1st. A sidereal day 1/110 of a second less than one rotation of the earth.
2nd. One rotation of the earth in 23 hours, 56 minutes and 4 seconds, nearly, of clock time.
3rd. One mean time clock day of 24 hours, which has been explained previously.
Now, isn't it remarkable that our 24-hour day is purely artificial, and that nothing in nature corresponds to it? Our real day of 24 hours is a _theoretical_ day. Still more remarkable, this theoretical day is the unit by which we express motions in the solar system. A lunar month is days--hours--minutes--and seconds of this theoretical day, and so for planetary motions. And still more remarkable, the earth's rotation which is _itself_ the foundation is expressed in this imaginary time! This looks like involution involved, yet our 24-hour day is as real as reality; and the man has not yet spoken who can tell whether a mathematical conception, sustained in practical life, is less real than a physical fact. Our legal day of practical life is therefore deduced from the day of a fraction _less_ than one earth rotation. In practice, however, the small difference between this and a rotation is often ignored, because as the tenth of a second is about as near as observations can be made it is evident that for single observations 1/110 of a second does not count, but for a whole year it does, and amounts to 3-1/3 seconds. Now as to the setting of our clocks. While the time measured by the point of the Spring equinox is what we must find it is found by noting the transits of fixed stars, because _the relation_ of star time to equinoctial time is known and tabulated. Remember we cannot take a transit of the equinoctial point, because there is nothing to see, and that _nothing_ is moving! But it can be observed yearly and astronomers can tell where it is, at any time of the year, by calculation. The stars which are preferred for observation are called "time stars" and are selected as near the celestial equator as possible. The earth's axis has a little wabbling motion called "nutation" which influences the _apparent_ motion of the stars near the pole; but this motion almost disappears as they come near the equator, because nutation gives the plane of the equator only a little "swashplate" motion. The positions of a number of "time stars" with reference to the equinoctial point, are known, and these are observed and the observations averaged. The distance of any time star from the equinoctial point, _in time_, is called its "right ascension." Astronomers claim an accuracy to the twentieth part of a second when such transits are carefully taken, but over a long period, greater exactness is obtained. Really, the time at which any given star passes the meridian is taken, _in practical life_, from astronomical tables in the Nautical Almanacs. Those tables are the result of the labors of generations of mathematicians, are constantly subject to correction, and cannot be made simple. Remember, the Earth's rotation is the only uniform motion, all the others being subject to variations and even compound variations. This very subject is the best example of the broad fact that science is a constant series of approximations; therefore, nothing is exact, and nothing is permanent but change. But you say that mathematics is an exact science. Yes, but it is a _logical abstraction_, and is therefore only the universal solvent in physical science.