is, they made the reckoning in strict conformity to the principle that the terminus is included. Thus 1 being leap-year, 4, 7, 10, &c., would be leap-years also: four counted from seven inclusive brings us to ten. As there is no question that this was set right by Augustus, and the superabundant intercalations introduced by the priests allowed for by a sufficient suppression of subsequent ones, it is presumed that we may reckon back as follows. When the Augustan period of correction was passed, it is certain that all the years divisible by four were leap-years. Accordingly A. D. 12 was leap-year, and A. D. 8, and so, it is said, would have been A. D. 4, according to the intention of Julius. The next preceding leap-year would have been B. C. 1, the year immediately preceding A. D. 1; the next before that B. C. 5, and so on, each year B. c. being leap-year which divided by four leaves a remainder 1. At this rate B. C. 45 would have been leap-year. Now B. C. 45 was the first year of the Julian reckoning: it is assumed then that Cæsar commenced with a leap-year. The great argument in favour of this is that by the number of leap-years thus introduced we are brought exactly back to what must have been the first of January, B. c. 45. For Cæsar commenced his year with a new moon :* and, just taking in such additional days as the preceding system of leap-years gives, we are brought, for this back-reckoned first of January, B. c. 45, to a day at which it was new moon at Rome at 11 o'clock in the evening.

We admit therefore the number of days introduced by the preceding hypothesis to be correct: so that there will be no dispute as to what was the actual day of the back-reckoning on which the first day of the first year of the Julian reform fell. But the supposition that Cæsar made a leap-year at the very commencement, is one of the most forced and unnatural that ever was pressed into the service of an explanation. For the preceding year, B. c. 46, thence called the year of confusion, had been made to consist of 445 days! No reason could ever be given why the additional day of February, which was made to allow for the odd six hours of the solar revolution as fast as they amounted to a day, should have been paid in advance: or why, after every thing had been upset by the year of confusion, any want of an additional day could have been felt in the first year, when, if ever, all was straight to begin with.

On looking at the manner in which the Augustan correction is generally stated, it appeared to us easy enough to explain

* Did the Egyptian astronomer know the very day of the full moon, when it happened at 11 o'clock in the evening? Are we to take it for granted that by two hours or more of error he might not throw it into the wrong day? The usual answer is affirmative. To us, however, the accordance with records of the system we put forward rather confirms the astronomer, than the converse.

the manner in which, without sacrificing a single day, the system of leap-years which lasted up to the Gregorian reformation was brought about, namely, that the years which are divisible by four became leap-years. In the following table, the explanation we propose is given on the right, and the most common one on the left. S stands for a sacerdotal leap-year, or one of those which were actually so: J stands for an intended leapyear of the Julian reformation: A stands for a leap-year after the Augustan edict. The years B. C. and A. D. are given, and also those of the Julian reckoning.

In both systems, the leap-years marked sacerdotal are actual; those marked Julian are only in the reputed intention of the reformer, except where also marked sacerdotal. Those marked Augustan are actual.

The difficulties of the two systems are as follows:—

Common system. First, the year of commencement is made to be leap-year, as already mentioned. Secondly, when Augustus ordained that there should be no leap-year for twelve years, he is made to have ordained that there should be none for fifteen years, in our way of reading, or for sixteen years in the Roman way. Thirdly, it is assumed that the Julian system began by paying (at the rate of six hours per annum) in advance, while, after the Augustan vacation, the payment was made only when due.

Proposed system. When Augustus ordained that there should be no leap-year for twelve years, he is made to have ordained, in our way of speaking, that there should be none for eleven years only.

Some persons may think that the final mode of correction is to be interpreted thus; that three sacerdotal (intended) leapyears should be omitted, and that then the reckoning should begin according to the Julian intention. But this would make A. D. 5 to be the first Augustan leap-year, and the common rule for determining leap-year would never have been established.

In asserting the probability of the system we have advanced, it will be observed that we maintain no leap-year for twelve years to be a phrase synonymous with leap-year in the twelfth year. This is the necessary consequence of a strict, but usual, rendering of the maxim, that the last of the old reckoning is the first of the new, to which Roman enumeration so strictly adhered that there is no first day before the Kalends except the day of the Kalends itself. Putting the difficulties of the two systems against each other, we think it

may be safely inferred that the one we propose is very much less than the cumulative amount of the three on the other side. Twelve cannot be twelve in our sense: shall it be our fifteen or sixteen under no rule at all, or shall it be our eleven under a practice which we know to have been common, and which we see in the divisions of the Roman month?

*

So much on the question of probability: we shall now look at the words of the historians who describe what actually took place. Of these there are three whose accounts are usually, and justly, preferred - Censorinus, A. D. 238; Solinus, probably his contemporary; and Macrobius, about A. D. 400. From Censorinus we learn nothing as to the mistake or the correction, only that the intercalary day was to be inserted after each elapsed period of four years, peracto quadriennii circuitu. All that has any allusion to the correction, is the information that the month Sextilis received the name of August when Martius Censorinus and C. Asinius Gallio were consuls; and as it is otherwise known that this change of name took place at the Augustan correction of the calendar, and that the abovenamed were consuls in the year 8 B. C., confirmation is given to the date of this correction. Solinus states that Cæsar added a quarter of a day in the year of confusion, which, as it is impossible to imagine a fraction of a day in any one year, we must take to mean that the year of confusion was considered as furnishing its quotum towards the first bissextile, so that the first bissextile would be the year 3 of the corrected calendar, or B. c. 43. Solinus further states that the priests made the error of adding the bissextile in the fourth year, instead of after the close of the fourth year; and that thus they added twelve days in the lapse of thirty-six years, while only nine ought to have been added, which fault Augustus reformed, and commanded that twelve years should run out without intercalation, jussit annos XII. sine intercalatione decurrere. Now observe, first, that in the system we propose, there are twelve, and should have been nine, sacerdotal leap-years preceding the intervention of Augustus, whereas, taking B. c. 45 as leap-year makes thirteen actual and ten intended leap-years. Secondly, in our system it takes the priests exactly thirty-six years to make this error; whereas, if B. c. 45 be taken as leap-year, they make the error described by Solinus in thirty-four years, and that which he should have described in thirty-seven years. The two isolated facts stated by this writer-first, that the year of confusion was considered as furnishing its quotum towards

* Some writers are very confused: Pliny, for example, interprets three leap-years omitted by Augustus into three new corrections upon corrections of the whole calendar by Sosigenes himself.

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an intercalation; secondly, that the total amount of the sacerdotal error accrued in thirty-six years-support one another. Macrobius repeats the statement of Solinus as to the thirtyyears, and tells the story of the correction of Augustus in very much the same manner. But he has one sentence more. Not being a Roman, and coming further from the events than his predecessors, it is likely that he should have searched for monuments. He mentions one of a remarkable character, a brass inscription ordained by Augustus for the perpetual preservation of the calendar; and we must presume that in mentioning the arrangement which this inscription perpetuated, he used its words. He tells us that, after commanding_that twelve years should expire without intercalation (annos XII. sine intercalari die transigi jussit), he directed that future intercalations should be made every fifth year, as Cæsar had ordained. Thus it appears that Augustus, finding the imported phrase of Sosigenes had been mistaken, substituted a more correct one to Roman ears. According to their counting, the selection of 8, 12, 16, &c., after 4 as a commencement, is the selection of every fifth number. This proof that the phrase first introduced was changed, in order that the direction might be given in the strictest Roman idiom, will justify us in asserting that every part of the direction, as given by Macrobius from the inscription, is to be as strictly rendered in the same way. Since, then, twelve years are to pass over without leapyear, we interpret it that the twelfth year was the next leapyear. To those who were well accustomed to begin new reckoning from the terminus at which they had arrived in the old one, it would not suggest itself as an impediment that there is logical absurdity in the last of the unintercalated years being the first of the intercalated ones. This brings the first Augustan, and thirteenth actual, leap-year, to A. D. 4, and the fourteenth actual leap-year to A. D. 8: being as if the Julian intention had been that B. C. 45 should have been leap-year. It is essential, as before explained, that the fourteenth actual intercalation should take place in A. D. 8: but the common system can only attain this by demanding that, under an edict of cessation of leap-year for twelve years, there should then be no leap-year until four more years had elapsed. This is an inconsistent way out of the difficulty, seeing that the way into it was a demand that intercalations should be considered as payable in advance.

We have not thought it necessary to trace out the origin of

"Post hoc unum diem secundum ordinationem Cæsaris quinto quoque incipiente anno intercalari jussit [Augustus], et omnem hunc ordinem æreæ tabulæ ad æternam custodiam incisioni mandavit."