54. Formula for finding the number of the solar epacts, and when leap year will fall
TRANSLATION
all respects, so that every year of this [solar] cycle becomes the ¢rst year of a 19-year cycle, and similarly, that each year of the 19-year cycle follows as the ¢rst of this [solar cycle]. Thus the entire series of the Easter observance will be ¢nished in no less that 532 years.47 /443/
54. FORMULA FOR FINDING THE NUMBER OF THE
SOLAR EPACTS, AND WHEN LEAP YEAR WILL FALL
Since the sequence of the [solar] epacts, that is, of the solar increment, coincides and is closely connected with the leap year, learn by means of this formula what is the state of both simultaneously.
If you wanttoknowwhentheleapyearwillbe,taketheyearoftheLord [725], divide it by 4, and if there is no remainder, it is leap year; if there is a remainder of 1, 2, or 3, it is 1, 2 or 3 years away from leap year.48 For instance, 46100 = 400, 4680 = 320, 461 = 4, with 1 left over, because it is the ¢rst year after the leap year. If you wish to know the solar increment, that is, the concurrent days of the week, take the number of years from the Lord's Incarnation, for instance 725 (which is the 8th indiction), and add to this one-fourth of these years, that is, at present, 181, which together [with 725] makes 906. Add four to this, which makes 910. Divide by 7: 76100 = 700, 7630 = 210, with no remainder. Therefore the solar epact or concurrent days of the week are 7.49
55. THE CYCLE OF BOTH EPACTS, AND HOW TO
CALCULATE THEM USING THE FINGERS
Because anyone who has memorized the cycle of the Sun and Moon grasps very readily the day of Easter and the recurrence of the other dates, assuming he knows how to multiply and divide these cycles by 19 and 28, every computist ought to remember that the same solar concurrents will follow in 30 years as will come next year, and in 60 years from now as will come in four years, and in 120 years from now as will come in eight years, and in 150 years from now as in ten years, and the others in like manner. And if you want to know the number of the concurrent
47 ''Take careful note . . . 532 years.'' Cf. cognate text in London, British Library
Cotton Caligula A.XV (s. VIII2) fols. 110r^117v.
48 Dionysius Exiguus, Argumenta titulorum paschalium 8 (76.27^77.1).
49 Ibid. 4 (76.4^10).
THE RECKONING OF TIME
for a year occurring at any interval, ¢nd out how many times 30 goes into that number [of years], and withthis 30th part doubled,you will know the harmony /444/ of the revolving years. For instance, 10 times 30 is 300, and hence the same concurrents occur in 300 years from now as will occur 20 years from now. If there is a remainder, add it in. In a similar manner, the order of the concurrents can be projected into the past.
Likewise, because the lunar cycle is 19 years long, the epact of the present year will occur 20 years from now; those for next year, 40 years from now; those of the third year, 60 years from now; those of the fourth year, 80 years from now; those of the ¢fth year, 100 years from now; those of the sixth year, 120 years from now; those of the seventh year, 140 years from now; those of the eighth year, 160 years from now; those of the ninth year, 180 years from now; those of the tenth year, 200 years from now, and so on for the rest. This should be understood to apply to the fourteenth Moon of Easter and everything else that is included in the 19-year cycle. The diligent calculator will in this manner be able to observe the recurrence of sequential dates that are still more distant from him, and by this means of these, he will consistently remember to bring even dates in the past into accord.
But it seems worth recalling that some people, in order to simplify calculation, have transferred both cycles, the lunar and the solar, onto the joints of the ¢ngers. Because the human hand has 19 joints if we include the tips of the ¢ngers, by applying each year to one of these joints, they begin the lunar cycle on the inside of the left hand, at the base of the thumb, and they ¢nish it on the inside, at the tip of the little
¢nger. Again, because the two hands together, if you do not count in the ¢ngertips, have 28 joints, they assign the years [of the solar cycle] to these, beginning at the little ¢nger of the left hand and ending on the thumb of the right. They do not do this, as in the lunar cycle, by proceeding through the ¢ngers in numerical order, but because of the calculation of the quarter-day [i.e. the leap-year increment], they mark out each four-year period by traversing the four ¢ngers, so that the six joints of the little ¢ngers contain all the leap years. Again, the six joints of the ¢ngers next to the little ¢ngers comprise all the years that follow immediately after the six leap years. By the same token,those of the second ¢nger
[from the little ¢nger] embrace the second years, and those of the third ¢nger the third years. On the other hand, the seventh leap year with the three years following it are appropriated to the four joints of the thumbs. Whether the computist wishes to avail himself of this or of