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Transcript
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.
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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
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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
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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
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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
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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
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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.