Lecture 10
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| Laplace |
On the Theory of Universal Gravitation
By observation and experiment we gain
the first materials of our knowledge. These are the foundations of all
philosophy. But facts however certain or numerous would by themselves be of
comparatively small importance if the inquiring spirit of Man did not seize him
to generalise his observations, and to form some theory to connect and account
for the various appearances presented to him by Nature. The very number of
these facts would soon overpower his memory unless he made use of method and
arrangement. Thus then the accumulation of his observations would necessarily
produce speculative knowledge and give rise to an attempt at theory.
It is to a legitimate use of theory
that the science of Astronomy is particularly indebted. She presents us with
some of its most happy illustrations. The indiscriminant zeal against
hypothesis so generally avowed by the followers of Bacon has been much
encouraged by the strong and decided forms in which they have been reprobated
by Newton. But the language of this great man must be qualified and limited by
the exemplification he has availed himself given of his general rules. And it
should be remembered that they were particularly directed against the vortices
of Descartes which were purely fictitious and were the prevailing doctrine of
the time. A very learned and acute writer has observed that "the votaries
of hypothesis have been challenged to shew one useful discovery in the work of
Nature that was ever made in that way." In reply to this challenge it will
be sufficient on the present occasion to maintain the Theory of Gravitation and
the Copernican System. Of the former I shall presently endeavour to prove from
a sketch of its history that it took its rise entirely from a conjecture, or
hypothesis suggested by analogy. Nor indeed could it be considered in any other
light until that period of Newton's life when, by a calculation founded in an
accurate measurement of the Earth by Picard, he evinced the evidence to be
true, even the law which regulates the fall of heavy bodies, that power which
retains the Moon in her orbit.
The
Copernican System offers however a still stronger case, inasmuch as the only
evidence which the author was able to offer was the advantage it possessed over
every other hypothesis in explaining with beauty and simplicity all the phenomena
of the heavens. In the mind of Copernicus therefore this system was nothing
more than a hypothesis, but it was an hypothesis conformable to the universal
law of nature, always accomplishing her ends by the simplest means.
Nor is the use of hypothesis
confined to these cases in which they have subsequently received confirmation.
It may be equally great where they have completely disappointed the
explanations of their authors. Indeed any hypothesis which possesses a
sufficient degree of plausibility to account for a number of facts will help us
to arrange those facts in proper order and will suggest to us proper
experiments either to confirm, or refute it. Nor is it solely by the erroneous
results of his own hypothesis that the philosopher is assisted in his enquiry
after truth.
Similar
lengths may often be collected by the errors of his predecessors: it was from a
review of the endless and hapless wanderings of preceding enquirers that Bacon
inferred the necessity of avoiding every beaten track, and it was this which
encouraged him with a confidence in his own powers, amply justified by the
event, to explore and to open a new path to the mysteries of Nature.
In this respect the maturity of
reason in the species is analogous to that in the individual. It is not the
consequence of any sudden or accidental cause but the fruit of re-iterated
disappointment connecting the mistakes of youth and inexperience. "There
is no subject," says [Bernard le Bouvier de] Fontenelle, "on which
men ever come to a reasonable opinion till they have once exhausted all absurd
views which it is possible to take of it. What follies" he adds,
"should we not be repeating this day if we had not been anticipated in so
many of them by the ancient philosophers."
Those
systems which are false are therefore by no means to be regarded as altogether
useless. That of Ptolemy, for example, as has well been observed by the elegant
historian of Astronomy, is founded on a prejudice so natural and so unavoidable
that it may be considered as a necessary step in the progress of astronomical
science, and, if it had not been proposed in ancient times, it would infallibly
have preceded among the moderns the system of Copernicus and have retarded the
period of its discovery.
Among the numerous discoveries which
have rendered illustrious the name of Kepler there are none more important than
those with which he enriched Astronomy at the commencement of the 17th century.
Galileo had begun the investigation and cleared away some of the difficulties
but it was Kepler who transported geometry to the heavens, who discovered the
laws of their movement. Let us for a moment follow Kepler in the ideas and
conjectures which led him to these memorable results.
By
comparing the different velocities of the Sun at different seasons of the year
with variation of its apparent diameter, he found that it was impossible to
account for the phenomenon by supposing the Earth to revolve [around the Sun]
in a circle. He therefore supposed that it might move in an oval, but of this
figure there are various kinds, and the first one he hit upon did not answer
his purpose. He was however more successful in the next trial and found that
the common ellipse would satisfy all the conditions. From this period we may
date the knowledge of the elliptic motion of the planets and the destruction of
the ancient prejudice which attributed to the heavenly bodies a uniform
circular motion which by its simplicity had seduced the ancient philosophers of
Greece.
After discovering that the planets made
an ellipse Kepler wished to find some law which might regulate their motions.
He knew that when they were nearest the Sun they moved fastest, but was not
acquainted with the law of the change of velocity. After numerous trials he
found out the following analogy. If we conceive a line drawn from the Sun to
the centre of any planet this line will always pass over equal areas in equal
time.
Kepler
observed that the more distant a planet was from the Sun the longer time it
required to perform its revolution round that body, and this led him to the
discovery of another law which prevails throughout the planetary system. He
found that the square of the time of any planet's revolution always bore a
certain proportion to the cube of its distance and that this ratio is constant.
Mercury [=
0.13050] 39 x 39 x 39
Venus [=
0.1356] 72 x 72 x 72
Earth [= 0.1332]
100 x 100 x 100
Mars [= 0.1344]
152 x 152 x 152
Jupiter [= 0.1335]
520 x 520 x 520
[Table of
the square of the number of days required by a planet to revolve round the Sun
divided by the cube of its mean distance from that body in millions of miles]
The laws
which I have just mentioned were discovered by Kepler after many trials and numerous
failures. They rested on no other foundations than experiment and were in
precisely a similar situation to the laws relating the planetary distances
which I endeavoured to explain in a former lecture. That is to say Mankind were
astonished at their coincidence with Nature, but were unable to divine the
cause which produced it.
Kepler is particularly distinguished
from the philosophers of his time by the great boldness and frequently by the
great correctness of his views in enquiring into the cause which produces the
phenomena of Nature. He considered the Sun as the supreme moderator of the
celestial bodies. "This star" says he, "is possessed with a power of moving bodies, which it
spreads with immense rapidity throughout all space and hence arise the motions
of the planets."
At one time he compares the weight
of heavy bodies on Earth to the gravity of the planets towards the Sun. In
another place he suspects that the combined action of the Sun and the Earth
produces the irregularities of the [motion of] the Moon. And he imagines that
the tides may possibly arise from the attractions of this body. One of his
fundamental doctrines is the motion of the Sun on its axis, an hypothesis which
was completely justified a few years after by the discovery of the spots which
cover its surface. These ideas and conjectures bear the evident stamp of
genius, the daring flight of a powerful and comprehensive mind, and they opened
to Newton that glorious path which led him to the most sublime discoveries.
Before
however we take the steps which led this great philosopher to his theory of
gravitation, it will be proper to pass in review the theory of Descartes, which
at the time universally prevailed. Philosophers of the highest antiquity had
recognised the existence of the heavenly bodies. They had each calculated their
distances and appreciated with some accuracy the motions by which they are
animated. But no one before the 17th century, had endeavoured to snatch from
Nature the secret mechanism by which they hold together the planetary system.
The honour of this bold enterprise was reserved for Descartes. Determined to
produce a system entirely novel in which everything should be reconciled with
his ideas of the harmony of Nature, he conceived himself at the formation of
the Universe, and thus presented to himself the spectacle of Creation -an
infinity of molecules of matter repose in the immensity of space. All possess
an extreme hardness, and their infinitely varied form victoriously opposes the
existence of a vacuum. "Creative power" says Descartes, "imposes
on them a force, which at the same time carries them forward in space, and
causes them to revolve on their axis. And certain laws are prescribed to them
by which their actions are to be regulated." Such is the chaos from which
Descartes conceives a universe like ours might arise, whose spectacle, although
habitual, daily excites fresh reason for surprise and admiration. It is
needless to pursue the speculations of this philosopher through their varied
course to the existence of vortices of extremely subtle matter, through whose
assistance the planets and satellites were conceived to revolve.
We have
already seen that they are totally devoid of proof, and are, in fact,
physically impossible. Yet were not the speculations of Descartes without their
use in the progress of philosophy? They were errors, but it must be confessed
that they were splendid ones, and that the mind which could frame such a theory
might under better guidance have arrived at more accurate results.
Kepler, Galileo and Descartes
contributed to dissipate the darkness which had for a long time enveloped
mankind. They became the benefactors of the human race, but for the shame of
the age which produced them, they received as the reward of their labours
nothing but injustice, persecution and disgrace. Galileo, whose brilliant
discoveries had merited a better fate, was dragged to an unworthy prison.
Kepler surrounded by the glory he had acquired by the his sublime views of
Nature experienced in his old age all the misery of want and indignance. And it
was not until 100 years after the death of Descartes that his grateful country
raised even a monument to his memory.
Doubtless
nothing would be more satisfactory to the mind than the physical system of Descartes
if it could sustain the process of examination and observation. Those vortices,
that is to say, those torrents of ethereal matter, which according to this
philosopher carry with them the planets round the Sun, present to the mind an
intelligible mechanism which enchants by its simplicity. But this theory, which
at first glance is so seducing, is subject to many difficulties. It is
unfortunately found to agree so little either with the phenomena or with the
laws of mechanics that notwithstanding the efforts of many ingenious writers,
it is universally allowed that the system of Descartes is not that of Nature.
Newton pursued a different course,
and on the ruins of this system he has erected a new, more solid one, which
presents every appearance of the greatest durability. In fact his system
exhibits a perpetual coincidence between theory and observation. Whether we
regard the grander laws which regulate the Universe, or whether we examine
these minute and almost insensible ramifications of observation and geometry
[it] has nothing to fear from the vicissitudes of time, or from the more
changeable opinions of men. The system of Newton is founded on the principles
of Universal Gravitation.
Every
particle of matter, whatever may be the mechanism or cause which produces this
effect, tends, according to this philosopher, to every other particle with a
force which decreases inversely as the square of the distance. This gravity
causes on the surface of the Earth the weight of a body, and among the celestial
bodies it is the source of the most complicated motions. I shall endeavour to
explain the proofs of this principle and the reasoning which lead Newton to it
after shortly stating what was known on the subject before the time of his
writings.
We find among the writings of the
ancients a glimpse of several of the most brilliant truths. This is
particularly the case with the principles of Universal Gravitation, of which we
discover some decided marks. Anaxagoras as we have already seen attributed to
all the heavenly bodies a tendency towards the Earth, and other traces of the
same opinion may be found among the writings of Democritus and Epicurus. It was
from this principle that Lucretius drew the bold conclusion that the world is
without bounds.
As soon
the true system of the world revived by Copernicus arose from its ashes that of
Universal Gravitation threw some rays of light. This celebrated astronomer
attributed the round figure of bodies to the attraction of their parts. He did
not extend this attraction from planet to planet. But Kepler more bold and more
systematic made this step. He attributed to the Moon a tendency towards the
Earth and said that they could meet in their common centre of gravity if they
were not prevented by their rotation. Nobody however before the time of Newton
so clearly perceived the principle of that attraction, or more nearly
approached in making a proper application of the system of the Universe than
Dr. Hooke. The philosophers whom we have mentioned had some of them seized one
branch, some another, but Hooke embraced it in all its generality.
His anticipation of that theory of
planetary motions which was soon after to present itself with increased and at
length demonstrative evidence to a still more powerful mind furnishes a
remarkable instance of this philosophical sagacity. This conjecture I shall
state in his own words, and it affords a decisive reply to the undistinguishing
censures which have so often been bestowed on the presumptuous vanity of
attempting by means of hypothesis to penetrate into the secrets of Nature.
"I will explain" says Dr. Hooke, in a communication made to the
P[resident of the Royal Society?] in 1666, "a system of the world very
different from any yet received. It is founded on the three following
positions."
"1st
that all the heavenly bodies have not only a gravitation of their parts to
their own centre, but that they mutually attract each other within their
spheres of action."
"2ndly that all bodies having a
simple motion will continue to move in a straight line unless continually moved
out of it by some extraneous force."
"3rdly that as this attraction
is so much greater as the bodies are nearer, as to the proportion in which
those forces diminish by an increase of distance."
Dr. Hooke
adds "I [on my] own have not discovered it although I have made some
experiments for this purpose. I leave this for others who have time and
knowledge sufficient for the purpose."
It should be observed that there is
a wide difference between the conjecture of Hooke and the proofs and sublime
demonstrations by which Newton supported this law of the Universe. Such however
was the state of the question when this profound philosopher appeared. It was
in 1666 that he first [began] to suspect the existence of this principle and to
endeavour to apply it to the motions of the heavenly bodies. He had retired
into the country to avoid the plague, which at that time prevailed in London.
His meditations were one day accidentally directed toward the weight of bodies.
His first
reflection was that the cause which produced the fall of heavy bodies always
acts upon them to whatever height we convey them. It may then be extended much
further than we think, possibly as far as the Moon or even beyond. From this he
conjectured that it might possibly be this same force which retains the Moon in
her orbit. At the same time he considered that though Gravity does not sensibly
alter at different heights to which we can attain, yet at greater distances it
may vary and these altitudes are too small to conclude that it is the same at
all distances. It now remained to discover the law by which it varied,. For
this purpose he argued that if gravity retained our Moon in her orbit round the
Earth, it must be a similar cause which retains the sattelites of Jupiter in
their orbits round that body, and by comparing the periods of these bodies with
their distances, he found that gravity must decrease inversely as the square of
the distance.
Newton
did not however rest here. He continued to examine an account of the Moon's
distance from the Earth: the force by which she is attracted will be 3,600
times less than that by which a body falls on the surface of the Earth. If we
can compare the space through which the Moon falls towards the Earth in a given
time with that through which a body on the Earth's surface falls, we shall have
a criterion by which to judge of the truth of our theory. But here arose a
difficulty, how shall we find how much the Moon falls towards the Earth in a
given time? This difficulty Newton overcame, and these are the means he made
use of. This had nearly overthrown his whole edifice. He supposed the
terrestrial degrees to contain 60 miles and in consequence of this the two
quantities (on whose relation the truth of his theory was to be tried) did not
afford a result favourable to it.
Many philosophers would have been
but little troubled by this disagreement and would have continued to construct
their theoretical edifice. But this incomparable man, whose object was the discovery
of truth and not the formation of a
system, when he found that a single fact overthrew all his conjectures which
had hitherto been so well founded, immediately relinquished them.
It was
not till 10 years after that he resumed the train of his ideas. In this
interval the opinion of Dr. Hooke had been published and a very important step
had been made by Ricard in ascertaining the magnitude of the Earth. From this
Newton learned that the terrestrial degree was nearly 70 miles in length, and
not 60 as he had considered it in his calculations. He now therefore again
returned to his theory and having calculated the magnitude of the lunar orbit,
he found to his great satisfaction that the space fallen through by the Moon
precisely agreed with what it ought from his theory. After this demonstration
Newton no longer hesitated to consider the force by which heavy bodies fall at
the surface of the Earth and that by which the Moon is retained in her orbit as
one and the same.
He assumed gravity as a well ascertained
fact and proceeded to reason upon it. He showed that it followed necessarily
from his theory that the planets move in ellipses and he demonstrated the laws
which Kepler had only found by induction. By a skillful application of
mathematical calculation to the phenomena of Nature aided by the theory of
gravity, he unravelled numberless irregularities to which the heavenly bodies
are subject. Yet such was the excessive modesty of this great man that it was
with the greatest difficulty that he was persuaded to publish his profound
discoveries.
At the
urgent request of his friend, Dr. Halley, and at the entreaty of the Royal
Society, he was persuaded to collect together his discoveries, which he did in
his Principia, a work which also is alone sufficient to immortalise its author.
This work however in which this great
man has built a new system of natural philosophy upon the most sublime geometry
did not at first meet with all that applause it deserved and was one day to
receive.
Two reasons concurred to produce
this effect. Descartes' system had at that time got full possession of the
world. His philosophy was indeed the creature of a fine imagination; he had
given her some of Nature's features and had painted the rest to a seeming
resemblance to her. Newton, on the otherhand, had with unparalleled penetration
and force of genius pursued Nature up to her most secret abode and was intent
to demonstrate her residence to others rather than anxious to describe
particularly the way by which he arrived at it himself. But at last that
approbation which had been so slowly gained became universal, and nothing was
heard from all quarters but one universal burst of applause [and] admiration:
"Does Mr. Newton eat, drink or sleep like other men?" said the
Marquis de l'Hospital, one of the most enlightened foreigners of the age, to
his English visitors, "I represent him to myself as a celestial genius
entirely disengaged from matter."
Yet in
the midst of these profound enquiries Newton had leisure for other pursuits. When
the privileges of the University [of Cambridge] were attacked by James II he
appeared as one of the most strenuous defenders. And he made a very successful
defence before the high commission court. He was also a member of the
Convocation Parliament in which he sat till it was dissolved.
In his private life Newton was
modest and unassuming in the highest degree. His temper was so mild and equal
that no accident could disturb it. He would have rather chosen to remain in obscurity
than to have the calm of life ruffled by the storms and disputes which genius
and learning so frequently draw on those who are eminent for them. From this
love of peace arose that unusual horror which he felt for all disputes, and
that steady unbroken attention which was his peculiar felicity; he knew and
well esteemed its value.
When some
objections hostily made to his discoveries concerning light and colours induced
him to lay aside his design of publishing his optical lectures, we find him reflecting
on that dispute into which he was unavoidably drawn in these terms: "I
blamed my own imprudence" said Newton, "for parting with so real a
blessing as quiet, to run after a shadow." Yet this shadow was one of the
most splendid and most original of these discoveries which have contributed to
make his name so illustrious.
In contemplating the genius of
Newton the penetration, the strength and the originality of his mind, in his
moral capacity the pre-eminent trait is his modesty and love of quiet. In his
intellectual character the most predominant feature is the astonishing power
which he possessed of concentrating his attention to the object on which he was
employed. It was to this almost supernatural power that he himself attributed
his profound discoveries. When he declared that if he had done the world any
service it was due to nothing but industry and patient thought; that he kept
the subject of consideration constantly before him and waited till the first
dawning opened gradually, by little and little into a clear and full light.
Such was Newton as a man, and as a philosopher both characters were tinged with
a similar colouring. As a man he did not possess that warmth and enthusiasm
which we should admire in a friend, but he displayed that calm unruffled
serenity, which we should reverence in superior beings. As a philosopher he was
not led away by the fire of genius whose too daring grasp by sometimes
fostering error, reminding us of his mortal origin. But he was the patient, the
accurate investigator of Nature, the deep, the profound philosopher.
To trace
the consequences of the law of Universal Gravitation through its numerous and
almost endless ramifications would lead us in succession through every branch
of astronomical science. Each seeming objection which has successively been
brought against its truths has furnished new arguments in its favour and
afforded new ground of triumph to the followers of Newton. It has frequently
anticipated observation and has predicted to future astronomers irregularities
that are yet to be recognised. To such a point of perfection has this science
been carried, that there does not now remain one single irregularity of the
heavenly bodies of any magnitude which does not follow as a consequence of this
law, and whose quantity cannot be calculated by it. It is obvious that an
enumeration of these varied irregularities would be of little improvement, but
there are some important questions which present themselves and which relate to
the law of gravity.
There are many of the planetary
irregularities, which increase and decrease alternately in longer and shorter
periods. Some however, since we have observations of them recorded, have been
found uniformly to decrease. Such for instance is the obliquity of the ecliptic.
If this obliquity were to decrease continually, the ecliptic would at last
coincide with the equator. This would not take place until after the lapse of
millions of years, but when it did the days and nights all over the world would
become equal; the Earth would possess a perpetual spring. Whether this change
would be for the advantage of the human race if it then existed, or whether it
would not render nearly half the globe uninhabitable, are doubtful questions;
yet remote as this change may be it is interesting to enquire whether it is
possible, because there are other irregularities which increase much more
rapidly and which might if they continued to increase totally derange the
system. The question then in the most general sense is to investigate whether
the irregularities of the planetary system will continually increase or whether
after attaining a certain point they will not decrease and return again in the
same order.
This
question which is a very important one has occupied some of the greatest philosophers
of the age. The progress was very gradual but the solution is complete. It is
to the united labours of Lagrange and Laplace that we are indebted for the
solution of this interesting question. It has been demonstrated that every
irregularity to which the planetary [system] is subject must from its nature be
periodical: that is, it will increase to a certain point and then decrease to
another point, between which two it will constantly oscillate, never exceeding
either of them, just in the same manner as a pendulum which constantly moves to
and fro, but never exceeds a certain fixed distance from its point of rest.
Almost everything in the system will therefore be in motion, but admidst this
universal change some few elements will remain constant: thus the mean distance
of each planet from the Sun will be unaltered. Thus then it appears that unless
some foreign force disturb the harmony of our System, it will forever continue
in its present arrangement, that it does not contain within itself the seeds of
destruction, but on the contrary that it is destined to an eternal duration,
unless the mechanical laws which govern matter be subverted or some influence
foreign to the System be exerted on it.
There
were, however, in its original constitution some conditions necessary, that the
inclination of the orbit of the planets to the Sun's equator be small, that the
ellipses which they describe should be nearly circular and that they should all
move round the Sun in the same direction. And if these conditions had not been
fulfilled it is possible that the beautiful system from its own action have
produced its destruction. If we enquire from the doctrine of chances whether it
is probable that the conditions should have been accidentally accomplished we
find that the contrary is indicated with the highest possible degree of
probability. In fact Laplace, when speaking on this subject, says that we have
stronger grounds for believing that the planetary motions and inclinations were
all influenced by the same primitive cause, than we have for giving credit to
any of the most authentic accounts in history. This wonderful contrivance of an
intelligent mind by which the permanence of our System is secured is highly
calculated to excite our admiration, yet it has been perverted to the worst,
the most unphilosophical purposes. It has been urged as the supporter of fate
and of necessity, and has been inconsiderately advanced as an argument against
the superintendence and existence of a first cause. It is a singular circumstance
that a fact which had a tendency so directly contrary should have been so
misunderstood, and it would perhaps be needless to refute it, but that it has
been said, though I believe falsely, to have received the sanction of one of
the most eminent of the continental philosophers. The difficulty is easily
removed. We have only to ask ourselves this question: which is the most skilful
artist? he who makes a clock which requires winding up every day and cleaning
every year, or he who contrives one which winds itself up and never requires
cleaning, to which
it may be
further added that among the infinite number of laws by which gravity might
act, all equally possible, this of Nature alone aided by the conditions already
specified will ensure the stability and permanence of the System. Had there
been any other originally established this Universe the beneficent result of
creative power would have long since have returned to its primitive chaos.
