## Abstract

Having been asked in February 1700 by The Royal Society to respond to G. W. Leibniz's letter from Hanover about the decision of the German states to accept a so-called ‘improved calendar’, Isaac Newton, then Master of the Mint, developed a proposal for the reform of the Julian and Ecclesiastical calendars that was later found among his unpublished manuscripts (now grouped as Yahuda MS 24). His calendar, if implemented, would have become for England a viable alternative to the Gregorian. Despite having a different algorithm, its solar part agrees with the latter until ad 2400 and is more precise in the long run, within a period of 5000 years. Although Newton's lunar algorithm is more elegant than the Gregorian, his Ecclesiastical calendar remained incomplete. We explain why blank spaces were left and data were changed in several of the manuscripts, discuss the time frame and the order in which Newton wrote different drafts of Yahuda MS 24, analyse their relation with three manuscripts from the Cambridge collection, and suggest a reason for Newton's delay and failure to press for the implementation of his calendar. Newton, as can be discerned from his statistical analysis of Hipparchus's equinox observations, can be credited, historically, with the first application of the technique known today as *linear regression analysis* and also with a remarkable guess about the ancient Greek observations of the equinoxes that recently has been confirmed.

## Introduction

At the turn of the eighteenth century, England was one of many Protestant countries that had not joined the calendar reform promulgated by Pope Gregory XIII in 1582 and had continued to use the Julian (J) calendar. The resulting 10-day difference in dates caused problems in trade with continental Europe. In 1700, the time disparity was expected to increase by yet another day, owing to the application of the Gregorian rule requiring that in years divisible by 100, but not by 400, a 29th day should not be appended to the month of February. The German Protestant states were actively discussing this problem. Despite the diversity of initial responses to the Gregorian reform during the sixteenth century, it had become clear by the beginning of the eighteenth century that it had largely succeeded. Several leading astronomers of the time—Tycho Brahe, Johannes Kepler and Ole Rømer—had endorsed it.1

In 1699, on 23 September (J), the council of German Protestant theologians (*Corpus Evangelicorum*) met at Regensburg [Ratisbon] and adopted a reform of the Julian calendar. In February 1700, this was put into effect in Denmark and all Protestant states in Germany. The new calendar, popularly known as the ‘improved calendar’ (*verbesserte Calender*), was synchronized with the Gregorian calendar by deleting 10 days between 18 February and 1 March 1700. It was kept in step by adopting the same leap day rule.2

However, there remained a difference in the Easter reckoning. At the recommendation of Erhard Weigel, Gottfried Wilhelm Leibniz and several other German astronomers and mathematicians, the date of Easter Sunday in the new calendar would no longer be derived from ‘cyclic’ ecclesiastical tables (based on the mean motions of the Sun and the Moon) but from the ‘most accurate astronomical calculations’ of the spring equinox and the following full moon. The astronomical calculations for determining these dates would be based on Johannes Kepler's 1627 *Tabulae Rudolphinae* and the meridian of Tycho Brahe's former observatory at Uraniborg, his castle on the Danish island, Hven.

In England, where local bishops had defeated calendar reform during the Elizabethan period, discussion of the problem was renewed in 1699 by an exchange of correspondence between the Archbishop of Canterbury and the mathematician John Wallis.3 The English translation of the Regensburg decision appeared in the *Philosophical Transactions of The Royal Society of London* under the title ‘The Conclusion of Protestant States of the Empire, of the 23^{rd} of Sept. 1699, concerning the Calendar’, having been communicated to The Royal Society by Mr Houghton, F.R.S., on 14 February 1700.4

A week later, The Royal Society received a letter from Leibniz asking for help in correcting the *Rudolphine* tables and seeking to enrol the Society among supporters of the reform in England.5 Two extracts from the Journal Books of The Royal Society of London comment on this:

21 February 1699, 1700. ‘A letter from Hanover to Dr. Sloane from Mr. Leibnitz was read, concerning the changing of the Stile. The Dr [Society's Secretary Alfred Sloane] said he heard Mr Newton made a very good calculation of the Year, and that the Settling yt affair might be helpt by yt. Dr Sloane was ordered to wait on Mr Newton about it.’

25 April 1700. ‘The Dr [Sloane] read an answer to Mr Leibniz's letter containing Mr Newton's opinion concerning the alteration of the stile. The V. P. [Sir Robert Southwell] said yt his opinion was yt this paper be sent to Mr Leibnitz, and in ye mean time he procure Mr. Flamsteed's and Dr Wallis's opinions, and send to him: also yt a copy of this be kept.’6

Sloane sent Newton's ‘paper’ to Leibniz on 4 July 1700. Newton's letter7 is very short and, indeed, is aimed at correcting Kepler's tables by providing new positions of the mean Sun and mean Moon at the Greenwich and Uraniborg meridians and suggesting a new maximal equation of centre 1° 56′ 20″ instead of Kepler's 2° 3′. Volume 4 of Newton's *Correspondence*8 lists two additional manuscripts from the same period—one dated 27 February (further: M) and another, undated (further: U)—M contains solar and lunar positions for 31 December 1680 and 1700 and explanations; U offers corrections to Flamsteed's tables for 1701. In letter and spirit they belong to the group of manuscripts we describe in this paper, known as Yahuda MS 24.

## Yahuda MS 24: a preview

The first brief description of the total pool of the Yahuda 24 manuscripts was made by David Castillejo, a Newton scholar, at the request of the Hebrew University in Jerusalem, which has possession of them. We have preserved his catalogue numbers—A, B, C and D—throughout the text, introducing additional pagination.9 Although Newton's proposal represents a mixture of ecclesiastical and astronomical programs, we will discuss each part separately. The most intriguing feature of Newton's manuscript is that he changed the length of the tropical year in the consecutive drafts A, and we suggest a reason. He also left several blank spaces in drafts A, and we suggest what was intended to be there.

In his 1855 *Life of Sir Isaac Newton*,10 Sir David Brewster mentioned two copies, one written specifically for publication, of *Considerations about rectifying the Julian Kalendar*. Explaining its content briefly, Brewster dates them to 1699 and quotes a passage from one of them (C1). This paper establishes that most of the Yahuda MS 24 drafts were written in February to April 1700, and the passage cited not only provides an incomplete picture of the Newton calendar, but also gives Newton's earlier approach to the matter, which he abandoned in the more mature drafts A. The first description of the lunar part of Newton's calendar was made in Robert Poole's *Time's alteration*,11 but it is incomplete.

As witnessed by Newton's remarks in drafts A2 and D1 about the length of the Gregorian lunar month, Newton was ignorant of certain nuances of the Gregorian calendar.12 Christopher Clavius, a Jesuit Professor of Mathematics and the first classical expositor of the Gregorian calendar, is completely unrepresented in Newton's library, as one can see from Harrison's Catalogue13 of Newton's library. Nor did Newton own *Elenchus et castigatio anni Gregoriani* (Lugduni Batavorum, 1595), an attack on Gregorian reform by its foremost critic, Joseph Julius Scaliger.14

By 1700, the weak points of the Gregorian calendar were glaring. The first was the length of the solar year, 365 days 5 h 49 min 12 s, chosen by the Papal Commission because it was close to the mean year given by the Alfonsine tables,15 which were considered the best by most astronomers at that time. Astronomers had already learned that its mean year was greater than the tropical solar year: according to Tycho Brahe, by 27 s; according to Kepler, by 14.5 s. Newton was the first to realize that a simple correction of the Gregorian algorithm could cure the problem, although the price, as expected, was a much longer cycle.

Reforming the Ecclesiastical calendar represented an even greater opportunity. The old epact system, supported by a so-called ‘Dionysius’ cycle,16 a partner of the Julian calendar, was gone; instead, a new, more elaborate, and somewhat cumbersome system of epacts around years *n*00, invented by Aloisius Lilius, had been introduced by the Papal Commission.17 The idea of the ‘most accurate astronomical calculation’ suggested in the Regensburg decision also was unsatisfactory. As Newton would have realized while working since 1694 on his Theory of the Moon's motion, the Rudolphine tables and many that followed were not perfect and could not be easily ‘corrected’, as Leibniz hoped. This fact would surely foster new debates on the ‘best astronomical calculations’ in some problematic cases, as indeed occurred in 1704 and could recur in 1780. The only way to circumvent the problem altogether would be a simple algorithm based on mean lunar parameters, and Newton found one. A provision in the decision of *Corpus Evangelicorum* that it was willing to accept in future a better ‘Cyclus’, if found, could have encouraged him in his search.

## Newton's calendar

### Solar part

The reform proposed by Newton would (i) discard February 29 in years divisible by 100 but not in those divisible by 500, and (ii) add February 30 in years divisible by 5000.18

The first rule made Newton's calendar year shorter than the Julian by 11 min 31 s; the second added back about 17 s. The final value, about 365 days 5 h 48 min 46.2 s, almost precisely matches the tropical year, *ca*. 1700, of 365 days 5 h 48 min 48.24 s (mean solar days). Such a calendar might win immediate support from the public. Because the delta-*t* problem19 was not yet known to the scientific community, everyone would agree that the calendar would perform perfectly over many millennia into the future; however, as Newton's own drafts show, there was no consensus in the scientific world on the length of the tropical year.

Interestingly, in the process of writing several drafts, Newton changed his opinion as many times on the value of the solar year. In the two drafts A, Newton declared that the Julian solar year is ‘too long by, about 11 1/4 minutes (A1) which he further converted to ‘by about 11 1/5 minutes’ (A2). In A3, however, Newton replaced the last fraction with ‘11 1/15m’, and in another place with ‘11 1/20 m’. The latter value, equal to ‘365 days 5 h 48 min 57 s’ also appeared in the margin of p. 23 of D1.

Further, in the text of D1, discussing the advantages of his calendar, Newton claimed that it would err by only a day in 10 000 years, whereas the Gregorian would err by a day in 5000 years. Because the Gregorian year is equal to 365 days 5 h 49 min 12 s, Newton here clearly assumed a tropical year of 365 days 5 h 48 min 55 s. This swift change of opinion demands an explanation.

### A1 and A2

The same expression—‘Pope Gregory XIII about 118 years ago ordained’—in all three drafts A implies that all three were written in 1700,20 and it is quite puzzling why, during such a short period of time, Newton changed his opinion three times on the length of a solar year, the major benchmark for any solar-based calendar. We found no due of how he decided on the length of the solar year in A1 and A2.

We hypothesize that when writing draft A1, Newton consulted the first book on his shelf and that this happened to be that of Tycho Brahe. A table out of *Annales de l'observatoire de Bruxelles*, taken from the NASA (National Aeronautics and Space Administration) Astrophysics Data System,21 provides the following data for the tropical year of various seventeenth-century astronomers (above the base of 365 days 5 h):

48 min 45.5 s for 1602 Tycho Brahe (

*Astronomiae instaurata progymnasmata*);48 min 57.6 s for 1627 Johannes Kepler (

*Tabulae Rudolphinae*);49 min 4.5 s for 1645 Ismael Boulliau (

*Astronomia Philolaica*);48 min 40 s for 1651 Fr. [Giovanni] Riccioli (

*Almagestum Novum*);48 min 8 s? for 1661 Thomas Streete (

*Astronomia Carolina*);48 min 57.5 s for 1687 John Flamsteed (Newton,

*Principia*?);2248 min 34.5 s? for 1719 Edmond Halley (

*Tabulae Astronomia*).

Although Harrison does not list in Newton's private library any books written by Tycho, Westfall23 brings to light one instance when one of Tycho's manuscripts was the focus of Newton's attention: in 1722 Newton presented Tycho's manuscript on comets to The Royal Society, and as president, ordered it printed. It is unclear whether it was his own manuscript, and if so, when he acquired it.

The next value, 365 days 5 h 48 min 48 s, is also so distant from the final one that one can think that it came from another book on astronomy, such as Thomas Streete's *Astronomia Carolina* (1661) or Vincent Wing's *Astronomia Britannica* (1669). According to Harrison, Wing's book [#1743] was heavily perused and dog-eared by Newton.24 Although we do not see the first edition (1661) of Streete's book in his private library, in his copy of the second edition (1710) of Streete [#1575] Newton changed the printed 41″ to 48″. Because there is neither 41″ nor 48″ among Streete's data, quoted by Leonhard Euler,25 we can assume that the change affected the seconds in the length of the solar year, which Newton could have remembered, while his first acquaintance with the book could have taken place either at Trinity College or in Barrow's library.26

### A3 and D1

The values, 365 days 5 h 48 min 56 s or 57 s, considered by Newton in A3, are more familiar, at first glance. As suggested by the *Annales* table, above, they encompass the year of 365 days 5 h 48 min 57.6 s ascribed to John Flamsteed, Astronomer Royal. Collaborating since 1694 with Flamsteed in constructing a new Theory of the Moon, it would be natural for Newton to pick up Flamsteed's parameters for a solar year. It is well known that after January 1699, Newton severed relations with Flamsteed.27 Besides, there is no evidence that Flamsteed used this value for year prior to 1700.

Whereas in A1 and A2 Newton might have acted as a ‘historian of astronomy’, in A3 and D1 he acted as an ‘astronomer’ and ‘statistician’. Pages 21 and 23 of D1 (see Appendix B) witness that he arrived at the value 365 days 5 h 48 min 57 s after many tedious computations. On p. 21, Newton embarked on comparing Hipparchus's times for autumnal and vernal equinoctial observations, made in the mid-second century bc and quoted by Ptolemy 300 years later in chapter III.1 of *Almagest*.28 Newton had the edition of *Almagest* in his library [#1357] with both Greek and Latin texts, edited by John Wallis, in which Ptolemy quoted Hipparchus's autumnal equinoxes in years 162, 159 and 158 bc and 147, 146, 143 bc, and vernal equinoxes in 146, 135 and 128 bc. Newton knew that two trios of Hipparchus's autumnal equinoxes are inconsistent; he therefore chose a statistical approach similar to the modern linear regression model. Not yet knowing today's most popular method, that of the ‘ordinary least square’ (OLS), he applied a quite different, indirect procedure that is reminiscent of his now-classical approximation method.29

Newton separated Hipparchus's six autumnal and three vernal equinoxes into two sets and picked up two middle ones: the autumnal equinox of 27 September 158 bc and the vernal equinox of 23 March 135 bc.30 Then he found the expected time of all other equinoxes if these two are true. For that he used the Horroxian year of 365 days 5 h 49 min.31 Taking the differences between expected times and Hipparchian observations, he found first the deviation for all individual observations and next the average deviation from Hipparchus's data for the autumnal (+2 h 5 min)32 and vernal (−35 min) equinoxes, separately. Correcting all of the equinoxes by these values, he found their expected times (figure 1*a*).33

Next, Newton undertook to find the times of the autumnal equinox of 158 and the vernal equinox of 146 bc by working backwards, using figures from Flamsteed's tables for ad 1700. Computing linearly their mean positions and solving Kepler's equations, he found their true positions (figure 1*b*). Newton's computations reveal several mistakes, not all of which were corrected. Most surprising was his mistake in computing the position of the solar apogee in 1 bc from Flamsteed's tables, and then back to 160 bc, where, instead of the apogee's proper speed (*ca*. 1° 45′), he used the speed of the equinoxes (1° 23′ 20″). Another error occurred when he erroneously added (instead of subtracting) the equation of time (7.3 min) to the position of the March equinox 146 bc (and additionally mistakenly assigned to it 33′ 52″ instead of just 33″) and then had to recompute the whole piece.

Two corrections were inserted later: one to the position of the mean sun (−20″) and another to the position of the solar apogee (+35′). They link p. 23 directly to three Cambridge manuscripts, discussed below.34

Finally, Newton found that the backward-computed autumnal equinox of 158 bc deviated by +21′ 9″ from the expected value, whereas the vernal equinox of 146 bc deviated by −13′ 50½″. Taking an average between these two, he found the asymmetry to be +3′ 39¼″. Newton's solution to compensate for this asymmetry, accumulated over 1850 years, was to increase the speed of the equinoxes or, equivalently, decrease the Horroxian year by 3 s.

Justifying asymmetry, Newton silently implicated Hipparchus's observational instrument, which could have been defective, resulting in bias of different signs for spring and autumn. He could have learned that from Ptolemy's record in *Almagest*35 of Hipparchus's testimony:

… instead of setting the instrument for a specific observation, it was set once for a long period of time; and the errors could have occurred unnoticed by some gradual displacement and been unnoticed over the period of time; and when he observed in Palaestra using bronze rings, which were supposed to be fixed in the plane of the equator, the distortion of their positioning, especially that of the larger and the older of the two, was apparent to such an extent that sometimes the direction of illumination of the concave surface in them shifts from one side to the other twice on the same equinoctial day.

In the twentieth century, analysing pre-Ptolemaic Greek observations, mainly those of Euctemon and Hipparchus, American astronomer Robert Newton36 confirmed Isaac Newton's remarkable guess:

The errors in equinoxes, as we saw earlier, are probably dominated by

*systematic errors* rather than random errors, and the errors at vernal and autumnal equinoxes have a strong tendency to be equal and opposite.

One can wonder whether, hesitating in A3 between subtracting 3 or 4 s from the Horroxian value, Newton performed in his mind the next step of his approximation procedure or simply wanted to attach more ‘weight’ to the (twice as numerous) autumnal equinoxes.37

The final result is so close (within half a second) to Kepler's value for the tropical year that one can wonder whether Leibniz's letter, with its request to correct the Rudolphine tables, came in the middle of his deliberations on the proper calendar year and made an imprint on his approach, prompting him to find the way from Tycho Brahe's value to Kepler's.38

### Five-thousand-year cycle

The choice of cycle also remains a puzzle. From D1 (see Appendix B), it seems that Newton first chose 5000 years as his solar cycle and looked only for the number of days to be deleted from it. He immediately discovered that the Gregorian calendar did not admit this cycle because a non-integer number (37.5) of days ought to be omitted from the 5000 years. His basic procedure of omitting 4 days out of 500 gave a 40-day skip. Two intermediate numbers, 38 days and 39 days, led to years of 365 days 5 h 49 min 4 s and 365 days 5 h 48 min 46 s, respectively. Somewhat surprisingly, Newton chose the lower one. True, the lower value is higher by only one second than that quoted in A1; however, starting his proposal on *Advantages of the Kalendar* on the bottom of D1, Newton implicitly referred to 365 days 5 h 48 min 55 s as a true solar year, whereas it lies exactly in between the two above.

There are two plausible explanations. Either Newton began writing A1 with Tycho's value, and computations at the top of p. 21 of D1 belong to the first stage of his work, whereas the first draft on *Advantages of the Kalendar* was written on the bottom of p. 21 later, or (if D1 was written all at once) Newton chose the lower value of the two on the basis of some external, non-astronomical, consideration.

The external consideration might have been theological: his desire that the calendar year be shorter than the true one. In that case, the equinoxes would slowly drift to the positions they occupied in the time of the historical Jesus. To repeat Newton's argument, in 30 000 years the vernal equinox would fall on 24 March, ‘as it was in the age of Christ’, and in 110 000 years, 1 January would fall on the winter solstice ‘as it ought to do.’ Newton, however, did not explain why 1 January ‘ought to’ fall on the winter solstice.39

In this case, computations in D1 were made later than A1 and A2, though in parallel with A3. A certain gap between the writing of A2 and A3 is seen from the fact that in A2 Newton considered the possibility of dropping 11 days in July (instead of in December), which he abandoned in A3. This could mean that A3 was written much later than the former two, probably in March or April, when already there was not enough time for pushing the reform in July 1700.

## Lunar part

Though the traditional Church calendar was concerned only with finding Easter, or, equivalently, with finding the first day of the first spring month, ‘Nisan’, in both drafts D Newton rejected the Gregorian calendar for its inability to provide an algorithm for finding the first days of all of the other lunar months. Newton suggested creating a Great Lunar cycle of 49 months, subdivided into three smaller cycles of 17, 17 and 15 months, respectively. In each of the three cycles, the odd months would have 30 days and the even, 29 days. Then 49 months would consist of 1447 days, and Newton's mean month would be equal to 29 days 12 h 44 min 4.9 s (in short: a Newtonian month).40

To organize the lunar calendar into a comparatively short 4000-year cycle, Newton proposed two different methods. The ‘deductive’ would discard four months in those Great cycles that embraced the end of 500, 1000, 1500, etc., years.41 The goal of this procedure is not clear.42

The other, an ‘additive’ method, was suggested in A2: add two months every 250 years, which is the same as adding 32 months every 4000 years. In A3, Newton suggested adding one more day to the last month of 29 days in 4000 years. To clarify his intentions, let us make several computations. In every 500 calendar years there areDivided by 1447, the last number gives 126 (Great cycles) and a remainder of 299 (days). In the four months that have to be added every 500 years there are 118 days. Subtracting 118 from 299 we get 181 days. To find the total remainder for the 4000-year period, we should multiply it by 8:or a Great cycle and a day. Overall, there arewhile 8×4=32 months remain outside. Therefore Newton's ‘master equation’ is

According to A3, the lone ‘1 day’ had to be absorbed into the last month of 29 days. This makes the mean month, m_month, equal to43

The last phrase in D1, that his calendar ‘would err only three hours in 500 years’ is ambiguous. It could refer either to the absorption of that single day in a 4000 year cycle, or it could equally refer to the difference between the Newtonian month of 29 days 12 h 44 min 4.9 s and the Horroxian mean month, 29 days 12 h 44 min 3.16 s, which he believed to be the true one.

In the computations in C2 (p. 18), intended to find the time of mean full moons for many years into the future,44 for the Epoch 1701.0 (=31 December (J), noon, 1700), Newton assigned a lunisolar difference in longitude of 24° 33′ 57″. The increment (*modulo* full circles) for 60 years gives 1^{s} 10° 14′ 12″. Therefore, a daily increase in elongation is 12° 11′ 26.7″, which leads to a synodic month of 29 days 12 h 44 min 3.16 s. The source for this synodic month seems to be Flamsteed's tables, which he developed in 1681 from the Horroxian lunar theory.45

## The Easter date

### Forty-nine Newtonian lunar cycles

Newton remarked in draft A that the lunar part of the Gregorian calendar, designed for the determination of Easter, is too hard to implement because it rests on three or four tables,46 so he suggested his own algorithm to find the Easter dates. Like the Gregorian one, it also has two principal parts: first, find the beginning of the lunar month ‘Nisan’, the first month that starts after 7 March; and second, add 13 days to get the date of the full moon and find the first Sunday after that.

The second part is the same as in the Gregorian lunar calendar. However, the first is essentially different and its algorithm deserves to be described in detail.

First, notice that for any two years, four years apart, the civil (Julian) date of the beginning of ‘Nisan’ in the latter one is 14 (=1461−1447) days earlier than in the former. If it falls on or before 7 March, the addition of 30 or 29 days should correct it. To find exactly 29 or 30 days is the major problem in the formalization of Newton's algorithm. It depends on the way in which his Great Lunar cycle is placed against the solar calendar, or rather, on its starting point.

Newton did not specify this point. In A1 to A3 he invariably left a blank space for the beginning of the cycle. Among his computations on p. 18 of C2, he considered, among others, two possibilities: 1 January 1701 (J) could be the second day of47 (i) the first month of the second small cycle (of 17 months)—written twice in the right margin—or (ii) the eighth month of the third small cycle (of 15 months)—written once in the middle of the page.

The particular choice of two cycles (especially of the second) shows that Newton could have considered starting his Great Lunar cycle not only at the beginning of one of three small cycles but also at the beginning of all 49 possibilities, or rather 49 different cycles, which we further denote N01, N02, …, N49, where the number indicates the month of the Great Lunar cycle, which had to begin on 31 December 1700. The two cycles that Newton mentioned explicitly in C2 (see above) are denoted N18 and N42.

We found Newtonian full moons for all 49 cycles by applying the following algorithm. In the first step we computed Julian days (JD) of Gregorian 7 Marches for all years from 1700 onwards. In the second step, for each cycle, in every year we computed the first new moon after 7 March and added 13 days to find the full moon. In the third step we converted JD to Gregorian dates. Finally, we obtained 49 Newtonian first spring full moon sequences and compared them with their Gregorian counterparts in the eighteenth century. The best scores were N09 (22 non-matches) and the worst were N38 (48 non-matches); N18 (26 non-matches) and N42 (37 non-matches) are in the middle of the list. Although the 100-year period was chosen somewhat arbitrarily, this still suggests a different reason for Newton's having chosen N18 and N42, namely that he was aiming at synchronization of his calendar with Gregorian or Protestant Easters, not with the full moons.48

### Easters in eighteenth-century Europe

As astronomers appointed to be in charge of the Ecclesiastical Protestant calendar49 rapidly discovered, Easter Sunday dates calculated according to the astronomical full moon would occasionally differ from the dates obtained from the Gregorian cyclical reckoning. Such occurrences (without taking the Passover postponement rule into account) would take place in 1700, 1724, 1744, 1778, 1798, etc., and lead to a one-week discrepancy between the two Easters.50 Another more serious ‘equinox crossing’ problem could cause a month's discrepancy. The full moons of 1704 and 1780 fell too close to the vernal equinox. The Gregorian calendar formally anchored the latter on 21 March and, because its cyclic first 1704 spring full moon fell on 21 March, placed Easter on 23 March. Half of the Newtonian calendars, like N18, had 1704 full moon and Easter on the same days as the Gregorian. The other half (those beginming with a 29-day month) had the corresponding 1704 full moon fall on 20 March, requiring postponement—their first spring full moon would fall on 19 April and Easter must be placed on 20 April 1704. This was the situation with N42, for example.

The Protestant Almanacs for 1704 finally settled their Easter on the Gregorian date.51 Uncertainty about the exact rules in February 1700, and/or anticipation of their changes in the near future, could have raised some questions in Newton's mind.52 Most of p. 18 of C2 he devoted to computations of the mean astronomical full moons in 1704 and 1780, making a mistake in the former by placing the full moon at 22:38 on 10 March (J), a day later than it should be.53 This mistake, discovered later, could have given him the idea that the true astronomical full moon would occur before the 1704 true vernal equinox (9 March (J), 13:50) and switched his attention from N18 to N42. However, Newton did not make any written conclusions and we can only guess about the direction of his thoughts.

### Conjectures

Newton could have looked for a cycle in which Easters would closely match Protestant Easters in the eighteenth century (see Appendix C). N42 agreed with all of these, including the controversial years 1724 and 1744, though it disagreed in the years 1720 and 1780. N18 did not score any better—it matches the Protestant Easters in 1704 and 1720, as well as in 1724 and 1744, but disagreed in 1751, 1771 and 1780. The ‘full moon champion’, N09, disagrees with Protestant Easters on four occasions: 1724, 1744, 1751 and 1771. The best matches—N46 and N48—agree with all of the Protestant Easters except in 1780.54 If Newton wanted the best match, he had to choose one of the latter two. Then N42 could be explained as a mere ‘slip of the pen’.

Yahuda MS 24, however, displays no sign of his concern with years 1724 and 1744. On the contrary, the bottom of p. 17 of C2 (see Appendix B) shows his computations of the 19-year cycle 1701–1719 with the Newtonian mean month, where he was uncertain which starting point to choose and where to place the full moon of 1704:55 on 21 March or 20 April. We finally conjecture that the problem of where to place Easter 1704, and therefore uncertainty about the choice of the cycle, was the main reason for Newton's uncertainty about the cycle, and this led him to leave blank spaces in all drafts A.56

## Division of the year and the Ecclesiastical calendar

In drafts A and C, Newton defended different ideas on how to divide the solar year. In A1 and A2, Newton complained that the Julian calendar was maimed by the Roman Senate reform, made in 8 bc to honour Augustus by adding a thirty-first day to the month named after him and depriving February of a day. He believed that Pope Gregory would have made a great achievement had he reversed that transfer and rearranged the months. In C1 Newton put forward quite a different idea: his close imitation of the contemporary seasonal divisions of the year—a temptation that later was attempted repeatedly by revolutionary nations. Newton suggested assigning 30 days to all winter months and 31 days to all summer months except the last one in the non-leap years.57 Because the proposal in drafts A asks for only minor correction (a reversal of the Roman Senate's decision), this shows that the text of C1 was also written earlier than those of drafts A.

In the calendar tables on pp. 15–16 that conclude C1, Newton suggested two quite different ways of dealing with the fixed feasts (Christmas, Lady Day, etc.). He arranged the year in two parallel columns, shifted by some 22–23 days against one another, marking along the way all the fixed feasts. The goal was to return all the feasts to their original positions, as set in the first century.58

The right column (with the length of the months remaining unchanged, and none of the days being omitted) suggested moving the fixed feasts to the contemporary cardinal points: Christmas would go to the winter solstice (11 December), Lady Day to the vernal equinox (11 March), etc. Thirteen days would be removed in October, from the 16th to the 28th.

The left column was arranged according to the new divisions of the year, as in C1. The date 31 September had to be an intercalary day. Christmas was set on 3 January, Lady Day on 3 April. For that, he was ready to omit 22–23 days from the calendar, which would move all the feasts eight or nine days earlier.

According to A1–A3 and C3, all the fixed feasts had to stay in place after the first 11 days of December 1700 were omitted. The historical preamble of drafts A resembles the economic concerns listed in C1. The theological erudition of C3 either remained unused in drafts A or belongs to a later period.

## Civil part

The major civil amendment propounded in the A drafts was that the year should begin on 1 January and not on 25 March.59 This would lead to the abolition of the peculiar old system in which February 24 was considered the true ‘leap’ day and was doubled in bissextile years.60 In draft B, however, Newton still discusses two different Sunday letters for 24 February. Therefore B is the earliest draft.

It seems that to prevent the government from losing 3% of its income in one year, Newton, the civil servant, suggests in the A drafts stretching tax deductions over three years.

## Discussion

### Relationship to three Cambridge manuscripts

Yahuda MS 24 C2 (p. 18) and D1 (p. 23) immediately predate three Cambridge manuscripts: M, U and L (memorandum, a future letter to Leibniz), cited in volume 4 of Newton's *Correspondence* under numbers 622 (February 27), 623 (undated) and 624 (25 April). The value of the tropical year found in D1 (365 days 5 h 48 min 57 s) had been accepted in M. The correction of −20″ to the Sun's mean position, applied in D1, is also seen in M, whereas in U the correction comes as a pair: −10″ or −20″.61

The role of U becomes particularly clear: it carries several corrections from D1 and C2 toward M's values. Although two corrections—(i) +16′ or +18′ 36″ for the motion of solar apogee for 100 years from 1° 23′ 20″ of D1 to 1° 44′ 00″ of M, and (ii) +2′ 10″ or 2′ 18″ for the Moon's mean motion from 10^{s} 15° 17′ 47″ of D1 and C2 combined62 to 10^{s} 15° 19′ 50″ of M—are close, but somewhat imprecise, the third one—(iii) +36′ 10″ for the position of solar apogee from 3^{s} 7° 08′ 20″ of D1 (recomputed backwards) to 3^{s} 7° 44′ 30″ of M—proves the case (see Appendix D).

The problem of dating U we resolve in the following manner. It carries two explicit references to Flamsteed's tables, which (references) are already wanting in M and in the letter to Leibniz.63 Seemingly, the first time that U's unique pair of corrections (−20″ or −10″) appears is in Flamsteed's letter of 18 June 1700 to Newton.64 But it was rightly doubted whether this lengthy letter by Flamsteed was ever sent.65 Besides, the memorandum of David Gregory of 7 July 1698 says, ‘Nor will be any mention of Flamsteed.’66 It seems that U was composed in February 1700 from Flamsteed's pre-1698 data.67

In Newton's 25 April memorandum (future letter to Leibniz), the mean place of the Sun for midday of 1 January (J) 1701, 9^{s} 21° 42′ 38″ is compatible with p. 23 and M's 9^{s} 20° 43′ 50″ for midday of 31 December 1700, having the −20″ correction already embedded. The position of the solar apogee (3^{s} 07° 44′ 30″) in the memorandum also matches M. Although the position of the mean Moon has a difference of 13″ from M, there is little doubt that the reply to Leibniz was designed first as a broad explanation, M, which was later condensed to a short memorandum.

This means that corrections were inserted into M no later than 25 April 1700, although the proper text was completed even before 25 March 1700, when David Gregory made a copy of it.68 Therefore, p. 23 of D1 with the same corrections was composed in parallel to M, at some time in April.

M's fate was to become the backbone of an entire chapter in Newton's next book. The letter with Newton's corrections that Sloane sent to Leibniz on 4 July 1700 concludes:

The Royal Society have labored to get his [Newton's] Theory of the Moon … but his excessive modesty has hitherto hindered him, but the Society will do what further they can with him.69

Although the Society's labours took two years to yield fruit,70 it is quite amusing that the beginning of Newton's intellectual effort was spurred by the calendar concerns and was partly provoked by his future adversary Leibniz.

### Dating Yahuda MS 24

Sloane's remark of 21 February that ‘he heard Mr Newton made a very good calculation of the Year’ could refer only to the drafts A1 and A2, and suggests that at least one of them was already written down. The chronology of drafts A can be deduced from Newton's consecutive proposal for when to discard 11 days: (i) the last 11 days of May in A1, (ii) the first 11 days of July or December in A2, and (iii) the first 11 days of December in A3.

From A1's vision of the reform at the end of May and assuming that it was written in mid-February, it follows that Newton expected at least three months would be necessary between promulgation and implementation of the calendar reform. This conjecture could date A2 and A3 as well. The hesitation between ‘July’ and ‘December’ suggests that A2 was composed sometime in March. In A3 only ‘December’ remained, which suggests some later time, most probably late April.

### Why did Newton fail to promote his calendar?

It is no surprise that Isaac Newton, a public servant, just appointed to be Master of the Mint, decided to try his hand at calendar reform. The calendar disparity with the rest of Europe seriously impeded trade.71 His decision, however, was Newtonian: his calendar would not be a copy of the Gregorian, but an improvement over it. A completely new and more elegant calendar would be an ingenious political solution. More particularly, it would justify the long-time English stand against the Gregorian reform and support English intellectual leadership in the pan-European war against France. As a byproduct, on the verge of the quickly approaching priority dispute with Leibniz about the discovery of calculus, it would discredit the latter's arguments and political efforts in favour of joining the reform. In future, while agreeing with the Gregorian calendar for a long period of time, eventually—with God's help!—the new calendar would push its rival out of the race.

Why did Newton fail to publish his calendar? The standard answer, in view of his character, is that he feared stirring up a public controversy. Another, according to Sloane, was ‘his excessive modesty’.72 Both would be a gross misrepresentation of his personality in 1700, after spending five years practically in charge of the Royal Mint. In that particular year, he had been promoted from Warden to Master of the Mint and was offered the Mastership of Trinity College, which he declined.73 Besides, Newton's reputation as the author of *Principia* would have given him added authority in future debates. Even if the 1704 Easter was his stumbling block, this cannot explain his later tactics. After all, he made sure that his highly controversial *Chronology of Ancient Kingdoms Amended* would be published after his death. Yet when this happened in 1728, his calendar was still buried in a pile of historical drafts.

The best explanation we can offer is his disillusionment with its major solar part. As we saw, in every consecutive draft A, Newton increased his estimate for the solar year from an initial 365 days 5 h 48 min 45 s to a final 365 days 5 h 48 min 57 s. Two years later, in his 1702 *Theory of Moon's Motion,* he increased it further by 0.6 s. With that, the Gregorian year was only 14.4 s greater than the ‘true’ one, whereas the Newtonian year was already about 11.4 s shorter: a difference of 3 s, which could hardly impress anyone.

## Conclusions

The following conclusions may be drawn.

Newton did not have first-hand knowledge about the Gregorian calendar.

The three major drafts A were written in February, March and late April, respectively.

The ‘problematic’ Easters of 1704 and 1780 caused Newton to postpone the choice of lunar cycle and leave blank spaces in all three drafts A.

Unlike Leibniz, Newton built his calendar on

*mean*motions of the sun and moon.Computing solar year from early astronomical observations by Hipparchus, Newton invented a certain technique known today as

*linear regression analysis*.The wrong opinion about the tropical year (365 days 5 h 48 min 57 s) forced Newton to withdraw publication of his calendar.

The question of the source of the solar value in A1 and A2 (Tycho Brahe, Streete or Halley?) is still open.

## Epilogue

Ironically, having decided to join the Gregorian Reform half a century later, England did not follow Newton even in the least important issue: 11 days were dropped in 1752, not in May, July or December, as Newton proposed in his drafts A, but in September.74 Newton's craft in constructing a new lunar and Ecclesiastical calendar also was in vain, but his failure was truly prophetic on the European scale: in 1775, in anticipation of the next disparity in 1778 between the Protestant and Gregorian Easters, the Prussian King Frederick II and the German Emperor Joseph II signed an order to use the Gregorian calendar, and only that, throughout their lands.

## Acknowledgements

We thank Ayval Leshem (Bar Ilan University) for help in transcription of Newton's original text; Heiner Lichtenberg (Bonn) for discussions about the lunar part of the Gregorian calendar and references to Clavius's works; Joan Griffith (Annapolis, MD) for polishing style and catching several errors in the transcript; John North (Oxford) for reading the first draft of our manuscript and for several helpful remarks; John Young (London) for help with quotations from Newton's library books; Robert van Gent (Utrecht) and Joachim Krueger (Bremen) for discussion on Easter 1704 in The Netherlands and the German Empire; Wolfgang Dick (Potsdam) and Klaus-Dieter Herbst (Technical University of Berlin) for information on Kirch's Almanac for 1704 and Kirch's biography; and Owen Gingerich (Harvard University) for information on Hoffman's 1704 Ecclesiastical calendar. A.B. thanks Yuval Roichman and Ron Adin for encouragement and financial support, Alex Friedman and Tamar Barzilai for technical support (all of Bar Ilan University).

## Appendix A Newton/Yahuda MS 2475

### A3 [pp. 8–10], with notes about A1 [pp. 1–3] and A2 [pp. 5–7]. Considerations about rectifying the Julian calendar

〚**p. 8**〛Times were at first reckoned by returns of day and night, new and full moon, summer and winter. Whence the oldest years consisted of lunar months and where twelve months were found too short a thirteenth was added to make up the year. These months began not at the conjunction of the Luminaries but at the first appearance of the new moon which used to be between 18 and 42 hours after the conjunction if the sky is clear.76 And because the new moon appeared at sunset the days of the lunar month began in the evening.

The just length of the summer and winter is the return of the Sun to the same equinox, that is 365 days and 6 hours wanting about 11 minutes and *3 or 4* *seconds*77 〚*11 1/4* in A1 and *11 1/5* in A2〛. And there being something more than 12 moons in summer and winter and something more than 29 days and half in a Moon, the first ages look at next round numbers of 30 days to a Month and 12 months to a year and so made the civil year to consist of 360 days, whence came the division of a circle into 360 degrees.

But this year being too short by five days and almost six hours the Egyptians added five days to the end of it and so made the year to consist of 12 lunar months and five days. And this year was in use in Egypt at least from the days of Amenophes the grandson of Sesostris and seems to have been received in the Assyrian and Persian Monarchies.

The Greeks used lunar months first of 30 days and then of 29 and 30 alternately, and contrived several ways to adapt those months to the year, the principal of which was in every 19 years to intercale 7 months, 78 whence came the golden number.

At length Julius Caesar 79 in lieu of the six hours added a day once in four years to the year of 365 days and by adapting this measure to the old Roman year made a new year of 12 months of various length without any good order or uniformity or agreement of the months with the stay of the sun in the twelve signs. And the Senate in honour of Augustus took a day from February and added it to August. 80 And so Caesar and the Senate together left us a year more irregular and intricate than the Egyptian, but better on this account that the same months keep better to the same seasons of the year. In the Kalendar of this year the Lunar years were supplied by setting the golden number to the days of the new Moons for 19 years together.

And because the Julian solar year proved too long by about *11 1/15* 〚*11* *1/4* in A1 and *11* *1/5* in A2〛 minutes, that is by a day in *130 years*81 〚*128 years* in A1, *128 or 129 years* in A2〛, Pope Gregory XIII about 118 years ago82 ordained that three days be taken from it in four hundred years by omitting the 29th day of February in the end of every 100 years excepting at the end of every 400. And to bring the vernal 〚**p. 9**〛 equinox to the 21st of March on which it fell in the time of the Council of Nicea83 he took 10 days from this year: whence arose the difference of 10 days between the old and new stiles 〚styles〛 in the century which is now expiring. And because the rule for finding the new moons by the Golden number erred about an hour and half in 19 years or a day in 312 years84 he corrected that rule every 300 years or thereabouts by the alteration of the day.85

Had Julius Caesar divided the year into four equal quarters according to the four cardinal periods of the solstices and mean equinoxes and then divided every quarter into three months as nearly equal as he could make them which he might have done by making the month of 30 and 31 days alternately and the last month of 31 days in leap years and 30 days in ordinary years so that in the leap year all the odd months should have 30 days and all the eaven 31, he would have made the Roman year of a regular and convenient form and well adapted to the motion of the sun and periods of summer and winter. And the Pope's correction would have made it lasting.

But without the consent of a good part of Europe I do not think it advisable to alter the number of the days in the months. The question is now whether the old stile should be retained in conformity with antiquity or the new received in conformity with the nations abroad. I press neither opinion but whenever the latter shall be resolved on I believe the best way may be to receive the new stile without the Gregorian calendar by an Act of Parliament to some such purpose as that which follows.

For avoyding the difference of recconing by the old and new stiles which is troublesome in commerce between this and other nations, it may be enacted that in the year of our Lord 1700 the first eleven days of December86 shall be omitted rejected and abolished out of that year and the twelfth day of the that month shall immediately succeed the month of November without any alteration in the days of the week or in the form of Julian calendar, excepting that the Golden number and epact may be omitted. And this accompl〚ishment〛 or stile shall thence forward in all his Majesty's Dominions be received used and understood in all Dates and recconings of time for keeping of set festivals fairs Birthdays and all other anniversary days and for performance of all covenants duties and services and payments of interest, rents, salaries, pensions, wages and all other debts and dues whatsoever with an abatement of interest rent salary pension or wages for and proportional unto eleven days in the first payment of interest rent salary pension or wages which shall by virtue of any covenant grant act or deed had made or done before the _______ day of _______ become due on or after the 12th day of December87 above mentioned, that is to say with an abatement of the hundredth part of three years interest88 rent salary pension and wages.

Provided nevertheless that all debts which ought to be paid and all things which ought to be done on any of the said eleven days of December which are hereby abolished, shall be paid and done on the same 〚**p. 10**〛 day or days on which they should have been done if this Act had never been made.

And for avoiding the double recconing by the civil and ecclesiastical years between the last day of December and the 25th day of March the ecclesiastical year shall in all his Majesty's Dominions from and after {the month of December of} the year of our Lord 1700 begin on the first day of January forever and be no longer dated from the 25th of March.

And that the year may be of a just length and the month remain constant to the seasons of summer and winter, it may be further enacted that the 29th day of February shall be omitted in the last year of every century excepting the last year of every fifth century and that in the last year of every fiftieth century a day shall be added to the end of February, that is to say the month of February in the years 1800, 1900, 2100 etc shall have 28 days and in the years 2000, 2500, 3000 etc each shall have 29 days and in the years 5000 and 10 000 etc (if the calendar should extend so far) it shall have 30 days.

And because the movable festivals and law-days depend upon the course of the Moon and the vulgar rule for determining that course needs frequent correction and is now grown very faulty, it may be further enacted that the lunar month shall be recconed to consist of 30 and 29 days alternately in three periods or cycles of months perpetually to succeed one another, each of which periods shall consist of an odd number of months, the two first of 17 and the third of 15 and the first and last month of each period shall contain 30 days so that all three periods summed up together shall make a larger period of 49 lunar months containing 1447 days or 4 solar years wanting a fortnight.89 〚In A1: And the period of 15 months once in every 1000 years that is to say next ensuing the years of the Lord 2000, 3000, 4000, 5000 etc. shall have eight months deducted from it, and shall consist of the seven remaining months and no more.〛 And the first day of January which shall be in the year of our Lord 1701 shall be the _______ day of the _______ month90 of the larger period of 49 months. And from thence forward the festival of Easter shall be kept on the Lord's day next after the 14th of that lunar month which shall begin next after the seventh day of March.

〚In A2: This rule for determining the course of the moon is much more simple and exact than that of the Golden number used by Pope Gregory for that rule errs an hour and an half in 19 ½ years91 or a day in 312 years and so needs frequent correction, this errs only a day in 4000 years. And if in the end of every 250 years the cicle of 15 months have two months of 29 and 30 days added to it so that all the three cycles do once consist of 17 months the rule will be much exacter.〛 And at the end of every four thousand years a day shall be added to the last lunar month of nine and twenty days.92

### B [pp. 11–12]. The use of the Kalendar for finding the Lord's day and the Moveable Feasts

Divide the year of our Lord by 28. Seek the remainder in the following table and you will find under it the Sunday Letter for that year. And in the third column of the Kalendar where you see that Sunday Letter the days are Sundays. In Leap year there are two Sunday Letters: the one counting till Feb. 24 and the other for the rest of the year.93

Divide the year of our Lord by 19 and the remainder increased by an unit shall be the Golden Number {or Prime} for that year. And in the first column of the Kalendar when you find that number the days are new moons … according to the calendar of the 14th day of moon is the Full Moon. 〚several lines crossed out〛 … Easter day is always the first Lord's day after the Full moon which happens upon or next after the one and twentieth day of March.

Advent Sunday is always the nearest Sunday to the Feast of St Andrew whether before or after.

〚**p. 12** full of computations; in the middle, twice: **The Lord Chief Justice Greby**〛

### C1 [pp. 13–14]

The Julian year now in use is very irregular. February has but 28 days and the other months 30 and 31 days without any regular order or reason for that irregularity.

The best form of the solar year is to divide it by 4 cardinal periods of the Equinoxes and Solstices into 4 quarters, so that the quarters of that year may begin with the Equinoxes and the solstices as they ought to do, and then to divide every quarter into 3 equal months which will be done by making the six winter months to consist of 30 days each and the six summer months of 31 days each excepting one of them suppose the last which in the leap year shall have 31 days in the other years only 30 days. Although the end of every hundred years omit the intercalary day in that leap year excepting at the end of every five hundred years. For this rule is exacter than the Gregorian of omitting it at the end of every hundred years excepting at the end of every 400 years. And this recconing by five hundreds and thousands of years is rounder than the other by four, eight and twelve hundreds. And this I take to be the simplest and in all respects the best form of the civil year that can be thought of.94 And this is all the reformation of the year which need be made at first.

As for the Ecclesiastical year if the fixed feasts shall be placed on the same of the months of this New Year as in the Julian year, they will come nearer to the truth than they do at present. For they are now become about 14 days later95 than they were in the first century in respect of summer and winter whereas in this new year they will be only eight or 9 sooner.96 So that the Calendar will be amended almost half (at best〚??〛) by this new year without translating the fixed feasts to other days of the months.

But if it may be allowed to translate them to other days of the months so as to bring them nearer to the places where they were in the first century in respect of Summer and Winter the Calendar made be still amended as follows.

Let Lady day {the first day of Ecclesiastical year} be removed from the 25th of March to the first of April and the Ecclesiastical year will begin at the Equinox and on the first day of the month as it ought to do, whereas in the present Julian year it begins neither at the Equinox nor on the first day of the month but on the 25th of March and 16 days after the Equinox.

Let Michaelmas be removed from the 29 of September to the 1st of October and these two principal days of payment will fall on the Equinoxes and on the first days of the month which begin the spring and autumn quarters of the year which is very proper and ready for recconing, and also more just for contracts. For the summer half year is eleven97 days longer than the winter half year in the vulgar Calendar but in this new one the difference will be but 5 days.

In like manner to regulate the days of quarterly payments let St John Baptist's day be removed from the 24th of June to the 4th of July and Christmas of 25th of December to the 1st of January, or perhaps to the 2nd that it will be distinguished from the New Years Day.

Thus will the year become fitter for civil uses and the festivals be reduced within a day or two to the places where they were in the first century in respect of summer and winter; whereas they now err 14 days98 from those places. And the like corrections may be made of 〚**p. 14**〛 all the other moveable festivals by setting them 7 or 8 days later.99

Easter is determined by making it the first Sunday after the first full moon after the first of April and the rest of the moveable feasts are determined by their distance from Easter as in the Vulgar Calendar.

The old Rule for finding Easter by the Prime of Dominical Letter is to be corrected at the end of every hundred or two hundreds years by ecclesiastical authority and so is the Rule of finding the new Moon by the Epact in the margin of the Calendar. And with such correction both Rules maybe retained for ever.

### New Ecclesiastical Calendar

〚**p. 15**: calendar for July–September. **p. 16**: calendar for January–June.〛

### C2 [pp. 17–18]. Computations

〚**p. 17**: computations of the mean full moons in January–April of years 1701–1704; on the bottom: computations of the Paschal full moons for years 1701–1719; **p. 18**: computations of the mean full moons for the years 1704, 1780, 1856, 1867.〛

### C3 [p. 19]. Notes about Ancient Chronology

The only feasts in the beginning till the reign of Trajan100 were the Lord day, Easter & Whitsunday. See Origen b 8 cont. Cels.101 Christmas began to be celebrated diverse places about the year 190 (Theophilus Caesariensis102 in epist. paschal.)

The Martyrs began to be commemorated on their passion days about the year 170 and these days at length were celebrated as feasts by the institution of Constantine the great (Euseb. in vit. Const. b.4) who also instituted the observation of Friday. Euseb. ibid. The heathens feasts turned into Christian. Theodoret b 8 de martyribus and Greg. M. b 9 Cap 71 citante Hospin. De Origin. Christ. Fest. p 15.

The Greeks celebrated the Epiphany or Baptism of Christ on the same day with his birth, {the Christmas on January 6} Hospin ad Jan 6.

Timothy martyred on Jan 24. Pauli Conversio Jan 25. The burning of light on Candlemas Day Feb 2 taken from the sacra of Ceres seeking her daughter Proserphina with light and torches. Feb 1. The Bacchanal rights 〚rites〛 turned into Christian carnivals in the first days of the Quinquagesima or week before Lent. Matthias Feb 24. Festus annunciationes Mariae March 25. St Mark martyred Apr 25. The Greeks celebrated {to all the Apostles} the feast of Peter and Paul Jan 29. The Latins that of Philip and James May 1. At length the day is left Phil. and James alone. Quadratus May 26. Nativity of John Baptist June 24. Peter and Paul on June 29 on which day the Heathens celebrated the feast of Hercules and the Muses. July 25 St James. Aug 24 St Bartholomew Sep 21 St Matthew. Sept 25 Cleopas. Sept 25 St Michael and all angels. Octob 18 St Luke. Octob 28 Simon and Jude. Novemb. 28 Adventus Domini. Nov 30 St Andrew.

### D1 [p. 21]. COMPUTATIONS

〚First four lines—the length of the year with 39, 40, 38, 37½ days deleted from 5,000 years.〛

OBSERVATIONES HYPPARCHI {autumnal equinoxes for years 162, 159, 158, 147, 146, 143 bc and vernal equinoxes for years 146, 135, 128 bc}

### [pp. 21–22]. Memorandum on the advantage of this Kalendar

And at the end of every 500 years the larger period of lunar months which shall or should be then running shall contain only 45 lunar months and the three lesser periods of which that larger period consists shall each of them contain only 15 lunar months, the two last months of the two periods containing 17 months being omitted.

The advantage of this Calendar above the Gregorian in respect of the solar year is that the solar year in the Gregorian errs a day in 5000 years and by that error recedes from the state it had in the age of Christ, in this it errs a day in 10,000 years and by that error approaches the state it had in the age of Christ so that in 30,000 years the equinox will fall on the 24th of March as it did in the age of Christ and in 110,000 years the beginning of January will fall on the winter solstice as it ought to do. Also the recconing by 500, 1000, 1500 etc runs in rounder and fewer numbers than by 400, 800, 1200, 1600 etc. And tho the Calendars differ yet they will agree in stile for 700 hundred years to come.103

〚**p. 22**〛 The advantage in respect of the Lunar year is much greater. For in the Gregorian Kalendar the full Moon on which Easter depends is not to be found without the help of three or four Tables, and when you have the full moon there is no rule in that Kalendar for finding the other full moons and the new moons throughout the year. But in this Kalendar all the new and full moons are found perpetually without any Tables or any other recconing then the continual addition of the 30 or 29 days 〚in D2: alternatively〛 which is so very easy a work that any Novice can perform it. And besides this rule is much exacter than the Gregorian for that errs three hours in 39 years,104 this errs but three hours in 500 years105 〚in D2: and may be corrected every 500 years to keep it exact〛.

### [p. 23] COMPUTATIONS

〚On the right margin: Annus equinoxiatis 365. 5h 48′ 57″ and below 365¼ −11′ 1/20〛

### D2 [p 24]

The advantage of this Kalendar … 〚see D1〛.

## Appendix B Four pages (figures A1–A4) with Newton's own handwriting from the Jewish National and University Library, Jerusalem

## Appendix C Three Newtonian Easter cycles versus the Gregorian one in the eighteenth century

The shadings in the table below have the following meanings:

## Appendix D Flamsteed's and Newton's Parameters of the Sun's and Moon's Motion

Flamsteed 1672–81: first appeared in 1672 Horroxian *Opera Posthuma* (see N. Kollerstrom, note 22 to the main text); republished in his own 1681 *Doctrine of the Sphere*, except for the solar apogee's position (see C. Wilson, n. 26 to the main text, p. 192).

Flamsteed *ca*. 1700: solar parameters that Astronomer Royal obtained in the 1680s to 1690s; L. Euler could quote them in 1745 (see note 25 to the main text) from the solar tables published in 1707 by W. Whiston in his *Praelectiones Astromomicae* (see C. Wilson, *ibid.*).

Newton 1702: parameters of Newton's *Theory of the Mean Motion*, published by D. Gregory in 1702 (see N. Kollerstrom, note 22 to the main text); they are practically the same as in ms. M, except for −20″ correction for the position of the mean sun (see note 70 to the main text).

- © 2005 The Royal Society

## Notes

- 1.↵
See extensive discussion in

*Gregorian reform of the calendar. Proceedings of the Vatican conference to commemorate its 400th anniversary* - 2.↵
Within the year they were joined by the six remaining Julian-observing parts of The Netherlands and the northern Swiss cantons. See

- 3.↵
See some peculiar details about Wallis' personality in

*Gregorian reform* - 4.↵
The full text is given in

- 5.↵
Full text and translation in

*The correspondence of John Flamsteed, the first Astronomer Royal*(compiled and edited - 6.↵
Quoted from

*The correspondence of Isaac Newton* - 7.↵
- 8.↵
- 9.↵
See Appendixes A and B. Compare with http://www.newtonproject.ic.ac.uk/catalogue/A02.htm (The Newton Project).

- 10.↵
- 11.↵
Poole

- 12.↵
The most important is its adaptability to possible changes in solar and lunar parameters suggested in the Pope's 1582 Bull,

*Inter Gravissimas*(see its first page in*Gregorian reform*and repeated

*verbatim*inSee extensive discussion in

where the boundaries for permissible parameters are established.

- 13.↵
- 14.↵
Although Julius Scaliger's

*De emendatione temporum*was found among Newton's books [*ibid.*, #1454], it does not touch on the Reform. - 15.↵
Compare

*Gregorian reform* - 16.↵
The 19-year cycle that equates 235 months with 6939¾ days.

- 17.↵
The Papal commission called the author of the reform by his Italian name, Luigi Giglio.

- 18.↵
Newton's calendar allows changes only in years

*n*00. This is in line with the Gregorian reform proper and with provisions contained in Clavius’*Explicatio* - 19.↵
The rotation of Earth is slowing down, and as a result the tropical year, expressed in days, is diminishing by about 5 s per 1000 years. See

- 20.↵
Brewster

- 21.↵
http://adsabs.harvard.edu//full/seri/AnOBN/0001//0000230.000.html

- 22.↵
We did not find in the first edition (1687) of

*Principia*any support for the value ascribed to Flamsteed, but this is exactly the value that Newton used for his*Theory of the Moon's motion*(1702). See N. Kollerstrom,*Newton's 1702 theory of the Moon's motion. A computer simulation*(http://www.ucl.ac.uk/sts/nk/newt-tmm/1702nwtn.zip).The value we deduced from the Horroxian tables is 365 days 5 h 49 min. From N. Kollerstrom, ‘Flamsteed's 1681 Horroxian lunar theory. A computer simulation’ (http://www.ucl.ac.uk/sts/nk/zip/flam1.zip). The value attributed to Halley seems plausible before 1700 but not afterwards. - 23.↵
- 24.↵
Unfortunately we did not have access to Wing's

*Astronomia Britannica*to verify our conjecture. - 25.↵
As can be seen from

- 26.↵
- 27.↵
Westfall,

- 2.↵
Ptolemy's Almagest (translated and annotated by

Fortunately, Newton did not use Ptolemy's own observations, thus avoiding a trap that many astronomers before him fell into. Seemingly, Jean-Baptiste

**-**Joseph Delambre was the first to notice a fallacy with Ptolemy's observations; see - 29.↵
To find the best-fitting line

*Y*=α+β*X to the set of observed data {*X*_{n},*Y*_{n}}, where*X*_{n}are years and*Y*_{n}are day times of the equinoxes, the OLS method claims that there is a unique pair of*α*and*β*such that the sum of the squared distances from every point to this line is minimal. Thus found,*β*is the best linear unbiased estimator of the length of the year. See, for example, R. Ramanathan,*Introductory econometrics with applications*, 5th edn (Mason, OH: South-Western, 2002, pp. 80–90). Newton's estimator resembles another one, β†=/, where and are the*averages*over*Y*_{n}and*X*_{n}, correspondingly; estimator β† is*less efficient*than β and*biased*when α≠0 (see*ibid*, pp. 123–4). - 30.↵
The vernal equinox of 23 March 146 bc, for which Hipparchus assigned two different times, 18 h and 23 h, gave him some trouble. Newton considered both data as valid and assumed the average value, 20:30 h. See

*Ptolemy's Almagest*, - 31.↵
Newton first tried to work with the year of 365 days 5 h 48 min 45 s, as seen from the 45 min differences between the equinoxes of the years lying four years apart (in the second row of p. 21 in D1); however, he did not complete these computations.

- 32.↵
The ‘5 min’ was a result of Newton's incorrect summation and must be removed.

- 33.↵
At this point, subtracting 2 h 16 min, Newton converted all the times from Alexandrian to Greenwich.

- 34.↵
The author of these corrections was the Astronomer Royal. According to Wilson,

*op. cit*. (note 26), p. 192, in ‘comparing his observations between 1679 and 1690, Flamsteed found a correction of 35′ for his earlier apogee's position’. - 35.↵
Ptolemy's Almagest,

- 36.↵
(note 28)

- 37.↵
Newton completely disregarded the problem, with which modern scholars have been much occupied, of whether the first trio of autumnal equinoxes was observed by Hipparchus himself. See

*Ptolemy's Almagest*,(note 28) n. 8.

- 38.↵
There is a slight possibility that the first value also came from computations, albeit erroneous. As we said above, incorrectly adjusting the equation of time for the 146 bc vernal equinox on p. 23, he first found the backward-computed value to be off the expected Hipparchian by +18′ 31″. With the above-mentioned +21′ 9″ error between the value for the 158 bc autumnal equinox and the backward-computed value, the average comes to 19′ 50″, or, translated in day-time, 15.5 s. This correction, if materialized in writing, would bring him close to the 365 days 5 h 48 min 45 s of A1.

- 39.↵
Newton's idea has several predecessors in the Papal Commission's writings. See

*Gregorian reform*, - 40.↵
It is more precise than the traditional ‘Dionysius’ 19-year cycles with mean lunar months of 29 days 12 h 44 min 25.5 s. However, the Gregorian calendar, in the first approximation, precipitates a removal of eight days from the lunar cycle every 2500 years. This leads to a mean month that is within half a second of the Horroxian mean month and the modern synodic mean month. See Lichtenberg,

.

- 41.↵
Either 8 months of a 15-month small cycle every 1000 years (A1), or the two last months in both 17-month small cycles every 500 years (D1).

- 42.↵
It seems that he took subtraction in the ‘additive’ procedure for the necessity to delete some months.

- 43.↵
This value is 3.5 s greater than the modern estimate of the mean month. Thus, the price of having a comparatively short cycle of 4000 years would be a two-day delay of Newton's calendar Moon against its mean position known today.

- 44.↵
He computed Easters in 1856 and 1867 and considered the sequence of 76 years apart: 1932, 2008 and 2084.

- 45.↵
Although we were unable to find Flamsteed's mean month in the literature, we compared the full moon positions in C2 against Flamsteed-Horroxian data by using Kollerstrom's (

*op. cit.*, note 22) computer program. The result (a difference of 3 seconds of arc) points (within negligible error) to an identical synodic month. The mean month of the 1702*Theory of Moon's motion*is less by 10 thirds: 29 days 12 h 44 min 3.0 s. - 46.↵
See these tables, for example, in Clavius,

- 47.↵
December 1700 (J) was a convenient date on which to start a lunar month. True, Newton did not explain at what time his lunar day begins. In the beginning of drafts A, however, he implicitly referred to Maimonides's visibility theory, which, in turn, implies 18:00 of the previous day (an average time for sunset). The lower boundary for the visibility of the Moon, 18 h between conjunction and sunset, fits the period around the vernal equinox, and the upper 42 h fits the period around the autumnal equinox. Both solstices fit the average value, 30 h, which in astronomical terms corresponds to

*ca.*15° of the true elongation between the Moon and the Sun. See details in - 48.↵
We decided to cut this line of investigation short and did not compare Newton's full moons with the Protestant (true astronomical) ones. Nowhere in Yahuda 24 does Newton compute positions of the true astronomical moons, nor does he make clear at which hour the lunar day starts.

- 49.↵
Gottfried Kirch (1639–1710), as Astronomer Royal in Berlin, and his family were granted the exclusive right to print calendars in Brandenburg state (http://www.gottfried-kirch-edition.de). Kirch's assistant since 1701, Johann Heinrich Hoffmann (1669–1716), the second Astronomer Royal after 1710, is also known to have issued his own calendars

*ca.*1704. - 50.↵
See Ginzel,

(note 2)

*Gregorian reform*,*op. cit.*(note 1), p. 253, or R. H. van Gent,*A perpetual Easter and Passover calculator*(http://www.phys.uu.nl/∼vgent/easter/eastercalculator.htm), ch. 3 (Anomalous Easter dates). The Protestant (astronomical) full moon in those years occurred on Saturdays, whereas the Gregorian took place on Sundays. As early as the first year of calendar reform (1700), the astronomical Easter Sunday could occur a week before the Gregorian Easter Sunday, but because it would then coincide with the Jewish Passover feast it was postponed to the next week (thus coinciding with the Gregorian date). Such occurrences (also possible in 1778 and 1798) van Gent calls ‘pseudo-anomalous Easters’. - 51.↵
There is a mistaken statement in the literature about the date of Protestant Easter in 1704. See

*Gregorian reform* - 52.↵
The final corrections to the Rudolphine tables were one problem. In manuscript U, Newton discussed even the Moon's parallax, apparent diameter, and positions in quadratures, although syzygies and even full moons were perfectly adequate for the purpose. The extra data were suppressed in the final letter to Leibniz. The geographical reference point, Uraniborg, was somewhat arbitrary and could be changed, for example, to Berlin after the Academy of Science opened. These would be important factors in 1704.

- 53.↵
He probably forgot it was a leap year.

- 54.↵
Excluding ‘pseudo-anomalous’ 1778 and 1798. Nowhere in Yahuda 24 MS did Newton pay any attention to this provision in the Regensburg decision.

- 55.↵
The problematic ‘year 13’ in Newton's 19-year table corresponds to year 1703. Because 21 March (J) in 1703 corresponds to 21 March Gregorian in 1704, Newton was aware of the problem. The starting point of his 19-year Easter cycle, the year 1690, matches (

*modulo*19) the classical Church ‘Dionysian’ Cycle, which was counted from the autumn of AD 284 (Diocletian Era). - 56.↵
Newton, as a statesman, may have taken into account a possibility that the German Protestants would consider postponing 1704 Easter for a month to align with England and Sweden, which still followed the Julian calendar and observed Easter in April.

- 57.↵
The only calendar to embed this idea was a medieval Persian one, designed by Omar Khayam in the 11th century (http://en.wikipedia.org/wiki/Persian_calendar#Background). In

*ca.*1700, the seasonal division was spring+summer+fall+winter=93 days+93.5 days+89.6 days+89.1 days, with roughly an eight-day difference between summer and winter. - 58.↵
Newton's brief historical discourse on the introduction of the feasts from Emperor Trajan's tenure on makes clear that only three major Christian feasts were established in the first century.

- 59.↵
This was later accepted. The tax year in England still begins on 5 April, which is 25 March, but in Gregorian disguise! What would the Master of the Mint say about such a turn of affairs?

- 60.↵
The term ‘bissextile’ came from the old Roman formula ‘double the sixth’ day before the March Kalends. J. R. Stockton, ‘Leap years’ (http://www.merlyn.demon.co.uk/leapyear.htm#1700).

- 61.↵
While

*Theory of Moon's motion*carries no correction at all, the −10″ correction appears again in the second and third editions of*Principia*.*Principia*'s Book III, Proposition XXXV, Scholium of Problem XVI, gives the mean longitude of the Sun for 1701.0 (31 December 1700, noon) as 9^{s}20° 43′ 40″, which is 10″ less than that of the Flamsteed tables. - 62.↵
C2 does not carry the Moon's longitude, only the Moon's elongation from the Sun: 24° 33′ 57′′.

- 63.↵
The reason that the Astronomer Royal ‘grumbled’ over reading the letter. See

*Flamsteed's**correspondence*, - 64.↵
See

*Newton's correspondence*, - 65.↵
See Scott's notes 1 and 2,

- 66.↵
- 67.↵
Although we did not have opportunity to check the Newton-Flamsteed correspondence prior to 1694 we would place the ‘U’ data in the early 1690s.

- 68.↵
See Gregory's lengthy note on the bottom of his copy of M in

*Newton's correspondence*, - 69.↵
- 70.↵
There are practically no discrepancies between M and

*Theory of Moon's Motion*. The only one is an absence in the latter of the −20″ correction to the mean Sun's position, which David Gregory, its editor, could skip because Newton himself hesitated on its precise value. - 71.↵
Curiously, one leaf in Yahuda MS 24 is directly related to the Mint—with a table of expenses: the price of copper and payments to different workers and clerks.

- 72.↵
Newton missed almost all meetings of the Society in 1696–1700. He was a busy man, after all, and Robert Hooke was still there! See Westfall,

- 73.↵
- 74.↵
On the whole, we failed to enlist any support for Poole's suggestion

- 75.↵
Transcribed from the originals in the National Library of the Hebrew University, Jerusalem, by the first author and Ayval Leshem. David Castillejo's cataloguing numbers—A, B, C, D—are preserved with additional pagination. ‘And’ was substituted for the symbol ‘&’ throughout the text. Double bracketed remarks [[]] and endnotes by the authors and Joan Griffith.

- 76.↵
This is related to Maimonides's lunar visibility theory, where both values are variable, depending on the season and the moon's latitude. The lower boundary, 18 h, fits the time of the vernal equinox with moon's large northern latitude. The upper boundary, 42 h, fits the time of the autumnal equinox and the moon's large southern latitude.

- 77.↵
Inserted instead of the crossed out ‘12’ before ‘seconds’.

- 78.↵
The so-called

*Metonic cycle*, ascribed to the Greek, Meton, 431 bc. - 79.↵
In 46 bc.

- 80.↵
In 8 bc.

- 81.↵
‘130’ is written above former ‘128’ while ‘129’ is crossed out.

- 82.↵
In October 1582.

- 83.↵
In ad 325.

- 84.↵
Or 22.39 s in a month, which is the difference between the Church (‘Dionysius’) mean month of 29 days 12 h 44 min 25.53 s and the Horroxian mean month of 29 days 12 h 44 min 3.16 s (adopted by Newton in C2 as a synodic month).

- 85.↵
The best correction of the lunar calendar is to take eight days out of the lunar cycle in 2500 years. This leads to 43 adjustments of the epact in the Gregorian calendar over the next 100 centuries. (See Christopher Clavius,

*Explicatio*, ed. 1603, p. 155: ‘Quoniam post 10000. annos ab anno 1700. elapsos facta est mutatio 43. literarum, …’.) - 86.↵
In A2, the word ‘July’ was written above ‘December’ as an equal option. In A1, the last 11 days of May must be deleted.

- 87.↵
See previous note.

- 88.↵
Since 3×365.25 days divided by 100 is 10.95 or approximately 11 days.

- 89.↵
The mean month coming from this procedure is 29 days 12 h 44 min 4.9 s (Newtonian month).

- 90.↵
On p. 18 of C2, Newton suggested,

*inter alia*, two choices: the 2nd day of the 18th month or of the 42nd month. - 91.↵
The ‘½’ is wrong and was omitted in A3.

- 92.↵
The last phrase is absent in A2. In A3 it makes no sense, as it stands now, without the second part of the bracketed paragraph from A2. Newton copied the first part of this paragraph in A3 earlier, but apparently forgot to insert the rest of it into A3.

- 93.↵
In Britain, until 1752, not 29 February, but 24 February was the true ‘leap’ day, as in the original Roman calendar. The term ‘bissextile’ (‘twice the sixth’) refers to the duplication of 24 February, the sixth day before March Kalends.

- 94.↵
The passage, beginning with the words ‘the best form’ up to ‘thought off’, was printed for the first time in Sir David Brewster's

*Life of Sir Isaac Newton*(1855). - 95.↵
‘13 days’ would be more precise than ‘14 days’.

- 96.↵
Newton's initial intention was to skip 22 days from the Julian calendar (instead of 11), placing the equinoxes and solstices on the first day of the corresponding months.

- 97.↵
Unclear: the modern calendar has a difference of only one day between the summer (April-September) and the winter (October-March) months.

- 98.↵
Again, ‘13 days’ is preferable to ‘14 days’.

- 99.↵
It was ‘8 or 9’ earlier in the text.

- 100.↵
ad 98–117.

- 101.↵
Origen,

*Against Celsus*, book 8, ch. XXII. The passage reads: ‘If it be objected to us on this subject that we ourselves are accustomed to observe certain days, as for example the Lord's day, the Preparation, the Passover, or Pentecost …’. - 102.↵
Bishop of Caesarea in the reign of Commodus (see Eusebius,

- 103.↵
The first difference between the Newtonian and Gregorian solar calendars appears in the year 2400.

- 104.↵
Or one day in 312 years—as in the ‘Dionysius’ 19-year cycle. Newton disregards the possibility of removing eight days in 2500 years from the lunar cycle in the Gregorian lunar calendar. With this correction, the Gregorian mean month comes within half a second of the Horroxian value and mean synodic month.

- 105.↵
It probably refers to the ‘lone day’ included in the 4000-year cycle (see the last phrase in A3) but can also refer to the 1.7 s difference between the Newtonian month and the Horroxian mean month.