yard is three feet, a mile eight furlongs, &c. But & 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 twenty-one 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.

(631.) 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 remarkable simplicity and neatness, 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.

(632.) The Gregorian rule is as follows:-The years are denominated from the birth of Christ, according to one chronological determination of that event. 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 year divisible by 100, but not by 400, again of 365; and every year divisible by 400, again of 366. For ex

ample, 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 1. Now, in 10000, the numbers not divisible by 4 will be 3 of 10000, or 7500 ; those divisible by 100, but not by 400, will in like manner be à 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, 365d 2425. The actual value of the tropical year (art. 327.) reduced into a decimal fraction, is 365-24224, so the error of the Gregorian rule on 10000 of the present tropical years is 206, or 24 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 might be 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.

(633.) 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 leap-days.

(634.) If the Gregorian rule, as above stated, had always been adhered to, 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, and that often to serve particular purposes and private interests. 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 was the first regular year, commencing 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 in