Monday, 11 November 2013

Lecture 10: On the Theory of Universal Gravitation

Lecture 10


 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.