depend all the more important arrangements and business of cultivated life is not conformable to such a multiple of the diurnal unit. Their return is regulated by the tropical year, or the interval between two successive arrivals of the sun at the vernal equinox, which, as we have seen (Art. 383.), differs from the sidereal year by reason of the motion of the equinoctial points. Now this motion is not absolutely uniform, because the ecliptic, upon which it is estimated, is gradually, though very slowly, changing its situation in space under the disturbing influence of the planets (Art. 640.). And thus arises a variation in the tropical year, which is dependent on the place of the equinox (Art. 383.). The tropical year is actually about 4.21 shorter than it was in the time of Hipparchus. This absence of the most essential requisite for a standard, viz. invariability, renders it necessary, since we cannot help employing the tropical year in our reckoning of time, to adopt an arbitrary or artificial value for it, so near the truth, as not to admit of the accumulation of its error for several centuries producing any practical mischief, and thus satisfying the ordinary wants of civil life; while, for scientific purposes, the tropical year, so adopted, is considered only as the representative of a certain number of integer days and a fraction— the day being, in effect, the only standard employed. The case is nearly analagous to the reckoning of value by guineas and shillings, an artificial relation of the two coins being fixed by law, near to, but scarcely ever exactly coincident with, the natural one, determined by the relative market price of gold and silver, of which either the one or the other whichever is really the most invariable, or the most in use with other nations, - may be assumed as the true theoretical standard of value.

(913). The other inconvenience of the tropical year as a greater unit is its incommensurability with the lesser. In our measure of space all our subdivisions are into aliquot parts: a yard is three feet, a mile eight furlongs, &c. But a year is no exact number of days, nor an integer number with any exact fraction, as one third or one fourth, over and above; but the surplus is an incommensurable fraction, composed of

hours, minutes, seconds, &c., which produces the same kind of inconvenience in the reckoning of time that it would do in that of money, if we had gold coins of the value of twentyone shillings, with odd pence and farthings, and a fraction of a farthing over. For this, however, there is no remedy but to keep a strict register of the surplus fractions; and, when they amount to a whole day, cast them over into the integer

account.

(914). To do this in the simplest and most convenient manner is the object of a well-adjusted calendar. In the Gregorian calendar, which we follow, it is accomplished with as much simplicity and neatness as the case admits, by carrying a little farther than is done above, the principle of an assumed or artificial year, and adopting two such years, both consisting of an exact integer number of days, viz. one of 365 and the other of 366, and laying down a simple and easily remembered rule for the order in which these years. shall succeed each other in the civil reckoning of time, so that during the lapse of at least some thousands of years the sum of the integer artificial, or Gregorian, years elapsed shall not differ from the same number of real tropical years by a whole day. By this contrivance, the equinoxes and solstices will always fall on days similarly situated, and bearing the same name in each Gregorian year; and the seasons will for ever correspond to the same months, instead of running the round of the whole year, as they must do upon any other system of reckoning, and used, in fact, to do before this was adopted as a matter of ignorant haphazard in the Greek and Roman chronology, and of strictly defined and superstitiously rigorous observance in the Egyptian.

(915.) The Gregorian rule is as follows:- The years are denominated as years current (not as years elapsed) from the midnight between the 31st of December and the 1st of January immediately subsequent to the birth of Christ, according to the chronological determination of that event by Dionysius Exiguus. Every year whose number is not divisible by 4 without remainder, consists of 365 days; every year which is so divisible, but is not divisible by 100, of 366; every ycar

again of 365; and every

divisible by 100, but not by 400, year divisible by 400, again of 366. For example, the year 1833, not being divisible by 4, consists of 365 days; 1836 of 366; 1800 and 1900 of 365 each; but 2000 of 366. In order to see how near this rule will bring us to the truth, let us see what number of days 10000 Gregorian years will contain, beginning with the year A. D. 1. Now, in 10000, the numbers not divisible by 4 will be of 10000 or 7500; those divisible by 100, but not by 400, will in like manner be of 100, or 75; so that, in the 10000 years in question, 7575 consist of 366, and the remaining 2425 of 365, producing in all 3652425 days, which would give for an average of each year, one with another, 365-2425. The actual value of the tropical year, (art. 383.) reduced into a decimal fraction, is 365-24224, so the error in the Gregorian rule on 10000 of the present tropical years, is 2-6, or 2d 14h 24m; that is to say, less than a day in 3000 years; which is more than sufficient for all human purposes, those of the astronomer excepted, who is in no danger of being led into error from this cause. Even this error is avoided by extending the wording of the Gregorian rule one step farther than its contrivers probably thought it worth while to go, and declaring that years divisible by 4000 should consist of 365 days. This would take off two integer days from the above calculated number, and 2.5 from a larger average; making the sum of days in 100000 Gregorian years, 36524225, which differs only by a single day from 100000 real tropical years, such as they exist at present.

(916.) In the historical dating of events there is no year A. D. 0. The year immediately previous to A.D. 1, is always called B. C. 1. This must always be borne in mind in reckoning chronological and astronomical intervals. The sum of the nominal years B. C. and A.D. must be diminished by 1. Thus, from Jan. 1. B. C. 4713, to Jan. 1. A. D. 1582, the years elapsed are not 6295, but 6294.

(917.) As any distance along a high road might, though in a rather inconvenient and roundabout way, be expressed without introducing error by setting up a series of milestones,

at intervals of unequal lengths, so that every fourth mile, for instance, should be a yard longer than the rest, or according to any other fixed rule; taking care only to mark the stones so as to leave room for no mistake, and to advertise all travellers of the difference of lengths and their order of succession; so may any interval of time be expressed correctly by stating in what Gregorian years it begins and ends, and whereabouts in each. For this statement coupled with the declaratory rule, enables us to say how many integer years are to be reckoned at 365, and how many at 366 days. The latter years are called bissextiles, or leap-years, and the surplus days thus thrown into the reckoning are called intercalary or leapdays.

(918.) If the Gregorian rule, as above stated, had always and in all countries been known and followed, nothing would be easier than to reckon the number of days elapsed between the present time, and any historical recorded event. But this is not the case; and the history of the calendar, with reference to chronology, or to the calculation of ancient observations, may be compared to that of a clock, going regularly when left to itself, but sometimes forgotten to be wound up; and when wound, sometimes set forward, sometimes backward, either to serve particular purposes and private interests, or to rectify blunders in setting. Such, at least, appears to have been the case with the Roman calendar, in which our own originates, from the time of Numa to that of Julius Cæsar, when the lunar year of 13 months, or 355 days, was augmented at pleasure to correspond to the solar, by which the seasons are determined, by the arbitrary intercalations of the priests, and the usurpations of the decemvirs and other magistrates, till the confusion became inextricable. To Julius Cæsar, assisted by Sosigenes, an eminent Alexandrian astronomer and mathematician, we owe the neat contrivance of the two years of 365 and 366 days, and the insertion of one bissextile after three common years. This important change took place in the 45th year before Christ, which he ordered to commence on the 1st of January, being the day of the new moon immediately following the winter solstice of the year before. We may

judge of the state into which the reckoning of time had fallen, by the fact, that to introduce the new system it was necessary to enact that the previous year, 46 B. C., should consist of 445 days, a circumstance which obtained for it the epithet of "the year of confusion."

(919.) Had Cæsar lived to carry out into practical effect, as Chief Pontiff, his own reformation, an inconvenience would have been avoided, which at the very outset threw the whole matter into confusion. The words of his edict, establishing the Julian system have not been handed down to us, but it is probable that they contained some expression equivalent to "every fourth year," which the priests misinterpreting after his death to mean (according to the sacerdotal system of numeration) as counting the leap year newly elapsed as No. 1. of the four, intercalated every third instead of every 4th year. This erroneous practice continued during 36 years, in which therefore 12 instead of 9 days were intercalated, and an error of three days produced; to rectify which, Augustus ordered the suspension of all intercalation during three complete quadriennia,-thus restoring, as may be presumed his intention to have been, the Julian dates for the future, and re-establishing the Julian system, which was never afterwards vitiated by any error, till the epoch when its own inherent defects gave occasion to the Gregorian reformation. According to the Augustan reform the years A. U. C. 761, 765, 769, &c., which we now call A. D. 8, 12, 16, &c., are leap years. And starting from this as a certain fact, (for the statements of the transaction by classical authors are not so precise as to leave absolutely no doubt as to the previous intermediate years,) astronomers and chronologists have agreed to reckon backwards in unbroken succession on this principle, and thus to carry the Julian chronology into past time, as if it had never suffered such interruption, and as if it were certain (which it is not, though we conceive the balance of probabilities to incline that way *) that Cæsar,

• With Scaliger, Ideler, and all the best authorities. Yet it has been argued that Cæsar would naturally begin his first quadriennium with three ordinary years, deferring the rectification of their accumulated error to the fourth, by