years, 3 days were ordered by Pope Gregory to be suppressed in every future 400 years, or (which is the same thing) those centenary years only are made bissextile or leap years the number of whose centuries is a multiple of 4; whereas, but for this order, every centenary year would have been reckoned a leap year, because it is always a multiple of 4-e.g. under the new rule (or style) 1700, 1800, and 1900 are ordinary years, 2000 is a leap year.

Correction of the Epact.

So little is said in the Prayer Book about the epact, it may seem unnecessary to follow it out to its correction, which, however, is as necessary as that of the place of the golden number itself; for, like it, being a lunar cycle, it is liable to the 'lunar error' before described, and being also accommodated to days of the solar year (i.e. representing the age of the moon on a day of the solar year), is liable also to the 'solar error.' It will now be evident, from what has been said, that those centenary years in which the epact has required, or will require, correction for solar error are (beginning from 1582) 1700, 1800, 1900, 2100, 2200, 2300, 2500, 2600, 2700, 2900, 3000, 3100, and so on-in other words, every centenary year that is not made bissextile-and that this error requires the subtraction of 1 (i.e. one day) every such hundredth year from the ordinary epact number.

1

It is also clear, from what has been said of the lunar error, that the centenary years which have required, or will require, correction on this account, counting from A.D. 1800, which Clavius, the real author of the Gregorian calendar, advised Gregory to take as the period of 25 centuries above referred to,1 are 2100, 2400, 2700, 3000, 3300, 3600, 3900, and so on; and that the correction of this error requires the addition of unity (that is, one day) seven times successively at the end of every 300, and once at the end of 400, years. When both errors concur, as in 2100, they neutralise each other, and no correction is needed; so also where neither error occurs, as in A.D. 2000. The lunar error thus requiring an addition, and the solar error a subtraction, the difference of the two will give the required correction. The formula for the former-i.e. the increase up to A.D. 4200-where Clavius stopped, as another day will not

be lost till then—is (—17)-5, where c equals the com

3

pleted centuries, 17 the century up to which all was set right,2 2 Woolhouse on Measures of Time, p. 153.

1 See supra.

w the integers, or whole numbers, which only are to be taken.

crease or subtraction on account of the solar error from 1700 to 4200 is

To apply the above formulæ to, e.g., A.D. 1942, what will be the corrected epact for that year?

And the ordinary new style epact = II

.. the corrected epact 14+8+ (1) ∞ + (-1) w

=14+8+6+4-19,

=14.

-C,

=13 (or a subtraction of unity

from the ordinary epact).

George, the second Earl of Macclesfield, son of the Lord Chancellor, in a paper read by him in 1750 at the Royal Society, of which he became President in 1752, puts the correction in his usual perspicuous and felicitous way:1

'It is plain that the lunar year will have lost one day more than ordinary' [i.e. will have lost twelve days in all] 'whenever the new moon shall have anticipated' i.e. fallen earlier by] 'a whole day, as it will have done at those times' [i.e. every 3107 years] 'when it is necessary that the golden numbers should be set back one day; and consequently the epact' .'which is the difference in whole days between the common Julian solar and the lunar year 'for that and the succeeding years must exceed by an unit the several corresponding epacts of the preceding 19 years.' "The augmenting of the epact is the same thing as setting back the golden number.'

12

And again, speaking generally of the comparative use of the epacts, and golden numbers under the Julian account

'If, instead of the golden numbers, the epacts of the several years were prefixed in the manner the Gregorians have done to the 1 Philosophical Transactions for 1750, vol. xliv. p. 121. 2 Ibid.

days of the calendar, in order to denote the days on which the new moons fall in those years whereof those numbers are the epacts, there would never be occasion to shift the places of those epacts in the calendar, since the augmentation, by an unit extraordinary, of those epacts would answer the purpose, and keep all tolerably right. But the regulating these things for those who use the Gregorian account is an affair of more intricacy.'1

And again

'The Gregorians order the epacts to have an additional augmentation of an unit 8 times in 2,500 years, beginning with 1800, as at the end of 400 years; to which 400 years, if there be added 3 times 700, or 2100 years, the period of 2500 will be completed in 3900; after which they do not make this extraordinary augmentation of an unit in the epacts till at the end of another 400 years, which defers that augmentation from 4200 to 4300.' 2

The above formulæ having given the quantum of correction, the following rule will show the student of ecclesiastical chronology (who will now understand the reason of it) when it is required :

correc

I. Years whose entire hundreds are exactly divisible by 3 (and therefore liable to tion for lunar error), but not by 4 (and therefore not liable to correction for solar error).

The Gregorian solar and the lunar year both lose one day.

2. Years whose entire hundreds are exactly divisible by 4, but not by three.

Neither the Gregorian solar nor the lunar year loses.

3. Years, whose entire hundreds are exactly divisible by both 4 and 3.

The Gregorian solar year goes on as usual.
The lunar year loses.

4. Years whose entire hundreds are ex-
actly divisible by neither 3 nor 4.
The Gregorian solar year loses.

The lunar year goes on as usual.

The epact The epact

the golden numbers are not number set nished nor augmented, and is augnished, and mented, the golden and the number set golden is dimiforward, one day.

back, one

The epact is neither dimi

shifted.

day.

And, inasmuch as the year is more often not divided

by 4 than it is divided by 3, the epacts are more often diminished than increased.

Those who wish to follow out yet further the solution of the

1 Philosophical Transactions for 1750, vol. xliv. p. 425.

2 Ibid.

Easter problem, and to calculate Easter without the necessity of employing either the golden numbers or the epacts, but from the sole datum of the number of the given year A.D., would do well to refer to the elaborate essay by the late Dr. Butcher at the head of our paper. The object of the Bishop is to connect the rule of the famous German mathematician Gauss with that of the equally famous French astronomer Delambre, and to show how that of the former follows from that of the latter-first considering the case of Easter in the Gregorian calculation, and then applying it to 'the particular and more simple case of Easter in the Julian calculation.'

ART. VI.-CONTINENTAL CULTURE.

I. German Home Life. (Longmans.)

2. French Home Life. (Blackwood.)

3. Kindergärten: a Visit to German Schools. By JOSEPH PAYNE. (Henry S. King.)

4. A French Eton. By the Rev. S. HAWTREY. (Hickesly.) 5. Round my House. By P. G. HAMERTON. (Seeley.)

MANY of our readers have no doubt become acquainted with Principal Shairp's Five Lectures on Culture and Religion, in which his great point is the failure of culture to raise or refine, except as the child of religion.

'Culture, though made our end never so earnestly, cannot shelter a man from thoughts about himself, cannot free him from that which all must feel to be fatal to high character-continual self-consciousness. The only forces strong enough to do this are great truths which carry him out of and beyond himself the things of the spiritual world, sought not mainly because of their reflex action on us, but for their own sakes, because of their own inherent worthiness. There is, perhaps, no truer sign that a man is really advancing than that he is learning to forget himself, that he is losing the natural thoughts about self in the thought of One higher than himself, to whose guidance he can commit himself and all men. This is no doubt a lesson not quickly learnt; but there is no help to learning it in theories of selfculture which exalt man's natural self-seeking into a specious and refined philosophy of life.'-Shairp, Lecture iii.

In other words, culture, unless subservient to religion, is mere exaggeration of selfishness in one form or another. It

leads either to isolated self-contemplation or to outward selfseeking, only so far respecting the rights of others as to avoid provoking them into returning evil for evil, thus achieving social law. Now the experiment which seems to be universal in the modern world, from the Ganges to the Mississippi, is that of putting culture foremost, religion as the mere accessory. We all know the stock argument-Sects cannot agree on a religious education. Therefore let the State enforce secular culture; and as to religion, chacun à son goût. A non-religious education is not irreligious.'

Therewith we are bidden to admire and follow the example of our neighbours. There are lands where the religion of culture has had full scope, and has educated the generation now acting and thinking. In France the Minister of Education can tell what every class in every school is doing at any given hour; and as nobody can get employment, enter any public office, or teach, without due diplomas, it is not possible to escape the regulation system. In like manner, in Germany, no one, from the prince to the ploughboy, can elude culture duly proportioned to his station in life. The Kindergarten fosters the infant's power of observation, definition, and drill; the Primary School impresses a thorough analytical knowledge of the mother tongue, the Real School educates the tradesman, the Classical School the scholar and the gentleman; the University crowns all, and there are throughout such discipline and thoroughness as render a German teacher one of the best of instructors.

In France, during the reign of the Citizen King, religion was entirely eliminated from all public instruction for the nobler sex, and was confined to seminaries for the priesthood. As we all know, it was the struggle of Montalembert's life to obtain the freedom of giving religious instruction in schools for laymen. He succeeded after the Revolution of 1848, but without achieving popularity for education based on religion; and though the religious orders, such as the Jesuits and the Frères Chrétiens, are not prohibited from teaching, the greater number of lads never come under their hands, and their tenure is insecure, although the pupils of the former are said to be eminently successful in competitions. As things stand at present in France, most boys of the upper classes are educated in Government lyceums, entirely in secular hands, and in the villages there is a schoolmaster, also entirely secular, and usually a hostile authority to the parish priest. The greater number of young ladies, on the other hand, if not brought up at home, go to the convents of the religious orders devoted