Fig. 28.Fig. 28.
The centrifugal force is increased as the velocity is increased. Thus, the parts of a millstone most remote from the centre sometimes acquire a centrifugal forceso much greater than the central parts, which move much slower, that the stone is divided, and the exterior portions are projected with great violence. In like manner, as the equatorial parts of the earth, in the diurnal revolution, revolve much faster than the parts towards the poles, so the centrifugal force is felt most at the equator, and becomes strikingly manifest by the diminished weight of bodies, since it acts in opposition to the force of gravity.
Although the foregoing law of motion, when first presented to the mind, appears to convey no new truth, but only to enunciate in a formal manner what we knew before; yet a just understanding of this law, in all its bearings, leads us to a clear comprehension of no small share of all the phenomena of motion. The second and third laws may be explained in fewer terms.
TheSECOND LAWof motion is as follows:motion is proportioned to the force impressed, and in the direction of that force.
The meaning of this law is, that every force that is applied to a body produces its full effect, proportioned to its intensity, either in causing or in preventing motion. Let there be ever so many blows applied at once to a ball, each will produce its own effect in its own direction, and the ball will move off, not indeed in the zigzag, complex lines corresponding to the directions of the several forces, but in a single line expressing the united effect of all. If you place a ball at the corner of a table, and give it an impulse, at the same instant, with the thumb and finger of each hand, one impelling it in the direction of one side of the table, and the other in the direction of the other side, the ball will move diagonally across the table. If the blows be exactly proportioned each to the length of the side of the table on which it is directed, the ball will run exactly from corner to corner, and in the same time that it would have passed over each side by the blow given in the direction of that side. This principle is expressed by saying, that a body impelled by two forces, actingrespectively in the directions of the two sides of a parallelogram, and proportioned in intensity to the lengths of the sides, will describe the diagonal of the parallelogram in the same time in which it would have described the sides by the forces acting separately.
The converse of this proposition is also true, namely, that any single motion may be considered as theresultantof two others,—the motion itself being represented by the diagonal, while the twocomponentsare represented by the sides, of a parallelogram. This reduction of a motion to the individual motions that produce it, is called theresolution of motion, or theresolution of forces. Nor can a given motion be resolved intotwocomponents, merely. These, again, may be resolved into others, varying indefinitely, in direction and intensity, from all which the given motion may be considered as having resulted. This composition and resolution of motion or forces is often of great use, in inquiries into the motions of the heavenly bodies. The composition often enables us to substitute a single force for a great number of others, whose individual operations would be too complicated to be followed. By this means, the investigation is greatly simplified. On the other hand, it is frequently very convenient to resolve a given motion into two or more others, some of which may be thrown out of the account, as not influencing the particular point which we are inquiring about, while others are far more easily understood and managed than the single force would have been. It is characteristic of great minds, to simplify these inquiries. They gain an insight into complicated and difficult subjects, not so much by any extraordinary faculty of seeing in the dark, as by the power of removing from the object all incidental causes of obscurity, until it shines in its own clear and simple light.
If every force, when applied to a body, produces its full and legitimate effect, how many other forces soever may act upon it, impelling it different ways, then it must follow, that the smallest force ought to move thelargest body; and such is in fact the case. A snap of a finger upon a seventy-four under full sail, if applied in the direction of its motion, would actually increase its speed, although the effect might be too small to be visible. Still it is something, and may be truly expressed by a fraction. Thus, suppose a globe, weighing a million of pounds, were suspended from the ceiling by a string, and we should apply to it the snap of a finger,—it is granted that the motion would be quite insensible. Let us then divide the body into a million equal parts, each weighing one pound; then the same impulse, applied to each one separately, would produce a sensible effect, moving it, say one inch. It will be found, on trial, that the same impulse given to a mass of two pounds will move it half an inch; and hence it is inferred, that, if applied to a mass weighing a million of pounds, it would move it the millionth part of an inch.
It is one of the curious results of the second law of motion, that an unlimited number of motions may exist together in the same body. Thus, at the same moment, we may be walking around a post in the cabin of a steam-boat, accompanying the boat in its passage around an island, revolving with the earth on its axis, flying through space in our annual circuit around the sun, and possibly wheeling, along with the sun and his whole retinue of planets, around some centre in common with the starry worlds.
TheTHIRD LAWof motion is this:action and reaction are equal, and in contrary directions.
Whenever I give a blow, the body struck exerts an equal force on the striking body. If I strike the water with an oar, the water communicates an equal impulse to the oar, which, being communicated to the boat, drives it forward in the opposite direction. If a magnet attracts a piece of iron, the iron attracts the magnet just as much, in the opposite direction; and, in short, every portion of matter in the universe attracts and is attracted by every other, equally, in an opposite direction. This brings us to the doctrine of universal gravitation, which is the very key that unlocks all the secrets of the skies. This will form the subject of my next Letter.
"To Him no high, no low, no great, no small,He fills, He bounds, connects, and equals all."—Pope.
"To Him no high, no low, no great, no small,He fills, He bounds, connects, and equals all."—Pope.
Wediscover in Nature a tendency of every portion of matter towards every other. This tendency is calledgravitation. In obedience to this power, a stone falls to the ground, and a planet revolves around the sun. We may contemplate this subject as it relates either to phenomena that take place near the surface of the earth, or in the celestial regions. The former,gravity, is exemplified by falling bodies; the latter,universal gravitation, by the motions of the heavenly bodies. The laws of terrestrial gravity were first investigated by Galileo; those of universal gravitation, by Sir Isaac Newton. Terrestrial gravity is only an individual example of universal gravitation; being the tendency of bodies towards the centre of the earth. We are so much accustomed, from our earliest years, to see bodies fall to the earth, that we imagine bodies must of necessity fall "downwards;" but when we reflect that the earth is round, and that bodies fall towards the centre on all sides of it, and that of course bodies on opposite sides of the earth fall in precisely opposite directions, and towards each other, we perceive that there must be some force acting to produce this effect; nor is it enough to say, as the ancients did, that bodies "naturally" fall to the earth. Every motion implies some force which produces it; and the fact that bodies fall towards the earth, on all sides of it, leads us to infer that that force, whatever it is, resides in the earth itself.We therefore call itattraction. We do not, however, say what attractionis, but what itdoes. We must bear in mind, also, that, according to the third law of motion, this attraction is mutual; that when a stone falls towards the earth, it exerts the same force on the earth that the earth exerts on the stone; but the motion of the earth towards the stone is as much less than that of the stone towards the earth, as its quantity of matter is greater; and therefore its motion is quite insensible.
But although we are compelled to acknowledge theexistenceof such a force as gravity, causing a tendency in all bodies towards each other, yet we know nothing of itsnature, nor can we conceive by what medium bodies at such a distance as the moon and the earth exercise this influence on each other. Still, we may trace the modes in which this force acts; that is, itslaws; for the laws of Nature are nothing else than the modes in which the powers of Nature act.
We owe chiefly to the great Galileo the first investigation of the laws of terrestrial gravity, as exemplified in falling bodies; and I will avail myself of this opportunity to make you better acquainted with one of the most interesting of men and greatest of philosophers.
Galileo was born at Pisa, in Italy, in the year 1564. He was the son of a Florentine nobleman, and was destined by his father for the medical profession, and to this his earlier studies were devoted. But a fondness and a genius for mechanical inventions had developed itself, at a very early age, in the construction of his toys, and a love of drawing; and as he grew older, a passion for mathematics, and for experimental research, predominated over his zeal for the study of medicine, and he fortunately abandoned that for the more congenial pursuits of natural philosophy and astronomy. In the twenty-fifth year of his age, he was appointed, by the Grand Duke of Tuscany, professor of mathematics in the University of Pisa. At that period, there prevailed in all the schools a most extraordinary reverence for the writings of Aristotle, the preceptor of Alexanderthe Great,—a philosopher who flourished in Greece, about three hundred years before the Christian era. Aristotle, by his great genius and learning, gained a wonderful ascendency over the minds of men, and became the oracle of the whole reading world for twenty centuries. It was held, on the one hand, that all truths worth knowing were contained in the writings of Aristotle; and, on the other, that an assertion which contradicted any thing in Aristotle could not be true. But Galileo had a greatness of mind which soared above the prejudices of the age in which he lived, and dared to interrogate Nature by the two great and only successful methods of discovering her secrets,—experiment and observation. Galileo was indeed the first philosopher that ever fully employed experiments as the means of learning the laws of Nature, by imitating on a small what she performs on a great scale, and thus detecting her modes of operation. Archimedes, the great Sicilian philosopher, had in ancient times introduced mathematical or geometrical reasoning into natural philosophy; but it was reserved for Galileo to unite the advantages of both mathematical and experimental reasonings in the study of Nature,—both sure and the only sure guides to truth, in this department of knowledge, at least. Experiment and observation furnish materials upon which geometry builds her reasonings, and from which she derives many truths that either lie for ever hidden from the eye of observation, or which it would require ages to unfold.
This method, of interrogating Nature by experiment and observation, was matured into a system by Lord Bacon, a celebrated English philosopher, early in the seventeenth century,—indeed, during the life of Galileo. Previous to that time, the inquirers into Nature did not open their eyes to see how the facts reallyare; but, by metaphysical processes, in imitation of Aristotle, determined how theyought to be, and hastily concluded that they were so. Thus, they did not study into the laws of motion, by observing how motion actually takesplace, under various circumstances, but first, in their closets, constructed a definition of motion, and thence inferred all its properties. The system of reasoning respecting the phenomena of Nature, introduced by Lord Bacon, was this: in the first place, to examine all the facts of the case, and then from these to determine the laws of Nature. To derive general conclusions from the comparison of a great number of individual instances constitutes the peculiarity of the Baconian philosophy. It is called theinductivesystem, because its conclusions were built on the induction, or comparison, of a great many single facts. Previous to the time of Lord Bacon, hardly any insight had been gained into the causes of natural phenomena, and hardly one of the laws of Nature had been clearly established, because all the inquirers into Nature were upon a wrong road, groping their way through the labyrinth of error. Bacon pointed out to them the true path, and held before them the torch-light of experiment and observation, under whose guidance all successful students of Nature have since walked, and by whose illumination they have gained so wonderful an insight into the mysteries of the natural world.
It is a remarkable fact, that two such characters as Bacon and Galileo should appear on the stage at the same time, who, without any communication with each other, or perhaps without any personal knowledge of each other's existence, should have each developed the true method of investigating the laws of Nature. Galileo practised what Bacon only taught; and some, therefore, with much reason, consider Galileo as a greater philosopher than Bacon. "Bacon," says Hume, "pointed out, at a great distance, the road to philosophy; Galileo both pointed it out to others, and made, himself, considerable advances in it. The Englishman was ignorant of geometry; the Florentine revived that science, excelled in it, and was the first who applied it, together with experiment, to natural philosophy. The former rejected, with the most positive disdain, the system ofCopernicus; the latter fortified it with new proofs, derived both from reason and the senses."
When we reflect that geometry is a science built upon self-evident truths, and that all its conclusions are the result of pure demonstration, and can admit of no controversy; when we further reflect, that experimental evidence rests on the testimony of the senses, and we infer a thing to be true because we actually see it to be so; it shows us the extreme bigotry, the darkness visible, that beclouded the human intellect, when it not only refused to admit conclusions first established by pure geometrical reasoning, and afterwards confirmed by experiments exhibited in the light of day, but instituted the most cruel persecutions against the great philosopher who first proclaimed these truths. Galileo was hated and persecuted by two distinct bodies of men, both possessing great influence in their respective spheres,—the one consisting of the learned doctors of philosophy, who did nothing more, from age to age, than reiterate the doctrines of Aristotle, and were consequently alarmed at the promulgation of principles subversive of those doctrines; the other consisting of the Romish priesthood, comprising the terrible Inquisition, who denounced the truths taught by Galileo, as inconsistent with certain declarations of the Holy Scriptures. We shall see, as we advance, what a fearful warfare he had to wage against these combined powers of darkness.
Aristotle had asserted, that, if two different weights of the same material were let fall from the same height, the heavier one would reach the ground sooner than the other, in proportion as it was more weighty. For example: if a ten-pound leaden weight and a one-pound were let fall from a given height at the same instant, the former would reach the ground ten times as soon as the latter. No one thought of making the trial, but it was deemed sufficient that Aristotle had said so; and accordingly this assertion had long been received as an axiom in the science of motion. Galileo ventured to appeal from the authority of Aristotle to thatof his own senses, and maintained, that both weights would fall in the same time. The learned doctors ridiculed the idea. Galileo tried the experiment in their presence, by letting fall, at the same instant, large and small weights from the top of the celebrated leaning tower of Pisa. Yet, with the sound of the two weights clicking upon the pavement at the same moment, they still maintained that the ten-pound weight would reach the ground in one tenth part of the time of the other, because they could quote the chapter and verse of Aristotle where the fact was asserted. Wearied and disgusted with the malice and folly of these Aristotelian philosophers, Galileo, at the age of twenty-eight, resigned his situation in the university of Pisa, and removed to Padua, in the university of which place he was elected professor of mathematics. Up to this period, Galileo had devoted himself chiefly to the studies of the laws of motion, and the other branches of mechanical philosophy. Soon afterwards, he began to publish his writings, in rapid succession, and became at once among the most conspicuous of his age,—a rank which he afterwards well sustained and greatly exalted, by the invention of the telescope, and by his numerous astronomical discoveries. I will reserve an account of these great achievements until we come to that part of astronomy to which they were more immediately related, and proceed, now, to explain to you the leading principles ofterrestrial gravity, as exemplified in falling bodies.
First,all bodies near the earth's surface fall in straight lines towards the centre of the earth. We are not to infer from this fact, that there resides at the centre any peculiar force, as a great loadstone, for example, which attracts bodies towards itself; but bodies fall towards the centre of the sphere, because the combined attractions of all the particles of matter in the earth, each exerting its proper force upon the body, would carry it towards the centre. This may be easily illustrated by a diagram. Let B, Fig. 29, page 140, be thecentre of the earth, and A a body without it. Every portion of matter in the earth exerts some force on A, to draw it down to the earth. But since there is just as much matter on one side of the line A B, as on the other side, each half exerts an equal force to draw the body towards itself; therefore it falls in the direction of the diagonal between the two forces. Thus, if we compare the effects of any two particles of matter at equal distances from the line A B, but on opposite sides of it, asa,b, while the force of the particle atawould tend to draw A in the direction of Aa, that ofbwould draw it in the direction of Ab, and it would fall in the line A B, half way between the two. The same would hold true of any other two corresponding particles of matter on different sides of the earth, in respect to a body situated in any place without it.
Fig. 29.Fig. 29.
Secondly,all bodies fall towards the earth, from the same height, with equal velocities. A musket-ball, and the finest particle of down, if let fall from a certain height towards the earth, tend to descend towards it at the same rate, and would proceed with equal speed, were it not for the resistance of the air, which retards the down more than it does the ball, and finally stops it. If, however, the air be removed out of the way, as it may be by means of the air-pump, the two bodies keep side by side in falling from the greatest height at which we can try the experiment.
Thirdly,bodies, in falling towards the earth, have their rate of motion continually accelerated. Suppose we let fall a musket-ball from the top of a high tower, and watch its progress, disregarding the resistance of the air: the first second, it will pass over sixteen feet and one inch, but its speed will be constantly increased, being all the while urged onward by thesame force, and retaining all that it has already acquired; so that the longer it is in falling, the swifter its motion becomes. Consequently, when bodies fall from a great height, they acquire an immense velocity before they reach the earth. Thus, a man falling from a balloon, or from the mast-head of a ship, is broken in pieces; and those meteoric stones, which sometimes fall from the sky, bury themselves deep in the earth. On measuring the spaces through which a body falls, it is found, that it will fall four times as far in two seconds as in one, and one hundred times as far in ten seconds as in one; and universally, the space described by a falling body is proportioned to the time multiplied into itself; that is, to the square of the time.
Fourthly,gravity is proportioned to the quantity of matter. A body which has twice as much matter as another exerts a force of attraction twice as great, and also receives twice as much from the same body as it would do, if it were only just as heavy as that body. Thus the earth, containing, as it does, forty times as much matter as the moon, exerts upon the moon forty times as much force as it would do, were its mass the same with that of the moon; but it is also capable ofreceivingforty times as much gravity from the moon as it would do, were its mass the same as the moon's; so that the power of attracting and that of being attracted are reciprocal; and it is therefore correct to say, that the moon attracts the earthjust as muchas the earth attracts the moon; and the same may be said of any two bodies, however different in quantity of matter.
Fifthly,gravity, when acting at a distance from the earth, is not as intense as it is near the earth. At such a distance as we are accustomed to ascend above the general level of the earth, no great difference is observed. On the tops of high mountains, we find bodies falling towards the earth, with nearly the same speed as they do from the smallest elevations. It is found, nevertheless, that there is a real difference; so that, in fact, the weight of a body (which is nothing morethan the measure of its force of gravity) is not quite so great on the tops of high mountains as at the general level of the sea. Thus, a thousand pounds' weight, on the top of a mountain half a mile high, would weigh a quarter of a pound less than at the level of the sea; and if elevated four thousand miles above the earth,—that is,twiceas far from the centre of the earth as the surface is from the centre,—it would weigh only one fourth as much as before; ifthree timesas far, it would weigh only one ninth as much. So that the force of gravity decreases, as we recede from the earth, in the same proportion as the square of the distance increases. This fact is generalized by saying, thatthe force of gravity, at different distances from the earth, is inversely as the square of the distance.
Were a body to fall from a great distance,—suppose a thousand times that of the radius of the earth,—the force of gravity being one million times less than that at the surface of the earth, the motion of the body would be exceedingly slow, carrying it over only the sixth part of an inch in a day. It would be a long time, therefore, in making any sensible approaches towards the earth; but at length, as it drew near to the earth it would acquire a very great velocity, and would finally rush towards it with prodigious violence. Falling so far, and being continually accelerated on the way, we might suppose that it would at length attain a velocity infinitely great; but it can be demonstrated, that, if a body were to fall from an infinite distance, attracted to the earth only by gravity, it could never acquire a velocity greater than about seven miles per second. This, however, is a speed inconceivably great, being about eighteen times the greatest velocity that can be given to a cannon-ball, and more than twenty-five thousand miles per hour.
But the phenomena of falling bodies must have long been observed, and their laws had been fully investigated by Galileo and others, before the cause of their falling was understood, or any such principle asgravity, inherent in the earth and in all bodies, was applied to them. The developement of this great principle was the work of Sir Isaac Newton; and I will give you, in my next Letter, some particulars respecting the life and discoveries of this wonderful man.
"The heavens are all his own; from the wild ruleOf whirling vortices, and circling spheres,To their first great simplicity restored.The schools astonished stood; but found it vainTo combat long with demonstration clear,And, unawakened, dream beneath the blazeOf truth. At once their pleasing visions fled,With the light shadows of the morning mixed,When Newton rose, our philosophic sun."—Thomson's Elegy.
"The heavens are all his own; from the wild ruleOf whirling vortices, and circling spheres,To their first great simplicity restored.The schools astonished stood; but found it vainTo combat long with demonstration clear,And, unawakened, dream beneath the blazeOf truth. At once their pleasing visions fled,With the light shadows of the morning mixed,When Newton rose, our philosophic sun."—Thomson's Elegy.
Sir Isaac Newtonwas born in Lincolnshire, England, in 1642, just one year after the death of Galileo. His father died before he was born, and he was a helpless infant, of a diminutive size, and so feeble a frame, that his attendants hardly expected his life for a single hour. The family dwelling was of humble architecture, situated in a retired but beautiful valley, and was surrounded by a small farm, which afforded but a scanty living to the widowed mother and her precious charge. The cut on page 144, Fig 30, represents the modest mansion, and the emblems of rustic life that first met the eyes of this pride of the British nation, and ornament of human nature. It will probably be found, that genius has oftener emanated from the cottage than from the palace.
Fig. 30.Fig. 30.
The boyhood of Newton was distinguished chiefly for his ingenious mechanical contrivances. Among other pieces of mechanism, he constructed a windmill so curious and complete in its workmanship, as to excite universal admiration. After carrying it a while by the forceof the wind, he resolved to substitute animal power, and for this purpose he inclosed in it a mouse, which he called the miller, and which kept the mill a-going by acting on a tread-wheel. The power of the mouse was brought into action by unavailing attempts to reach a portion of corn placed above the wheel. A water-clock, a four-wheeled carriage propelled by the rider himself, and kites of superior workmanship, were among the productions of the mechanical genius of this gifted boy. At a little later period, he began to turn his attention to the motions of the heavenly bodies, and constructed several sun-dials on the walls of the house where he lived. All this was before he had reached his fifteenth year. At this age, he was sent by his mother, in company with an old family servant, to a neighboring market-town, to dispose of products of their farm, and to buy articles of merchandise for their family use; but the young philosopher left all these negotiations to his worthy partner, occupying himself, mean-while, with a collection of old books, which he had found in a garret. At other times, he stopped on the road, and took shelter with his book under a hedge, until the servant returned. They endeavored to educate him as a farmer; but the perusal of a book, the construction of a water-mill, or some other mechanical or scientific amusement, absorbed all his thoughts, when the sheep were going astray, and the cattle were devouring or treading down the corn. One of his uncles having found him one day under a hedge, with a book in his hand, and entirely absorbed in meditation, took it from him, and found that it was a mathematical problem which so engrossed his attention. His friends, therefore, wisely resolved to favor the bent of his genius, and removed him from the farm to the school, to prepare for the university. In the eighteenth year of his age, Newton was admitted into Trinity College, Cambridge. He made rapid and extraordinary advances in the mathematics, and soon afforded unequivocal presages of that greatness which afterwards placed him at the head of the human intellect. In 1669, at the age of twenty-seven, he became professor of mathematics at Cambridge, a post which he occupied for many years afterwards. During the four or five years previous to this he had, in fact, made most of those great discoveries which have immortalized his name. We are at present chiefly interested in one of these, namely, that ofuniversal gravitation; and let us see by what steps he was conducted to this greatest of scientific discoveries.
In the year 1666, when Newton was about twenty-four years of age, the plague was prevailing at Cambridge, and he retired into the country. One day, while he sat in a garden, musing on the phenomena of Nature around him, an apple chanced to fall to the ground. Reflecting on the mysterious power that makes all bodies near the earth fall towards its centre, and considering that this power remains unimpaired at considerable heights above the earth, as on the tops of trees and mountains, he asked himself,—"May not the same force extend its influence to a great distance from the earth, even as far as the moon? Indeed, may not this be the very reason, why the moon is drawn away continually fromthe straight line in which every body tends to move, and is thus made to circulate around the earth?" You will recollect that it was mentioned, in my Letter which contained an account of the first law of motion, that if a body is put in motion by any force, it will always move forward in a straight line, unless some other force compels it to turn aside from such a direction; and that, when we see a body moving in a curve, as a circular orbit, we are authorized to conclude that there is some force existing within the circle, which continually draws the body away from the direction in which it tends to move. Accordingly, it was a very natural suggestion, to one so well acquainted with the laws of motion as Newton, that the moon should constantly bend towards the earth, from a tendency to fall towards it, as any other heavy body would do, if carried to such a distance from the earth. Newton had already proved, that if such a power as gravity extends from the earth to distant bodies, it must decrease, as the square of the distance from the centre of the earth increases; that is, at double the distance, it would be four times less; at ten times the distance, one hundred times less; and so on. Now, it was known that the moon is about sixty times as far from the centre of the earth as the surface of the earth is from the centre, and consequently, the force of attraction at the moon must be the square of sixty, or thirty-six hundred times less than it is at the earth; so that a body at the distance of the moon would fall towards the earth very slowly, only one thirty-six hundredth part as far in a given time, as at the earth. Does the moon actually fall towards the earth at this rate; or, what is the same thing, does she depart at this rate continually from the straight line in which she tends to move, and in which she would move, if no external force diverted her from it? On making the calculation, such was found to be the fact. Hence gravity, and no other force than gravity, acts upon the moon, and compels her to revolve around the earth. By reasonings equally conclusive, it was afterwards proved, that a similar force compels allthe planets to circulate around the sun; and now, we may ascend from the contemplation of this force, as we have seen it exemplified in falling bodies, to that of a universal power whose influence extends to all the material creation. It is in this sense that we recognise the principle of universal gravitation, the law of which may be thus enunciated;all bodies in the universe, whether great or small, attract each other, with forces proportioned to their respective quantities of matter, and inversely as the squares of their distances from each other.
This law asserts, first, that attraction reigns throughout the material world, affecting alike the smallest particle of matter and the greatest body; secondly, that it acts upon every mass of matter, precisely in proportion to its quantity; and, thirdly, that its intensity is diminished as the square of the distance is increased.
Observation has fully confirmed the prevalence of this law throughout the solar system; and recent discoveries among the fixed stars, to be more fully detailed hereafter, indicate that the same law prevails there. The law of universal gravitation is therefore held to be the grand principle which governs all the celestial motions. Not only is it consistent with all the observed motions of the heavenly bodies, even the most irregular of those motions, but, when followed out into all its consequences, it would be competent to assert that such irregularities must take place, even if they had never been observed.
Newton first published the doctrine of universal gravitation in the 'Principia,' in 1687. The name implies that the work contains the fundamental principles of natural philosophy and astronomy. Being founded upon the immutable basis of mathematics, its conclusions must of course be true and unalterable, and thenceforth we may regard the great laws of the universe as traced to their remotest principle. The greatest astronomers and mathematicians have since occupied themselves in following out the plan which Newton began, by applying the principles of universal gravitation to all the subordinateas well as to the grand movements of the spheres. This great labor has been especially achieved by La Place, a French mathematician of the highest eminence, in his profound work, the 'Mecanique Celeste.' Of this work, our distinguished countryman, Dr. Bowditch, has given a magnificent translation, and accompanied it with a commentary, which both illustrates the original, and adds a great amount of matter hardly less profound than that.
Fig. 31.Fig. 31.
We have thus far taken the earth's orbit around the sun as a great circle, such being its projection on the sphere constituting the celestial ecliptic. The real path of the earth around the sun is learned, as I before explained to you, by the apparent path of the sun around the earth once a year. Now, when a body revolves about the earth at a great distance from us, as is the case with the sun and moon, we cannot certainly infer that it moves in a circle because it appears to describe a circle on the face of the sky, for such might be the appearance of its orbit, were it ever so irregular a curve. Thus, if E, Fig. 31, represents the earth, and ACB, the irregular path of a body revolving about it, since we should refer the body continually to some place on the celestial sphere, XYZ, determined by lines drawn from the eye to the concave sphere through the body,the body, while moving from A to B through C, would appear to move from X to Z, through Y. Hence, we must determine from other circumstances than the actual appearance, what is the true figure of the orbit.
Fig. 32.Fig. 32.
Were the earth's path a circle, having the sun in the centre, the sun would always appear to be at the same distance from us; that is, the radius of the orbit, orradius vector, (the name given to a line drawn from the centre of the sun to the orbit of any planet,) would always be of the same length. But the earth's distance from the sun is constantly varying, which shows that its orbit is not a circle. We learn the true figure of the orbit, by ascertaining therelative distancesof the earth from the sun, at various periods of the year. These distances all being laid down in a diagram, according to their respective lengths, the extremities, on being connected, give us our first idea of the shape of the orbit, which appears of an oval form, and at least resembles an ellipse; and, on further trial, we find that it has the properties of an ellipse. Thus, let E, Fig. 32, be the place of the earth, anda,b,c, &c., successive positions of the sun; therelativelengths of the lines Ea, Eb, &c., being known, on connecting thepointsa,b,c, &c., the resulting figure indicates the true figure of the earth's orbit.
These relative distances are found in two different ways; first,by changes in the sun's apparent diameter, and, secondly,by variations in his angular velocity. The same object appears to us smaller in proportion as it is more distant; and if we see a heavenly body varying in size, at different times, we infer that it is at different distances from us; that when largest, it is nearest to us, and when smallest, furthest off. Now, when the sun's diameter is accurately measured by instruments, it is found to vary from day to day; being, when greatest, more than thirty-two minutes and a half, and when smallest, only thirty-one minutes and a half,—differing, in all, about seventy-five seconds. When the diameter is greatest, which happens in January, we know that the sun is nearest to us; and when the diameter is least, which occurs in July, we infer that the sun is at the greatest distance from us. The point where the earth, or any planet, in its revolution, is nearest the sun, is called itsperihelion; the point where it is furthest from the sun, itsaphelion. Suppose, then, that, about the first of January, when the diameter of the sun is greatest, we draw a line, Ea, Fig. 32, to represent it, and afterwards, every ten days, draw other lines, Eb, Ec, &c.; increasing in the same ratio as the apparent diameters of the sun decrease. These lines must be drawn at such a distance from each other, that the triangles, Ea b, Eb c, &c., shall be all equal to each other, for a reason that will be explained hereafter. On connecting the extremities of these lines, we shall obtain the figure of the earth's orbit.
Similar conclusions may be drawn from observations on the sun'sangular velocity. A body appears to move most rapidly when nearest to us. Indeed, the apparent velocity increases rapidly, as it approaches us, and as rapidly diminishes, when it recedes from us. If it comes twice as near as before, it appears to move not merely twice as swiftly, but four times as swiftly; if itcomes ten times nearer, its apparent velocity is one hundred times as great as before. We say, therefore, that the velocity varies inversely as the square of the distance; for, as the distance is diminished ten times, the velocity is increased the square of ten; that is, one hundred times. Now, by noting the time it takes the sun, from day to day, to cross the central wire of the transit-instrument, we learn the comparative velocities with which it moves at different times; and from these we derive the comparative distances of the sun at the corresponding times; and laying down these relative distances in a diagram, as before, we get our first notions of the actual figure of the earth's orbit, or the path which it describes in its annual revolution around the sun.
Having now learned the fact, that the earth moves around the sun, not in a circular but in an elliptical orbit, you will desire to know by what forces it is impelled, to make it describe this figure, with such uniformity and constancy, from age to age. It is commonly said, that gravity causes the earth and the planets to circulate around the sun; and it is true that it is gravity which turns them aside from the straight line in which, by the first law of motion, they tend to move, and thus causes them to revolve around the sun. But what force is that which gave to them this original impulse, and impressed upon them such a tendency to move forward in a straight line? The nameprojectileforce is given to it, because it is the sameas thoughthe earth were originally projected into space, when first created; and therefore its motion is the result of two forces, the projectile force, which would cause it to move forward in a straight line which is a tangent to its orbit, and gravitation, which bends it towards the sun. But before you can clearly understand the nature of this motion, and the action of the two forces that produce it, I must explain to you a few elementary principles upon which this and all the other planetary motions depend.
You have already learned, that when a body is acted on by two forces, in different directions, it moves in the direction of neither, but in some direction between them. If I throw a stone horizontally, the attraction of the earth will continually draw it downward, out of the line of direction in which it was thrown, and make it descend to the earth in a curve. The particular form of the curve will depend on the velocity with which it is thrown. It will alwaysbeginto move in the line of direction in which it is projected; but it will soon be turned from that line towards the earth. It will, however, continue nearer to the line of projection in proportion as the velocity of projection is greater. Thus, let A C, Fig. 33, be perpendicular to the horizon, and A B parallel to it, and let a stone be thrown from A, in the direction of A B. It will, in every case, commence its motion in the line A B, which will therefore be a tangent to the curve it describes; but, if it is thrown with a small velocity, it will soon depart from the tangent, describing the line A D; with a greater velocity, it will describe a curve nearer the tangent, as A E; and with a still greater velocity, it will describe the curve A F.
Fig. 33.Fig. 33.
As an example of a body revolving in an orbit under the influence of two forces, suppose a body placed at any point, P, Fig. 34, above the surface of the earth, and let P A be the direction of the earth's centre; that is, a line perpendicular to the horizon. If the body were allowed to move, without receiving any impulse,it would descend to the earth in the direction P A with an accelerated motion. But suppose that, at the moment of its departure from P, it receives a blow in the direction P B, which would carry it to B in the time the body would fall from P to A; then, under the influence of both forces, it would descend along the curve P D. If a stronger blow were given to it in the direction P B, it would describe a larger curve, P E; or, finally, if the impulse were sufficiently strong, it would circulate quite around the earth, and return again to P, describing the circle P F G. With a velocity of projection still greater, it would describe an ellipse, P I K; and if the velocity be increased to a certain degree, the figure becomes a parabola, L P M,—a curve which never returns into itself.
Fig. 34.Fig. 34.
In Fig. 35, page 154, suppose the planet to have passed the point C, at the aphelion, with so small a velocity, that the attraction of the sun bends its path very much, and causes it immediately to begin to approach towards the sun. The sun's attraction will increase its velocity, as it moves through D, E, and F, for the sun's attractive force on the planet, when at D, is acting in the direction D S; and, on account of the small angle made between D E and D S, the force acting in the line D S helps the planet forward in the path D E, and thus increases itsvelocity. In like manner, the velocity of the planet will be continually increasing as it passes through D, E, and F; and though the attractive force, on account of the planet's nearness, is so much increased, and tends, therefore, to make the orbit more curved, yet the velocity is also so much increased, that the orbit is not more curved than before; for the same increase of velocity, occasioned by the planet's approach to the sun, produces a greater increase of centrifugal force, which carries it off again. We may see, also, the reason why, when the planet has reached the most distant parts of its orbit, it does not entirely fly off, and never return to the sun; for, when the planet passes along H, K, A, the sun's attraction retards the planet, just as gravity retards a ball rolled up hill; and when it has reached C, its velocity is very small, and the attraction to the centre of force causes a great deflection from the tangent, sufficient to give its orbit a great curvature, and the planet wheels about, returns to the sun, and goes over the same orbit again. As the planet recedes from the sun, its centrifugal force diminishes faster than the force of gravity, so that the latter finally preponderates.