History of the Intellectual Development of Europe, Volume II (of 2)

Steady advance of the Copernican system. The result of the discoveries of Copernicus and Galileo was thus to bring the earth to her real position of subordination and to give sublimer views of the universe. Mœstlin expresses correctly the state of the case when he says, "What is the earth and the ambient air with respect to the immensity of space? It is a point, a punctule, or something, if there be any thing, less." It had been brought down to the condition of one of the members of a family – the solar system. And since it could be no longer regarded as holding all other bodies in submissive attendance upon it, dominating over their movements, there was reason to suppose that it would be found to maintain interconnexions with them in the attitude of an equal or subordinate; in other words, that general relations would be discovered expressive of the manner in which all the planetary members of the solar system sustain their movements round the sun.

Kepler, his mode of inquiry. Among those whose minds were thoroughly occupied with this idea, Kepler stands pre-eminently conspicuous. It is not at all surprising, considering the tone of thought of those times, that he regarded his subject with a certain mysticism. They who condemn his manner of thus viewing things do not duly appreciate the mental condition of the generation in which he lived. Whatever may be said on that point, no one can deny him a marvellous patience, and almost superhuman painstaking disposition. Guess after guess, hypothesis after hypothesis, he submitted to computations of infinite labour, and doubtless he speaks the melancholy truth when he says, "I considered and reflected till I was almost mad." Yet, in the midst of repeated disappointment, he held, with a truly philosophical determination, firmly to the belief that there must be some physical interconnexion among the parts of the solar system, and that it would certainly be displayed by the discovery of laws presiding over the distances, times, and velocities of the planets. In these speculations he was immersed before the publications of Galileo. In his "Mysterium Cosmographicum" he says, "In the year 1595 I was brooding with the whole energy of my mind on the subject of the Copernican system."

Discovery of Kepler's laws. In 1609 he published his work entitled "On the Motion of Mars." This was the result of an attempt, upon which he had been engaged since the beginning of the century, to reconcile the motions of that planet to the hypothesis of eccentrics and epicycles. It ended in the abandonment of that hypothesis, and in the discovery of the two great laws now known as the first and second laws of Kepler. They are respectively that the orbits of the planets are elliptical, and that the areas described by a line drawn from the planet to the sun are proportional to the times.

In 1617 he was again rewarded by the discovery which passes under the designation of Kepler's third law: it expresses the relation of the mean distances of the planets from the sun with the times of their revolutions – "the squares of the periodic times are in the same proportion as the cubes of the distances." In his "Epitome of the Copernican Astronomy," published 1622, he showed that this law likewise holds good for the satellites of Jupiter as regards their primary.

His remonstrance with the Church. Humboldt, referring to the movement of Jupiter's satellites, remarks: "It was this which led Kepler, in his 'Harmonices Mundi,' to state, with the firm confidence and security of a German spirit of philosophical independence, to those whose opinions bore sway beyond the Alps, 'Eighty years have elapsed during which the doctrines of Copernicus regarding the movement of the earth and the immobility of the sun have been promulgated without hindrance, because it was deemed allowable to dispute concerning natural things and to elucidate the works of God, and, now that new testimony is discovered in proof of the truth of those doctrines – testimony which was not known to the spiritual judges, ye would prohibit the promulgation of the true system of the structure of the universe.'"

Rectification of the Copernican theory. Thus we see that the heliocentric theory, as proposed by Copernicus, was undergoing rectification. The circular movements admitted into it, and which had burdened it with infinite perplexity, though they had hitherto been recommended by an illusive simplicity, were demonstrated to be incorrect. They were replaced by the real ones, the elliptical. Kepler, as was his custom, ingenuously related his trials and disappointments. Alluding on one occasion to this, he says: "My first error was that the path of a planet is a perfect circle – an opinion which was a more mischievous thief of my time, in proportion as it was supported by the authority of all philosophers, and apparently agreeable to metaphysics."

The philosophical import of these laws. The philosophical significance of Kepler's discoveries was not recognized by the ecclesiastical party at first. It is chiefly this, that they constitute a most important step to the establishment of the doctrine of the government of the world by law. But it was impossible to receive these laws without seeking for their cause. The result to which that search eventually conducted not only explained their origin, but also showed that, as laws, they must, in the necessity of nature, exist. It may be truly said that the mathematical exposition of their origin constitutes the most splendid monument of the intellectual power of man.

Necessity for mechanical science. Before the heliocentric theory could be developed and made to furnish a clear exposition of the solar system, which is obviously the first step to just views of the universe, it was necessary that the science of mechanics should be greatly improved – indeed, it might be said, created; for during those dreary ages following the establishment of Byzantine power, nothing had been done toward the acquisition of correct views either in statics or dynamics. It was impossible that Europe, in her lower states of life, could produce men capable of commencing where Archimedes had left off. She had to wait for the approach of her Age of Reason for that.

Leonardo da Vinci. The man of capacity at last came. Leonardo da Vinci was born A.D. 1452. The historian Hallam, enumerating some of his works, observes, "His knowledge was almost preternatural." Many of his writings still remain unpublished. Long before Bacon, he laid down the maxim that experience and observation must be the foundation of all reasoning in science; that experiment is the only interpreter of nature, and is essential to the ascertainment of laws. Unlike Bacon, who was ignorant of mathematics, and even disparaged them, he points out their supreme advantage. Seven years after the voyage of Columbus, this great man – great at once as an artist, mathematician, and engineer – gave a clear exposition of the theory of forces obliquely applied on a lever; a few years later he was well acquainted with the earth's annual motion. He knew the laws of friction, subsequently demonstrated by Amontons, and the principle of virtual velocities; he described the camera obscura before Baptista Porta, understood aerial perspective, the nature of coloured shadows, the use of the iris, and the effects of the duration of visible impressions on the eye. He wrote well on fortification, anticipated Castelli on hydraulics, occupied himself with the fall of bodies on the hypothesis of the earth's rotation, treated of the times of descent along inclined planes and circular arcs, and of the nature of machines. He considered, with singular clearness, respiration and combustion, and foreshadowed one of the great hypotheses of geology, the elevation of continents.

Stevinus continues the movement in Natural Philosophy. This was the commencement of the movement in Natural Philosophy; it was followed up by the publication of a work on the principles of equilibrium by Stevinus, 1586. In this the author established the fundamental property of the inclined plane, and solved, in a general manner, the cases of forces acting obliquely. Six years later Galileo's treatise on mechanics appeared, a fitting commencement of that career which, even had it not been adorned with such brilliant astronomical discoveries, would alone have conferred the most illustrious distinction upon him.

Discovery of the laws of motion. The dynamical branch of Mechanics is that which is under most obligation to Galileo. To him is due the establishment of the three laws of motion. They are to the following effect, as given by Newton:

(1.) Every body perseveres in its state of rest or of uniform motion in a right line unless it is compelled to change that state by forces impressed thereon.

(2.) The alteration of motion is ever proportional to the motive force impressed, and is made in the direction of the right line in which that force is impressed.

(3.) To every action there is always opposed an equal reaction, or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Up to this time it was the general idea that motion can only be maintained by a perpetual application, impression, or expenditure of force. Galileo himself for many years entertained that error, but in 1638 he plainly states in his "Dialogues on Mechanics" the true law of the uniformity and perpetuity of motion. Such a view necessarily implies a correct and clear appreciation of the nature of resistances. No experimental motion that man can establish is unrestrained. But a perception of the uniformity and perpetuity of motion lies at the very basis of physical astronomy. With difficulty the true idea was attained. The same may be said as respects rectilinear direction, for many supposed that uniform motion can only take place in a circle.

Establishment of the first law of motion, The establishment of the first law of motion was essential to the discovery of the laws of falling bodies, in which the descent is made under the influence of a continually acting force, the velocity increasing in consequence thereof. Galileo saw clearly that, whether a body is moving slowly or swiftly, it will be equally affected by gravity. This principle was with difficulty admitted by some, who were disposed to believe that a swiftly moving body would not be as much affected by a constant force like gravity as one the motion of which is slower. With difficulty, also, was the old Aristotelian error eradicated that a heavy body falls more swiftly than a light one.

and of the second, The second law of motion was also established and illustrated by Galileo. In his "Dialogues" he shows that a body projected horizontally must have, from what has been said, a uniform horizontal motion, but that it will also have compounded therewith an accelerated motion downward. Here again we perceive it is necessary to retain a steady conception of this intermingling of forces without deterioration, and, though it may seem simple enough to us, there were some eminent men of those times who did not receive it as true. The special case offered by Galileo is theoretically connected with the paths of military projectiles, though in practice, since they move in a resisting medium, the air, their path is essentially different from the parabola. Curvilinear motions, which necessarily arise from the constant action of a central force, making a body depart from the rectilinear path it must otherwise take, are chiefly of interest, as we shall presently find, in the movements of the celestial bodies.

and of the third. A thorough exposition of the third law of motion was left by Galileo to his successors, who had directed their attention especially to the determination of the laws of impact. Indeed, the whole subject was illustrated and the truth of the three laws verified in many different cases by an examination of the phenomena of freely falling bodies, pendulums, projectiles, and the like. Among those who occupied themselves with such labours may be mentioned Torricelli, Castelli, Viviani, Borelli, Gassendi. Through the investigations of these, and other Italian, French, and English natural philosophers, the principles of Mechanics were solidly established, and a necessary preparation made for their application in astronomy. By this time every one had become ready to admit that the motion of the planetary bodies would find an explanation on these principles.

Application of Mechanics to the celestial motions. The steps thus far taken for an explanation of the movements of the planets in curvilinear paths therefore consisted in the removal of the old misconception that for a body to continue its motion forward in a straight line a continued application of force is necessary, the first law of motion disposing of that error. In the next place, it was necessary that clear and distinct ideas should be held of the combination or composition of forces, each continuing to exercise its influence without deterioration or diminution by the other. The time had now come for it to be shown that the perpetual movement of the planets is a consequence of the first law of motion; their elliptic paths, such as had been determined by Kepler, a consequence of the second. Several persons almost simultaneously had been brought nearly to this conclusion without being able to solve the problem completely. Thus Borelli, A.D. 1666, in treating of the motions of Jupiter's satellites, distinctly shows how a circular motion may arise under the influence of a central force; he even uses the illustration so frequently introduced of a stone whirled round in a sling. In the same year a paper was presented to the Royal Society by Mr. Hooke, "explicating the inflection of a direct motion into a circular by a supervening attractive principle." Huygens also, in his "Horologium Oscillatorium," had published some theorems on circular motions, but no one as yet had been able to show how elliptical orbits could, upon these principles, be accounted for, though very many had become satisfied that the solution of this problem would before long be given.

Newton; publication of the "Principia." In April, 1686, the "Principia" of Newton was presented to the Royal Society. This immortal work not only laid the foundation of Physical Astronomy, it also carried the structure thereof very far toward its completion. It unfolded the mechanical theory of universal gravitation upon the principle that all bodies tend to approach each other with forces directly as their masses, and inversely as the squares of their distances.

Propounds the theory of universal gravitation. To the force producing this tendency of bodies to approach each other the designation of attraction of gravitation, or gravity, is given. All heavy bodies fall to the earth in such a way that the direction of their movement is toward its centre. Newton proved that this is the direction in which they must necessarily move under the influence of an attraction of every one of the particles of which the earth is composed, the attraction of a sphere taking effect as if all its particles were concentrated in its centre.

Preparation for Newton. Galileo had already examined the manner in which gravity acts upon bodies as an accelerating force, and had determined the connexion between the spaces of descent and the times. He illustrated such facts experimentally by the use of inclined planes, by the aid of which the velocity may be conveniently diminished without otherwise changing the nature of the result. He had also demonstrated that the earth's attraction acts equally on all bodies. This he proved by inclosing various substances in hollow spheres, and showing that, when they were suspended by strings of equal length and made to vibrate, the time of oscillation was the same for all. On the invention of the air-pump, a more popular demonstration of the same fact was given by the experiment proving that a gold coin and a feather fall equally swiftly in an exhausted receiver. Galileo had also proved, by experiments on the leaning tower of Pisa, that the velocity of falling bodies is independent of their weight. It was for these experiments that he was expelled from that city.

Extension of attraction or gravity. Up to the time of Newton there were only very vague ideas that the earth's attraction extended to any considerable distance. Newton was led to his discovery by reflecting that at all altitudes accessible to man, gravity appears to be undiminished, and that, therefore, it may possibly extend as far as the moon, and actually be the force which deflects her from a rectilinear path, and makes her revolve in an orbit round the earth. Admitting the truth of the law of the inverse squares, it is easy to compute whether the moon falls from the tangent she would describe if the earth ceased to act upon her by a quantity proportional to that observed in the case of bodies falling near the surface. In the first calculations made by Newton, he found that the moon is deflected from the tangent thirteen feet every minute; but, if the hypothesis of gravitation were true, her deflection should be fifteen feet. It is no trifling evidence of the scrupulous science of this great philosopher that hereupon he put aside the subject for several years, without, however, abandoning it. At length, in 1682, learning the result of the measures of a degree which Picard had executed in France, and which affected the estimate of the magnitude of the earth he had used, and therefore the distance of the moon, he repeated the calculations with these improved data. It is related that "he went home, took out his old papers, and resumed his calculations. As they drew to a close, he became so much agitated that he was obliged to desire a friend to finish them." The expected coincidence was verified. And thus it appeared that the moon is retained in her orbit and made to revolve round the earth by the force of terrestrial gravity.

The cause of Kepler's laws. These calculations were founded upon the hypothesis that the moon moves in a circular orbit with a uniform velocity. But in the "Principia" it was demonstrated that when a body moves under the influence of an attractive force, varying as the inverse square of the distances, it must describe a conic section, with a focus at the centre of force, and under the circumstances designated by Kepler's laws. Newton, therefore, did far more than furnish the expected solution of the problem of elliptical motion, and it was now apparent that the existence of those laws might have been foreseen, since they arise in the very necessities of the case.

Resistless spread of the heliocentric theory. This point gained, it is obvious that the evidence was becoming unquestionable, that as the moon is made to revolve round the earth through the influence of an attractive force exercised by the earth, so likewise each of the planets is compelled to move in an elliptical orbit round the sun by his attractive force. The heliocentric theory, at this stage, was presenting physical evidence of its truth. It was also becoming plain that the force we call gravitation must be imputed to the sun, and to all the planetary bodies as well as to the earth. Accordingly, this was what Newton asserted in respect to all material substance.

Perturbations accounted for. But it is a necessary consequence of this theory that many apparent irregularities and perturbations of the bodies of the solar system must take place by reason of the attraction of each upon all the others. If there were but one planet revolving round the sun, its orbit might be a mathematically perfect ellipse; but the moment a second is introduced, perturbation takes place in a variable manner as the bodies change their positions or distances. An excessive complication must therefore be the consequence when the number of bodies is great. Indeed, so insurmountable would these difficulties be, that the mathematical solution of the general problem of the solar system would be hopeless were it not for the fact that the planetary bodies are at very great distances from one another, and their masses, compared with the mass of the sun, very small.

Results of the theory of gravitation. Taking the theory of gravitation in its universal acceptation, Newton, in a manner that looks as if he were divinely inspired, succeeded in demonstrating the chief inequalities of the moon and planetary bodies; in determining the figure of the earth – that it is not a perfect sphere, but an oblate spheroid; in explaining the precession of the equinoxes and the tides of the ocean. To such perfection have succeeding mathematicians brought his theory, that the most complicated movements and irregularities of the solar system have been satisfactorily accounted for and reduced to computation. Trusting to these principles, not only has it been found possible, knowing the mass of a given planet, to determine the perturbations it may produce in adjacent ones, but even the inverse problem has been successfully attacked, and from the perturbations the place and mass of a hitherto unknown planet determined. It was thus that, from the deviations of Uranus from his theoretical place, the necessary existence of an exterior disturbing planet was foreseen, and our times have witnessed the intellectual triumph of mathematicians directing where the telescope should point in order to find a new planet. The discovery of Neptune was thus accomplished.

It adds to our admiration of the wonderful intellectual powers of Newton to know that the mathematical instrument he used was the ancient geometry. Not until subsequently was the analytical method resorted to and cultivated. This method possesses the inappreciable advantage of relieving us from the mental strain which would otherwise oppress us. It has been truly said that the symbols think for us. The "Principia;" its incomparable merit. Mr. Whewell observes: "No one for sixty years after the publication of the 'Principia,' and, with Newton's methods, no one up to the present day, has added any thing of value to his deductions. We know that he calculated all the principal lunar inequalities; in many of the cases he has given us his processes, in others only his results. But who has presented in his beautiful geometry or deduced from his simple principles any of the inequalities which he left untouched? The ponderous instrument of synthesis, so effective in his hands, has never since been grasped by any one who could use it for such purposes; and we gaze at it with admiring curiosity, as on some gigantic implement of war which stands idle among the memorials of ancient days, and makes us wonder what manner of man he was who could wield as a weapon what we can hardly lift as a burden."

Philosophical import of Newton's discoveries. Such was the physical meaning of Newton's discoveries; their philosophical meaning was of even greater importance. The paramount truth was resistlessly coming into prominence – that the government of the solar system is under necessity, and that it is mathematically impossible for the laws presiding over it to be other than they are.

Thus it appears that the law of gravitation holds good throughout our solar system. But the heliocentric theory, in its most general acceptation, considers every fixed star as being, like the sun, a planetary centre. Unity of idea in the construction of the universe. Hence, before it can be asserted that the theory of gravitation is truly universal, it must be shown that it holds good in the case of all other such systems. The evidence offered in proof of this is altogether based upon the observations of the two Herschels on the motions of the double stars. Among the stars there are some in such close proximity to each other that Sir W. Herschel was led to suppose it would be possible, from observations upon them, to ascertain the stellar parallax. While engaged in these inquiries, which occupied him for many years, he discovered that many of these stars are not merely optically in proximity, as being accidentally in the same line of view, but are actually connected physically, revolving round each other in regular orbits. The motion of these double suns is, however, in many instances so slow as to require many years for a satisfactory determination. Gravitation of double stars. Sir J. Herschel therefore continued the observations of his father, and with other mathematicians, investigated the characteristics of these motions. The first instance in which the true elliptic elements of the orbit of a binary star were determined was given by M. Savary in the case of chi Ursæ Majoris, indicating an elliptic orbit of 58 ¼ years. But the period of others, since determined, is very much longer; thus, in sigma Coronæ, it is, according to Mr. Hind, more than 736 years. From the fact that the orbits in which these stars move round each other are elliptical, it necessarily follows that the law of gravitation, according to the inverse square, holds good in them. Considering the prodigious distances of these bodies, and the departure, as regards structure of the systems to which they belong, from the conditions obtaining in our unisolar system, we may perhaps assert the prevalence of the law of gravitation throughout the universe.

Coloured light of double stars. If, in association with these double suns – sometimes, indeed, they are triple, and occasionally, as in the case of epsilon Lyræ, quadruple – there are opaque planetary globes, such solar systems differ from ours not only in having several suns instead of a single one, but, since the light emitted is often of different tints, one star shining with a crimson and another with a blue light, the colours not always complementary to one another, a wonderful variety of phenomena must be the result, especially in their organic creations; for organic forms, both vegetable and animal, primarily depend on the relations of coloured light. How varied the effects where there are double, triple, or even quadruple sunrises, and sunsets, and noons; and the hours marked off by red, or purple, or blue tints.

Grandeur of Newton's discoveries. It is impossible to look back on the history of the theory of gravitation without sentiments of admiration and, indeed, of pride. How felicitous has been the manner in which have been explained the inequalities of a satellite like the moon under the disturbing influence of the sun; the correspondence between the calculated and observed quantities of these inequalities; the extension of the doctrine to satellites of other planets, as those of Jupiter; the determination of the earth's figure; the causes of the tides; the different force of gravity in different latitudes, and a multitude of other phenomena. The theory asserted for itself that authority which belongs to intrinsic truth. It enabled mathematicians to point out facts not yet observed, and to foretell future events.

And yet how hard it is for truth to force its way when bigotry resists. In 1771, the University of Salamanca, being urged to teach physical science, refused, and this was its answer; "Newton teaches nothing that would make a good logician or metaphysician; and Gassendi and Descartes do not agree so well with revealed truth as Aristotle does."

The earth in time. Among the interesting results of Newton's theory may be mentioned its application to secular inequalities, such as the acceleration of the moon's mean motion, that satellite moving somewhat quicker now than she did ages ago. Laplace detected the cause of this phenomenon in the influence of the sun upon the moon, combined with the secular variation of the eccentricity of the earth's orbit. Moreover, he showed that this secular inequality of the motion of the moon is periodical, that it requires millions of years to re-establish itself, and that, after an almost inconceivable time, the acceleration becomes a retardation. In like manner, the same mathematician explained the observed acceleration in the mean motion of Jupiter, and retardation of that of Saturn, as arising from the mutual attraction of the two planets, and showed that this secular inequality has a period of 929 ½ years. With such slow movements may be mentioned the diminution of the obliquity of the ecliptic, which has been proceeding for ages, but which will reach a limit and then commence to increase. These secular motions ought not to be without interest to those who suffer themselves to adopt the patristic chronology of the world, who suppose that the earth is only six thousand years old, and that it will come to an end in about one thousand years more. They must accept, along with that preposterous delusion, its necessary consequences, that the universe has been so badly constructed, and is such a rickety machine, that it can not hold together long enough for some of its wheels to begin to revolve. Astronomy offers us many illustrations of the scale upon which the world is constructed as to time, as well as that upon which it is constructed as to space.

Dominion of law in the universe. From what has been said, the conclusion forces itself upon us that the general laws obtaining as respects the earth, hold good likewise for all other parts of the universe; a conclusion sustained not only by the mechanism of such motions as we have been considering, but also by all evidence of a physical kind accessible to us. The circumstances under which our sun emits light and heat, and thereby vivifies his attendant planets, are indisputably the same as those obtaining in the case of every fixed star, each of which is a self-luminous sun. There is thus an aspect of homogeneousness in the structure of all systems in the universe, which, though some have spoken of it as if it were the indication of a uniformity of plan, and therefore the evidence of a primordial idea, is rather to be looked upon as the proof of unchangeable and resistless law.