* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project
JHA, xli (2010) ISAAC NEWTON’S HISTORIA COMETARUM AND THE QUEST FOR ELLIPTICAL ORBITS J. A. RUFFNER It is necessary to have a record of the rising of comets in times past. Because of their rarity, their path cannot yet be understood, nor can it be determined whether they maintain sequences and whether a definite pattern causes them to reappear at a particular time. Seneca1 Introduction In 1680, astronomers were in general agreement that comets roughly follow an apparent great circle arc across the sky while actually moving amongst the planets. Beyond this consensus there were many questions. Do they dissipate or move on unseen into space? Are the paths straight or nearly so, or are they circular or nearly so? If the latter, do they return and with what period? Johannes Kepler and Johannes Hevelius believed they move in straight or nearly straight lines and dissipate. Pierre Gassendi favoured uniform motion in straight lines indefinitely. Giovanni Domenico Cassini imagined circles so vast the visible portion near the Earth could be treated as straight but with periods of only a few years.2 John Flamsteed suggested a path much like that of the planets, with the comet of 1677 having a period of about 12 years.3 Robert Hooke complained that any number of “lame shifts” could account for their motion but believed Christopher Wren’s rectilinear method, also advocated by John Wallis, gave the best results, with due allowances for an attenuated gravitational principle in comets and the effects of fluid resistance.4 From about 1684 to the end of his life Isaac Newton sought demonstrations that the orbits are ellipses with due allowance for perturbations. But first, in 1680/1, he followed the path trod by Kepler, Wren, and Hooke. The comet of 1680 elicited far-ranging technical accounts funnelled to interested parties through Cassini at the Paris Observatory, Flamsteed at the Greenwich Observatory, and others. It was seen in November moving toward the Sun although weather conditions in northern Europe generally prevented detailed measurements. Two sets of observations at Avignon and Rome, first by Jean Charles Gallet and later by Marco Antonio Cellio, in due course reached Flamsteed through the aid of Edmond Halley who at the time was assisting Cassini at the Paris Observatory. During his trip to Paris, in what proved to be one of the most important observations, Halley had seen the tail projecting above the horizon early in the morning of 8 December. Two days later, first a tail and then an even more spectacular comet appeared in the evening. The tail soon reached an apparent length of 70 arc degrees.5 This comet was visible by naked eye until the end of January and by telescope until last observed 0021-8286/10/4104-0425/$10.00 © 2010 Science History Publications Ltd 426 J. A. Ruffner by Newton on 9 March. It moved away from the Sun along an apparent track quite different from the approach. Flamsteed and Newton disagreed sharply as to whether the appearances were of one comet or two different ones. Newton knew little about the November appearances before receiving in midFebruary 1680/1 a bungled list of Gallet’s observations.6 Alerted by colleagues to the new appearances, however, he joined their observation on 15 December. By the end of December he was fully engaged, making detailed observations and pursuing an independent research program. By about the end of January, he had derived a rectilinear solution for December and early January 1680/1.7 So far he had received from Flamsteed only a few preliminary December observations and a boast of having foreseen the reappearance of the November comet.8 Around February 1680/1, Newton prepared a list of memorable comets and related phenomena.9 The phenomena listed, parallax from annual motion, lack of sensible parallax, deviations from great circle paths, and the approximate anti-solar nature of the tail, presaged arguments placing comets in the midst of the planetary region. Since Flamsteed and nearly all of their contemporaries agreed with that placement, it is likely Newton was contemplating a private demonstration against those who had ignored the evidence before their eyes and placed them in the solar vortex beyond Saturn or previously lodged them below the Moon. Alerted in late November by an assistant, Flamsteed predicted the comet would be seen moving away from the Sun. He was rewarded with an appearance of the tail in the evening of 10 December. He continued observations into early February, distributing to Newton and others hastily reduced positions, later revisions, data from the Continent including a brief account of Halley’s sighting on 8 December, and a physical theory of the comet’s motion.10 According to this theory, the comet was initially attracted by the Sun and repulsed without passing around it forming a u-shaped curve trajectory. Newton urged Flamsteed to revise the solution before publication by dropping the proposed magnetic mechanism and allowing the comet to pass around the Sun. Flamsteed retained a less complex form of magnetism as the physical cause and at first was reluctant to allow passage around the Sun because of problems with implied speeds and the actual length of the tail. Before Newton could respond, Flamsteed adopted such a passage as if there never had been any doubt.11 Newton took the opportunity to query Flamsteed on details related to his own work.12 He also may have studied Hooke’s Cometa.13 He was particularly interested in data on the orientation of the tail and the positions of the termini, which he plotted on a celestial globe. Consequently, he proposed a law that the apparent end points follow some regular relation such as an arithmetic or geometric progression.14 In this regard he wished for greater details than those provided by Flamsteed about Halley’s observation of 8 December. When he returned to his rectilinear solution for the path in December and early January, additional data forced him to admit a very slight concave bend as the comet passed by the Sun having emerged from the depths of space. He imagined the comet seen in November continued in the other direction along a trajectory that also was very nearly straight. As for Flamsteed’s Isaac Newton’s Historia Cometarum 427 view that the comet was initially attracted and then repulsed, Newton pointed out in an unused draft that the comet would have been continuously accelerated. The postulated magnetic force in the Sun must be continuously attractive in order to fetch the comet around it and retard the egress. Although the argument was suppressed, some scholars think Newton was contemplating a gravity-driven parabolic solution. Results based on his now slightly curved path, however, followed immediately and were repeated in the actual letter. He was merely trying to clarify Flamsteed’s ideas, should he persist in positing a magnetic force and a single highly curved path. In the end, Newton argued the path, whether or not corrected to allow for such a passage, left technical difficulties in accounting for the date of the ascending node. Assuming the early portion of the path fell exactly along a great circle, he must have calculated the apparent position of the node or, better, found it graphically on a celestial globe (as he had done for the envelop of the tail termini) and then determined the time required to reach it along a path with minimum curvature. He further posited a heliocentric longitude for the node that despite some uncertainty must have been based on the intersection with his modified trajectory extended to that time. As if to doubt this logic, he continued: “whatever there be in these difficulties, to make ye Comets of November & December but one is to make that one paradoxical. Did it go in such a bent line other comets would do ye like & yet no such thing was ever observed in them but rather the contrary.”15 Four years later a fresh study of the historical record found several precedents. Newton abandoned his comet research program of 1680/1, in part because of dis illusionment in the quality of all but one of his early observations and insufficiently accurate star catalogues.16 He may also have feared entanglement as with Hooke a year earlier if he continued to seek needed support. He resumed his chemical experiments and other topics of a philosophical or theological nature. Comet data continued to arrive sporadically and in 1682 Newton made a short series of observations of the comet enshrined under Halley’s name. The subject lay fallow until about the winter of 1683/4 when he studied the works of ancient authors in association with the early stages of writing Theologiae gentilis origines philosophicae (Philosophical origins of gentile theology).17 He pieced together widespread evidence of the degradation of knowledge, claiming later that the true system of the world had become hidden in allegories. Among the alleged ancient gems were beliefs that planets revolve around the Sun in almost concentric orbits and the comets in very eccentric orbits, and other important principles such as mutual gravitation, perturbations of orbits, precession of the equinoxes, lunar inequalities, the equality of falling bodies in vacua, and tides.18 Of particular interest are two rough accounts culminating in facts known since his student days but now taking on greater moment. The main points indicated that celestial matter is the same sort of material as terrestrial matter, the heavens are fluid, the Sun is in the centre, and comets are a kind of planet. These views, attributed variously to Egyptians, Chaldeans, and Pythagoreans, had been lost everywhere by the time of Plato and Eudoxus. Comets became atmospheric phenomena caged below the Moon by solid spheres.19 Once he 428 J. A. Ruffner was convinced that all bodies mutually gravitate, the close approach to the Sun of the comet of 1680 in December required much greater acceleration than he previously contemplated and made it likely the November appearances were of the same body. Soon after Newton wrote this early work on Origines philosophicae, Halley arrived in Cambridge around August 1684 to pose the famous problem of planetary motion under an inverse-square distance central force. Arguably, comets were also an early matter of discussion. Details Newton wished for in April 1681 about Halley’s observation on 8 December are found at the end of the related notes concerning the shape and extent of tails, most likely written in March 1680/1. A new sentence is also found in the corresponding entry at the beginning of these notes. These new details were attributed to what “Halley told me”.20 For one thing, Halley indicated the head of the comet was hard by the Sun as it came into view. If comets were indeed a topic of their first meeting in 1684, such a detail could have been a factor in linking the November and December appearances, if he had not already done so. Whatever the case, comets were again in the forefront of Newton’s mind. Newton’s response to Halley’s problem (after failing to produce the proper papers) was a nine-page manuscript, De motu corporum in gyrum (On the motion of bodies in orbit), including the problem of comet motion. Even without actual statements about the species of orbit, Newton could proceed with what he demonstrated anew in De motu. The orbits of the planets were ellipses with small eccentricities. The task remaining was to demonstrate that comets that appeared only at long intervals when they came close to the Sun must have travelled in highly eccentric ellipses. He proposed to use a rectilinear approximation to provide a distance and speed vector from which an ellipse could be determined. By reiteration the ellipse could be made to fit the observations well enough. Then, by comparing the elliptical elements of many comets, it would be possible “to ascertain … whether the same comet returns with some frequency to us”.21 Almost immediately, around December 1684, a revised version, De motu sphaericorum corporum in fluidis (On the motion of spherical bodies in fluids), introduced the problem of perturbation by which the Sun, Moon, and planets mutually interact with one another so that they “neither move exactly in an ellipse nor revolve twice in the same orbit”.22 The celestial fluid was not part of the problem as it offered no sensible resistance. He noted that all the sounder astronomers, who knew how to calculate the approximate distances from their parallax, place comets below the orbit of Saturn, where they are “carried with immense speed indifferently in all directions in all parts of the heavens yet do not lose their tail nor the vapour surrounding their heads [by having them] impeded or torn away by the resistance of the ether”. Moreover, “the planets actually have now persisted in their motion for thousands of years, so far are they from experiencing any resistance”.23 Correspondence with Flamsteed in December 1684 and January 1684/5 showed equal intention “to determine the lines described by ye Comets of 1664 & 1680 according to ye principles of motion observed by ye Planets” as to get to the bottom “of ye influences of ye Planets one upon an another”.24 He had already consulted Isaac Newton’s Historia Cometarum 429 Kepler’s Rudolphine tables, Mercator’s Institutionum astronomicarum and one or two other nameless astronomy texts that left him no wiser. His first request had been for accurate star positions to complete reduction of his micrometer-based observation made in 1680/1 between 25 February and 9 March. Secondarily, he sought accurate measures of the sizes and periods of Jupiter, Saturn, and their satellites.25 By late February 1684/5, Newton indicated he had already invested a lot of time, “a great deale of it to no purpose”.26 The unsuccessful details are lost but it is reasonable to suppose an attempt to calculate an elliptical orbit according to the principles of the original De motu tract was among the casualties. Comets might even have been set aside as Newton became more engrossed with preparing definitions and conducting experiments related to what would take shape as De motu corporum liber primus and finally Book One of the Principia. In due course, he embarked on a second book, De motu corporum liber secundus (hereafter DMCII), intended as a popular System of the World including empirical evidence for the inverse-square law, basic problems of perturbation, lunar theory, tide formation, and comets.27 By late summer he determined to fashion a new Book Two on the motion of bodies under resistance, and a new more technical System of the World as a third book. DMCII, Section 1 recounted the debasement of ancient truths recast from the Yahuda manuscripts cited above. The key note would restore free spaces to the heavens, and comets to the status of a kind of planet moving in highly eccentric orbits, in accordance with the older and sounder philosophies of the Chaldeans and Pythagoreans.28 Nearly forty percent of the text starting at Section 58 was devoted to comets, which even at that was unfinished. Nearly eighty percent of the treatment on comets provided analysis of what can be learned about their nature and location from phenomena discerned by naked eye.29 The basic point of these sections, devoid of gravitational considerations, was to show how the wisest of the ancients had it right without the aid of telescopes or full knowledge of the principles of motion, and how contrary views from Aristotle to Descartes missed the evidence right before their eyes. Historia Cometarum Although no preliminary materials for the dynamical portions of DMCII have been identified, a worksheet labelled Historia cometarum (The history of comets)30 (see Figures 1–4) illustrates some of the bright ideas and blind alleys experienced in developing the more general parts of the account. Rough work and simplified models characteristic of his early approach to problems are evident. One notable entry, 1680 = 1681, unified the November appearances with those of December and later. Three works by Hevelius were used extensively in developing this document and a fourth one in writing DMCII.31 Cometographia provided detailed descriptions of historical comets up to 1665 and a table of lunar distances. Select descriptions were summarized in this document and in some cases quoted verbatim in DMCII. Newton previously used Riccioli’s account for historical data and continued to consult it in writing DMCII.32 Descriptio cometae provided added details about the comparative 430 J. A. Ruffner Fig. 1. Cambridge University Library Add Ms 3965.11, f. 173v. This and the following figures are reproduced by permission of the Syndics of Cambridge University Library. Isaac Newton’s Historia Cometarum Fig. 2. Cambridge University Library Add Ms 3965.11, f. 172r. 431 432 J. A. Ruffner Fig. 3. Cambridge University Library Add Ms 3965.11, f. 172v. Fig. 4. Cambridge University Library Add Ms 3965.11, f. 173r. sizes of planets and the comet of 1665. Mercurius in sole provided extensive data on the apparent sizes of key stars and the planets. Similar annotations were added to his copy of Mercator that he had already studied to little effect.33 Annus climactericus provided details on lunar librations, the diameter of Mars at perihelion, and the comets of 1680, 1682 and 1583 for use in DMCII. Tycho’s star catalogue in the Rudolphine tables and Wing’s solar theory were the probable sources of coordinates underlying most calculations.34 The Waste Book section on comets was complete by this time, except for some of Cassini’s coarser observations of 1680 which were possibly squeezed in later. Ponteo’s edition of the observation of 1680 made at Rome was also at hand in writing DMCII.35 Further particulars are given below. The initial entries were brief descriptions of 30 comets starting with 371 b.c. with only an incomplete date entry for 1401 (see Figure 1). The source was Hevelius’s Cometographia which listed over five times as many in the same time-period.36 The comets selected had sufficient information, such the zodiac sign and season or months in which they appeared, to permit location with respect to the Sun. Abandoning this list, Newton completed a more systematic arrangement according to the same selection process (see top of Figure 2). Fifty-two comets were observed in the solar hemisphere and thirteen in the opposite hemisphere. The summary that there are four or five times more comets near the Sun than opposite the Sun was used as evidence that they are not located at great distances from the Sun during their period of visibility. Similarly Isaac Newton’s Historia Cometarum 433 to the earlier memorable list, comets were also grouped according to comets with very spectacular tails, comets with long tails, comets whose heads decreased and tails increased as they moved toward the Sun, comets that reversed their apparent motion in the last observations, and years with two comets. Various tail particulars were intended to demonstrate close approaches to the Sun. Deviations in the final observations were used as indicators of annual parallax and hence distance from the Sun. The years with two comets were probably selected for similarity to the situation in 1680. Incidental to these compilations, a table of lunar distances was copied from Cometographia. Instead, DMCII, Section 10 listed Riccioli’s mean lunar distances supplemented by Flamsteed’s value.37 Descartes contended: “it is observed that comets pass, one through one region of the heaven, and another through a different region, without following any rule known to us.”38 He depicted vortices around each star, with comets wandering near the outer edge of their vortex to be swept randomly into an adjacent one where they were no longer visible. Newton was convinced that there were rules, once known in Antiquity, that had been developed from phenomena visible by naked eye. He assumed that orbital periods and other significant information could be inferred from the phenomena without detailed knowledge of the comets’ motions. In particular, he identified numerous comets that might have been previous appearances of the comet of 1680 (see middle of Figure 2). In previous visits the comet also would have passed close to the Sun, hence his list excluded comets known to have appeared only in the opposite hemisphere. The comets selected, with several exceptions, had bright large heads and a splendid tail. Cometographia provided a sufficient source. The main list was tightened to exclude several less spectacular examples. It was then recast to provide the time period between the comet of 1680 and each of the others (see foot of Figure 2). These intervals were divided into as many circuits as needed to yield base periods between about 70 and 140 years. This process formed a matrix of almost every possible period between the comet of 1680 and other notable comets. A second matrix provided possible periods between many of these comets and the comet of 1618. Comets underlined here and there in the document may have been highlighted as work progressed, to indicate prospective earlier appearances of the comet of 1680. Thus, 1456 and 1338 were underlined, suggesting base periods of 112 and 171 years. The select list ending with the notation “1680 = 1681 (perihelion in 26)” includes underlining for 1264, 1401, and 1527. The comet of 1264 (which, as will be seen, Newton had reason to consider congruent with the comet of 1680) stands out as having a base period of 104 years or 139 years. The comets of 1401 and 1527 suggest base periods of 139 and 153 years, respectively. Finally, the matrix included underlining for intervals of 153 and 279 (2 × 139½) years. Table 1 illustrates sequences of any known comet in Cometographia that most nearly matches various prospective sequences with summary comments from their descriptions. For periods of twice the length, merely drop every other one. These examples are only a few possible arrangements. For instance, if the base period 434 J. A. Ruffner Table 1. Example sequences of possible appearances of comet of 1680. Approximate Period 104 yrs 112 yrs 139 yrs 153 yrs 171 yrs Theory Actual Theory Actual Theory Actual Theory Actual Theory Actual 1680 1680 1680 16801680 1680 1680 1680 1680 1680 1576 1578 long tail 1568 1569 very bright color 1541 1542 fiery 1527 1526 flaming sword 1509 1511 sanguine colour 1472 1472 wonderful greatness 1456 1456 2 comets, tails 60° 1402 1401 terrible, splendid 1374 1375 no details 1368 1362 seen all night 1344 1347 lasted 2 months 1338 1338 no details 1264 1264 1263 1264 signal magnitude 1232 1238 vast magnitude 1221 1223 dreadful appearance 1160 1165 1167 1165 two comets together? 1124 1132 stupendous 1120 1113 a huge comet 1068 1066 at first equal to moon 1056 1058 long tail 1008 1005 terrible aspect 996 999 stupendous magnitude 985 983 no details 952 962 unusual grandeur 915 908 no details 896 902 lasted 40 days 848 844 846 844 “above Venus” 825 830 no details 784 ? nothing listed 762 761 no details 744 745 no detail 707 ? nothing listed 672 676 lasted 3 months, great flame 654 ? nothing listed 640 633 lasted 30 days 609 604 very bright 568 570 no details 560 556 dreadful comet 483 488 unusual figure and magnitude 456 454/457 wonderful magnitude 536 541 dreadful 448 448 lasted many days 432 442 or 423 429 423 442 lasted many days/423 horrible 336 335 immense & terrible grandeur 312 306 or 305 no details 303 306 or 305 nothing listed 290 ? nothing listed 224 218 224 218 lasted 18 days 151 145 150 145 141 145 lasted 6 nights 120 128 lasted 39 nights 112 ? nothing listed Isaac Newton’s Historia Cometarum 435 Table 1, continued] Approximate Period: 104 yrs 112 yrs 139 yrs 153 yrs 171 yrs Theory Actual Theory Actual Theory Actual Theory Actual Theory Actual 16 14 flaming torch 12 10 flaming torch 1 bc 1 ad 4 bc 1 ad flaming torch 31 bc 29 bc lasted 90 days 89 bc ? nothing listed 113 bc 110 bc conspicuous light 128 bc 122 bc lasted 80 days 157 bc 154 bc no details 193 bc 194 bc extraordinary magnitude 202 bc 196 bc stupendous magnitude 225 bc 220 bc lasted 2 days 267 bc ? nothing listed 297 bc ? nothing listed 310 bc ? nothing listed 337 bc 339 bc no details 373 bc 371 bc a beam over 1/3 of heaven 401 bc 411 bc 406 bc 411 bc no details were 114 instead of 112 years, the sequence could include 371 b.c. with no larger deviations. Gaps could be attributed to poor observing conditions or unavailable chronicles. Very large deviations could also be attributed to inaccurate dates, many of which were uncertain. We do not know have far Newton followed this type of analysis for 1680 or how he would have settled on any given sequence. Newton proceeded to indicate an actual sequence for the great comet of 1618. Table 2 lists the choices against a period of about 105 years.39 Reports in Hevelius’s account that noted apparent motion against the order of the signs as in 1618 was the sure basis for inclusion, but most would have been selected only on the basis of near agreement with the expected mean period. The deviations generally were less than two or three years, but much larger where the record was sparse and probably Table 2. Prospective sequence for the comet of 1618 at an interval of 105 years. Expected Newton’s Choices 1618 1513 1408 1303 1198 1093 988 883 778 673 568 463 358 1618, initially moved contrary to order of signs in the early morning hours 1511, 1512, 1513 (moved in order of the signs), 1516 1407 or 1408, several comets of which we have no particular descriptions 1301, 1304, 1305, 1307; 1301 moved contrary to order of the signs; no useful details for the others 1200, moved contrary to order of signs, appeared larger than Venus 1096, no useful description 983, no useful description 882, 875, 876. 882 tail of great length, 875 long hairy beams, 876 no useful description 763, terrible to behold 669 or about 673, no useful description about 570, no useful description 457, of wonderful magnitude 363 or 367, lasted many days 436 J. A. Ruffner incomplete. This amount of variation would call for very strong accelerations or retardations in the remote portions of the orbit. Newton’s progress in understanding the magnitude of perturbations deserves separate study. De motu in gyrum promised to determine whether comets return by solving the orbital motion of many comets and finding those with similar elements. Here, Newton imagined matrices of bright comets would highlight the most likely candidates. The selected intervals even if not definitive were long enough to demonstrate that the orbits must be highly eccentric — given his principles of motion — ellipses. He placed great faith in the lawful order produced by arithmetic or geometric progressions and he may have regarded matrices as another tool for producing order. Newton probably imagined the Pythagoreans (or others in Antiquity who had correct views) developed them from a more detailed data base available in Antiquity and hoped a modern list could repeat the process. Nothing more was seen of the matrix concept. Was it a bright idea that failed and abandoned at this point? Or did he move, confident he could find common periods of 100 to 300 years or so that would place the aphelion distances for comets moving close to the Sun no greater than about 40 to 90 a.u.? Whatever the particulars, in DMCII, Section 30 the aphelia of comets were placed at modest distances beyond the planets — far closer than the stars but too far to be seen from Earth — where their slow elliptical motion allowed them to spend almost all their time and their collective effect could account for slight alterations in the orbits of the outer planets.40 The idea was soon dropped. Book 3 of the Principia indicated, “from the actions of the revolving planets and comets upon one another some inequalities will arise, which, however, are so small they can be ignored here”.41 Having investigated the matrices of major comets with an eye toward finding common periods, Newton flipped the folded sheet again to make notes about four similar comets with estimates of the heliocentric coordinates at the time they disappeared (see Figure 3). It is reasonable to assume that Newton was still crafting arguments to demonstrate, contrary to Descartes, that comets traverse the planetary region in well ordered paths. Although these examples were abandoned they represent a fascinating attempt to show what can be known approximately without exact knowledge of their motion. Newton assumed that the brilliant appearances and long tails of these comets indicated a passage very close to the Sun. The spectacular comet of 1106 highlighted on the other side of the sheet lacked detailed observations for similar treatment. No worksheets for any of these calculations have been found. The work probably was based on the “parallax” method hinted in DMCII, Section 58. The method was described in detail in the Principia.42 Newton may have thought the technical details inappropriate for the original popular treatment. A key component from an alleged lecture purported “to determine the position of a comet proceeding uniformly in [a] straight line … from three observations of its course”.43 The Earth was presumed to be at rest. Now he undertook to exploit this flaw to determine the distance at the time of its disappearance, or presumably other times. A fourth or final position would have Isaac Newton’s Historia Cometarum 437 been extrapolated from three earlier observations. This ‘expected’ position would then be compared with the actual observed position, leading to a graphical determination of its distance from the Earth. The heliocentric parameters would follow by triangulation. I have not however achieved conclusive agreement using this method and further study is warranted. Newton’s results, in any case, are inconsistent. The difficulty of reconstructing Newton’s results stems not only from uncertainty of the vanishing date and time chosen but also from fundamental problems with the extrapolation technique compounded by difficulties with the final graphical solution. The protocol in the Principia is not entirely clear as to which three initial observations to select. The first and third observations establish the graphical orientation but the resolution varies with different middle observations. Given this caveat, consider Newton’s examples. The Comet of 1680 Newton’s work on the comet of 1680 before September 1685 utilized the set of Flamsteed’s observations recorded in the Waste Book.44 The data chosen for its last position of naked eye visibility are ambiguous. According to notes in the Waste Book, the head of the comet appeared as a seventh magnitude star or not much greater on 25 and 26 January.45 It ceased as a seventh magnitude star on 30 January with traces of the tail still seen by naked eye. From that point to 10 February he studied the tail with a telescope.46 Newton’s use of a seventh magnitude designation is not entirely clear. By long standing convention the least visible naked eye stars were designated sixth magnitude. Hevelius who supposedly made observations only with open sights, however, listed positions of a few seventh magnitude stars. Newton seems to be indicating the head was just barely visible to the unaided eye as late as 26 January. The last available position for a time when traces of the tail were still visible to the naked eye would have been Flamsteed’s observation for 30 January, 8:07 p.m. The difference between this observation and one extrapolated from three earlier positions starting with the first one available provides the basis for the final graphical resolution of both geocentric and heliocentric coordinates. This result could stand or be further extrapolated to whatever time Newton chose from his own experience. As noted, the position extrapolated from three observations differs slightly according to the choice of the middle one. For example, taking 30 January as Flamsteed’s observation nearest the vanishing point, 12 December as the first observation available and 13 January as the third choice, the extrapolated position for 30 January varies by nearly one degree when the second position is chosen as 24, 26 or 29 December. A narrow reading of the protocol in the Principia suggests using the first three available observations, in this case, 12, 21, and 24 December, but this yields a heliocentric longitude of about 5°, some 9 degrees less than Newton’s value. The problem of the middle observation arises for other combinations. Be that as it may, I find no graphical resolution with other combinations that yield Newton’s heliocentric coordinates for a vanishing date around 30 January. 438 J. A. Ruffner Taking at face value Newton’s heliocentric result of 14°, or rather 14½° to allow for its having been rounded off, the heliocentric distance using Flamsteed’s 30 January observation would be slightly less than 2.0 a.u. with latitude 15° N. This distance accords with the assertion in DMCII, Section 71 of the comet’s having scarcely doubled its distance since exiting the Earth’s orbit. The stated heliocentric latitude of 17° N evidently came from assuming rough equality of distances from the Earth and Sun rather than from calculation. Newton’s actual procedure remains an open question. The method I have adopted works tolerably well, however, in yielding his heliocentric longitude for the comet of 1618. The Comet of 1618 The tail of the final comet of 1618 reached the extraordinary apparent length of 104°, indicating a very close encounter with the Sun. One of Newton’s student notebooks included observations at Leiden by Snell up to 24 December.47 Kepler at Linz followed the comet with his telescope until 7 January.48 His ephemeris included places extrapolated to 20 January but actual observations were listed only until 28 December.49 Cysatus at Ingolstadt continued observations to 21 January. It equalled a fifth magnitude star on 20 December but by the 24th its light was greatly attenuated and on the 25th it was obscured by the Moon.50 Given this information, the choice of a date when it “just ceases to be seen” could be 24 December or, in the absence of the Moon, 28 December for naked eye views or 21 January for telescopic views. The sources from this material might have been Riccioli’s Almagestum novum, Hevelius’s Cometographia or, less likely, the original texts of Cysatus and Kepler. Each source handled the data differently, inviting different selections. Newton would have needed to calculate the solar positions corresponding to Kepler’s data; the solar location for the Cysatus data in Riccioli would have needed to be converted from distances to standard coordinates; or he might have calculated his own values. Cysatus’s observations with corresponding solar positions as provided by Hevelius51 diverge from the posited heliocentric longitude after 7 January, indicating that Newton did not use the final telescopic observation for the vanishing point. Kepler’s observations and Wing’s solar data with 24 December as the vanishing date yield results close to Newton’s heliocentric value of Leo 7° (longitude 127°). The heliocentric latitude, however, differs significantly from Newton’s 30º or 35º N (see Table 3). TR and SR are calculated distances in astronomical units reduced to the plane of the ecliptic between the comet R, the Earth T, and the Sun S. CR is the comet’s distance above the ecliptic plane, thereby fixing the latitude. Cysatus’s dates are equivalent to 1618 December 23, 18h, etc. “Aristotle’s” Comet The entry with the blank date related to the great comet described by Aristotle that disappeared in Orion’s belt.52 The earthquake in Achaea and other associated events indicated it appeared in the winter of year 4, Olympiad 101. According to the Julian 439 Isaac Newton’s Historia Cometarum Date (NS) Loc. Sun Kepler/Wing data Dec 24, 18h 273.51° Dec 28, 16½h 277.49° Cysatus/Hevelius data Dec 24, 6 a.m. 272.52º Dec 28, 6 a.m. 276.55º Jan 2, 6 a.m. 282.1º Jan 7, 6 a.m. 287.13º Table 3. Loc. Comet TR CR SR Heliocentric Lat. 0.84 1.01 1.44 1.89 1.47 44° N 1.75 47° N 58.72º N 0.83 60.97º N 0.97 62.6 º N 1.48 62.47º N 4.41 1.37 1.75 2.86 8.46 1.43 1.69 2.33 5.33 168° 58.73° N 156.33° 62° N 170º 158.42º 143.5º 131.63º 43° N 46° N 51° N 58º N calendar, the first appearance would be dated December 373 b.c. or January 372 b.c. Circumstances detailed by Diodorus Siculus date what is assumed to be the same comet in the following year 1, Olympiad 102 (372/371 b.c.).53 Riccioli dated it 372 or 373 b.c., Hevelius 371 b.c.54 A less detailed earlier document with Riccioli as the probable source shows Newton accepting a 372 b.c. date.55 This document, at least as far as the matrices, followed Hevelius’s choice of 371 b.c. The matrix listing of 2051 should denote 372 b.c. but for this and other b.c. dates, Newton evidently simply added the dates, failing to allow for the lack of a year zero, so that supposedly 430 b.c. + 1680 = 2110, 371 b.c. + 1680 = 2051, and so forth, although 122 b.c. plus 1680 is not 1800. The date he actually used for this exercise remains ambiguous. None was provided in DMCII. Newton first added a date for this comet as year 4, Olympiad 101 in his interleaved copy of Principia’s second edition.56 The larger issue is that the comet disappeared in Orion in southern latitudes. Newton’s conclusion that it was seen during January and February and disappeared with heliocentric coordinates of 15º, 24º N is as puzzling as it is fascinating. Did he slip in thinking it really had north latitude and was a possible appearance of the 1680 comet? It was included in the matrix of what evidently were possible orbital periods for the comet of 1680. Newton’s analysis was modelled on that comet. Both appeared in the winter almost in conjunction with the Sun. Long tails were seen first just behind the setting Sun followed a day or so later by the least possible angular distance from the Sun to be seen. In both cases, the apparent motions were direct. These features indicate Sun-grazing comets in almost the same celestial setting with similar orbits. Given that naked eye views of the comet of 1680 ended about 54 or 55 days after passing the Sun at an elongation of about 81½° or slightly less, similar circumstances were assumed for Aristotle’s comet. Tycho’s catalogue of star coordinates for the end of 1600 listed the middle star of Orion’s belt at longitude 17º54′, latitude 24º38′ S. Allowing for precession, the comet, whether last seen in 371 or 372 b.c., would have disappeared at roughly longitude 51º ( 21°) with about the same southern latitude. Lacking further detailed observations, Newton must have simply used parameters obtained for the comet of 1680 at the time of its naked eye disappearance, about 1.9 a.u. and 2.0 a.u. from the Earth and Sun, respectively. Recall 440 J. A. Ruffner that these distances were based on the assumption of uniform motion. Arbitrarily dating perihelion at 1 January, a transit time of about 54 or 55 days (as in 1680/1) placed the Sun on 23 or 24 February at about 1º for an elongation of about 80°. This elongation was slightly less than in 1680/1 but consistent with the greater latitude. Here the difficulty arises. The heliocentric coordinate was altered to read 15°. The original heliocentric coordinates seem to have been 26° or 20°.57 Strictly calculated, a coordinate of 26° leads to distances that are too small while 20° yields distances too large. An intermediate value would be just right but there is no possible trace of a different numeral. If the distances were not reduced to the plane of the ecliptic, however, the heliocentric longitude would be roughly 20°. The heliocentric latitude of 24° N (sic) assumed nearly equal distances from the Earth and Sun as in the case of 1680. No matter. Something other than a simple arithmetic error led him to make a drastic change. Working backwards from the comet’s newly posited heliocentric longitude of 15º, the distances are cut almost in half. The distance from the Earth in the trajectory is about 1.0 a.u. The corresponding heliocentric distance is about 1.3 a.u. with the heliocentric latitude reduced to about 19º [N? S?]. He was beginning to revise the results and must have abandoned the work before changing the latitude and possibly correcting the N/S designation. The basis for starting the revision remains uncertain. The Comet of 1264 The treatment of the comet of 1264 in DMCII, Section 63 cited “Append. Matt. Paris Historiae Anglia p. 967” as found in Hevelius’s account.58 The apparent motion of the comet was exactly opposite that of 1680. The comet of 1264 approached the Sun in the evening and exited a few days later, rising after Venus in the constellation Taurus. Both comets were very conspicuous with long tails reaching at times to the middle of the sky. The morning risings following conjunction would have been near the eastern limit of the constellation Taurus somewhere around zodiacal sign Gemini 20º or 25º. These appearances were just behind the morning star, which reached Gemini 20º on 15 July. It moved slowly westward, staying within the limits of constellation Taurus before disappearing famously on 7 October, the day Pope Urban IV died. Thus, the comet stayed in the same constellation for about 11 weeks, an exceptionally slow apparent motion. According to DMCII, Section 58 and later Principia, Book 3, an exceptionally slow motion indicated both the Earth and the comet were moving in the same direction and the comet was between the Earth and the Sun. Thus, the true motion of the comet of 1264 was direct, and having swung around the Sun its spectacular appearance and long tail indicated a close approach just as in 1680. Newton continued to entertain the notion these two comets were identical with a period of 416 years or integral fraction which for a Sun-grazing comet placed the aphelion less than 100 a.u. from the Sun. Isaac Newton’s Historia Cometarum 441 The Comet of 1106 A listing on the other side of the sheet provided an approximate perihelion position for the comet of 1106 which, from its spectacular appearance and long tail, would also have been regarded as a Sun-grazing comet. Insufficient information was available to include it among these examples. It was observed for some twenty-five days, first in the evening during the early part of Lent and later in the morning before disappearing on Good Friday. Since the Sun would have just entered Aries when the comet vanished, conjunction would have occurred somewhat earlier in Pisces (sign 12). Again, assuming the date of perihelion was little different from conjunction for comets that passed very close to the Sun, it would have been in or near Pisces or Virgo depending on whether the apex faced the Earth or faced away on the other side of the Sun. Newton originally wrote Aquarius or Pisces (sign 11 or 12), allowing for perihelion to have been reached shortly before conjunction. This orientation may have been a slip caused by his continuing to think in geocentric coordinates. He would have realized the comet was coming into view from the depth of space on the far side of the Sun and was beginning to swing around it with a direct motion. Otherwise an earlier position would place it closer to the Earth where it would have been seen. Also, the axis would have to be aligned slightly before the longitude of conjunction to ensure that both the earlier and later positions were not detectible. The note was corrected to indicate perihelion (as viewed from the Sun) was in Leo or Virgo (sign 5 or 6). These examples were not used in his subsequent text, except as they indicated a close approach to the Sun or provided an estimate of the heliocentric distance at the time of disappearance. He made no attempt to provide the other heliocentric coordinates that follow naturally. The subsequent claim that he probably based on these and a few other examples was only that the method yielded vanishing points “ordinarily lower than the orbit of Jupiter”.59 Such a claim was a tacit indication of the crude nature of the method, because he was aware it required more than ten months from perihelion to reach such a distance, longer than any comet for which adequate data existed. Newton undoubtedly worked quickly with round numbers, made some mental calculations, and drew some rough sketches. He admitted such an initial procedure when discussing a proposed method for determining a parabolic orbit, working “first by graphic procedure in a rough and hasty way; then by a new graph with greater care; and lastly by an arithmetical computation”.60 Changes made for Aristotle’s comet may indicate he was beginning to make revisions and decided against using these examples. Perhaps he realized that speed requirements precluded further worthwhile analysis. According to Table l in Section 74, it would take a Sun-grazing comet almost three months to reach a distance 2 a.u. from the Sun, whereas these parameters indicate such a distance is reached in about two months. No further development of these ideas has been found. These are beautiful examples of Newton’s working where no one had gone before, and of what a fertile mind could wring out of extremely limited 442 J. A. Ruffner data. We can imagine that Newton’s eagerness to pile up evidence against Descartes carried him a little farther than the evidence justified. The final additions to the worksheet likely were notes on the apparent diameters of major stars and the planets extracted without credit from Hevelius’s Mercurius in Sole visus Gedani (Gdansk, 1662) (see Figure 4). This section was followed by a description of the comet of 1665 credited to Hevelius’s Descriptio cometae anno aerae Christ. M.DC.LVX. exorti … (Gdansk, 1666). A different set of notes on the diameter of star and planets was added to his copy of Mercator.61 These annotations included his first known interest in the intensity of light reflected from Saturn developed further in CUL Add Ms 3965.11, f. 175r and DMCII, Section 57. DMCII used data from different sources yet to be identified. Newton was carefully picking and choosing his supporting data. From Historia Cometarum to Liber Secundus A number of indeterminate steps preceded the final text for DMCII. Numerous technical calculations on other sheets are lost or unidentified. Many details were extracted from notes in his Waste Book. The Yahuda material was significantly rewritten for DMCII, Section 1. An early draft of major parts of DMCII, Sections 57, 58, 63 and 64 was greatly expanded in the final text.62 One notable feature of this early draft was an argument against placing comets at the edge of the solar vortex. Was this only an allusion to Descartes’s theory or did Newton have a lingering belief in vortices? A reformulation in keeping with the terminology laid out in DMCII, Section 2 indicated comets could not whirl inside the sphere of the circumsolar force at a distance 10 to 20 times the distance of Saturn from the Sun.63 If the rebutted vortex statement were only an account of Descartes’s belief, there would have been no need to substitute circumsolar language which may represent a turning point in Newton’s views. Neither formulation was used in DMCII. Another key source was the list of propositions on comets.64 This list included principles that can now be dated after the revised De motu tract of late 1684. Propositions that discuss mutual gravitation and place the Sun “nearly in the focus” of a comet’s trajectory would have postdated introduction of the concept of perturbation. The proposition on celestial matter revolving around the centre of the cosmos predated the introduction of the circumsolar force terminology in 1685. The components of Historia cometarum had various fates. As noted, Newton used a different set of lunar distances and apparent sizes of the planets. The spectacular tails of several comets highlighted for similarity with the comet of 1680 were used as evidence of close approaches to the Sun. The supposed astrological significance was dismissed as “fancies of the vulgar”.65 The matrix concept vanished and no attempt was made to provide an estimated period of any comet. Indeed, while he held on to the prospect of elliptical orbits, computational efforts were directed at parabolic approximations of the portion near the Sun. The various listings for comets that departed from great circle paths or reversed directions near the vanishing point were Isaac Newton’s Historia Cometarum 443 subsumed under general arguments about parallax distances. The text in DMCII, Sections 58 and 59 alluded to rough determinations of parallax based on such departures. DMCII, Section 60 alluded to his work with the Wren/Wallis rectilinear hypothesis and the work of Kepler, Hevelius, and others who proposed various trajectories largely within the orbit of Mars or somewhat further. The brightness and apparent length of tail were other indicators of this truth spelled out in great detail in Sections 61 to 64. Section 65 reinforced the conclusion by appeal to the summary statement that four or five times more comets had been seen in the solar hemisphere than the opposite. The logical consequence was that comets could not be located beyond the sphere of Saturn, for otherwise those opposite the Sun would be closer and hence brighter and more numerous in appearance. This consequence also helped establish the point that the trajectories must be highly eccentric. This high eccentricity prepared the way to substitute parabolas for the elliptical portion near the Sun. DMCII, Section 66 took information about the length of tails to the next level, noting that the tails of comets “descending towards the sun always appear short and rare, and are seldom said to have exceeded 15 or 20 deg. in length; but in recess … often shine like fiery beams, and soon after reach to 40, 50, 60, 70 deg. in length or more”.66 The empirical basis might have gone something like this. Comets seen in the evening with motion in the order of the signs or in the morning moving contrary to the order of signs were in recess. Conversely the comets would be in descent towards the Sun. The results for comets with sufficient detail found in Riccioli, Hevelius, or later notes to determine these conditions are shown in Table 4. Anomalies such as 1590 and 1661 for those in recess might be attributed to unfavourable geocentric positions. The long and radiant tale of 1532 was accompanied by an unusually large head, three times larger than Jupiter, consistent with a favourable position very close to Earth. This analysis affirmed the close relation between a tail and the heat of the Sun and set up a series of Sections from 67 to 71 on the nature and rate of growth of tail. General support was provided for Kepler’s views and contrary to those of Descartes and Hooke. Included was a proposed chimneylike mechanism for lifting tail particles that gave Halley, Gregory, Pemberton (and everybody since) pause.67 DMCII, Sections 72 and 73 finally began to connect comets with motion in conic sections. In Section 65, Newton originally drafted, but then cancelled, a passage indicating that comets frequent the planetary regions, but that it was not yet established that the orbits were ellipses rather than hyperbolas or parabolas. Part of this cancelled material became Section 72, setting the stage to argue in Section 73 that, so far as he hitherto had “observed”, comets move in very eccentric ellipses very nearly approaching to parabolas. He added the questionable assertion that various parallax determinations coupled with apparent arc measures yielded speeds near the required parabolic limit. In Sections 74–76, he developed along dynamical lines a heuristic method to determine the time required for an object moving at parabolic speed to travel from various perihelion distances to the Earth’s mean distance from the Sun.68 These results provided the first empirical evidence that the November 444 J. A. Ruffner Table 4. Apparent length of tail. Comets in Recess 371 bc 729 ad 1042 1106 1264 1577 1590 1607 1618 #2 1618 #3 1618 #4 1661 1680 Dec Length/Description of Tail 60º like a flaming torch long flaming hairs a splendour like a great beam long and broad 30º long, 5º broad 10º long and thick 45º long 104º greater than 6º 70º Comets in Descent 1532 1533 1580 1618 #1 1647 1665 1672 1677 1680 Nov 1682 1683 Length/Description of Tail long and radiant 15º rare short and broad 12º 25º almost imperceptible 6º 20º 6º short and December comets were the same object. He still entertained in Section 76 the possibility that the speed was slightly less than required for a parabola and the true orbit of 1680 was an ellipse. Section 77 purported to demonstrate that the comets of 1618, 1607, 1665, 1682, and 1472 also entered and exited the sphere of the Earth’s orbit at a time interval consistent with parabolic speed. These results were flawed by a misrepresentation of his table relating perihelion distance and transit time.69 Newton proceeded with improved methods of interpolation (stemming from an earlier method) to track the motion of a comet using three observations. There was as yet no hint of the method of differences for this purpose. The text offered a general account of how to determine the first approximation of an orbit as if he had already accomplished that process, noting “all this I do, first by graphical procedure in a rough and hasty way; then by a new graph with greater care; and lastly an arithmetic computation”.70 The text ended with an explanation of what must be done to correct that result by successive iterations as needed. No actual trajectory, parabolic, elliptical or otherwise, was provided. Unfortunately no worksheets for any of these claims have been located. The Principia In the first edition of the Principia it was still only a hope to demonstrate the true paths were elliptical, Newton noting, “Unless I am mistaken, comets are a kind of planet and revolve in their orbits with a continual motion…”.71 As proof, various lines of evidence indicated the bodies do not dissipate quickly but are “solid, compact, fixed, and durable, like the bodies of planets”.72 It was also possible to conceive of their travelling in straight lines or open curves through a sufficiently large universe that was only a few thousand years old and might soon end. Similarly, the restorative role imagined for comets required only a sufficiently large number of them.73 But continuing, he argued, “if comets revolve in orbits, these orbits will be ellipses … so Isaac Newton’s Historia Cometarum 445 near to parabolas that parabolas can be substituted for them without sensible error”.74 The single solution presented was a parabola for the comet of 1680 that saved select observations with errors that range from –10½′ to +2′. As for the true ellipse, in the first edition he indicated, “I leave the transverse diameters of the orbits and the periodic times to be determined by comparing comets that return in the same orbits after long intervals of time”.75 He concluded with an untested procedure by which the parabolic approximation could be corrected to determine the actual transverse diameter. The historical method of comparison was realized by Halley. Starting in the mid1690s with suggestions from Newton, he calculated parabolic elements for 24 comets observed between 1337 and 1698. The main outcomes were parabolic elements for three comets, 1682, 1607, and 1531 that differ by no more than about 1½ degrees or 3½% in perihelion distance before corrections for perturbations. He predicted it would return in 1758.76 Further work added the comet of 1456. Using principles similar to those laid out by Newton in DMCII and again in Book III of Principia, Halley determined it also was retrograde, “which tho’ nobody made Observations upon it … I cannot think was different from those I have just mentioned”, adding, “and since looking over the Histories of Comets I find, at an equal Interval of Time, a Comet to have been seen about Easter in the 1305, which is another double period of 151 years before the former”.77 The only other comets among the 24 solutions in even rough agreement were those of 1532 and 1661, but the observations “concerning the first of these Comets, are too rude and inaccurate for any thing of certainty to be drawn from them, in so nice a manner”.78 The second edition of the Principia (1713) introduced Halley’s proof of periodicity for the comet of 1682 and hence evidence that its orbit must be elliptical but without a confirming demonstration. Again, in the third edition (1726), Newton noted, once the period and other elements are known, “it will not be at all difficult to determine the elliptical orbit of the comet”.79 He was well aware that perturbations would alter the eccentricities and periodic times but argued that the aphelia of comets were distributed through all regions of the heavens so that “they may be as far distant from one other as possible and may attract one other as little as possible”.80 Thus, ellipses would be sufficient when only the disturbances of other comets were involved. The perturbing effects of the planets were also generally considered to be small.81 Still, no actual demonstration was provided. Some time before 1717, although only published posthumously in 1752, Halley had indeed calculated elliptical orbits for the comet of 1682 and its earlier appearance in 1607 and 1531. He achieved fits better than 3 arc minutes for 1682 with significantly larger errors for the cruder measures of 1607 and 1532.82 Newton evidently was never informed about Halley’s elliptical orbit for the comet of 1682, even when they corresponded in 1725 about an elliptical orbit for the comet of 1680. Inconclusive work in the 1690s by Newton on the direct determination of an elliptical orbit for the comet of 1680 had provided indications of periods up to perhaps 3000 years.83 For the second edition he speculated only that the period for the comet of 1680 may be something more than 500 years.84 For his part, Halley argued 446 J. A. Ruffner the equality of period and similar appearance of the comets of a.d. 1680/1, 1106 and 531, and 44 b.c. made it “not improbable but this Comet may have four times visited us at intervals of about 575 years”.85 Since no useful data were available to determine elements for the earlier comets to verify this period, Halley inverted the required procedure. Using the corresponding transverse diameter and other assumed elements, he calculated an ellipse that agreed within 2′31″ or better for the full range of observations for the comet of 1680. Newton claimed that “this agreement provides proof that it [the comet of 1680] was one and the same comet which appeared all this time and that it was one and that its orbit has been correctly determined here”.86 Close agreement from one arbitrarily chosen sequence, however, does not provide proof. As the early work of Newton showed, several other long interval series were possible that undoubtedly would have saved the observations to within a few arc minutes. Modern calculations have placed the period for the comet of 1680 on the order of 10,000 years. Conclusion Isaac Newton developed an extensive but little-known research program on comets in 1680/1 in conjunction with the comet(s) of that year. Flamsteed’s intervention with a single comet solution left a very different record of correspondence that has obscured other important documents. Newton thought he had empirical evidence against comets’ making sharp turns about the Sun. Contrarily, he supported trajectories that were nearly straight with some slight curvature induced by motion across the flow of the vortex or perhaps, even as Hooke thought, by a gravitational principle that was not completely destroyed. A likely turning point, a few months before Halley’s momentous visit in 1684, was the fresh emphasis given to the ancient view that comets are a kind of planet travelling freely through space with the Sun fixed at the centre. The idea was commonplace and certainly not new to him. The new element was the concept of gravitation between all bodies that allowed the same principles of motion to be applied to planets and comets. Historia cometarum almost seems to be a retrograde document. Memorable comets in this document and an earlier one were enlisted for the purpose of demonstrating the placement of comets among the planets but from entirely different perspectives of motion. Matrices were set up with the apparent hope of finding previous returns of the comets of 1680 and 1618 and hence an indication of motion in ellipses, without the computations envisaged in the De motu tracts but in accordance perhaps with an ancient practice. The more general goal was to amass evidence that placed comets close to the Sun while visible with recourse only to simple models involving uniform motion and exact solar opposition of the tail. By far the largest segments of DMCII were devoted to such evidence. The expectation in the De motu tracts of being able to determine the true ellipse by successive approximation gave way in DMCII for a plan to substitute a parabolic approximation. After many false starts, such a solution was realized for the comet Isaac Newton’s Historia Cometarum 447 of 1680. The first edition of the Principia could only offer the durability of a comet under the extreme heat of the Sun as an indication they can return again and again. Halley’s example of a single comet, that of 1682 with a period of about 75 years and a presumed aphelion distance of about 35 a.u., provided the only demonstration in the second edition. While Halley calculated ellipses for this comet, all that Newton had from him for the third edition was an ellipse for the comet of 1680 based on an assumed period of 575 years that fell short of proof. The Principia provided a brilliant breakthrough for a rational theory of comets, but left Newton’s dream of a direct demonstration of elliptical orbits unfulfilled, with much work yet to be done.87 There is much research still needed on the genealogy of the Principia. Acknowledgements The suggestions of several anonymous referees are greatly appreciated. REFERENCES 1. Lucius Annaeus Seneca, Quaestiones naturales, transl. by Thomas H. Corcoran (Cambridge, MA, 1972), Bk 7.3.1, 231–3. 2. J. A. Ruffner, “The curved and the straight: Cometary theory from Kepler to Hevelius”, Journal for the history of astronomy, ii (1970), 178–94. 3. Flamsteed to Towneley, 11 May 1677, The correspondence of John Flamsteed the first Royal Astronomer, ed. by E. G. Forbes et al. (3 vols, Bristol and Philadelphia, 1995–2002; hereafter: Flamsteed correspondence), i, 552–4. 4. Robert Hooke, Cometa (London, 1678), in R. T. Gunther (ed.), Early science in Oxford (15 vols, London, 1967–68), viii, 209–71, pp. 251 seq. 5. Understand that apparent length provides at best a rough indication of the true length depending on perspective and other factors. 6. Flamsteed’s copy of Gallet’s observations mixed old and new style dates as if they were all old style. They would have been included in his letter of 12 February 1680/1 sent to James Crompton for Newton. The full letter was lost or discarded, with an abstract added to a blank page of Flamsteed’s 7 March 1680/1 letter. The correspondence of Isaac Newton, ed. by H. W. Turnbull et al. (7 vols, Cambridge, 1959–77; hereafter: Correspondence), ii, 336, 343, 348–9. Other material was entered in Newton’s Waste Book, see ref. 10. Flamsteed tried to blame Newton for the confusion. The same misrepresentation was in the copy sent to Towneley, 7 February 1680/1, Flamsteed correspondence, i, 756. Flamsteed also indicated that Gallet’s observations were made at Rome, rather than at Avignon. 7. Cambridge University Library (CUL) Add Ms 3965.11, f. 153. Reel 4 in the Chadwick-Healy microfilm edition of Sir Isaac Newton manuscripts and papers available at many institutions or through interlibrary loan. Some scholars will have access to digital scans of documents in CUL Add Ms 3965 through the RLG Cultural Materials Project available at a few subscribing libraries. The document used initial data from Flamsteed, later corrected in his letter of 12 February 1680/1. When plotted, the calculated parameters map a straight line. 8. Flamsteed to Crompton for Newton, 15 December 1680/1; Flamsteed to Crompton, 3 January 1680/1, Correspondence, ii, 315–17, 319–20. 9. CUL Add Ms 3965.14, 614r. Reel 5. The list is consistent with what could have been compiled from Giovanni Battista Riccioli, Almagestum novum (2 parts, Bologna, 1651), ii, 2–20. The memorable listings followed an account by an anonymous observer for 15 December 1680/1 cited in a letter of 28 February 1680/1 and preceded a list of his observations, one of which (for 11 January 448 J. A. Ruffner 1680/1) was revised in a letter of 16 April 1681. By that time he largely discounted the rest as insufficiently accurate. Correspondence, ii, 346, 365–6. 10. The theory would have been part of Flamsteed’s letter of 12 February 1680/1 but not deemed worth saving. See ref. 6. The details are known from Flamsteed’s letter to Halley, 17 February 1680/1 and Newton’s response of 28 February 1680/1. Correspondence, ii, 336–47. The letter of 12 February also included the first full set of observations from Greenwich and a partial set from Paris. These observations were preserved in Newton’s Waste Book along with a corrected copy of Gallet’s observations, and further observations of the November comet made by Tho[mas] Hill at Canterbury and Marco Antonio Cellio at Rome (from Flamsteed’s letter of 7 March). CUL Add Ms 4004, f. 98v. Newton copied the “former” set of Greenwich observations from Flamsteed’s letter of 12 February in preference to the revised set of 7 March. This former set originally listed the longitude for 10 January as Aries 20º49½′ altered sometime after the 16 April letter to 20º41½′ in closer agreement with revised set. See Correspondence, ii, 354, 365. 11. Flamsteed to Towneley, 22 March 1689/1, Flamsteed correspondence, i, 781. See also The Gresham lectures of John Flamsteed, ed. by E. G. Forbes (London, 1975), 30, 105–17. 12. Newton to Flamsteed, 28 February 1680/1, Correspondence, ii, 346. 13. Compare Newton’s point of agreement with Flamsteed’s view “that ye atmosphere about ye head [of a comet] shines also the suns light, though perhaps not altogether by it” (ibid.) with one aspect of Hooke’s analysis, “the light of the Comet did not depend wholly from the Sun beams”, Cometa (ref. 4), 261. But see also Newton’s notes “ex Hookii Cometa edito ann 1678” (“from Hooke’s Cometa published in 1678”). CUL Add Ms 4004, ff. 103r–104r. Reel 14. Judged on the basis of the immediately surrounding material, these notes were made in March 1680/1 when Newton began to compile information on comets from a variety of sources. When he first read the work is at issue. 14. CUL Add Ms 4004, f. 101r. 15. Newton to Flamsteed, 16 April 1681, Correspondence, ii, 364. 16. Ibid., ii, 366. 17. R. S. Westfall, Never at rest: A biography of Isaac Newton (Cambridge, 1980), 353. The title comes from Jewish National and University Library, Yah. Ms. Var 1 Newton Ms 16.2. Reel 39. Hereafter: Yahuda Ms. 18. Simon Schaffer, “Comets and idols: Newton’s cosmology and political theology”, in P. Theerman and A. F. Seef (eds), Action and reaction (Cranbury, NJ, 1993), 183–231, p. 220. From an unused preface for the third edition. The mathematical papers of Isaac Newton, ed. by D. T. Whiteside et al. (8 vols, Cambridge, 1967–81; hereafter: Mathematical papers), vii, 495. See also Nicholas Fatio de Duillier to Christiaan Huygens, 29 February 1691, Correspondence, iii, 193. 19. Yahuda Ms 17.2, ff. 18r–19r. Reel 39. Two paragraphs on f. 20r are repeated in “Philosophical origins”, Yahuda Ms 16.2, f. 1. Westfall indicates the basic argument of Yahuda Ms 16.2 is sketched in Yahuda Ms 17.2, f. 14. He dates the material to late 1683 or early 1684, Never at rest (ref. 17), 351–2, n. 55. See also R. Iliffe, “Is he like other men?”, in G. Maclean (ed.), Culture and society in the Stuart Restoration (Cambridge, 1995), 159–176, see pp. 164–70, esp. notes 19, 21, 23, 24, 28. 20. CUL Add Ms 4004, f. 101v. The added sentence is at f. 99r. Reel 14. Some scholars think Newton learned these details at a meeting in 1682 when Halley returned from the Continent. I am unaware of any independent evidence for such a meeting. 21. CUL Add Ms 3965.7, ff. 55–62. Reel 4. Mathematical papers, vi, 59. John Herivel, The background to Newton’s Principia (London, 1965), 257–92, p. 285. 22. CUL Add Ms 3965.7, ff. 40–54. Reel 4. Herivel, op. cit. (ref. 21), 301. 23. Ibid., 302. 24. Newton to Flamsteed, 12 January 1684/5, Correspondence, ii, 413. 25. Newton’s request is lost. The contents are known from Flamsteed’s response of 27 December 1684. Correspondence, ii, 403–6. 26. Newton to Aston, 24 February 1684/5, Correspondence, ii, 415. Isaac Newton’s Historia Cometarum 449 27. CUL Add Ms 3990. Reel 12. Published posthumously as A treatise of the system of the world. Translated into English (London, 1728; 2nd edn, 1731; reprinted 1969). Also Sir Isaac Newton’s Mathematical principles of natural philosophy and his System of the world, transl. by Andrew Motte, revised by Florian Cajori (Berkeley, 1930, etc.; hereafter: Newton’s system, Cajori edn), 549–626. Section numbers follow those in this edition. A more faithful critical translation is needed. 28. Newton’s system, Cajori edn, Section l, 549–50. 29. Early drafts of DMCII, Sections 57, 58, 63, and 64 are found in CUL Add Ms 3965.11, ff. 175–76. Reel 4. 30. CUL Add Ms 3965.11, ff. 172r–173v. Reel 4. 31. Johannes Hevelius, Cometographia (Gdansk, 1668); Descriptio cometae anno aerae Christ. M.DC.LVX. exorti… (Gdansk, 1666); Mercurius in sole visus Gedani (Gdansk, 1662); Annus climactericus (Gdansk, 1685). 32. Riccioli, Almagestum novum (ref. 9). 33. Nicolas Mercator, Institutionum astronomicarum libri duo (London, 1675). 34. Johannes Kepler, Tabulae Rudolphinae (Ulm, 1627), in Gesammelte Werke, ed. by Max Caspar and Franz Hammer (Munich, 1937– ), x, 104–42; Vincent Wing, Astronomia Britannica (London, 1669), tables, 70, passim. 35. Cometae observationes cometae habitae ab Academia Physicomathematica Romana anno 1680 et 1681 (Rome, 1681), ed. by Giuseppe Ponteo. This book was probably first seen by Newton in 1685. 36. Hevelius, Cometographia, Bk 12, 719–913. Stanislaw Lubienski, Theatrum cometarum pars posterior (Amsterdam, 1666) might have been a supplemental source but I find no specific trace of its use at this time. 37. Riccioli, Almagestum novum (ref. 9), i, 226. 38. René Descartes, The principles of philosophy, transl. by V. R. Miller and R. P. Miller (Dordrecht, 1983), sections 128, 157. 39. There was confusion whether four or better three different comets appeared in 1618. Newton followed Riccioli and Hevelius in considering this comet to be the fourth. No period has been established for it. Gary W. Kronk, Cometography: A catalog of comets (4 vols, Cambridge, 1999–2008), i, 338–41. 40. Newton’s System, Cajori edn (ref. 27), 576 . 41. Isaac Newton, Philosophiae naturalis principia mathematica (London, 1687; hereafter: Principia (1687)), 420; The Principia, a new translation by I. Bernard Cohen and Anne Whitman (Berkeley, 1999; hereafter: Principia (1999)), 819. 42. Principia (1687), 474; (1999), 888–90. 43. Lucasian Lecture 9, 1676, problem 16, CUL Dd 9.68, f. 200; Mathematical papers, v, 210–13. Similar methods were used by Kepler and others. Variations were used in DMCII, lemmas 3, 4, and 5 for interpolation. Newton’s System, Cajori edn (ref. 27), 623–4. 44. CUL Add Ms 4004, f. 98v. See ref. 10. He sought Flamsteed’s latest revisions as he prepared to construct an actual orbit in September 1685. Correspondence, ii, 419. 45. CUL Add Ms 4004, f. 101r. Reel 14. 46. Ibid. 47. J. E. McGuire and Martin Tamny, Certain philosophical questions: Newton’s Trinity notebook (Cambridge, 1983), 412–13. The dates for this comet are new style. 48. Johannes Kepler, De cometis libelli tres (Augsberg, 1619–20) in Gesammelte Werke (ref. 34), iv, 196. 49. Ibid., 195, 200–1. Kepler’s account mixed his observations with those of Cysatus and several other astronomers, making it difficult to determine which were which. I have followed Riccioli’s tabulations, Almagestum novum (ref. 9), ii, 20–1. 50. Kepler, op. cit. (ref 48), iv, 194. 51. Hevelius, Cometographia (ref. 31), 627. 450 J. A. Ruffner 52. Aristotle, Meteorologica, transl. by H. P. D. Lee (Cambridge, MA, 1952), Bk 1.6, 343b, 44–7. 53. Diodorus Siculus, Historical library, transl. by C. L. Sherman (Cambridge, MA, 1971), Bk 15.2, 88–91. 54. Hevelius, Cometographia (ref. 31), 796–7, 904. 55. CUL Add Ms 3965.14, f. 613v. Reel 5. 56. Isaac Newton’s Philosophiae naturalis principia mathematica: The third edition (1726) with variant readings, assembled by Alexandre Koyré, I. B. Cohen, and Anne Whitman (2 vols, Cambridge, 1972; hereafter: Principia, Koyre/Cohen), ii, 747. 57. The Gemini sign is followed by a 2 and what seems to be a 6 although the loop still evident may be a smaller than usual 0 rather than the bottom part of a 6. Notice what happens if a bold 5 is inscribed over a 6. 58. Hevelius, Cometographia (ref. 31), 827, 905. 59. Newton, Principia (1687), 476; (1999), 890. 60. Newton’s System, Cajori edn (ref. 27), 625. 61. Mercator, Institutionum astronomicarum (ref. 33). Newton’s copy is in the library of Trinity College, Cambridge, NQ 10.152. The holograph annotations are on page 213. 62. CUL Add. Ms 3965.11, ff. 175r–176v. Reel 5. Another source for Section 64 was Extracta ex Hevelii anno climacterico. CUL Add Ms 3965.14, ff. 581r–582r. Reel 5. Hevelius, Annus climactericus (ref. 31). A deletion in Section 64 (CUL Add Ms 3990, f. 41) indicates the use of Ponteo’s edition of the Roman observations of 1680 and 1681 (ref. 35) A set of Ponteo’s November observations is in CUL Add Ms 4004, f. 98r. 63. CUL Add Ms 3965.11, f. 176v. 64. CUL Add Ms 3965.14, f. 613v. J. A. Ruffner, “Newton’s propositions on comets: Steps in transition, 1681–84”, Archive for history of exact sciences, liv (2000), 259–77. 65. For a different interpretation, see Sara J. Schechner, Comets, popular culture, and the birth of modern cosmology (Princeton, 1997), especially p. 142. The passages were taken from Hevelius who in turn quoted from older accounts. Cometographia (ref. 31), 797, 822, 827, 836, 844–5, 876–77, 880–1, 883. A passage about Justin was more likely from Riccioli, Almagestum novum (ref. 9), ii, 4. See also Iliffe’s discussion of Newton’s antagonism to astrological signs, “Other men” (ref. 19), 168–9. 66. Newton’s System, Cajori edn (ref. 27), 606. 67. For a detailed discussion of Newton’s views about tails as finally represented in Principia, third edition, see Tofigh Heidarzadeh, A history of physical theories of comets, from Aristotle to Whipple (Dordrecht, 2008), 96–101, 111–17. 68. See Whiteside’s calculus-based derivation, Mathematical papers, vi, 484–5. 69. Newton’s System, Cajori edition (ref. 27), 618–19. The times cited for perihelion distances of 122, 350, and 390 parts (a.u. = 1000) were said to be 30, 33½, and 34 days, respectively. In fact his Table I attributed these transit times to perihelion distances half those values. For a method of calculating intermediate values, see Mathematical papers, vi, 484, n. 10. 70. Newton’s System, Cajori edition (ref. 27), 625. 71. Principia (1687), 480; (1999), 895. 72. Principia (1687), 498; (1999), 918. 73. Principia (1687), 506; (1999), 926. 74. Principia (1687), 480; (1999), 895. 75. Principia (1687), 508–9; (1999), 928–9. 76. Edmond Halley, “Astronomiae cometicae synopsis”, Philosophical transactions, xxiv (1705), 1882–99, pp. 1886, 1897; idem, A synopsis of the astronomy of comets, in William Whiston, Sir Isaac Newton’s mathematick philosophy more easily demonstrated (London, 1716, reprinted 1972; hereafter: Whiston/Halley), 409–44, pp. 414, 439. A full discussion of the different editions of Halley’s “Synopsis” is David W. Hughes, “Edmond Halley: His interest in comets”, in N. J. Thrower (ed.), Standing on the shoulders of giants (Berkeley, 1990), 324–72. Isaac Newton’s Historia Cometarum 451 77. Whiston/Halley (ref. 76), 439. It is not clear why Halley did not include the comet of 1380 prominently listed by both Hevelius and Lubieniecki. 78. Ibid., 440. 79. Principia (1999), 936. 80. Ibid., 937. 81. Principia (1687), 420; (1999), 819. In 1695, responding to a query by Halley, he suggested the influence of Jupiter or Saturn might increase or decrease a comet’s period by a day up to a year or more. Correspondence, iii, 181. 82. Alan Cooke, Edmond Halley: Charting the heavens and the seas (Oxford, 1998), 214. 83. David Gregory’s memoranda of March 1702/3. Correspondence, iii, 402–3. 84. Isaac Newton, Philosophiae naturalis principia mathematica, editio secunda (Cambridge, 1713), 464–5; Principia, Koyre/Cohen (ref. 56), 733. 85. Whiston/Halley (ref. 76), 441. Newton had noted similarities between the comets of 1680 and 1106 but Hevelius did not list the comet of 531 and dated the comet of 44 b.c. associated with the death of Julius Caesar as 41 b.c. Halley’s source could have been Lubieniecki, op. cit. (ref. 31). 86. Principia (1999), 911. The table is on page 912. The result is also highlighted in the Preface, p. 400. 87. Simon Schaffer, “Halley, Delisle, and the making of the comet”, in Thrower (ed.), op. cit. (ref. 76), 254–98.