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 expla nation 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.

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 thirtysix years, 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."

the palpably absurd statement which is found in various works in general estimation, namely, that the first of the Augustan leap-years was A. D. 7, after which they proceeded without mistake. How the leap-years afterwards obeyed the rule of falling into dates which are divisible without remainder by 4, is a mystery to those who adopt the statement, and think about it.

When a reckoning is made from 1, the century terminates at 100; but when it is made from 0, through 1, 2, &c., it terminates with 99. About the year 1799, there was discussion whether the eighteenth century terminated at the end of 1799, or at the end of 1800. This was equivalent to a discussion whether the usual reckoning had a year 0, or began with 1. It so happens that the history of our mode of reckoning has been made to have a point of obscurity which may tend to prolong this discussion; and perhaps some may be found to doubt whether this present year 1850, ends the first half of the nineteenth century, or begins the second.

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A century is any collection of one hundred; its restriction to collection of years is modern. Most readers remember the century of inventions," and many remember that they thought at first it was the account of some inventive century. Bale's work on English writers is divided into centuries, not of years, but of scholars; and centuries have been published of nativities, and of other things.

A century of years may begin or end with any year, just as a year of days may begin or end with any day; and as the year ending April 7 began at the preceding April 8, so the century ending 1745 began with 1646. But, in like manner as the year of reckoning (as distinguished from a year-space of measurement of time) begins with January 1, so it is presumed that a century is also a unit of reckoning, and has a definite commencement: and that it is so is clear, as to modern times, from the constant phraseology of writers, who talk of the twelfth century, the nineteenth century, &c. But it generally happens that, in speaking of centuries, writers are using a rough denomination thus no one who finds a paragraph which alludes to the religious troubles of the sixteenth century, can possibly guess whether that century be meant to begin with 1500 or 1501.

There is no ancient usage as to the beginning of centuries,

It is to be regretted that we are obliged to talk of centuries under numeral figures which contradict the dates. Fourteen hundred and twenty is in the fifteenth century. We are always obliged to pause a moment before we put a year into its century: and even practised historical writers sometimes make a slip. The second edition of Mr. Macaulay's essays is their third impression; and yet (vol. ii. p. 15) it is said, "We know that, during the fierce contests of the sixteenth century, both the hostile parties spoke of the time of Elizabeth as a golden age." The italics, of course, are our own.