We must premise that the whole theory of taking time by sidereal observations is based on angular motion, the mensuration of one of the angles of motion giving a measurement of space, so that to say space, or distance, is equivalent to saying time. From noon of one day to noon of another is the whole problem to be solved by correct division. The astronomical day begins at noon, but in civil law the day is dated from midnight. So in the year the astronomical day is dated December 31, while in common reckoning the 1st of January is the initial point. This day is divided into twenty-four hours, counted in England, America, and the most of the Continental nations of Europe, by twelve and twelve. The French astronomers, however, adopted the decimal system, for ease in the computation. Thus they divided the day into ten hours, the hour into one hundred minutes, and the minute into one hundred seconds. This plan was in conformity with the French system of decimal weights and measures. Again, in Italy, the day was divided into twenty-four hours, but counting from one to twenty-four o’clock. The French system presents some features well worthy of adoption, as it gives results so much more easy in computation--a facility unattainable in the common division; yet it did not come into general use in other countries, and although some French astronomers still hold to the system, it is gradually dying out. At one time during the Revolution in France a clock in the gardens of the Tuileries was regulated to show time by the decimal system. For the Horologist the mean length of the day is sufficient to show the rate of his instrument for that particular day, but the astronomical and civil division requires a much longer period of observation. This is obtained by the position of the mean annual equinoxes or solstices, and is estimated from the winter solstice, the middle of the long annual night under the North Pole; and the period between this solstice and its return is a natural cycle, peculiarly suited for a standard of measurement. Even with such a standard as the civil year of 365d. 5h. 48m. 49.7s., the incommensurability that exists between the length of the day and the real place of the sun makes it very difficult to adjust the ratio of both in whole numbers. Were we to return to the point in the earth’s orbit in exactly 365 days, we would have precisely the same number of days in each year, and the sun would be at the same point on the ecliptic at the same second at the beginning and end of the year. There is, however, a fraction of a day, so that a solar year and civil are not of equal duration.