If two points A,, A, pass across B at an interval which is an ascertained fraction of the period of A,, with respect to B, then, knowing nothing else about them, an hypothesis is required, in order to tell from this observation what fraction of four right angles A, subtends at o. If A, rotate as if rigid and uniformly, the fraction is the same as the fraction of the period. If they ro tate independently, say under the action of the same acceleration to the centre o, no conclusion can be drawn without further data. Generally it must be remembered that in order to define the mo tion, e.g., in its relation to the right angle, it would be requisite to define first the metric of the space in which it takes place in addition to assigning its own "law." But in the absence of special statement, rigidity and uniform rotation will be presupposed, making the time interval and the angle A,OA, equivalent measures. Units of Time.—The practical standard for measuring time is the rotation of the earth, and is very close to the ideal one. If we take the earth as the body A and the sphere of stars as the body B enclosing it, their relative period gives an observable unit of time, the so-named sidereal day. It differs from the mean solar day employed in civil reckoning in a way defined below. The stars are, of course, not quite fixed with respect to one another, but as they are their own standard of reference, their mean is fixed, and there appears to be no meaning to be attached to the idea that it is in motion, especially if we allow in the case of each star for all systematic apparent motions such as those due to the earth and the sun, and the individual motions peculiar to each star, leaving• only an accidental residuum to cancel itself out. Taking then the mean of the sphere of stars as the zero, and supposing the local Euclidean geometry continued outwards to mark the positions of the stars, the rotation of the earth with respect to the sphere of stars becomes an absolute quantity, that is to say, a quantity completely defined. In describing it so, some practical qualifications are subsumed. Thus, small allowances varying in different latitudes and different times of the day and year require to be made on account of the finite velocity of light by which the stars are recognized. Also, it is convenient to measure from a slowly moving zero, the equinox, which is easier to identify than the mean of the stars. Reference is made to these and others below. The principle is unchanged.
A smaller unit than the day is provided by the pendulum clock, of which one beat or a semi-oscillation equals a second, of which there are 24X 6o X 6o= 86,40o in the day. Much loose writing is found in relativist literature about "clocks," as though the clock were a simple instrument which it was unnecessary to define further. Such is not the case. We may regard the clock as a machine to imitate the rotation of the earth. If the 24-hours dial of a sidereal clock is imagined as set in a plane parallel to the equator, with the right side up, its index will continue to point always to the same star. The theory of the clock is complicated and has nothing primitive about it. It indicates, however, that the construction of a clock that shall go uniformly is possible ; but the practical execution of clocks, though enormously improved recently, is not perfect. The going is liable to derangement from a number of causes, not all of which are clear. In short, the clock is in the position of a secondary standard only, which must be adjusted to the primary standard, viz., the sidereal day or ob served rotation of the stars. The same remarks apply with in creased force to portable watches and the marine chronometer, in which the beats are given by a balance wheel actuated by a spiral spring. The method by which the clock is adjusted to the rotation of the earth is described below. To subdivide the second, given by the beat of the clock, various devices are employed which are described more particularly in the article CHRONOGRAPH. In
principle, they all rely upon converting time measures into space measures, by introducing a machinery that shall move a re cording paper at a sufficiently uniform speed, making the clock mark seconds upon this, or, alternatively, making a tuning fork or other rapid vibrator impress closer marks, and subdividing the distance between the marks by any device suitable for space measurement. When we go beyond times which the chronograph can record, we have passed out of the region of actually countable steps of time.
For a larger measure than the sidereal day, we turn to the year. The year can only be defined precisely by means of a curiously elaborate theory. The year being the period of the revolution of the earth about the sun, we must define its motion and the point to which it returns. Since the motion is perturbed by the moon and the other planets, the orbit does not repeat itself, and any specification has no more than an "instantaneous" value at an assigned epoch. The process is to form a "theory of the sun," as signing its motion, for we may just as well contemplate geo metrically a motion of the sun round the earth, as of the earth round the sun. To do this, all the observations of the sun's posi tion are assembled, over the whole period of time for which they can be considered sufficiently exact, and the orbit is determined of a character assigned by theory, and agreeing with the observa tions as well as they allow over the whole period. The outcome is an ellipse of changing shape in a changing plane. The changes of course are small and slow, but by no means negligible. The mean motion with respect to the stars may, however, be taken as unchanged, and this defines the length of the sidereal year. But as remarked elsewhere, a return relative to the stars is not con venient to identify. It is more convenient to count from the equinox, or point where the equator intersects the sun's path. A return to this point corresponds also with a return of the seasons. Hence we define the tropical year, which is slightly shorter than the other because the equinox is in regressive motion.
From the theory of the sun comes also the definition of a deriv ative but indispensable measure, viz., mean solar time. The mean solar day is the mean period of rotation of the earth with respect to the sun. The sidereal day is the period of rotation of the earth with respect to the equinox. If the sun makes a circuit of its orbit starting from the equinox, in a tropical year, the earth will have rotated just one time more with respect to the equinox than with respect to the sun in that period. The tropical year is found to contain 366.2422 sidereal days. Hence it contains 365.2422 mean solar days, and we have two parallel sets of units, days, hours, minutes and seconds, in the ratio of 1.002738 :1, or i the solar unit being the greater. Moreover, the mean solar day starts at a different point from the sidereal day. The latter starts when the equinox is on the meridian. The former starts when an artificially defined "mean sun" is on the meridian. The "mean sun" is an imaginary body defined for the purpose of keeping time, in such a way that it moves in the equator in agree ment with the apparent sun's mean motion in the ecliptic. It is the transit of this body across the meridian of Greenwich that gives the zero for "Greenwich mean time." As, of course, the "mean sun" cannot be observed, its reputed place with respect to the stars is calculated from the theory of the sun, in advance, in the ephemerides, and this permits us to observe other stars instead of it and to deduce the moment of its transit. By these means any observatory, of known longitude, constructs for itself "Greenwich mean time" from its own observations. In respect to the measure ment of time the relations of the year with the system of mean time are more prominent than its use as a major unit of time. In the latter connection it belongs to the subjects of th6 CALENDAR and CHRONOLOGY.