CHAPTER XIV.

UNITS AND STANDARDS OF MEASUREMENT.

As we have seen, instruments of measurement are only means of comparison between one magnitude and another, and as a general rule we must assume some one arbitrary magnitude, in terms of which all results of measurement are to be expressed. Mere ratios between any series of objects will never tell us their absolute magnitudes; we must have at least one ratio for each, and we must have one absolute magnitude. The number of ratios n are expressible in n equations, which will contain at least n + 1 quantities, so that if we employ them to make known n magnitudes, we must have one magnitude known. Hence, whether we are measuring time, space, density, mass, weight, energy, or any other physical quantity, we must refer to some concrete standard, some actual object, which if once lost and irrecoverable, all our measures lose their absolute meaning. This concrete standard is in all cases arbitrary in point of theory, and its selection a question of practical convenience.

There are two kinds of magnitude, indeed, which do not need to be expressed in terms of arbitrary concrete units, since they pre-suppose the existence of natural standard units. One case is that of abstract number itself, which needs no special unit, because any object which exists or is thought of as separate from other objects (p. 157) furnishes us with a unit, and is the only standard required. Angular magnitude is the second case in which we have a natural unit of reference, namely the whole

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revolution or perigon, as it has been called by Mr. Sandeman.1 It is a necessary result of the uniform properties of space, that all complete revolutions are equal to each other, so that we need not select any one revolution, but can always refer anew to space itself. Whether we take the whole perigon, its half, or its quarter, is really immaterial; Euclid took the right angle, because the Greek geometers had never generalised their notions of angular magnitude sufficiently to treat angles of all magnitudes, or of unlimited quantity of revolution. Euclid defines a right angle as half that made by a line with its own continuation, which is of course equal to half a revolution, but which was not treated as an angle by him. In mathematical analysis a different fraction of the perigon is taken, namely, such a fraction that the arc or portion of the circumference included within it is equal to the radius of the circle. In this point of view angular magnitude is an abstract ratio, namely, the ratio between the length of arc subtended and the length of the radius. The geometrical unit is then necessarily the angle corresponding to the ratio unity. This angle is equal to about 57°, 17, 44′′8, or decimally 57°295779513... 2 It was called by De Morgan the arcual unit, but a more convenient name for common use would be radian, as suggested by Professor Everett. Though this standard angle is naturally employed in mathematical analysis, and any other unit would introduce great complexity, we must not look upon it as a distinct unit, since its amount is connected with that of the half perigon, by the natural constant 3'14159... usually denoted by the letter π.

When we pass to other species of quantity, the choice of unit is found to be entirely arbitrary. There is absolutely no mode of defining a length, but by selecting some physical object exhibiting that length between certain obvious points-as, for instance, the extremities of a bar, or marks made upon its surface.

1 Pelicotetics, or the Science of Quantity; an Elementary Treatise on Algebra, and its groundwork Arithmetic. By Archibald Sandeman, M.A. Cambridge (Deighton, Bell, and Co.), 1868, p. 304. 2 De Morgan's Trigonometry and Double Algebra, p. 5.

Standard Unit of Time.

Time is the great independent variable of all change-that which itself flows on uninterruptedly, and brings the variety which we call motion and life. When we reflect upon its intimate nature, Time, like every other element of existence, proves to be an inscrutable mystery. We can

only say with St. Augustin, to one who asks us what is time, "I know when you do not ask me." The mind of man will ask what can never be answered, but one result of a true and rigorous logical philosophy must be to convince us that scientific explanation can only take place between phenomena which have something in common, and that when we get down to primary notions, like those of time and space, the mind must meet a point of mystery beyond which it cannot penetrate. A definition of time. must not be looked for; if we say with Hobbes,1 that it is "the phantasm of before and after in motion," or with Aristotle that it is "the number of motion according to former and latter," we obviously gain nothing, because the notion of time is involved in the expressions before and after, former and latter. Time is undoubtedly one of those primary notions which can only be defined physically, or by observation of phenomena which proceed in time.

If we have not advanced a step beyond Augustin's acute reflections on this subject, it is curious to observe the wonderful advances which have been made in the practical measurement of its efflux. In earlier centuries the rude sun-dial or the rising of a conspicuous star gave points of reference, while the flow of water from the clepsydra, the burning of a candle, or, in the monastic ages, even the continuous chanting of psalms, were the means of roughly subdividing periods, and marking the hours of the day and night.3 The sun and stars still furnish the standard of time, but means of accurate subdivision have become requisite, and this has been furnished by the pendulum

English Works of Thos. Hobbes, Edit. by Molesworth, vol. i. p. 95. 2 Confessions, bk. xi. chapters 20--28.

3 Sir G. C. Lewis gives many curious particulars concerning the measurement of time in his Astronomy of the Ancients, pp. 241, &c.

and the chronograph. By the pendulum we can accurately divide the day into seconds of time. By the chronograph we can subdivide the second into a hundred, a thousand, or even a million parts. Wheatstone measured the duration of an electric spark, and found it to be no more than one 115,200th part of a second, while more recently Captain Noble has been able to appreciate intervals of time not exceeding the millionth part of a second.

When we come to inquire precisely what phenomenon it is that we thus so minutely measure, we meet insurmountable difficulties. Newton distinguished time according as it was absolute or apparent time, in the following words:" Absolute, true, and mathematical time, of itself and from its own nature, flows equably without regard to anything external, and by another name is called duration; relative, apparent and common time, is some sensible and external measure of duration by the means of motion."1 Though we are perhaps obliged to assume the existence of a uniformly increasing quantity which we call time, yet we cannot feel or know abstract and absolute time. Duration must be made manifest to us by the recurrence of some phenomenon. The succession of our own thoughts is no doubt the first and simplest measure of time, but a very rude one, because in some persons and circumstances the thoughts evidently flow with much greater rapidity than in other persons and circumstances. In the absence of all other phenomena, the interval between one thought and another would necessarily become the unit of time, but the most cursory observations show that there are changes in the outward world much better fitted by their constancy to measure time than the change of thoughts within us.

The earth, as I have already said, is the real clock of the astronomer, and is practically assumed as invariable in its movements. But on what ground is it so assumed? According to the first law of motion, every body perseveres in its state of rest or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon. Rotatory motion is subject to a like

1 Principia, bk. i. Scholium to Definitions. Translated by Motte, vol. i. p. 9. See also p. 11.

condition, namely, that it perseveres uniformly unless disturbed by extrinsic forces. Now uniform motion means

motion through equal spaces in equal times, so that if we have a body entirely free from all resistance or perturbation, and can measure equal spaces of its path, we have a perfect measure of time. But let it be remembered that this law has never been absolutely proved by experience; for we cannot point to any body, and say that it is wholly unresisted or undisturbed; and even if we had such a body, we should need some independent standard of time to ascertain whether its motion was really uniform. As it is in moving bodies that we find the best standard of time, we cannot use them to prove the uniformity of their own movements, which would amount to a petitio principii. Our experience comes to this, that when we examine and compare the movements of bodies which seem to us nearly free from disturbance, we find them giving nearly harmonious measures of time. If any one body which seems to us to move uniformly is not doing so, but is subject to fits and starts unknown to us, because we have no absolute standard of time, then all other bodies must be subject to the same arbitrary fits and starts, otherwise there would be discrepancy disclosing the irregularities. Just as in comparing together a number of chronometers, we should soon detect bad ones by their going irregularly, as compared with the others, so in nature we detect disturbed movement by its discrepancy from that of other bodies which we believe to be undisturbed, and which agree nearly among themselves. But inasmuch as the measure of motion involves time, and the measure of time involves motion, there must be ultimately an assumption. We may define equal times, as times during which a moving body under the influence of no force describes equal spaces; but all we can say in support of this definition is, that it leads us into no known difficulties, and that to the best of our experience one freely moving body gives the same results as any other.

1

When we inquire where the freely moving body is, no perfectly satisfactory answer can be given. Practically the rotating globe is sufficiently accurate, and Thomson

Rankine, Philosophical Magazine, Feb. 1867, vol. xxxiii p. 91.