The Length of the Year in the Original Proposal for the Gregorian Calendar

Swerdlow N. , “The origin of the Gregorian Civil Calendar”, Journal for the history of astronomy, v (1974), 48–49 with references to earlier literature. Since that date, some interesting papers have appeared, in particular, one by O'Connell D. J. K. , “Copernicus and calendar reform”, Studia Copernicana, xiii (1975), 189–202, and several in the collection Gregorian Reform of the calendar: Proceedings of the Vatican Conference to commemorate its 400th anniversary, 1582–1982, ed. by Coyne G. V. Hoskin M. A. and Pedersen O. (Vatican City, 1983). I wish to thank Grafton A. T. and Neugebauer O. for their comments on this paper.

Cited as Clavius. Originally published as Romani calendarii a Gregorio XIII P.M. restituti explicatio (Rome, 1603), and reprinted in Clavius C. , Operum mathematicorum tomus quintus (Mainz, 1612). The proposal of 1577, the so-called Compendium, is printed at pp. 3–12. It may always have been scarce since Clavius, p. 71, says that “because it will frequently be mentioned, and lest the memory of it be lost, we have placed it at the beginning of this work in exactly the same words in which it was sent to the Princes, nothing having been changed”.

Pitatus P. , Compendium … super annua Solaris, atque Lunaris Anni quantitate, … Romanique Calendarii instauratione … (Verona, 1560), Prop. II. I do not know whether Pitatus was the original inventor of the omission of three out of four centennial intercalations—he proposes it as though it were his own invention—but his tract, printed three times (Verona, 1560, Venice, 1564 (which I have not seen), and Basel, 1568) must have been the source for later writers, including of course Lilius. In any case, it appears that Pitatus should probably receive the credit for the best-known, although not the most important, innovation in the new calendar. The proposal to move the calendrical equinox to 25 March, while historically correct for the time of the Crucifixion—concerning the true date of which Pitatus wrote a tract—was too radical (heretical?) a departure from the Nicaean Calendar to be adopted for it would have necessitated entirely new tables and limits for the date of Easter.

Alfonso X , Tabulae astronomici divi Alfonsi… (Venice, 1518/21); Copernicus N. , De revolutionibus orbium coelestium… (Nuremberg, 1543; repr. New York, 1965); Reinhold E. , Prutenicae tabulae coelestium motuum… (Tübingen, 1551).

This was given incorrectly in my earlier note as 365; 14,33,11,12…^d. The maximum and minimum values of the variable tropic years, rounded to three-places, are: De revolutionibus 365;14,48,49^d 365;14,17,37^d Prutenic tables 365; 14,49, 4^d 365; 14,17,15^d All three mean tropical years will of course produce the common rough estimate that the Julian year is in excess by one day in 134 years, e.g. from the Alfonsine tables 1/(365;15–365;14,33,10) ≈ 134; 10, while from De revolutionibus the figure is 134;24 and from the Prutenic tables 134;7.

Clavius , 11–12: “Quod si alicui Alfonsi calculi incertiores esse videbuntur, quam ut illis fidendum putet, potiusque recentioribus adhaerendum existimet, is profecto intelliget eam esse huius artificiosi Cycli tabulaeque Epactarum a Lilio excogitatae dispositionem ac digestionem, ut nullo negotio siue Copernici, siue cuiusuis alterius calculis possit aptari, si tabella aequationis ex illis confecta, pro ea, quam ad marginem scripsimus, substituatur: Ueluti haec, quam exempli gratia a Copernici ratione non multum distantem apposuimus.”

There is a numerical analysis and tabulation of the relation of Copernicus's rate of precession and length of the tropical year in Swerdlow N. , “Long-period motions of the Earth in De revolutionibus”, Centaurus, xxiv (1980), 215–45, pp. 221–2, 241–3.

Plausible results can also be found by beginning the computation at 300, but not later. While it might at first seem strange that slightly less than 3 centennial intercalations are required up to 1500, a moment's reflection will show that the Gregorian intercalation, were it in use, would have indicated only 3 for that entire period, the 4th not occurring until 1600. Further, the fastest part of Copernicus's precession occurs about 800, and the tropical year is quite short for about 500 years on each side, so no centennial intercalation is required for 1000 years.

Clavius , 67, 71–73, raises numerous objections to a variable intercalation even if, as he admits is entirely possible, the tropical year is non-uniform, but he also astutely remarks that just as the tables of Ptolemy and King Alfonso have with time been found in error, so one should not believe that the Prutenic tables, based on the theory of Copernicus, will last forever. Yet it must have served as some kind of confirmation of the Alfonsine tropical year that the mean tropical years of Copernicus and Reinhold agreed with it, and, as we shall see, the mean synodic month of the calendar must be that of the Prutenic tables.

Clavius , 67. This, and much of what follows, is from chap. IV, “Why the Church, disregarding true or apparent motions, uses only mean or uniform [motions], or preferably cycles, in the celebration of the movable feasts”. The principles explained in this chapter by themselves preclude the use of the variable intercalation for the simple reason that the very essence of a sensible calendar must be mean periods and cycles since periodic inequalities pose unnecessary complications and in any case have no permanent effects.

Clavius , 79. Chap. VI, “On the period of the anomaly of the equinoxes and of the inequality of [tropical] years according to the theory of Nicholaus Copernicus”, provides a detailed investigation of the entire question.

Clavius , 71.

Clavius , 74. He computes the error as follows: If one bissextile day is omitted in 134 years, then in two years the omission is 2^d/134= 1^d/67, which is the error between the 400-year cycle of the calendar and 3·134 = 402 years, and an error of 1^d/67 in 400 years accumulates to 1 day in 67·400 = 26800 years. A better estimate is simply 1^d/0;0,0,10^d/_y = 21600^y, indicating that one day should be added in 21600 years. Clavius's computations and citations of parameters often contain small errors, which are generally of no significance, although one should check before relying upon his numbers for further computations.

Clavius , 73. It is interesting, and not generally known, that Clavius, in what amounts to the official exposition of the calendar, was entirely open to later adaptations taking account of more accurate values for the length of the tropical year, although for the reasons just given not to cycles for periodic inequalities. Hence, since in fact the Gregorian year is too long by one day in about 3600 years, the common proposal for omitting one intercalation in 3600 years or, for the sake of consistent rules, 4000 years, would seem almost to have been sanctioned in advance. Of course, the fact that Clavius also gives tables of moveable feasts up to the year 5000, tables of correction of the epacts up to 303,300, and tables for finding Golden Numbers, Dominical Letters, and epacts for literally hundreds of millions, even billions, of years seems to show that he also believed the calendar would not require modification for some time.

Clavius , 86–87. O'Connell , op. cit., 200, has drawn attention to Clavius's use of the mean synodic month from the Prutenic tables.

There is of course nothing new about a correction to a lunar cycle, such as the 19-year cycle, of one day in something over 300 years, for it is at least as old as Hipparchus's cycle of 304 years = 3760 months = 111035 days, which was based upon the Babylonian System B mean synodic month of 29;31,50,8,20 days and intended as a correction of-1 day to four Callippic Cycles of 76 years = 940 months = 27759 days. In his treatise on the calendar Pitatus proposes a lunar correction of 23;29 hours in 304 julian years, from which the underlying synodic month is found to be ((304.365¹/₄^d) — 23;29^h)/3760 = 29;31,50,7,37^d, obviously derived from the Alfonsine tables.