Authors born between 1665 and 1700 CE
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Scope of Newton's Principia
Space and Time
Axioms, or Laws of Motion
Rules of Reasoning in Philosophy
Orbits of Planets Around the Sun
Gravity, a Universal Force
The Center of the System of Worlds
The Unknown Nature of Gravity
Isaac Newton (1642-1727), the son of a farmer, was born in Woolsthorpe, Lincolnshire, England. He rose to be head boy at the grammar school of Grantham, a town about six miles from his small hamlet. When his mechanical ingenuity and interest in mathematical problems caused relatives to recognize he would not be happy in farming, he was sent to Trinity College, Cambridge. Seeking information on trigonometry he put aside Euclid’s Elements as "a trifling book" and turned instead to Descarte’s Geometry. When he did badly in an examination on the Elements he reviewed it more carefully and concluded it was a work of considerable merit. In 1667, he took the degree of Master of Arts and was subsequently elected a fellow of Trinity College. In the following years he engaged in optical and chemical experiments and began his development of a theory of fluxions, later known as the calculus; he also investigated other branches of pure mathematics. Newton became Lucasian professor of mathematics in 1669, lecturing primarily on optics.
Newton was due to lose his position of Lucasian professor in 1675, as he had not met the religious requirements of the post. However, a patent from the crown enabled him to retain his position without the obligation to take holy orders, removing a source of financial anxiety. Although Newton did not meet the requirements of religious orthodoxy, he maintained a deep interest in theology, the history of religion, biblical scripture and prophesies, the chronology of ancient kingdoms, and in numerology. He devoted much time also to the study of chemistry and the works of the alchemists.
It is popularly believed that it was under an apple tree at Woolthorpe in 1666 that Newton’s thoughts turned to the mathematics of gravity, an anecdote credited to Voltaire. It is possible that the tendency of an apple to fall to earth, no matter from what height, led Newton to the idea that the moon remained in orbit by falling toward the earth. He calculated the orbit of the moon on this basis, but found the result was in error. It subsequently turned out that the error was in the value for the size of the earth as it was known at that time.
Newton was elected to the Royal Society in 1672 at a meeting where his invention of the reflecting telescope was described. His first paper read before the society concerned the composition of white light, which had led him to recognize that different colors came to a focus at different distances from a lens. This led to his proposal for a mirror in a telescope as an alternative. He further concluded that color was an inherent property of different types of light, later publishing many papers on various aspects of optics.
When a more accurate value for the size of the earth was established by P. Picard in 1674, a recalculation by Newton validated his original idea of gravitation determining the moon’s orbit. In 1684, when a discussion about gravity arose, Newton was able to draw on previous calculations to explain that a planet moving in the sun’s gravity traveled in an ellipse. In 1685 he sent a long paper to the Royal Society, describing this finding and summarizing his theory of gravity, which became the core of his book entitled Philosophiae Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy), published in 1686 and 1687. This was the superb culmination of the remarkable insights into motion gained by Galileo earlier in the century. Galileo found that an object continues to move uniformly in a straight line if not acted upon by a force. If acted on by a force, it accelerates: the product of its mass and acceleration equals the force. To these two laws, Newton added a third: if one object exerts a force on another, the second exerts an equal and opposite force on the first. From these laws Newton was able to show how Kepler’s three laws of planetary motion arose, to calculate corrections to Kepler’s orbits, and to describe many other effects such as the precession of the equinoxes. Newton’s analysis of motion in the universe was unsurpassed until the advent of Einstein’s theory of relativity. It is still accurate for the vast majority of calculations of motion.
In 1687 Newton was drawn into politics. In direct violation of the law, King James II issued a mandate that a Benedictine monk should be admitted as master of arts at the University of Cambridge. The king had previously made John Massey dean of Christ Church College at Oxford university, on the sole justification that he was a Roman Catholic. Cambridge refused to follow a similar course, and sent a deputation that included Newton to the court. Newton’s role in this matter probably had some part in his election to parliament in 1688 as one of the representatives of the university. In the year he held his seat (after which parliament was dissolved) he brought forward a bill to settle the charters and privileges of the university of Cambridge. (In 1689 James II fled to France and William of Orange became king of England.)
In 1690 Newton wrote a Historical Account of Two Notable Corruptions of the Scripture. Shortly afterwards, he realized that it discredited two Biblical passages supporting the Christian doctrine of the trinity, and the consequences this could have in the England of his day alarmed him. The practice of burning heretics alive continued well into the Eighteenth Century. He therefore sought to have this book published in France. When a version in Latin, the common language of Europe, was proposed, Newton withdrew the manuscript from publication. In analyzing the mathematics of the solar system, Newton clearly places the sun at the center of planetary orbits; but he takes care to remark that others may envision the planets as revolving around the earth.
In his early fifties, Newton had received no promotion to higher ranks in the university, because he had not entered priestly orders. Eminent friends such as John Locke were uneasy that Newton should be so poorly recognized and forced to subsist on the meager compensation of a professor and fellow. When Charles Montague, a friend of Newton and another fellow of Trinity College, was appointed Chancellor of the Exchequer in 1694, he appointed Newton to be Warden of the Mint. This was at the time of the issue of new coins, to which Newton’s mathematical and chemical knowledge made important contributions. He wrote an official report on the coinage and developed an extensive table of assays of foreign coins. He was promoted to Master of the Mint at the turn of the century, and resigned his professorship and membership at Trinity in 1701. Newton was elected President the Royal Society in 1703, holding the office for 25 years. During this period he continued his mathematical studies and did much to set the course of experimental research in the 18th century. He worked on two revised editions of the Principia and in 1704 published Optics, which included a description of his calculus and its applications. He also made recommendations before parliament on alternative means of finding longitude accurately at sea, an important question at that time.
The extracts of Newton’s work given here are taken from the Principia, omitting the more mathematical aspects of that work. While these are the outstanding contribution that Newton made to our understanding of the universe, they do not lend themselves readily to summary.
1 Since the ancients (as we are told by Pappus), made great account of the science of mechanics in the investigation of natural things, and the moderns, laying aside substantial forms and occult qualities, have endeavored to subject the phenomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy. The ancients considered mechanics in a twofold respect; as rational, which proceeds accurately by demonstration; and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical; what is less so, is called mechanical. . . .
Our design is not concerned with arts but philosophy, and our subject is not manual but natural powers, we therefore consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive; and thus we offer this work as the mathematical principle of philosophy. For all the difficulty of philosophy seems to consist in this—from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena. To this end the general propositions in the first and second book are directed. In the third book we give an example of this in explaining the system of the world. For by the propositions mathematically demonstrated in the former books, we in the third derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, we deduce the motions of the planets, the comets, the moon, and the sea.
I wish we could derive the rest of the phenomena of nature by the same kind of reasoning from mechanical principles; for I suspect for many reasons that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other. These forces being unknown, philosophers have hitherto attempted the investigation of nature in vain; but I hope the principles here laid down will afford some light on either this or some truer method of philosophy.
The Author’s Preface
2 Definition I. The quantity of matter is the measure of the same, arising from its density and bulk together. Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction; and of all bodies that are by any causes whatever differently condensed. (I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies.) It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be described later.
3 Definition II. The quantity of motion [momentum] is the measure of the same, arising from the velocity and quantity of matter together. The motion of the whole is the sum of the motions of all the parts; and therefore in a body doubled in quantity, with equal velocity, the motion is doubled; with twice the velocity, it is quadrupled.
4 Definition III. The innate force of matter [inertia], is a power of resisting, by which every body, as much as in it lies, endeavors to persevere in its present state, whether it be of rest, or of moving uniformly forward in a straight line. This force is always proportional to the body whose force it is: and differs in no way from the inactivity of the mass, except in our way of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita may, by a most significant name, be called inertia, or the force of inactivity. . .
5 Definition IV. An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a straight line. This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires by its inertia only. Impressed forces are of different origins—as from percussion, from pressure, or from centripetal force.
6 Definition V. A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a center. Of this sort is gravity, by which bodies tend to the center of the earth; magnetism, by which iron tends to the loadstone; and that force, whatever it is, by which the planets are perpetually drawn aside from the motion in a straight line that they would otherwise pursue, and made to travel in curved orbits. A stone whirled about in a sling, endeavors to recede from the hand that turns it. By that endeavor, it stretches the sling, and that with a greater force as it is revolved with greater velocity. As soon as it is let go, it flies away. That force which opposes itself to this endeavor, and by which the sling perpetually draws back the stone towards the hand, and retains it in its orbit—because it is directed to the hand as the center of the orbit—I call the centripetal force. And the same thing is to be understood of all bodies revolved in any orbits. They all endeavor to recede from the centers of their orbits; and were it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would fly off in straight lines with an uniform motion.
7 A projectile, if it was not for the force of gravity, would not deviate towards the earth, but would go off from it in a straight line, and that with an uniform motion, if the resistance of the air was taken away. It is by its gravity that it is drawn aside perpetually from its rectilinear course, and made to deviate toward the earth, more or less, according to the force of its gravity, and the velocity of its motion. The less its gravity is, for the quantity of its matter, or the greater the velocity with which it is projected, the less will it deviate from a straight line, and the farther it will go. Suppose a leaden ball, projected from the top of a mountain by the force of gunpowder with a given velocity, and in a direction parallel to the horizon, is carried in a curved line to the distance of two miles before it falls to the ground. The same ball, in the absence of resistance from the air, if given twice or ten times the velocity, would fly twice or ten times as far. And by increasing the velocity, we may at pleasure increase the distance to which it might be projected, and reduce the curvature of the line that it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls. Or lastly, with even higher velocity it might never fall to the earth, but go forward into the celestial spaces, and proceed to infinity in its motion.
8 And after the same manner that a projectile, by the force of gravity, may be made to revolve in an orbit and go round the whole earth, the moon also, either by the force of gravity (if it is affected by gravity) or by any other force that impels it towards the earth, may be perpetually drawn aside towards the earth, out of the rectilinear or straight-line path that its innate force [inertia] would take it. It would be made to revolve in the orbit that it now describes. Nor could the moon without some such force be retained in its orbit. If this force were too small, it would not sufficiently turn the moon out of a rectilinear course: if it were too great, it would turn it too much, and draw down the moon from its orbit towards the earth. It is necessary, that the force be of the right amount, and it belongs to the mathematicians to find the force that may serve exactly to retain a body in a given orbit, with a given velocity; and vice versa, to determine the curvilinear path into which a body projected from a given place, with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force.
The quantity of any centripetal force [gravity] may be considered as of three kinds; absolute, accelerative, and motive
9 Definition VI. The absolute quantity of a centripetal force is the measure of the same proportional to the efficacy of the cause that propagates it from the center, through the spaces round about. Thus the magnetic force is greater in one load-stone and less in another according to their sizes and strength of intensity.
10 Definition VII. The accelerative quality of a centripetal force is the measure of the same, proportional to the velocity that it generates in a given time. Thus the force of the same load-stone is greater at a less distance, and less at a greater. Also the force of gravity is greater in valleys, less on tops of exceeding high mountains. The same force is even less (as shall be shown later) at greater distances from the body of the earth. On the other hand, at equal distances it is the same everywhere; because (taking away, or allowing for, the resistance of the air) it equally accelerates all falling bodies, whether heavy or light, great or small.
11 Definition VIII. The motive quality of a centripetal force is the measure of the same proportional to the motion which it generates in a given time. Thus the weight is greater in a greater body, less in a less body; and in the same body it is greater near to the earth, and less at remoter distances. This sort of quantity is the centripetency, or propensity of the whole body to move towards the center—or, as I may say, its weight. It is always measurable as the quantity of an equal and contrary force just sufficient to hinder the descent of the body.
These quantities of forces, we may, for brevity's sake, call by the names of motive, accelerative, and absolute forces; and, for distinction's sake, consider them, with respect to the bodies that tend to the center; to the places of those bodies; and to the center of force towards which they tend. That is to say, I refer the motive force to the body as an endeavor and propensity of the whole towards a center, arising from the propensities of the several parts taken together. I refer to the accelerative force to the place of the body, as a certain power or energy diffused from the center to all places around to move the bodies that are in them. And I refer the absolute force to the center, as endued with some cause, without which those motive forces would not be propagated through the spaces round about; whether that cause be some central body (such as is the load-stone, in the center of the magnetic force, or the earth in the center of the gravitating force), or anything else that does not yet appear. For I intend here to give only a mathematical notion of those forces, without considering their physical causes and places of origin.
12 Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be understood in the following discourse. I do not define time, space, place and motion, as. they are well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to perceptible objects. And it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.
I. Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration. Relative, apparent, and common time, is some perceptible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.
II. Absolute space in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some measurable dimension or measure of the absolute spaces; which our senses determine by its position with respect to bodies; and which is vulgarly taken for immovable space; such is the dimension of a subterranean, an areal, or a celestial space, determined by its position in respect of the earth. Absolute and relative space are the same in figure and magnitude; but they do not remain always numerically the same. For instance, if the earth moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes; at another time it will be another part of the same. Absolutely understood, it will be perpetually mutable.
III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space—not the situation, nor the external surface of the body. For the places of equal solids are always equal; but their surfaces, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same thing as the sum of the motions of the parts. That is, the translation of the whole, out of its place, is the same thing as the sum of the translations of the parts out of their places. Therefore the place of the whole is the same thing as the sum of the places of the parts, and for that reason, it is internal, and in the whole body.
13 IV. Absolute motion is the translation of a body from one absolute place into another; and relative motion, the translation from one relative place into another. Thus in a ship under sail, the relative place of a body is that part of the ship in which the body is situated; or that part of its interior that the body fills, and which therefore moves together with the ship. Relative rest is for the body to remain in the same part of the ship, or of its interior. But real, absolute rest is for the body to remain in the same part of that immovable space, in which the ship itself, its interior, and all that it contains, is moved. Thus, if the earth is really at rest, the body, which relatively rests in the ship, will really and absolutely move with the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body will arise, partly from the true motion of the earth, in immovable space and partly from the relative motion of the ship on the earth. And if the body moves also relatively in the ship, its true motion will arise, partly from the true motion of the earth, in immovable space, and partly from the relative motions as well of the ship on the earth and the body in the ship; and from these relative motions will arise the relative motion of the body on the earth. . .
As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they will be moved (if the expression may be allowed) out of themselves. For times and spaces are, as it were, the places as well of themselves as of all other things. All things are placed in time as to order of succession; and in space as to order of situation. It is from their essence or nature that they are places. And that the primary places of things should be moveable, is absurd. These are therefore the absolute places; and translations out of those places, are the only absolute motions.
But because the parts of space cannot be seen, or distinguished from one another by our senses, we therefore in their stead use perceptible measures of them. For from the positions and distances of things from any body considered as immovable, we define all places; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in philosophical discussions, we ought to step back from our senses, and consider things themselves, distinct from what are only perceptible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred.
Scholium to Definitions
14 Law I. 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. Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve the motions both progressive and circular for a much longer time.
15 Law II. The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the straight line in which that force is impressed. If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.
16 Law III. 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. Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the stretched rope, endeavoring to relax or unbend itself, will draw the horse as much towards the stone as it does the stone towards the horse. It will obstruct the progress of the one as much as it advances that of the other.
If a body impinge upon another and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change in its own motion towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.
Laws of Motion
17 Hitherto I have laid down such principles as have been received by mathematicians, and are confirmed by abundance of experiments. By the first two Laws and the first two Corollaries, Galileo discovered that the descent of bodies observed the duplicate ratio of the time, and that the motion of projectiles was in the curve of a parabola. Experience confirms both, except so far as these motions are a little retarded by the resistance of the air. When a body is falling, the uniform force of its gravity acting equally impresses, in equal particles of time, equal forces upon that body, and therefore generates equal velocities; and in the whole time impresses a whole force and generates a whole velocity proportional to the time. And the spaces described in proportional times are as the velocities and the times conjunctly; that is, in a duplicate ratio of the times. And when a body is thrown upwards, its uniform gravity impresses forces and removes velocities proportional to the times; and the times of ascending to the greatest heights are as the velocities to be removed, and those heights are as the velocities and the times conjointly, or in the duplicate ratio of the velocities. . .
Scholium to Laws of Motion
18 In the preceding Books [I & II] I have laid down the principles of philosophy, principles not philosophical but mathematical; namely, such as we may build our reasoning upon in philosophical inquiries. These principles are the laws and conditions of certain motions, and powers or forces, which chiefly have respect to philosophy. But, in case they should appear dry and barren in themselves, I have illustrated them here and there with some philosophical scholia. These give an account of such things as are of more general nature, which philosophy seems chiefly to be founded on—such as the density and the resistance of bodies, spaces void of all bodies, and the motion of light and sounds. It remains for me to demonstrate an outline of the system of the world based on the same principles. . . [in] the form of propositions (in the mathematical way). This should be read by only those who have first made themselves masters of the principles established in the preceding Books . . . [notably] the Definitions, the Laws of Motion and the first three Sections of the first Book. . .
Introduction to Book III
19 Rule I. We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. To this purpose the philosophers say that nature does nothing in vain, and more is in vain when less will serve; for nature is pleased with simplicity, and affects not the pomp of superfluous causes.
20 Rule II. Therefore to the same natural effects we must, as far as possible, assign the same causes.
As to respiration in a man and in a beast; the descent of stones in Europe and in America; the light of our culinary fire and of the sun; the reflection of light in the earth and in the planets.
21 Rule IlI. The qualities of bodies, which admit neither intension nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever. For since the qualities of bodies are only known to us by experiments, we are to hold for universal all such as universally agree with experiments; and such as are not liable to diminution can never be quite taken away. We are certainly not to relinquish the evidence of experiments for the sake of dreams and vain fictions of our own devising; nor are we to recede from the analogy of nature, which remains simple and always consonant to itself. We know the extension of bodies in no other way than by our senses, and our senses do not make us aware of the extent of all bodies. But because we perceive extension in all that affect the senses, we therefore ascribe it universally to all other bodies as well.
That many bodies are hard, we learn by experience. And because the hardness of the whole arises from the hardness of the parts, we therefore justly infer the hardness of the undivided particles not only of the bodies we feel but of all others. That all bodies are impenetrable, we gather not from reason, but from sensation. The bodies which we handle we find impenetrable, and thence conclude impenetrability to be an universal property of all bodies whatsoever. That all bodies are moveable, and endowed with certain powers (which we call inertia) of persevering in their motion, or in their rest, we only infer from the like properties observed in the bodies which we have seen. The extension, hardness, impenetrability, mobility, and inertia of the whole, result from the extension, hardness, impenetrability, mobility, and inertia of the parts. And so we conclude the least particles of all bodies to be also all extended, and hard and impenetrable, and moveable, and endowed with their proper inertia. And this is the foundation of all philosophy.
22 Moreover, that the divided but contiguous particles of bodies may be separated from one another, is matter of observation; and, in the particles that remain undivided, our minds are able to distinguish yet lesser parts, as is mathematically demonstrated. But whether the parts so distinguished, and not yet divided, may, by the powers of nature, be actually divided and separated from one another, we cannot certainly determine. Yet had we the proof of but one experiment that any undivided particle, in breaking a hard and solid body suffered a division, we might by virtue of this rule conclude that the undivided as well as the divided particles may be divided and actually separated to infinity.
23 Lastly, if it universally appears, by experiments and astronomical observations, that all bodies about the earth gravitate towards the earth, and that in proportion to the quantity of matter which they severally contain; that the moon likewise, according to the quantity of its matter, gravitates towards the earth; that, on the other hand, our sea gravitates towards the moon; and all the planets mutually one towards another; and the comets in like manner towards the sun; we must, in consequence of this rule, universally allow that all bodies whatsoever are endowed with a principle of mutual gravitation. For the argument from the appearances concludes with more force for the universal gravitation of all bodies than for their impenetrability; of which, among those in the celestial regions, we have no experiments, nor any manner of observation. Not that I affirm gravity to be essential to bodies: by their innate force I mean nothing but their inertia. This is immutable. Their gravity is diminished as they recede from the earth.
24 Rule IV. In experimental philosophy we are to look upon propositions collected by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.
This rule we must follow to ensure that the argument of induction may not be evaded by hypotheses.
Rules of Reasoning in Philosophy
25 That the five primary planets, Mercury, Venus, Mars, Jupiter, and Saturn, with their several orbits, encompass the sun. That Mercury and Venus revolve about the sun, is evident from their moon-like appearances. When they shine out with a full face, they are, in respect of us, beyond or above the sun. When they appear half full, they are about the same height on one side or other of the sun. When horned, they are below or between us and the sun; and they are sometimes, when directly under, seen like spots traversing the sun's disk. That Mars surrounds the sun, is as plain from its full face when near its conjunction with the sun, and from the gibbous figure which it shows when in its quadratures. And the same thing is demonstrable of Jupiter and Saturn, from their appearing full in all situations; for the shadows of their satellites that appear sometimes upon their disks make it plain that the light they shine with is not their own, but borrowed from the sun.
26 That the fixed stars being at rest, the periodic times of the five primary planets, and (whether of the sun about the earth, or) of the earth about the sun, are in, the sesquiplicate proportion of their mean distances from the sun. This proportion, first observed by Kepler is now received by all astronomers; for the periodic times are the same, and the dimensions of the orbits are the same, whether the sun revolves about the earth, or the earth about the sun. And as to the measures of the periodic times, all astronomers are agreed about them. But for the dimensions of the orbits, Kepler and Bullialdus, above all others, have determined them from observations with the greatest accuracy; and the mean distances corresponding to the periodic times differ but insensibly from those which they have assigned, and for the most part fall in between them . .
27 That the primary planets, by radii drawn to the earth, describe areas in no way proportional to the times; but that the areas which they describe by radii drawn to the sun are proportional to the times of description. For to the earth they appear sometimes direct, sometimes stationary, even sometimes retrograde. But from the sun they are always seen direct, and to proceed with a motion nearly uniform, that is to say, a little swifter in the perihelion and a little slower in the aphelion distances, so as to maintain an equality in the description of the areas. This a noted proposition among astronomers, and particularly demonstrable in Jupiter, from the eclipses of his satellites. By the help of these eclipses, as we have said, the heliocentric longitudes of that planet, and its distances from the sun, are determined.
28 Suppose several moons to revolve about the earth, as in the system of Jupiter or Saturn. The periodic times of these moons (by the argument of induction) would observe the same law which Kepler found to obtain among the planets. Therefore their centripetal forces would vary reciprocally as the squares of the distances from the center of the earth, by Prop. I, of this Book. Now if the lowest of these were very small, and were so near the earth as almost to touch the tops of the highest mountains, its centripetal force retaining it in its orbit would be very nearly equal to the weights of any terrestrial bodies found upon the tops of those mountains, as may be known by the foregoing computation. Therefore if the same little moon should be deserted by its centrifugal force that carries it through its orbit, and so be disabled from going onward therein, it would descend to the earth. It would do so with the same velocity as a heavy body does actually fall at the tops of those very mountains; because of the equality of the forces that oblige them both to descend. And if the force by which that lowest moon would descend were different from gravity, and if that moon were to gravitate towards the earth, as we find terrestrial bodies do upon the tops of mountains, it would then descend with twice the velocity, as being impelled by both these forces conspiring together. Therefore since both these forces, that is, the gravity of heavy bodies, and the centripetal forces of the moons, are directed to the center of the earth, and are similar and equal between themselves, they will (by Rule I and II) have one and the same cause. And therefore the force which retains the moon in its orbit is that very force which we commonly call gravity; because otherwise this little moon at the top of a mountain must either be without gravity, or fall twice as swiftly as heavy bodies are wont to do.
Scholium, Proposition IV. Theorem IV.
29 That the planets around Jupiter gravitate towards Jupiter; the planets around Saturn towards Saturn; the planets around the sun towards the sun; and by the forces of their gravity are drawn off from rectilinear motions, and constrained in curvilinear orbits. The revolutions of the planets around Jupiter, around Saturn, and of Mercury and Venus and the other planets around the sun, appear to be like the revolution of the moon about the earth. Therefore, by Rule II, these revolutions must be attributed to the same sort of causes, especially since it has been demonstrated that the forces upon which those revolutions depend tend to the centers of Jupiter, of Saturn, and of the sun. Furthermore, those forces, in receding from Jupiter, from Saturn, and from the sun, decrease in the same proportion, and according to the same law, as the force of gravity does in receding from the earth.
Corollary 1: There is, therefore, a power of gravity tending to all the planets; for, doubtless, Venus, Mercury, and the rest, are bodies of the same sort with Jupiter and Saturn. And since all attraction (by Law III) is mutual, Jupiter will therefore gravitate towards his own satellites, Saturn towards his, the earth towards the moon, and the sun towards all the primary planets.
Corollary 2: The force of gravity which tends to any one planet is reciprocally as the square of the distance of places from that planet's center.
Corollary 3: All the planets do mutually gravitate towards one another, by Corollary I and 2. And hence it is that Jupiter and Saturn, when near their conjunction, by their mutual attractions sensibly disturb each other's motions. So the sun disturbs the motions of the moon; and both sun and moon disturb our sea, as we shall later explain.
Scholium: The force which retains the celestial bodies in their orbits has been hitherto called centripetal force; but it being now made plain that it can be no other than a gravitating force, we shall hereafter call it gravity. For the cause of that centripetal force which retains the moon in its orbit will extend itself to all the planets by Rule I, II, and IV.
Proposition V. Theorem V
30 That all bodies gravitate towards every planet ; and that the weights of bodies towards acting towards the same planet, at equal distances from the center of the planet, are proportional to the quantities of matter which they severally contain. It has been, now of a long time, observed by others, that all sorts of heavy bodies (allowance being made for the inequality of retardation which they suffer from a small power of resistance in the air) descend to the earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy, by the help of pendulums. I experimented with gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I provided two wooden boxes, round and equal: I filled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the center of oscillation of the other. The boxes hanging by equal threads of 11 feet made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forward and backward, for a long time, with equal vibrations. And therefore the quantity of matter in the gold (by Corollary I and 6, Prop. XXIV, Book II) was to the quantity of matter in the wood as the action of the motive force (or vis motrix) upon all the gold to the action of the same upon all the wood; that is, as the weight of the one to the weight of the other. The like happened in the other bodies. By these experiments, in bodies of the same weight, I could manifestly have discovered a difference of matter less than the thousandth part of the whole, had any such been.
But, without all doubt, the nature of gravity towards the planets is the same as towards the earth. For, should we imagine our terrestrial bodies removed to the orbit of the moon, and there, together with the moon, deprived of all motion, to be let go, so as to fall together towards the earth, it is certain, from what we have demonstrated before, that, in equal times, they would describe equal spaces with the moon, and of consequence are to the moon, in quantity of matter, as their weights to its weight. Moreover, since the satellites of Jupiter perform their revolutions in times which observe the sesquiplicate proportion of their distances from Jupiter's center, their accelerative gravities towards Jupiter will be reciprocally as the squares of their distances from Jupiter's center; that is, equal, at equal distances. And, therefore, these satellites, if supposed to fall towards Jupiter from equal heights, would describe equal spaces in equal times, in like manner as heavy bodies do on our earth. And, by the same argument, if the circumsolar planets were supposed to be let fall at equal distances from the sun, they would, in their descent towards the sun, describe equal spaces in equal times. But forces which equally accelerate unequal bodies must be as those bodies: that is to say, the weights of the planets towards the sun must be as their quantities of matter.
Proposition VI. Theorem VI
31 Corollary 1: Hence the weights of bodies do not depend upon their forms and textures; for if the weights could be altered with the forms, they would be greater or less, according to the variety of forms, in actual matter; altogether against experience.
Corollary 2: Universally, all bodies about the earth gravitate towards the earth; and the weights of all, at equal distances from the earth's center, are as the quantities of matter which they severally contain. This is the quality of all bodies within the reach of our experiments; and therefore (by Rule III) to be affirmed of all bodies whatsoever. . .
Corollary 3: All spaces are not equally full; for if all spaces were equally full, then the specific gravity of the fluid which fills the region of the air on account of the extreme density of the matter, would fall nothing short of the specific gravity of quicksilver, or gold, or any other the most dense body; and, therefore, neither gold, nor any other body, could descend in air; for bodies do not descend in fluids, unless they are specifically heavier than the fluids. And if the quantity of matter in a given space can, by any rarefaction, be diminished, what should hinder a diminution to infinity?
Corollary 4: If all the solid particles of all bodies are of the same density, nor can be rarefied without pores, a void, space, or vacuum must be granted. By bodies of the same density, I mean those whose inertias are in the proportion of their bulks.
Corollary 5: The power of gravity is of a different nature from the power of magnetism; for the magnetic attraction is not as the matter attracted. Some bodies are attracted more by the magnet; others less; most bodies not at all. The power of magnetism in one and the same body may be increased and diminished. It is sometimes far stronger, for the quantity of matter, than the power of gravity; and in receding from the magnet decreases not in the duplicate but almost in the triplicate proportion of the distance, as nearly as I could judge from some rough observations.
Corollaries to Proposition VI. Theorem VI
32 That there is a power of gravity tending to all bodies, proportional to the several quantities of matter which they contain. That all the planets mutually gravitate one towards another, we have proved before; as well as that the force of gravity towards every one of them, considered apart, is reciprocally as the square of the distance of places from the center of the planet. And thence (by Prop. LXIX, Book I, and its Corollaries) it follows, that the gravity tending towards all the planets is proportional to the matter which they contain.
Moreover, since all the parts of any planet A gravitate towards any other planet B; and the gravity of every part is to the gravity of the whole as the matter of the part to the matter of the whole; and (by Law III) to every action corresponds an equal re-action; therefore the planet B will, on the other hand, gravitate towards all the parts of the planet A; and its gravity towards any one part will be to the gravity towards the whole as the matter of the part to the matter of the whole. Q.E.D.
Corollary 1: Therefore the force of gravity towards any whole planet arises from, and is compounded of, the forces of gravity towards all its parts. Magnetic and electric attractions afford us examples of this; for all attraction towards the whole arises from the attractions towards the several parts. The thing may be easily understood in gravity, if we consider a greater planet, as formed of a number of lesser planets, meeting together in one globe; for hence it would appear that the force of the whole must arise from the forces of the component parts. If it is objected, that, according to this law, all bodies with us must mutually gravitate one towards another, whereas no such gravitation anywhere appears, I answer, that since the gravitation towards these bodies is to the gravitation towards the whole earth as these bodies are to the whole earth, the gravitation towards each other them must be far less than to be observable to our senses.
Corollary 2: The force of gravity towards the several equal particles of any body is reciprocally as the square of the distance of places from the particles; as appears from Corollary 3, Prop. LXXIV, Book I.
Proposition VII. Theorem VII
33 That the motions of the planets in the heavens may subsist an exceedingly long time. . . .It is shown in the Scholium of Prop. XXII, Book II, that at the height of 200 miles above the earth the air is more rare than it is at the superficies of the earth in the ratio of 30 to 0.0000000000003999, or nearly 75,900,000,000,000 to 1. And hence the planet Jupiter, revolving in a medium of the same density with that superior air, would not lose by the resistance of the medium the 1,000,000th part of its motion in 1,000,000 years. In the spaces near the earth the resistance is produced only by the air, exhalations, and vapors. When these are carefully exhausted by an air pump from under a receiver, heavy bodies fall within the receiver with perfect freedom, and without the least sensible resistance; gold itself, and the lightest down, let fall together, will descend with equal velocity; and though they fall through a space of four, six, and eight feet, they will come to the bottom at the same time; as appears from experiments. And therefore the celestial regions being perfectly void of air and exhalations, the planets and comets meeting no sensible resistance in those spaces will continue their motions through them for an immense tract of time.
Proposition X. Theorem X.
34 That the center of the system of worlds [solar system] is immovable. This is acknowledged by all, while some contend that the earth, others that the sun, is fixed in that center. Let us see what may from hence follow.
35 That the common center of gravity of the earth, the sun, and all the planets, is immovable. For (by Corollary 4 of the Laws) that center either is at rest, or moves uniformly forward in a right line; but if that center moved, the center of the world would move also, against the hypothesis.
Proposition XI. Theorem XI.
36 That the sun is agitated by a perpetual motion, but never recedes far from the common center of gravity of all the planets. For since (by Corollary 2, Prop. VIII) the quantity of matter in the sun is to the quantity of matter in Jupiter as 1067 to 1; and the distance of Jupiter from the sun is to the semi-diameter of the sun in a proportion but a small matter greater, the common center of gravity of Jupiter and the sun will fall upon a point a little without the surface of the sun. By the same argument, since the quantity of matter in the sun is to the quantity of matter in Saturn as 3021 to 1, and the distance of Saturn from the sun is to the semi-diameter of the sun in a proportion but a small matter less, the common center of gravity of Saturn and the sun will fall upon a point a little within the surface of the sun. And, pursuing the principles of this computation, we should find that though the earth and all the planets were placed on one side of the sun, the distance of the common center of gravity of all from the center of the sun would scarcely amount to one diameter of the sun. In other cases, the distances of those centers are always less; and therefore, since that center of gravity is in perpetual rest, the sun, according to the various positions of the planets, must perpetually be moved every way, but will never recede far from that center.
Corollary: Hence the common center of gravity of the earth, the sun, and all the planets, is to be esteemed the center of the system of worlds; for since the earth, the sun, and all the planets, mutually gravitate one towards another, and are therefore, according to their powers of gravity, in perpetual agitation, as the Laws of Motion require, it is plain that their moveable centers cannot be taken for the immovable center of the world. If that body were to be placed in the center, towards which other bodies gravitate most (according to common opinion), that privilege ought to be allowed to the sun; but since the sun itself is moved, a fixed point is to be chosen from which the center of the sun recedes least, and from which it would recede yet less if the body of the sun were denser and greater, and therefore less apt to be moved.
Proposition XII. Theorem XII
37 . . .Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This much is certain—that it must proceed from a cause that penetrates to the very centers of the sun and planets, without suffering the least diminution of its force; that operates not according to the quantity of the surfaces of the particles upon which it acts (as mechanical causes usually do), but according to the quantity of the solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always in the duplicate proportion of the distances. Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body of the sun is composed; and in receding from the sun decreases accurately in the duplicate proportion of the distances as far as the orb of Saturn, as evidently appears from the quiescence of the aphelions of the planets; nay, and even to the remotest aphelions of the comets, if those aphelions are also quiescent. But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses. For whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction. Thus it was that the impenetrability, the mobility, and the impulsive force of bodies, and the laws of motion and of gravitation, were discovered. And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea. . .
Adapted from Philosophiae Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy), by Isaac Newton, translated by Andrew Motte.
Authors born between 1665 and 1700 CE
Introduction and adaptation of extracts Copyright © Rex Pay 2003