Kepler. This astronomer first saw it on the evening of the twenty-sixth
of September, when it had the appearance of a star of the first
magnitude, and, to his vision, was without a tail; but the friends who
accompanied him had better sight, and distinguished the tail. Before
three o'clock the following morning the tail had become clearly visible,
and had acquired great magnitude. Two days afterwards, the comet was
observed by Longomontanus, a distinguished philosopher of the time. He
describes its appearance, to the naked eye, to be like Jupiter, but of a
paler and more obscured light; that its tail was of considerable length,
of a paler light than that of the head, and more dense than the tails of
ordinary comets.
The next appearance, and that which was observed by Halley himself, took
place in 1682, a little before the publication of the '_Principia_.' In
the interval between 1607 and 1682, practical astronomy had made great
advances; instruments of observation had been brought to a state of
comparative perfection; numerous observatories had been established, and
the management of them had been confided to the most eminent men in
Europe. In 1682, the scientific world was therefore prepared to examine
the visitor of our system with a degree of care and accuracy before
unknown.
In the year 1686, about four years afterwards, Newton published his
'_Principia_,' in which he applied to the comet of 1680 the general
principles of physical investigation first promulgated in that work. He
explained the method of determining, by geometrical construction, the
visible portion of the path of a body of this kind, and invited
astronomers to apply these principles to the various recorded
comets,--to discover whether some among them might not have appeared at
different epochs, the future returns of which might consequently be
predicted. Such was the effect of the force of analogy upon the mind of
Newton, that, without awaiting the discovery of a periodic comet, he
boldly assumed these bodies to be analogous to planets in their
revolution round the sun.
Extraordinary as these conjectures must have appeared at the time, they
were soon strictly realized. Halley, who was then a young man, but
possessed one of the best minds in England, undertook the labor of
examining the circumstances attending all the comets previously
recorded, with a view to discover whether any, and which of them,
appeared to follow the same path. Antecedently to the year 1700, four
hundred and twenty-five of these bodies had been recorded in history;
but those which had appeared before the fourteenth century had not been
submitted to any observations by which their paths could be
ascertained,--at least, not with a sufficient degree of precision, to
afford any hope of identifying them with those of other comets.
Subsequently to the year 1300, however, Halley found twenty-four comets
on which observations had been made and recorded, with a degree of
precision sufficient to enable him to calculate the actual paths which
these bodies followed while they were visible. He examined, with the
most elaborate care, the _courses_ of each of these twenty-four bodies;
he found the exact points at which each one of them crossed the
ecliptic, or their _nodes_; also the angle which the direction of their
motion made with that plane,--that is, the _inclination of their
orbits_; he also calculated the nearest distance at which each of them
approached the sun, or their _perihelion distance_; and the exact place
of the body when at that nearest point,--that is, the _longitude of the
perihelion_. These particulars are called the _elements_ of a comet,
because, when ascertained, they afford sufficient data for determining a
comet's path. On comparing these paths, Halley found that one, which had
appeared in 1661, followed nearly the same path as one which had
appeared in 1532. Supposing, then, these to be two successive
appearances of the same comet, it would follow, that its period would be
one hundred and twenty-nine years, reckoning from 1661. Had this
conjecture been well founded, the comet must have appeared about the
year 1790. No comet, however, appeared at or near that time, following a
similar path.
In his second conjecture, Halley was more fortunate, as indeed might be
expected, since it was formed upon more conclusive grounds. He found
that the paths of comets which had appeared in 1531 and 1607 were nearly
identical, and that they were in fact the same as the path followed by
the comet observed by himself in 1682. He suspected, therefore, that the
appearances at these three epochs were produced by three successive
returns of the same comet, and that, consequently, its period in its
orbit must be about seventy-five and a half years. The probability of
this conclusion is strikingly exhibited to the eye, by presenting the
elements in a tabular form, from which it will at once be seen how
nearly they correspond at these regular intervals.
=====================================================================
Time.|Inclination of|Long. of the |Long. Per.|Per. Dist. |Course.
|the orbit. |node. | | |
=====================================================================
1456 | 17°56´ | 48°30´ |301°00´ | 0°58´ |Retrograde.
1531 | 17 56 | 49 25 |301 39 | 0 57 | "
1607 | 17 02 | 50 21 |302 16 | 0 58 | "
1682 | 17 42 | 50 48 |301 36 | 0 58 | "
=====================================================================
So little was the scientific world, at this time, prepared for such an
announcement, that Halley himself only ventured at first to express his
opinion in the form of conjecture; but, after some further investigation
of the circumstances of the recorded comets, he found three which, at
least in point of time, agreed with the period assigned to the comet of
1682. Collecting confidence from these circumstances, he announced his
discovery as the result of observation and calculation combined, and
entitled to as much confidence as any other consequence of an
established physical law.
There were, nevertheless, two circumstances which might be supposed to
offer some difficulty. First, the intervals between the supposed
successive returns were not precisely equal; and, secondly, the
inclination of the comet's path to the plane of the earth's orbit was
not exactly the same in each case. Halley, however, with a degree of
sagacity which, considering the state of knowledge at the time, cannot
fail to excite unqualified admiration, observed, that it was natural to
suppose that the same causes which disturbed the planetary motions must
likewise act upon comets; and that their influence would be so much the
more sensible upon these bodies, because of their great distances from
the sun. Thus, as the attraction of Jupiter for Saturn was known to
affect the velocity of the latter planet, sometimes retarding and
sometimes accelerating it, according to their relative position, so as
to affect its period to the extent of thirteen days, it might well be
supposed, that the comet might suffer by a similar attraction an effect
sufficiently great, to account for the inequality observed in the
interval between its successive returns: and also for the variation to
which the direction of its path upon the plane of the ecliptic was found
to be subject. He observed, in fine, that, as in the interval between
1607 and 1682, the comet passed so near Jupiter that its velocity must
have been augmented, and consequently its period shortened, by the
action of that planet, this period, therefore, having been only
seventy-five years, he inferred that the following period would probably
be seventy-six years, or upwards; and consequently, that the comet ought
not to be expected to appear until the end of 1758, or the beginning of
1759. It is impossible to imagine any quality of mind more enviable than
that which, in the existing state of mathematical physics, could have
led to such a prediction. The imperfect state of mathematical science
rendered it impossible for Halley to offer to the world a demonstration
of the event which he foretold. The theory of gravitation, which was in
its infancy in the time of Halley's investigations, had grown to
comparative maturity before the period at which his prediction could be
fulfilled. The exigencies of that theory gave birth to new and more
powerful instruments of mathematical inquiry: the differential and
integral calculus, or the science of fluxions, as it is sometimes
called,--a branch of the mathematics, expressed by algebraic symbols,
but capable of a much higher reach, as an instrument of investigation,
than either algebra or geometry,--was its first and greatest offspring.
This branch of science was cultivated with an ardor and success by
which it was enabled to answer all the demands of physics, and it
contributed largely to the advancement of mechanical science itself,
building upon the laws of motion a structure which has since been
denominated 'Celestial Mechanics.' Newton's discoveries having obtained
reception throughout the scientific world, his inquiries and his
theories were followed up; and the consequences of the great principle
of universal gravitation were rapidly developed. Since, according to
this doctrine, _every body in nature attracts and is attracted by every
other body_, it follows, that the comet was liable to be acted on by
each of the planets, as well as by the sun,--a circumstance which
rendered its movements much more difficult to follow, than would be the
case were it subject merely to the projectile force and to the solar
attraction. To estimate the time it would take for a ship to cross the
Atlantic would be an easy task, were she subject to only one constant
wind; but to estimate, beforehand, the exact influence which all other
winds and the tides might have upon her passage, some accelerating and
some retarding her course, would present a problem of the greatest
difficulty. Clairaut, however, a celebrated French mathematician,
undertook to estimate the effects that would be produced on Halley's
comet by the attractions of all the planets. His aim was to investigate
_general rules_, by which the computation could be made arithmetically,
and hand them over to the practical calculator, to make the actual
computations. Lalande, a practical astronomer, no less eminent in his
own department, and who indeed first urged Clairaut to this inquiry,
undertook the management of the astronomical and arithmetical part of
the calculation. In this prodigious labor (for it was one of most
appalling magnitude) he was assisted by the wife of an eminent
watchmaker in Paris, named Lepaute, whose exertions on this occasion
have deservedly registered her name in astronomical history.
It is difficult to convey to one who is not conversant with such
investigations, an adequate notion of the labor which such an inquiry
involved. The calculation of the influence of any one _planet_ of the
system upon any other is itself a problem of some complexity and
difficulty; but still, one general computation, depending upon the
calculation of the terms of a certain series, is sufficient for its
solution. This comparative simplicity arises entirely from two
circumstances which characterize the planetary orbits. These are, that,
though they are ellipses, they differ very slightly from circles; and
though the planets do not move in the plane of the ecliptic, yet none of
them deviate considerably from that plane. But these characters do not
belong to the orbits of comets, which, on the contrary, are highly
eccentric, and make all possible angles with the ecliptic. The
consequence of this is, that the calculation of the disturbances
produced in the cometary orbits by the action of the planets must be
conducted not like the planets, in one general calculation applicable to
the whole orbits, but in a vast number of separate calculations; in
which the orbit is considered, as it were, bit by bit, each bit
requiring a calculation similar to the whole orbit of the planet. Now,
when it is considered that the period of Halley's comet is about
seventy-five years, and that every portion of its course, for two
successive periods, was necessary to be calculated separately in this
way, some notion may be formed of the labor encountered by Lalande and
Madame Lepaute. "During six months," says Lalande, "we calculated from
morning till night, sometimes even at meals; the consequence of which
was, that I contracted an illness which changed my constitution for the
remainder of my life. The assistance rendered by Madame Lepaute was
such, that, without her, we never could have dared to undertake this
enormous labor, in which it was necessary to calculate the distance of
each of the two planets, Jupiter and Saturn, from the comet, and their
attraction upon that body, separately, for every successive degree, and
for one hundred and fifty years."
The attraction of a body is proportioned to its quantity of matter.
Therefore, before the attraction exerted upon the comet by the several
planets within whose influence it might fall, could be correctly
estimated, it was necessary to know the mass of each planet; and though
the planets had severally been weighed by methods supplied by Newton's
'Principia,' yet the estimate had not then attained the same measure of
accuracy as it has now reached; nor was it certain that there was not
(as it has since appeared that there actually was) one or more planets
beyond Saturn, whose attractions might likewise influence the motions of
the comet. Clairaut, making the best estimate he was able, under all
these disadvantages, of the disturbing influence of the planets, fixed
the return of the comet to the place of its nearest distance from the
sun on the fourth of April, 1759.
In the successive appearances of the comet, subsequently to 1456, it was
found to have gradually decreased in magnitude and splendor. While in
1456 it reached across one third part of the firmament, and spread
terror over Europe, in 1607, its appearance, when observed by Kepler and
Longomontanus, was that of a star of the first magnitude; and so
trifling was its tail that, Kepler himself, when he first saw it,
doubted whether it had any. In 1682, it excited little attention, except
among astronomers. Supposing this decrease of magnitude and brilliancy
to be progressive, Lalande entertained serious apprehensions that on its
expected return it might be so inconsiderable, as to escape the
observation even of astronomers; and thus, that this splendid example
of the power of science, and unanswerable proof of the principle of
gravitation, would be lost to the world.
It is not uninteresting to observe the misgivings of this distinguished
astronomer with respect to the appearance of the body, mixed up with his
unshaken faith in the result of the astronomical inquiry. "We cannot
doubt," says he, "that it will return; and even if astronomers cannot
see it, they will not therefore be the less convinced of its presence.
They know that the faintness of its light, its great distance, and
perhaps even bad weather, may keep it from our view. But the world will
find it difficult to believe us; they will place this discovery, which
has done so much honor to modern philosophy, among the number of chance
predictions. We shall see discussions spring up again in colleges,
contempt among the ignorant, terror among the people; and seventy-six
years will roll away, before there will be another opportunity of
removing all doubt."
Fortunately for science, the arrival of the expected visitor did not
take place under such untoward circumstances. As the commencement of the
year 1759 approached, "astronomers," says Voltaire, "hardly went to bed
at all." The honor, however, of the first glimpse of the stranger was
not reserved for the possessors of scientific rank, nor for the members
of academies or universities. On the night of Christmas-day, 1758,
George Palitzch, of Politz, near Dresden,--"a peasant," says Sir John
Herchel, "by station, an astronomer by nature," first saw the comet.
An astronomer of Leipzic found it soon after; but, with the mean
jealousy of a miser, he concealed his treasure, while his contemporaries
throughout Europe were vainly directing their anxious search after it to
other quarters of the heavens. At this time, Delisle, a French
astronomer, and his assistant, Messier, who, from his unweared assiduity
in the pursuit of comets, was called the _Comet-Hunter_, had been
constantly engaged, for eighteen months, in watching for the return of
Halley's comet. Messier passed his life in search of comets. It is
related of him, that when he was in expectation of discovering a comet,
his wife was taken ill and died. While attending on her, being withdrawn
from his observatory, another astronomer anticipated him in the
discovery. Messier was in despair. A friend, visiting him, began to
offer some consolation for the recent affliction he had suffered.
Messier, thinking only of the comet, exclaimed, "I had discovered
twelve: alas, that I should be robbed of the thirteenth by
Montague!"--and his eyes filled with tears. Then, remembering that it
was necessary to mourn for his wife, whose remains were still in the
house, he exclaimed, "Ah! this poor woman!" (_ah! cette pauvre femme_,)
and again wept for his comet. We can easily imagine how eagerly such an
enthusiast would watch for Halley's comet; and we could almost wish that
it had been his good fortune to be the first to announce its arrival:
but, being misled by a chart which directed his attention to the wrong
part of the firmament, a whole month elapsed after its discovery by
Palitzch, before he enjoyed the delightful spectacle.
The comet arrived at its perihelion on the thirteenth of March, only
twenty-three days from the time assigned by Clairaut. It appeared very
round, with a brilliant nucleus, well distinguished from the surrounding
nebulosity. It had, however, no appearance of a tail. It became lost in
the sun, as it approached its perihelion, and emerged again, on the
other side of the sun, on the first of April. Its exhibiting an
appearance, so inferior to what it presented on some of its previous
returns, is partly accounted for by its being seen by the European
astronomers under peculiarly disadvantageous circumstances, being almost
always within the twilight, and in the most unfavorable situations. In
the southern hemisphere, however, the circumstances for observing it
were more favorable, and there it exhibited a tail varying from ten to
forty-seven degrees in length.
In my next Letter I will give you some particulars respecting the late
return of Halley's comet.
LETTER XXVI.
COMETS, CONTINUED.
"Incensed with indignation, Satan stood
Unterrified, and like a comet burned,
That fires the length of Ophiucus huge
In the Arctic sky, and from his horrid train
Shakes pestilence and war."--_Milton._
AMONG other great results which have marked the history of Halley's
comet, it has itself been a criterion of the existing state of the
mathematical and astronomical sciences. We have just seen how far the
knowledge of the great laws of physical astronomy, and of the higher
mathematics, enabled the astronomers of 1682 and 1759, respectively, to
deal with this wonderful body; and let us now see what higher advantages
were possessed by the astronomers of 1835. During this last interval of
seventy-six years, the science of mathematics, in its most profound and
refined branches, has made prodigious advances, more especially in its
application to the laws of the celestial motions, as exemplified in the
'Mecanique Celeste' of La Place. The methods of investigation have
acquired greater simplicity, and have likewise become more general and
comprehensive; and mechanical science, in the largest sense of that
term, now embraces in its formularies the most complicated motions, and
the most minute effects of the mutual influences of the various members
of our system. You will probably find it difficult to comprehend, how
such hidden facts can be disclosed by formularies, consisting of _a_'s
and _b_'s, and _x_'s and _y_'s, and other algebraic symbols; nor will it
be easy to give you a clear idea of this subject, without a more
extensive acquaintance than you have formed with algebraic
investigations; but you can easily understand that even an equation
expressed in numbers may be so changed in its form, by adding,
subtracting, multiplying and dividing, as to express some new truth at
every transformation. Some idea of this may be formed by the simplest
example. Take the following: 3+4=7. This equation expresses the fact,
that three added to four is equal to seven. By multiplying all the terms
by 2, we obtain a new equation, in which 6+8=14. This expresses a new
truth; and by varying the form, by similar operations, an indefinite
number of separate truths may be elicited from the simple fundamental
expression. I will add another illustration, which involves a little
more algebra, but not, I think, more than you can understand; or, if it
does, you will please pass over it to the next paragraph. According to a
rule of arithmetical progression, _the sum of all the terms is equal to
half the sum of the extremes multiplied into the number of terms_.
Calling the sum of the terms _s_, the first term _a_, the last _h_, and
the number of terms _n_, and we have _(1/2)n(a+h)=s_; or _n(a+h)=2s_; or
_a+h=2s/n_; or _a=(2s/n)-h_; or _h=(2s/n)-a_. These are only a few of
the changes which may be made in the original expression, still
preserving the equality between the quantities on the left hand and
those on the right; yet each of these transformations expresses a new
truth, indicating distinct and (as might be the case) before unknown
relations between the several quantities of which the whole expression
is composed. The last, for example, shows us that the last term in an
arithmetical series is always equal to twice the sum of the whole series
divided by the number of terms and diminished by the first term. In
analytical formularies, as expressions of this kind are called, the
value of a single unknown quantity is sometimes given in a very
complicated expression, consisting of known quantities; but before we
can ascertain their united value, we must reduce them, by actually
performing all the additions, subtractions, multiplications, divisions,
raising to powers, and extracting roots, which are denoted by the
symbols. This makes the actual calculations derived from such
formularies immensely laborious. We have already had an instance of this
in the calculations made by Lalande and Madame Lepaute, from formularies
furnished by Clairaut.
The analytical formularies, contained in such works as La Place's
'Mecanique Celeste,' exhibit to the eye of the mathematician a record of
all the evolutions of the bodies of the solar system in ages past, and
of all the changes they must undergo in ages to come. Such has been the
result of the combination of transcendent mathematical genius and
unexampled labor and perseverance, for the last century. The learned
societies established in various centres of civilization have more
especially directed their attention to the advancement of physical
astronomy, and have stimulated the spirit of inquiry by a succession of
prizes, offered for the solutions of problems arising out of the
difficulties which were progressively developed by the advancement of
astronomical knowledge. Among these questions, the determination of the
return of comets, and the disturbances which they experience in their
course, by the action of the planets near which they happen to pass,
hold a prominent place. In 1826, the French Institute offered a prize
for the determination of the exact time of the return of Halley's comet
to its perihelion in 1835. M. Pontecoulant aspired to the honor. "After
calculations," says he, "of which those alone who have engaged in such
researches can estimate the extent and appreciate the fastidious
monotony, I arrived at a result which satisfied all the conditions
proposed by the Institute. I determined the perturbations of Halley's
comet, by taking into account the simultaneous actions of Jupiter,
Saturn, Uranus, and the Earth, and I then fixed its return to its
perihelion for the seventh of November." Subsequently to this, however,
M. Pontecoulant made some further researches, which led him to correct
the former result; and he afterwards altered the time to November
fourteenth. It actually came to its perihelion on the sixteenth, within
two days of the time assigned.