CHAPTER XIV.

UNITS AND STANDARDS OF MEASUREMENT.

INSTRUMENTS of measurement are, as we have seen, 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 quantity. 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, weight, mass, 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, except two, cases absolutely arbitrary in point of theory, and its selection a question of practical convenience.

Of the two cases in which a natural standard unit is ready made for us, one case is that of number itself. Abstract number needs no special unit; for any object by existing or being thought of as separate from other objects (p. 176), furnishes us with a unit, and is the only standard required.

Angular magnitude is the second case in which we have a natural and almost necessary unit of reference, namely, the whole revolution or perigon, as it has been called by Mr. Sandemana.

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, and 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 generalized their notions of angular magnitude sufficiently to conceive clearly angles of all magnitude, or of unlimited quantity of revolution. But Euclid defines a right angle as half that made by a line with its own continuation, not called by him an angle, and which is of course equal to half a revolution. In mathematical analysis, again, 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. This angle, called by De Morgan the arcual unit, is equal to about 57°, 17', 44"8, or decimally 57°295779513 . and is such that the half revolution contains 314159265.... such units. Though this standard angle is naturally employed in mathematical analysis, and any other unit would introduce needless complexity, we must not look upon it as a distinct unit, since its amount is connected with that of the half perigon, by a natural constant 3.14159 ..... ... usually signified by the letter.

When we pass to other species of quantity, the choice of unit is found to be entirely arbitrary. There is abso

a-Pelicotetics, or the Science of Quantity; an Elementary Treatise on Algrim and its groundwork Arithmetic.' By Archibald Sandeman, M.A. Cambridge Deighton, Bell, and Co.) 1868, p. 304.

D. Mergas · gonometry and Double Algebra,' p. 5.

lutely 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.

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 life and motion. 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, 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

c English Works of Thos. Hobbes,' Edit. by Molesworth, vol. i. p. 95.

wonderful advances which have been made in the practical measurement of its efflux. 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 equable chanting of psalms, gave the means of roughly subdividing periods, and marking the hours of the day and night. 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 and the chronoscope. 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

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15,200 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: -

Alsolute, 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". Though we are perhaps obliged to assume the existence of a uniformly increasing quantity which we call time,

Nu (towis gives many curious particulars concerning the measurement of pony Astronomy of the Ancients,' pp. 241, &c.

Scholium to Definitions.' Translated by Motte,

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 and change 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. 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 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 at the same time, 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 entirely 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 theoretically speaking use them to prove the uniformity of their own movements, which would amount to a petitio principii. Our experience amounts to this, that when we examine and compare the movements of bodies which seem to us nearly free from disturbance, we find them give nearly harmonious measures of time. If any one