No CrossRef data available.
Article contents
Principles of Theoretical Physics*
Published online by Cambridge University Press: 03 November 2016
Extract
Physical Magnitude is of a variety of “kinds”: not of only one or two kinds, nor, on the other hand, of an infinity—or an unlimited number—of kinds, but of a very considerable (though not perhaps quite definite) number of kinds. A catalogue of these—not set out for just that purpose—is to be found on pp. 3-7 of Kaye and Laby (1st Edn.).
- Type
- Research Article
- Information
- Copyright
- Copyright © Mathematical Association 1942
Footnotes
*
In 1939 the writer delivered a Presidential Address to the Mathematical Association of Victoria, on “The Theoretical Structure of Physics” (see Gazette, July, 1940, p. vi.). This was being written up for publication when the outbreak of war interrupted such work. This present paper, on somewhat different lines, was written recently—at the request of Professor T. H. Laby, F.R.S.—for inclusion in the typed Lecture Notes in Natural Philosophy of the University of Melbourne. Some mathematical points elaborated in the Address are referred to briefly in footnotes.
References
page no 87 note † “Physical and Chemical Constants” by Kaye, G. W. C. and Laby, T. H. (Longmans, Green & Co.)Google Scholar.
page no 88 note * See Gazette, July, 1938, xxii, 250, pp. 225-233.
page no 89 note * Because they necessarily involve the theory of the Real Numbers; but the practically important cases of p, q integral or fractional numbers are comparatively simple.
page no 89 note † xxii, 250, p. 229, §4.
page no 89 note ‡ See footnote to § 5, infra, on the fundamental theorems of Joint Proportion.
page no 90 note * The fundamental theoretical propositions of Proportion are, of course, geometrical and are bound up with the fundamental relation of “number” (as ratio) to both Length and Angle. They are: (1) The proportionality of the (variable) lengths OP, OQ, if PQ remains constant in direction; (2) the proportionality of arc-length to angle, for sectors of a given circle; (3) the proportionality of arc-length to radius (length), for sectors of given angle; (4) the proportionality of area to base-length, for triangles (or parallelograms) of given altitude.
page no 90 note † The basic case of Joint Proportion (in terms of the basic kinds of Magnitude, Length and Angle) is: (1) arc-length of circle-sector jointly proportional to angle and radius (length); expressible, conversely, in the form: (2) angle of sector jointly proportional, directly to arc-length, inversely to radius (length). Only secondary to these is: (3) area of rectangle jointly proportional to the two lengths of the sides. These propositions of Geometry appear to be basic, as the concrete foundation of the whole Proportion-structure of Theoretical Physics. The proposition (2) leads to Circular Measure of Angle; (3), to the “ derived unit ” of Area. (See § 7, infra, for the latter; and § 8, infra, with footnote, for the former.)
page no 90 note ‡ Tacit to the proof of this type of proposition is the basic assumption (see § 2, supra) that the kinds of magnitude in question are “continuous”, and that there is a quantity P corresponding to any given set of quantities of the other kinds in question; e.g. in the simple cases of the preceding footnote, P′ corresponding to X 2 and V 1—so that (P′: P 1) = (X 2: X 1) and (P 2: P′) = V 2: V 1, or to V 1: V 2; the theorem in question then resulting from (P 2: P 1) = (P 2: P′) × (P′: P 1)—for which cf. § 3, supra, with footnotes.
page no 91 note * Time, as basic, is in actual fact bound up with Angle, as geometrically basic. See § 4, supra and § 8 (i), infra, with footnote.
page no 92 note * Which has commonly been taken as the fundamental form for the type of Physical functional relation in question. The technique of §§ 5, 6, supra, seems to give true perspective to this theory.
page no 92 note † Here from, say, A 2: A 1 = (S 2: S 1)/(R 2: R 1), since R and S are of the same kind (both lengths), the general forms of §§ 7, 8 reduce to
(1) (A:A)=(S:L)/(R:L)=(S:R)
—the theorem of Circular Measure, the unit angle A corresponding to S = R( = L), but being in fact independent of the unit length, L; (2) (A′: A) = (L′: L)0 = 1— again expressing that independence.
page no 93 note * This relation is exhibited in the use of common terms (second, minute) for both angles and times, and in the whole technique of Spherical Astronomy. Proportionality of time to length in uniform straight motion is, by comparison, clearly not of general practical utility; the moving particle does not remain “at hand”
page no 93 note † The importance of these applications—of sub-section (ii)—of “Dimensions”, by comparison with their straightforward use (as in (i)) in determination of change of derived units, tends to be exaggerated.
page no 94 note * Use of the theoretical system of units is so general that the terms length, time, mass, velocity, acceleration, force, ...., are commonly used to mean the measures (all, of course, “numbers”) of quantities properly so named, in terms of the units of the theory. The loose type of phrase exemplified by “length in feet”, “force in dynes”, arises out of that kind of usage.
page no 94 note † This is, in fact, a most useful symbol. Both symbols are of the “symmetric” type, like the sign of equality (and unlike the signs of inequality), and are, from that point of view, not quite well “designed”
page no 94 note ‡ “Speed” is defined, primarily, for a finite motion. Speed-at-an-instant, the important limiting case, is of course secondary to that. To call the former “the average speed” is, therefore, something like “putting the cart before the horse” The same is true of other similar cases, such as that of curvature.
page no 95 note * It is not, of course, quite so straightforward as this: seeing that “multiplicate proportion” is a kind of corollary to joint proportion, arrived at through the mathematical expression (see § 6, supra). It should be noted that the basic case of multiplicate proportion is the relation of the “derived unit” of area to the unit of length (i.e. the relation of the area of a square to the length of its side).
page no 95 note † When we say that the area of a rectangle is the “product” of the two lengths of its sides, we are in fact making justifiable generalisation of definition of the term “product” And similarly in the other cases in question. But the significance of this is obscured by the practice (referred to below) of ignoring the distinction between physical quantities and their measures.
page no 95 note ‡ There are distinguished authorities who take the opposite view. But it is of the essence of the treatment here given that that view cannot finally be maintained. It appears to arise out of the fact that the theoretical system of units is not given its due place of absolute and essential importance in Physical theory; and to consequent inadequate realisation of the significance of the above-mentioned “convenient practice”
page no 95 note § For instance in elementary presentation of the fundamentals of Relativity, where the ordinary canons of measurement may not be assumed.
page no 95 note ‖ The fact that this system of units is not merely (as is often stated) “the most convenient ”, but is an essential of Physical theory, cannot be too strongly emphasised.