anything that is measurable on the scale in question has a definite objective measure on that scale. The condition b) ensures that on any given scale of measurement it is always at least logically possible to measure something that has never been measured before; and to make a mistake in the process. Possession of such a determinative rule is certainly a necessary condition for having a scale of measurement. And, it seems also to be a sufficient condition. In any case, it is a considerably more restrictive condition than Stevens'.

Let us consider now what is meant by 'a scale for the measurement of a given quantity q'. I have discussed the concept of a quantity in detail elsewhere, [4] and I need not repeat the discussion here. I shall merely draw upon the conclusions of this discussion. Firstly, I argued that a quantity exists if and only if a set of ordering relationships exists; and, hence, that wherever there are such ordering relationships, we may say (analytically) of any two things A and B connected by them that A is greater than, equal to, or less than, B in respect of some quantity.' Secondly, that given any set of ordering relationships (defining a quantity q), then we may (at least theoretically) arrange those things connected by these relationships in a linear order. For example, we may arrange them so that if A and B are any two things possessing the quantity q, (i.e. which are connected by these relationships), then A occurs higher

1 I am now less sure that the existence of a set of ordering relationships is a sufficient co.-idition for the existence of a quantity. But it would take us too far afield to discuss what further restrictions are necessary.



ON THE NATURE OF DIMENSIONS 359

than, at the same level as, or lower than B, according as A is greater than, equal to, or less than B in q. I called this order 'the order of the quantity q'. Thirdly, I argued that it often happens that the same order may be generated by two or more logically independent sets of ordering relationships; and that where it appears that the same order is generated in all physically possible circumstances by the different relationships, then we regard them as alternative ways of ordering things in respect of the same quantity. I concluded, therefore, that it is the order which provides us with our criteria for the identity of quantities, and not the ordering relationships.

In view of these conclusions, I now wish to suggest that we have a scale for the measurement of any given quantity q, if and only if we have a determinative rule for assigning numerals to things which possess q, such that if these things are arranged in the order of the numerals assigned to them in accordance with this rule, they are also arranged in the order of q.

It may be objected that this is unduly restrictive, since it does not allow for the possibility of nominal scales ([2] and [3]) for the measurement of any given quantity q. But it is doubtful whether there is anything which would be described as a nominal scale for the measurement of a quantity. Nominal scales have indeed been described. But these scales cannot be said to be scales for the measurement of anything except, perhaps, sameness and difference. I cannot, for example, be said to measure anything in any respect if I merely assign numerals to the various players in a football team. Yet, in many discussions, ([2] and [3]) this example serves as the paradigm of a nominal scale.

It may also be objected that my criterion for saying that we have a scale for the measurement of any given quantity q is not sufficiently restrictive since, for example, it allows for the possibility of scales of mass and length which are dissimilar (i.e. not related by similarity transformations2) to our ordinary scales for the measurement of these quantities. But it seems to me that this is something which we should allow for. There is already a considerable literature on time scales which are dissimilar to our ordinary time scales.3 And no-one, to my knowledge, has questioned the legitimacy of calling such scales 'time scales'. Why, then, should we not adopt a similar attitude to, say, length scales? Why should we not say that we have a length scale if and only if we have a determinative rule for assigning numerals to things, such that if these things are arranged in the order of the numerals thus assigned, they are also arranged in the order of length?

I can think of no good reason why we should not speak in this way, and one very good reason why we should. It is that by adopting my criterion for saying that we have a scale for the measurement of any given quantity, we are in a position to adopt a uniform criterion for the identity of quantities. For we may now say, quite generally, that two sets of ordering relationships are ordering relationships for the same quantity if and only if they would always generate the same order among the same things. Under any other conditions, we would have to distinguish different kinds of quantities and adopt different criteria for their identity and difference. Thus, we would have to say that a given scale is not a scale of length (but, presumably, a scale of something else) if it is dissimilar to our metre scale, but that a given scale may yet be a scale

2 For a discussion of the concept of a scale transformation, and for an explanation of the term 'similarity transformation', see [7] and [8].

3 See, for example, the issue of the British Journal for the Philosophy of Science devoted to 'The Age of the Universe', Vol. V, No. 19, Nov., 1954.



360 BRIAN ELLIS

of time-interval even though it is dissimilar to our second scale. And, hence, we would have to make out a case for saying that time-interval is a different kind of quantity from length.

Finally, let us consider the criterion for the identity of scales. It is clear that the same quantity may often be measured on the same scale by a variety of different procedures. For example, we may measure the mass of a given object in grams in any of the following ways:

(1) By means of a beam balance, using objects whose mass in grams is already known;

(2) by placing the object in the vicinity of another object whose mass in grams is known, and measuring the mutually induced accelerations; or

(3) by allowing the object to collide with another object whose mass in grams is known, and measuring the respective velocity changes.

These various ways are indeed theoretically linked. Nevertheless, they are logically independent, and the question therefore arises; why should they be regarded as alternative procedures for determining the mass of any given object in grams?

At least part of the answer lies in the fact that we have good reasons (both theoretical and experimental) for believing that where they are applicable, these alternative procedures would always lead (within the limits of experimental error) to the same numerical assignments to the same things under the same conditions. For, if we had any good reason for believing otherwise, these procedures would not be acceptable as alternative procedures for determining the mass of any given object in grams. Hence, sameness of numerical assignment provides at least a necessary condition for the identity of scales.

In my view, sameness of numerical assignment is also a sufficient condition for the identity of scales. I cannot produce a conclusive argument for this. I can only say that it is not clearly inconsistent with our ordinary usage of 'same scale.' It is true that some people would wish to say that Kelvin's Thermodynamic Scale of Temperature is not the same scale as the Perfect Gas Scale, even though we have good reason to believe that the criterion of sameness of numerical assignment is satisfied. But there are as many people who would wish to say that those scales are identical, and who would make no distinction between different kinds of degree Absolute. It therefore seems to me that we are free to say what we like. And for the sake of uniformity of usage, I shall say that two procedures are alternative procedures for measuring on the same scale if and only if, wherever both are applicable, they would always lead to the same numerical assignments to the same things under the same conditions.

2. Fundamental Scales. These scales are set up by what we may call fundamental measuring operations. These operations have the following characteristics. Let p be any quantity for which there exists a fundamental measuring operation 0. Such a quantity, we shall say, is fundamentally measurable. Let '> p', p' and '< p' signify 'is greater in p than', 'is equal in p to' and 'is less in p than', respectively.4 Then Q must be an operation performable on any two systems S1 and T1 possessing p such that the resulting system O(S1, T,) also possesses p, and is such that:

4For a discussion of the formal properties of these relationships, and of the conditions under which things may be said to be connected by such relationships, see [4].




(a) O(S1, T1) = p O(S2, T2) = p O(T2, S2) where S1 = 2,S2 and T1 ==J2

(b) O(S1, T1) > ,S1 for all S1, T1 (c) 0 [O(S1, T1), V1] = 2 0 [S1, O(T1, V1)] for all Sl, T1, V1. (d) lf Sl = WS2 S3 p ..... and T is any other system possessing p such that

S1 <P, T then there is a number N such that for all n > N, if the systems

51 C S S3 .... Sn are combined successively by the operation 0, then the

composite system thus formed O(S1, S2, S31 ..... S) > J.5

Now it is not difficult to show that our ordinary fundamental measuring operations for mass, length, time-interval (and several other quantities) satisfy these require- ments. But, in general, there may be more than one fundamental measuring operation for any given quantity p. In the case of length, for example, the right-angled addition operation illustrated in Fig. 1 has as much right to be described as a fundamental measuring operation for length as the ordinary linear 'stepping off' procedure.6 Thus in setting up a scale of length, there arises, firstly, the problem of choice of fundamental measuring operation. For although we may begin with the same initial standard, we may yet set up very different scales for the measurement of the quantity concerned, according to which fundamental measuring operation we select.

But even if there were no such problem of choice, our scale of p would still not be determined by our choice of initial standard. For in setting up a scale we must adopt a determinative rule for making numerical assignments, and the mere choice of an initial standard still leaves open a great many possibilities. To show this, consider the following mathematical problem. Let it be required to find a function f such that:

(a') f(x,y) = f(y,x) (b') f(x,y) > x

(c') f(f(x,y),z) = f(x,f(y,z))

(d') If f (2) (x,x) = Dff(X,X)

f 31 (x.x.x) = Dff(x,f 12) (x,x)) (^

f(n) (x,x,x,...) = Dff(x,f(n?1) (x,x,x .... ) and f(2) (a,a) < b then there is an N such that for all n > N

f (n) (a,a, a.... ) > b.

(Note that these conditions onf are formally analogous to those placed on fundamental measuringoperations.) It is readily seen that this problem has no unique solution. In fact, there is a wide variety of solutions. FIG. 1

The following three are of some interest:

f(x,y)z rx+y, (x>O,y>O) (1)

f(x,y) =x.y,(x> 1,y> 1) (2) f(x,y) (I V7I + VI A)2, (X > 0, y > 0) (3)

5 Campbell states the first three of these conditions correctly, but this condition (d) is also necessary to ensure that any given system possessing p can be matched in this respect. See

([3], Ch. 2.) 6 For a full discussion of this example, and for a justification of this statement, see [4].



362 BRIAN ELLIS

Now, because this problem has no unique solution, we cannot determine a scale of p merely by stipulating that a certain object S1 possessing p is to be assigned the numeral 'n', that the numerals 'a' and 'b' to be assigned to any two objects Si and S1 possessing p are to be such that:

a S b according as Si ; X Sj, and that if the objects Si and Sj possessing p are assigned the numerals 'a' and 'b' respectively, then O(Si,Sj) is to be assigned the numeral f(a,b). In other words, it is possible to interpret the physical operation 0 used in fundamental measurement by any of a number of different arithmetical operations in order to obtain a scale of p. If solution (1) is chosen, then we get a normal additive scale (of the kind ordinarily used in the measurement of, say, mass, length or time-interval). But if we select solution (2), we get what we might call a multiplicative scale. For, on the resulting scale, it is the arithmetical operation of multi- plication (rather than addition) which corresponds to the physical operation 0.

It is instructive to see what a multiplicative scale of, say, length would look like. Suppose we select some object S1 to act as an initial standard and assign to it the numeral '2'. Then, according to the multiplicative interpretation of 0, a composite object O(S1,S2), where S1 = S2, must be assigned the numeral '22'. Likewise an object O(10(S2,S3)), where S, -S2 = AS3 (which in view of condition (c) we may write simply as 0(3) (S1,S2,S3)), must be assigned the numeral '23'. And, generally, an object Q(n) (S1,S2,S3,... S,) must be assigned the numeral '2n'. Now this would yield a very different kind of scale from any we are used to. But obviously it is a legitimate scale. And numerical assignments on such a scale would clearly contain as much information as those on an additive one. Any law which could be expressed with respect to additive scales of length could just as well be expressed with respect to multiplicative scales.

The third solution (3) is interesting for another reason. If this interpretation for 0 were adopted, then we should obtain a scale which is, as it were, the square of an ordinary additive scale. For suppose, as before, we select some object S, as initial standard and assign to it the numeral l'1. Then, according to (3), an object O(S1, S2)

D

FIG\ l \ ~I /

\I

FIG. 2 0 A




must be assigned the numeral '22a, an object O(3) (S1, S2, S3) the numeral '32', and generally, an object On (S., S2, S3 ...... Sn) must be assigned the numeral n2. This solution is particularly interesting because, at least in the case of length (and also force), there exists an operation Q1, different from the usual fundamental measuring operation 0 for this quantity which would make the 'square' scale additive. This is the right-angled addition operation to which I have already referred. (The reader will readily verify that if the successive diagonals OA, OB, OC,... in Fig. 2 are assigned the numerals Cl', '2', C3',..., then the successive lines OA1, OB1, OC' ... in Fig. 3 must be assigned the numerals C12', '22', '32' ..... , assuming that S1 = A AS3

0 A B 'C

FIG. 3

We come then to the conclusion that in setting up a fundamental scale there are a number of degrees of freedom. A fundamental scale for the measurement of a given quantity cannot be specified simply by the nomination of an initial standard. In addition, we must specify the fundamental measuring operation, and choose some principle of correlating numerals with composite objects produced by means of this operation.

Now it is clear, I think, that if different scales for the measurement of the same quantity, based on the same initial standard, but utilizing different fundamental measuring operations, or different principles of correlation, were in practical use, then different unit names would be required. Consequently, in any general theory of measurement, unit names in fundamental measurement cannot be regarded as the names of the initial standards. Rather unit names must be regarded as the names of scales. And hence the criterion for the identity of fundamental units must be simply the criterion for the identity of fundamental scales. And that criterion I take to be simply sameness of numerical assignment. If two procedures always lead to the same numerical assignments to the same things under the same conditions, then they are different procedures for measuring on the same scale. Or, alternatively, we may say that they are different procedures for measuring in the same units.

3. Associative Scales. Associative scales are set up in a rather different way from fundamental scales. Let p be any quantity for which there are direct criteria7 for quantitative equality and inequality. Let us, as before, signify these relationships of equality and inequality in p by C> p,' C p' and C< p'. Let us suppose, further, that there is associated with p, another quantity q, which is independently (e.g. fundamentally) measurable, and which is such that for any two systems A and B of a certain kind, maintained under certain precisely specified conditions, A SP B according as A q B. Then, we may take the measure of q (on some selected scale) for systems of this kind maintained under these conditions, as the measure of p, and thus define a scale of p. (The quantities p and q would not, of course, be the same quantity, unless the orders of p and q always coincided for all kinds of systems under all physically possible conditions.) A scale set up in this way will here be described as an associative scale.

7 i.e. criteria which do not depend upon previous measurement.



364 BRIAN ELLIS

Our ordinary scales of temperature8 are examples of associative scales. In each case they are set up by taking some thermometric property (e.g. volume, electrical resistance, thermo-e.m.f., efficiency of a perfectly reversible heat engine) which under certain circumstances varies with temperature. The measure of this property (on some selected scale) under precisely specified conditions then defines our temperature scale.

It should be noted that to make associative measurement possible we only need one associated and independently measurable quantity. If, for example, electrical resistance were the only known thermometric property, then a scale of temperature could be set up. It is doubtful, of course, whether we would ever form a concept of temperature under these conditions. We should probably prefer to relate the variation of the electrical resistance of an object directly with the circumstances in which it is placed.9 But nevertheless it would be logically possible to form a concept of temperature and set up a scale for measuring it if only one such thermometric property were known.

Now it is obvious that in setting up associative scales there are two areas of choice. Firstly, we may usually choose any of a number of different associated quantities and conditions. And, secondly, we have a choice of scale for the measurement of any given associated quantity (or of principle correlating the measure of this associated quantity on any given scale with the quantity we are dealing with). Hence, as with fundamental scales, associative scales may be of many different kinds, and the unit names in associative measurement must be taken to have precisely the same significance, i.e., they must be taken to be the names of scales.

4. Derivative Scales. Our discussion of derivative scales must be prefaced by a discussion of numerical laws. For derivative measurement is characteristically the evaluation of constants in such laws. Let us suppose that under certain conditions C1 a physical system As of specifications S possesses the quantities p, q, r..., where these quantities are independently measurable (e.g. on fundamental or associative scales). Let these quantities be measured on scales X, Y, Z, ... respectively, and let x1, yl, z1, ... be the respective results of measurements in X, Y, Z, ... made on A s under the conditions C1. Now let us suppose that the conditions C1 may be changed successively to the conditions C2, C3, C4, ... in such a way that p, q, r, ... are seen to vary, while the specifications S of A, remain unchanged. Let measurements of p, q, r, ... in X, Y, Z, ... be made after each change, and the results of these measurements be x2, x3, x4, *... Y2 Y3, Y4, .... z2, z3, z4, ..., respectively. Then, if we can find some simple function f such that:

f(xi,Yi, Zi, ..*) = (1)

holds approximately'0 (where k, is a constant whose value may or may not depend

8 For a fuller discussion of the logic of temperature measurement see [5]. But the reader should note that when writing this paper I was not clear about the distinction I now wish to draw between associative and derivative measurement.

I For example, we recognize the existence of thermal equilibrium by the fact that after sufficiently long isolation from heat sources and sinks the thermometric properties of objects cease to vary. But if electrical resistance were the only known thermometric property, we should probably prefer to say simply that the electrical resistance of an object ceases to vary under these conditions.

10 The degree of simplicity and approximation required are matters for the scientific profession to decide in the particular case.




on S), we may say that we have discovered (empirically) a numerical law relating the quantities p, q, r, ... possessed by A,.

If this law is found to be valid for all systems of a class A, differing only in their specifications S, and the value of k, is found only to depend upon the choice of specifications S, then k, may be described as a system-dependent constant, and it may be taken to be the measure of some quantity possessed by the system A, (and all systems of the class A) on a derivative scale dependent on the fundamental and associative scales X, Y, Z, ... If otherwise, the value of k. is found to be independent of S, then k. may be described as a universal constant, and the subscript may be omitted.

To illustrate this, let A be any gas sample in a container of variable volume. Let

A. be any sample of hydrogen of mass m grams maintained at temperature T?A in such a container. Now let the pressure of the surrounding atmosphere be varied and, after each successive change, let measurements of the pressure and volume of the hydrogen sample be taken. Then, if these measurements were made on scales similar to our ordinary pressure and volume scales, we should find that p.v. k. In this case, ks would be found to be a system-dependent constant. For it would be found that as the specifications of A,, are varied, a relation of the form p.v.-k, still held, but that the value of ks depends upon the particular gas, the mass of the sample, and its Absolute temperature. If, however, k8 were found to be independent of these specifications, then we should say that k, is a universal constant.

Now there are several ways in which the law (1) may be expressed. Firstly, we may choose to express it (as we have done), with respect to the particular scales X, Y, Z, ... And, in that case, these particular scales must be mentioned in the statement of the law.11 For, generally, the value of the constant (and even the form of the law) will be seen to depend upon the initial choices of scales for p, q, r, ... But this is not the only, or even the most usual way of expressing our numerical laws. More often, especially in mechanics, we choose to express our laws in a form which is relatively independent of the initial choices of scales X, Y, Z, ...

To see how this is possible, I must first say something about scale transformations. Let X and X' be any two scales for the measurement of the same quantity, and let x and x' be the results of any two measurements (in X and X' respectively) made on the same particular under the same conditions. Then generally

x= g(x)

where 'g' is a strictly monotonic increasing function. Let us call g the transformation function from X to X'.

Now, in view of our discussion of fundamental and associative scales, it should be clear that no a priori restrictions can be placed on the form which g may take. If the choice of initial standard were the only arbitrary element in fundamental measurement, this would not be the case. The transformation functions for different fundamental scales would have to be of the form:

x = mx

That is, to borrow a term from Stevens, the transformation would have to be a

11 Thus in electrostatics and electromagnetics, laws are usually expressed with respect to particular scales (units), and the expressions '(e.s.u.)' and '(e.m.u.)' often written after such laws indicate which particular scales have been chosen.



366 BRIAN ELLIS

'similarity' transformation. But such is not the case and, hence, in a general theory of measurement, we must suppose that other transformations are possible.

It is a fact, however, that the various fundamental scales in common use (at least in mechanics) for the measurement of any given quantity are all similar to one another,'2 i.e., they are all connected by similarity transformations. And this fact is important for two reasons. Firstly, it has enabled scientists to express the laws of mechanics without apparent reference to the scales on which the mechanical quantities are measured. And, secondly, it has concealed an important convention governing the expression of our laws-a convention which has important consequences for the theory of dimensions.

Now, in the light of this, let us see how the numerical law (1) may be otherwise expressed. Let [X], [Y], [Z] ... denote the classes of scales similar to (i.e. related by similarity transformations to) X, Y, Z, ... respectively. Let X', Y', Z' ... be any other scales of the classes [X], [Y], [Z], ... such that the transformation or conversion factors from X to X', Y to Y', Z to Z', ... are m, n, o ... respectively. Then, if these scales X', Y', Z' ... had been chosen initially, the empirically discovered relationship would have been:

f ( x ,

,

' X = ks (2) m n o i

wheref is the same function, and k, is the same constant as in equation (1). If, therefore, we are prepared to introduce as many scale-dependent constants m, n, o, ...; into our equation as there are independent variables, we may always express the law in a form which is valid for all scales of the classes [X], [Y], [Z], ..., viz., in the form (2).

Fortunately, however, it is not often necessary to do this. (There would be little advantage in it if it were.) For often the various scale-dependent constants m, n, o, ... can be brought together into a single scale-dependent constant which depends in a complex way on the choice of scales within the classes [X], [Y], [Z], ... And in that case the law may simply be expressed by the formula:

f(x,y,z,...) Cs (3)

where cs is the only scale-dependent constant which appears. In mechanics, at any rate, this represents our usual practice. It seems that in this

field we have adopted the convention of expressing our laws with respect to classes of similar scales. Other conventions could certainly have been adopted but, for whatever reason, they have not been. We could, for example, have chosen to express our laws with respect to classes of scales linear with respect to each other. In general, more scale-dependent constants would then have been required to express the same law. But, at the same time, the expression would have given us information concerning a wider field of scale transformations. Again, we could have chosen to express our law with respect to classes of scales which are, as it were, powers of each other, i.e., which are related by scale transformations of the form:

x' =X m, y n = ,z' -= zo

12 Not quite. Angle is measured on a variety of dissimilar scales. (The tangent, sine, cosine, and radian scales of angle are mutually dissimilar.) But then our laws involving angle are always stated with reference to particular scales of angle.




And, in that case, the law (1) would have to have been written:

(xm., yn, ? k.,

where m, n, o ... are again scale-dependent constants. But it seems that such alternative modes of expression are never used. Our practice is always to express our laws, either with respect to particular scales (as in electromagnetism and heat theory), or else with respect to classes of similar scales (as in mechanics).

Let us then examine some of the consequences of this convention governing the expression of mechanical laws. First, we should note that an empirical law cannot always be expressed in the form (3). This is possible if and only if:

f(x, y,z , ...) = const xa * yP * lz . ..... 13

That is, if we adopt the convention of expressing our laws with respect to classes of similar scales, then the only possible numerical laws in which single scale dependent constants occur must be of the form:

Xa *yP * fl . .....z . Cs (4)

where c8 is in general both a system and a scale-dependent constant. Now, assuming that f(x, y, z, ...) is a function of the required kind, let us see how

the value of c8 must depend upon the choices of independent scales within classes [X], [Y], [Z] . ..... We have as our starting point that for the system A8, and the initial choices of scales X, Y, Z ..... , cs has the value ks. As before, let xi, yi, zi . ..... be results of measurements made under the conditions Ci on the system As in X, Y, Z, ..... respectively. Then we have immediately (from (4)):

xol *yz * z. ... == ks (5)

Let X', Y', Z' ...... be another set of scales of the classes [X], [Y], [Z] . .... respectively, and xi', yi', zi' ... be the results of the corresponding measurements made on these scales on A. under the conditions C?. Then we have (from (4)):

xt * .y * - *y -k't (6)

where ks' is the value of cs for the system As and the independent scales X', Y', Z', ..... Now, let, m, n, o, ... be the conversion factors from X to X', Y to Y', Z to Z', ..... respectively. Then:

x' mx.

yi fnY. (7) z = oz.

And substituting (7) in (6), we get

mnx * x. * nOyhozv *k

Whence, from (5)

k' (ma * nf * oy - )k8 (8)

13 For a proof of this proposition, see [6], pp. 33-35. The original proof appears in Bridgman's Dimenzsional Analysis, but Focken's presentation of it is clearer.



368 BRIAN ELLIS

That is, the two derivative scales dependent on X, Y, Z . ..... and X', Y', Z' ...... on which the numerical assignments ks and k,' have been made, are themselves similar scales. And denoting these derivative scales by K and K' respectively, we may say that the conversion factor from K to K' is a product of powers of the conversion factors from X to X', Y tot Y', Z to Z' ....... , the particular powers being determined by the form of the la-w on which these derivative scales are defined.

We can, however, do better than use simple unit names like "K" and "K"' to signify these derivative scales. Such names are in use (e.g., 'erg', 'dyne', 'poundal'), but we can give much more information concerning these derivative scales by using complex unit names. For example, if instead of K, we use the complex name ,Xx Y?l* Z ......', then by this means we can indicate:

(1) the particular independent scales upon which this derivative scale depends, (2) the manner of the dependence of the various similarly defined derivative scales

upon the initial choices of independent scales, and (3) the form of the law on which the derivative scale is defined.

All this information is, for example, contained in such complex unit names as 'gm.cm. sec-2' and 'cm. sec-l'.

It is true that by adopting this procedure we obtain unit names which may not be uniquely referring. That is, it is possible that we may have scales for the measurement of different quantities designated by one and the same complex unit name. The situation is, in this respect, reminiscent of chemistry. There we have adopted individuating names for the chemical elements, and certain rules for naming com- pounds formed of these elements. But in doing so, we obtain substance names which may or may not be uniquely referring. However, if molecular formulae are substance names, then in a similar sense the complex unit names of derivative measurement are the names of derivative scales. And hence we may conclude, quite generally, that unit names, whether simple or complex, whether of fundamental, associative or derivative measurement, are the names of scales.

Part II: DIMENSIONS

It has been shown that numerical laws may be expressed in a variety of different ways, leading to a variety of different kinds of derivative scales. But that, conventionally, we nearly always (especially in mechanics) choose to express our laws with respect to classes of similar scales. This has important consequences. For, if

f(x, y, Z.) Cs

is a numerical law, expressed in this way, where C, is the only scale dependent constant which occurs, then this law must be of the form:

xc * yf * zl . ...z.. CS

And, as we shall see, this proposition is of fundamental importance in the theory of dimensions.

I now wish to examine the nature of dimensions and to consider the significance of dimensional symbols and formulae. I shall argue that dimension names (symbols, formulae) must be regarded as the names of classes of similar scales, and I shall try




to show that dimensional analysis depends upon the convention, already mentioned, that our laws be expressed with respect to classes of similar scales.

1. The Concept of a Dimension. It is difficult to say how dimensions are usually regarded. No one seems to have any clear conception. Generally, I suppose, dimensions are tied to quantities. We speak simply of the dimensions of p, where "p" is a quantity name, and we usually suppose that dimension names (e.g. dimensional formulae) are somehow characteristic of quantities, i.e., that to each quantity there corresponds one and only one dimension or dimensional formula.

But this account is soon seen to be inadequate. For if we examine the way that dimension names are actually assigned, we see that the dimensional formula for a given quantity q is determined firstly by the choice of the numerical law on which our derivative scales for the measurement of q are to be defined and, secondly, on the choice of classes of similar independent scales with respect to which this law is to be expressed. To see the first of these points, consider the following simple case:

We know that weight and mass are independently measurable. Weight is easily measured on an associative scale using the extension of a spring as associated quantity, and mass is fundamentally measurable on a beam balance. Let w, be the result of any measurement on a scale W (which, for convenience, we shall suppose is similar to our dyne scale), and let m, be the result of any measurement, made on the same object under the same conditions (e.g., in the same place) on a scale M (which, we shall assume, is similar to our gram scale). Then clearly we could discover empirically the law:

mS 7US-8 (1)

where g, is a constant whose value depends on the place where the measurements are carried out. Accordingly, we may take g, to be the measure of some quantity q, characteristic of spatial location, on a scale M-1W. Expressing the law with respect to the classes of similar scales [M] and [W], we have:

m-1w = g (2)

where g is now a dimensional or scale-dependent constant whose dimensional formula is M-'W (according to the usual conventions). We should, therefore, say that q has the dimensions M-1W.

But it so happens that there is another law, independent of this one, on which similar derivative scales for ql4 may be defined. This is Galileo's Law of Free Fall, which, when expressed with respect to classes of scales [L] and [T], similar to our centimetre and second scales respectively, assumes the form:

It-2 = g (3)

And, according to this law, expressed in this way, q should have the dimensional formula LT-2. Consequently, the dimensional formula which we should assign to q depends upon the choice of the numerical law on which our derivative scales for the measurement of q are to be defined. (In fact we have chosen Galileo's Law of Free Fall and, accordingly, q is called "gravitational acceleration".)

But even when our choice of numerical law has been made, and the convention

14 We know that the same quantity is involved, since the same order among spatial locations is generated.



370 BRIAN ELLIS

that our laws be expressed with respect to classes of similar scales adopted, the dimensional formula for a given quantity q, measured on a derivative scale defined in terms of this law, is still not determined. For, as we have seen, the same law may be expressed with respect to different classes of similar scales. And the dimensional formula for q will depend upon these choices of classes of similar scales.

Consider again Galileo's Law of Free Fall. Expressed with respect to the classes of scales [L] and [T] similar to our centimetre and second scales, respectively, it assumes the form (3). But length could also be measured (even fundamentally) on a scale which is, as it were, the square of our centimetre scale, i.e., on a scale L' which is related to our ordinary centimetre scale L by a transformation function of the form:

1' _ 12

And, if we were to choose the classes of similar scales [L'] and [T] as our basic reference classes, Galileo's Law of Free Fall would assume the form:

1't-4 - g' (4)

So that, under these conditions, it seems, q must be assigned the dimensional formula L'T-4.

It is evident, therefore, that dimension names are not characteristic of quantities. The same quantity q, measured derivatively, may be assigned any of a number of different dimensional formulae, according to the conventions which we adopt in defining our derivative scales for q. Moreover, even those quantities which are measured fundamentally cannot be assigned a unique dimensional symbol. The dimensional symbol L' obviously cannot have the same significance as the symbol L, yet they are both dimensional symbols for length. There are, therefore, at least two dimensions of length, L and L', and indeed, by the same reasoning, there are as many dimensions of length as there are mutually dissimilar scales of length. No satisfactory account of dimensions as characteristic of quantities can, therefore, be given.

How then should dimension names be regarded? In my view the answer is simply that dimension names must be regarded as the names of classes of similar scales. In the following subsections I shall examine the consequences of this assumption, and show how the theory of dimensional analysis, as classically presented by P. W. Bridgman [1] may be raised on this foundation.

2. Classes of Similar Scales. Let us begin by examining some of the properties of classes of similar scales. Let X1 and X2 be any two scales belonging to the class of similar scales [X]. Let xl and x2 be the results of measurements made on some system S under the same conditions on the scales X1 and X2 respectively. Then,

X2 = mXI

where m is the conversion factor from X1 to X2. Now let x1' and x2' be the results of measurements on X1 and X2 respectively, made on some other system S' under the same conditions. Then also, x- mx1

x2=m

Hence X 2 xl

X2 x1

That is, relative magnitude is invariant within any given dimension.




Let Y be any scale dissimilar to X1 (for the measurement of the same quantity), and y and y' be the results of measurements on the systems S and S' respectively under the same conditions. Then

and g(x1)

y'g(xs)

where g is the transformation function from X1 to Y. But if

Y' XI Y xi

then

g(xj) = X i.e. xj) _ g(x1) == const. g(xl) X1 X; X

That is, g must be a similarity transformation (which is contrary to the original supposition). Hence relative magnitude is only invariant within any given dimension.

I conclude, therefore, that sameness of relative magnitude is both a necessary and a sufficient condition for sameness of dimension. Two scales X1 and X2 for the measurement of the same quantity q belong to the same dimension if and only if relative measures of q in X1 are the same as relative measures of q in X2, the measurements being made on the same pairs of systems under the same conditions. This result is quite general. It applies whether the scales concerned are fundamental, associative or derivative.

In developing his theory of dimensions, Bridgman took, as his starting point, the requirement of the absolute significance of relative magnitude. Concerning fundamental scales he remarked:

... the ratio of the [measures] of any two particular objects has an absolute significance independent of the size of the units." ([1], p. 19)

And concerning derivative scales he said:

"Now there is a certain definite restriction on the rules of operation which we are at liberty to set up in defining secondary quantities. We make the same requirement that we did for primary quantities, namely, that the ratio of the numbers measuring any two concrete examples of a secondary quantity shall be independent of the size of the fundamental units used in making the required primary measurements." ([1], P. 19)

The requirement, therefore, amounts to the demand that our scales for the measurement of any given quantity, whether they be fundamental or derivative scales, should always belong to the same dimension.

This requirement is, of course, necessary if we are to be able to speak unambiguously of the dimensions of any given quantity. But why should that be a consideration ? What is it which constrains our liberty in setting up derivative (or fundamental) scales ? Is it the force of some argument? If so, no argument is given. Is it the force of logic ? No, for mutually dissimilar scales for the measurement of any given quantity can always be set up. Is it Nature herself which acts as the constraining force ? But how, then, can Nature constrain us to adopt one mode of arithmetical representation rather than another ?



372 BRIAN ELLIS

The answer is, of course, that we are not so constrained. We are at liberty to set up any scales for the measurement of any quantity, and if our concept of a dimension is not sufficiently flexible to cope with unorthodox scales, it is our concept which is at fault. I believe that the conception of a dimension as a class of similar scales has the required flexibility, and that by regarding dimensions in this way we are able to see more clearly the nature and scope of dimensional analysis. It will be seen that the invariance of relative magnitude with iso-dimensional changes of scale can serve every purpose of Bridgman's apparently metaphysical assumption of the absolute significance of relative magnitude. The conception, therefore, loses nothing in power, but it gains a great deal in both clarity and flexibility.

3. Similar Scale Systems. We have already noted that similarly defined derivative scales (i.e., scales defined on the same numerical law and referred to similar independent scales) are themselves similar to one another. This fact is of some importance. In the first place, since most of our derivative scales for the measurement of any given quantity are in fact similarly defined, it follows that Bridgman's requirement of the invariance of relative magnitude is usually satisfied for quantities measured derivatively. Hence we have no need for any special restrictions concerning derivative scales (such as Bridgman suggests). And, in the second place, it means that classes of similarly defined derivative scales may function as basic reference classes for the expression of numerical laws. For they are themselves classes of similar scales. Thus, the scale classes [XI, [Y], [Z], ..., to which the numerical law

f(x,y,z,...) = Co

is referred, may, indifferently, be classes of similar fundamental or associative scales, or classes of similarly defined derivative scales.

This means that, provided sufficiently many numerical laws are known, the number of independent scales required to set up a suitable scale system for the measurement of the various quantities relevant to a given area of scientific discourse may be considerably reduced. We need only specify a certain number of basic independent scales for the measurement of a limited number of quantities, define other scales on the basis of laws referred to these scales, and define yet other scales in terms of these.

This is, in fact, the usual procedure in mechanics. We select certain independent scales (usually of mass, length and time interval), define derivative scales of velocity, angle, acceleration, angular velocity and many other quantities on the basis of laws connecting these three quantities and expressed with reference to the chosen independent scales, and then use the system of dependent and independent scales so obtained to define derivative scales for yet other quantities (e.g., force, momentum, moment of inertia, energy, pressure, and density). We thus obtain a complete scale- system for the field of mechanics. And, the definitions being agreed upon, we need only specify the initially chosen scales of mass, length and time interval in order to denote this system. Thus we have the M.K.S., the F.P.S. and the C.G.S. systems of scales for mechanical measurement.

These systems may be described as similar scale systems. They have similar independent scales for the measurement of the same quantities, and their derivative scales are similarly defined. Consequently, the derivative scales for the measurement of any given quantity within similar scale systems are also similar to one another. And, hence, if it is our convention always to express our numerical laws with respect to the classes of similar scales belonging to a certain class of similar scale systems, then to each




quantity there will correspond a unique dimension (i.e., class of similar scales). So that, under these circumstances, we may speak unambiguously of the dimension of any given quantity q (relevant to the given field).

It is important to realize that this depends upon a convention. We could, equally well, choose to express our numeral laws with respect to a different class of similar scale systems, e.g. by taking independent scales dissimilar to the usual ones. And, in that case, a different set of dimensions would be associated with the various quantities relevant to the given field. We could choose a more highly centred scale system. Mass, for example, could be measured on a derivative scale dependent upon Newton's Law of Gravitation. And thus we could set up a scale system centred around independent scales for the measurement of length and time interval only. We could choose a less highly centred scale system. For other quantities besides mass, length and time-interval are independently measurable (force, for example). Or, finally, we could choose to express our numerical laws with respect to a wider class of scale systems, e.g., the class of scale systems linear with respect to our ordinary ones. We can only speak unambiguously of the dimension of any given quantity q just because there is no such diversity in our practice. Just why we should adopt the particular conventions which we do is not clear. But, if, for whatever reason, we assume that the convention has been adopted that we should express our laws with respect to some particular class of similar scale systems, then the whole theory of dimensions which B3ridgman develops may be built upon an alternative foundation. For it follows that the only permissible changes of scale (consistent with this convention) are changes within some definite dimension. And since, as we have seen, relative magnitude is invariant with iso-dimensional changes of scale, there is no need to assume that relative magnitude has an absolute significance. Invariance of relative magnitude (under this convention) is all that is needed for Bridgman to carry through his analysis.

It would be tedious and pointless to show this in detail. My analysis would be parallel to Bridgman's in every respect. So, for the remainder of this paper, I shall simply assume the various results which Bridgman obtained. In particular, I shall assume the "algebra" of dimensional formulae as fully established.

4, Dimension Names. Just as we need names for particular scales (unit names) we need names for particular dimensions. And the obvious naming procedure seems to be to follow that uised for the naming of scales, i.e., to select simple dimension names for classes of similar independent scales, and to use complex dimension names for classes of similarly defined derivative scales-thus showing the manner of their dependence. Now in fact we have no such names, or, at least, none conceived in this way. But we do use simple dimensional symbols in association with those quantities usually measured on independent scales, and complex of dimensional symbols (or dimensional formulae) for those quantities measured derivatively. And it is my view that these can (and should) be regarded as the names of the classes of similar (independent or dependent) scales on which these quantities are usually measured.

Regarding dimension names in this way, then, let us see what information the dimensional formula [X]%[Y]I3[Zvv..... contains. In the first place, it tells us that within the class of similar scale systems to which we have chosen to refer our numerical laws, the quantity q with which the above formula is associated is measured on a certain class of similarly defined derivative scales. In the second place, it tells us how the derivative scales of this class depend upon the choices of independent scales



374 BRIAN ELLIS

within the classes [X], [Y], [Z] ...... Thus, if the independent scales XX, Y1, Z1. are chosen, and appropriate measurements carried out, q would be measured in the units X1 . Y - Z]..... And if now we wished to calculate the measure of q which would have been obtained had the independent scales X2, Y2, Z2 . ..... been chosen, we know that the first result must be multiplied by the factor ma . - -Y..... where m, n, 0,..... are the conversion factors from X1 to X2, Y1 to Y2, Z1 to Z2 .... respectively. And, in the third place, provided that the dimensional formula is not degenerate,15 it tells us the form of the numerical law (with respect to the given classes of similar scales) on which our derivative scales for the measurement of q are defined. Thus, our scales for q must be defined on a numerical law of the form:

xc.yi. z).. = k

It can be seen, then, that complex dimension names are very informative. They contain a great deal of information, both empirical and formal. And, as we shall see, this fact is very important for the understanding of dimensional analysis.

5+ "Dimensionless" Scales. It sometimes appears that a derivative scale is, nevertheless, an independent scale within our scale system. Our radian scale of angle, for example, is such a scale. It is defined on the numerical law (expressed with respect to our ordinary dimensions of length):

sr- = 0

where s is the length of the arc, and r is the radius of any segment of anly given class of similar circular segments. It is thus a derivative scale. But, on the other hand, since s and r are both measures of length, they are always determined on the same scale. Hence, the complex unit name for 0 should be of the form XX-1,, and the corresponding dimensional formula [X][X]-1. But this means that our measure of angle is independent of our choice of length scale (provided that it is a choice from within the class [X]). And hence our Radian scale is an independent scale.

Now traditionally we say that angle is a "dimensionless" quantity. The term is a misnomer, for there certainly are dimensions of scales of angle. But, nevertheless, it has its point. For, conventionally, we express our laws with respect to particular scales of angle (radian, degree, sine, cosine, etc ...... ), and not, as we might, with respect to a class of similar angle scales. And hence no dimension of angle is involved in the expression of our numerical laws.

The case could well be otherwise. Let us consider just one example. In a wire of circular cross-section of radius r and of length 1, the couple or torque T required to produce a twist 0 (in radians) is given by the relation:

T - 2 nr4f-10 (1)

where n is the modulus of rigidity of the material. (The law is, of course, expressed with respect to the usual dimensions of length and torque. But note that 0, unlike the other variables, is specifically stated to be in radians.) Now the modulus of rigidity (or shear modulus) is defined by another law involving angle. If a homogeneous cube of material is subjected to a uniform shearing stress S parallel to one of the faces

15 In the way, for example, that our ordinary dimensional formula for the rigidity modulus is degenerate. This example will be discussed more fully presently.




of the cube, then the cube will be transformed into a parallelepiped, with the faces originally parallel to S now being parallelograms. And if b (in radians) is the angular displacement of the sides resulting from this transformation then

S = no (2)

where S is determined on any stress scale of the usual dimension. Accordingly, on the usual analysis, n has the dimensional formula [M][L]-1[T]2.

But clearly the law (2) could be expressed with respect to the class [A] of angle scales similar to our radian scale. And, in that case, the value of the constant (call it n' instead of n) would obviously depend upon our choice of scale from the class [A]. In fact n' must be assigned the dimensional formula [M][L]-1[T]-2[A]-1. And now it can be seen that the law (1) expressed in the form

vT r4 T == 2ni -

Of (3) 2 1

must hold for all scales of angle similar to our radian scale. The difference may seem very slight. In form, the law appears identical. But in

fact there is a great deal of difference. For our dimensional formula for the rigidity modulus now contains information about the form of the law on which the rigidity modulus is defined. And, armed with this extra information, dimensional analysis can now be applied to tell us in advance something about the way in which 0 must enter into the law of torsion.16 It appears, therefore, that there may be some advantage in adopting this second mode of expression, i.e., in choosing to express our numerical laws involving angle with respect to classes of similar angle scales. To call angle a "dimensionless" quantity may, therefore, be highly misleading.

Similar considerations apply, I believe, to other so-called "dimensionless" quantities (and constants). The derivative scales on which they are measured turn out to be particular independent scales. Then, conventionally, we choose to express our numerical laws with respect to these particular scales. Thus, no dimensions of scales of these quantities are involved in the ordinary expression of these laws. Hence, we may say that the quantities concerned are dimensionless. But other conventions could certainly be adopted, and our laws could be expressed with respect to classes of similar scales for the measurement of these quantities. And then we should no longer be tempted to say that these quantities (or constants) were dimensionless.

6. Dimeinsiona! A alysis. Several hints have already been given as to the nature of dimensional analysis. It has been pointed out that dimensional formulae are not empiricially empty. They contain information concerning the forms of the basic laws on which our derivative scales are defined. It is not surprising, therefore, that by using this information we may be able to say something about the forms of other laws assumed to be derivable from these basic ones. Dimensional analysis is not, as it sometimes appears to be, a genuine a priori way of doing physics. It is, as Bridgman rightly remarks "an analysis of an analysis". ([1], p. 52) In the complete analysis of a physical problem the boundary conditions are stated, the fundamental laws applied, and the particular derivative law deduced. But in a dimensional analysis of this same problem, we merely assume that the particular law we want is derivable

16 This will be shown presently.



376 BRIAN ELLIS

from these fundamental laws, and utilise the information concerning the forms of these laws contained in our various dimensional formulae to make predictions concerning the form of the derivative law.

To illustrate this, consider the law of torsion discussed earlier. We want to know the relationship between the torque applied, the angle of twist, the length and radius of the wire, and the elastic constants of the material. A complete analysis would evidently require some complex mathematical manipulations. But it is evident that any given element of the wire is subject to a pure shear strain, and we may reasonably suppose that the law is derivable from the fundamental law relating shear stress and shear strain, viz.:

S = nO

where the stress S is determined on any stress scale of the usual dimension, 0 is the strain measured in radians, and n is the modulus of rigidity (or shear modulus) of the material. We may, therefore, assume that the required elastic constant is the modulus of rigidity, and that this is the only scale-dependent constant which appears in the law we are looking for. Other scale-dependent constants could, of course, appear. But then the law could not be derived from the law relating shear stress and shear strain, and dimensional analysis would be powerless to yield us any information at all.

Let us then set out the dimensional analysis in the usual way.

Quantity Symbol Dimension Torque T [F][L] Length of wire 1 [L] Radius of wire r [L] Angle of twist 0 Modulus of rigidity n [F][L]-2

(Here I have denoted the class of similar force scales by the term "[F]" rather than the usual "[M][L][T]-2". The reason is simply that the problem is a static one, and we have no reason to suppose that the laws of dynamics (on which our derivative force scales are defined) would feature in the detailed analysis.) Now since the derivative law must hold for all scales of the given dimensions, it must be of the form:

T = - - r- ny f(6)

and [F][L] - [L]x [L]l [F]v [L]-2v

so that y 1, and cx + P= 3.

The required law must, therefore, be of the form:

T = n la - r3 f(6), where cx + =3.

This result is, of course, in agreement with the empirical form:

T-7T * n * r4 .l-l . 0

But the information is not very complete. One reason for this is that r and I are always measured on the same scale, and hence dimensional analysis is unable to distinguish between them. But why have we been able to get no information concerning 6? After all, angle is one of the variables related in the fundamental law on which the rigidity




modulus is defined. The reason is simply that we have chosen to express this law with respect to a particular scale of angle, and hence the resulting dimensional formula for n tells us nothing whatever about the way in which the strain 6 enters into this law.

Had we chosen otherwise, dimensional analysis would have been more fruitful. Expressing the fundamental law with respect to the class [A] of scales of angle similar to our radian scale, we have:

S = !'- 6t

where n' not only depends upon our choice of stress scale, but also on our choice of angle scale. And the rigidity modulus} thus defined, has the dimensional formula [FJ[L]-2[A]-1. Now repeating the dimensional analysis, we have:

Quantity Symbol Dimension

Torque T [F][L] Length of wire I [L] Radius of wire r [L] Angle of twist 0' [A] Modulus of rigidity nt [F][L1]2[A]-'

And since T - c l- r* n'v O'c then

Hence, [F][L] - [L]OI[L]$[F]Y[L]-2V[A]X]Y[A]

Therefore, the law of torsion must be of the form:

T = clrn'O' where a + P = 3

And this result is considerably more informative. The example illustrates the nature of dimensional analysis. It shows how, in the

first place, dimensional analysis depends upon the convention that we should express our numerical laws with respect to classes of similar scales. Where this convention is not adopted no dimensional analysis is possible. Where it is only partially adopted, its power is considerably weakened. But where it is fully adopted, it may be a very powerful tool indeed. The example also shows how it is the information which we have put into our dimensional formulae which counts in the end. To apply dimensional analysis we must, of course, exercise our judgement as to which laws are likely to be used in a full and detailed analysis. And this demands a good deal of knowledge and experience of the field. But it is evident that the more information concerning these laws we can put into our dimensional formulae, the more we can get out of them later.

7. The Scope of Dimensional Analysis. If my account of the nature of dimensional analysis is correct, then it follows that dimensional analysis must be completely useless for the discovery of fundamental laws. And this is in fact the situation. To Galileo, looking for the form of the law of free fall, or to Newton looking for the form of the law of gravitation, a knowledge of the techniques of dimensional analysis would have been valueless.

Let us see what happens if we naively try to find the form of the law of gravitation using dimensional analysis. As before, we set down the quantities which we think may be relevant.



378 BRIAN ELLIS

Quantity Symbol Dimension Force of attraction f [M][L][T]-2 Mass of body A MA [M] Mass of body B MB [Ml Distance AB r [IL Relative velocity AB v [L][TJ-1

We suppose (without knowing why) that the law must be of the form:

f C c MAX MB ry Va

And hence,

[M][L][T]-2 _ [M]0[M11[L][LJ1[T]-

By symmetry we suppose o t , and then we get:

y =2

Thus:

f const. MAMv

The result is, of course, quite erroneous. And the reason is simply that we have no reason whatsoever for supposing that the law of gravitation is derivable from the laws of motion. On the contrary, the laws of motion are supposed to tell us how a body would move under any given law of force. Yet this is the implicit assumption behind the above abortive analysis.

We must conclude, therefore, that dimensional analysis can only yield information concerning derivative laws. It is completely powerless to tell us anything at all concerning fundamental laws.

8. Summary. The existence of a quantity entails and is entailed by the existence of an objective linear order, (the physical order). We have a scale for the measurement of a given quantity if and only if we have a determinative rule for making numerical assignments to things that possess the quantity such that the resulting numerical order coincides with the physical one. If two or more logically independent measuring procedures (i.e. determinative rules) always lead to the same numerical assignments to the same things under the same conditions, then we may regard them as logically independent ways of measuring on the same scale. Sameness of numerical assignment thus provides us with our criterion for sameness of scale.

The simple unit names of fundamental measurement, such as 'gram' and 'metre' cannot be regarded as the names of standard masses or measuring rods; nor of any properties possessed by these standards. For, radically different fundamental scales can be based upon the choice of the same standard (e.g. by using different fundamental measuring operations). But, we cannot measure the same quantity on the same scale in different units, or on different scales in the same units. Hence, a fundamental unit name cannot be defined just by choosing a standard; and fundamental unit names must be regarded not as the names of standards, but of scales.




For similar reasons, the unit names of associative and derivative measurement must also be regarded as the names of scales-although, the complex unit names of derivative measurement (cf. molecular formulae) may not be uniquely referring.

Derivative measurement involves the evaluation of 'single' constants in numerical laws. The simplest kind of numerical law refers to a particular system and relates the results of measurements made on that system. But, by introducing system-dependent constants into our symbolism, more general numerical laws, referring to a whole class of systems are expressible. Laws of either kind may be expressed with respect to particular scales of measurement. But, by introducing scale-dependent constants, we may always express these laws so that they are valid for all choices of scales within any previously defined classes of scales of measurement. Derivative measurement is, typically, the evaluation of 'single' constants that are both system- and scale-dependent.

Now, it is conventional that we usually express our laws with respect to particular classes of mutually similar scales, (i.e. scales related to each other by 'similarity' transformations). In mechanics, especially, most of our laws are expressed in this way. It is also conventional that we use only 'similarly defined' derivative scales for any quantity that we wish to measure derivatively. These two conventions have a number of important consequences. For, provided that they are always adhered to, a class of similar scales will be uniquely associated with each quantity, (since similarly defined derivative scales must be similar to each other). Also, classes of similar scales have precisely the same formal properties as dimensions. For example, relative magnitude is invariant within and only within classes of similar scales (c.f. Bridgman's postulate of 'the absolute significance of relative magnitude'). We have thus found a satisfactory denotation for dimension names.

Simple dimension names are the names of classes of similar fundamental or associative scales. Complex dimension names (dimensional formulae) are the names of classes of similarly defined derivative scales. This conception has a number of important consequences concerning the nature and scope of dimensional analysis.

a) Dimensional analysis is simply the process of extracting from our dimensional formulae the information that we have put into them regarding the forms of the basic numerical laws upon which our derivative scales are defined; and using this information to say something about the forms of any laws that may be derivable from these basic ones.

b) Dimensional analysis is useless for deriving information about any laws that we cannot suppose to be derivable from laws that are already known to us.

c) There are no such things as 'dimensionless' quantities. It is true, for example, that we express our laws involving angular displacement with respect to particular scales of angle (e.g. the Radian Scale). But the power of dimensional analysis can be increased by expressing all of our laws involving angle with respect to the class of scales similar to our Radian Scale and introducing a dimension name '[A]' to refer to this class.

d) The scope of dimensional analysis can be considerably enlarged by adopting uniformly throughout physics the two basic conventions for expressing numerical laws that we appear to have adopted (unwittingly) in mechanics.

6



380 BRIAN ELLIS

REFERENCES

[1] BRIDGMAN, P. W. Dimensional Analysis, Yale, 1931. [2] CAMPBELL, N. R. Foundations of Science, Dover, 1957. [3] CAMPBELL, N. R. An Account of the Principles of Measurement and Calculation, Longmans

Green, 1928. [4] ELLIS, B. "Some Fundamental Problems of Direct Measurement", Australasian Journal of

Philosophy, Vol. 38, No. 1, May, 1960. [5] ELLIS, B. "Some Fundamental Problems of Indirect Measurement", Australasian J7ournal

of Philosophy, Vol. 39, No. 1, May, 1961. [6] FOCKEN, C. M. Dimensional Methods and their Applications, London, 1953. [7] STEVENS, S. S. "On the Theory of Scales of Measurement", Science, Vol. 103, No. 2684, 1946. [8] STEVENS, S. S. "Measurement, Psychophysics and Utility", Measurement: Definitions and

Theories, edited by C. W. Churchman and P. Ratoosh, Wiley, 1959 (pp. 18-63).



Documents

On the Nature of Dimensions