Sir Isaac Newton was born at Woolsthorpe, Lincolnshire, England,
Dec. 25, 1642, the son of a small landed proprietor. For the famous
episode of the falling apple, Voltaire, who admirably explained his
system for his countrymen, is responsible. It was in 1680 that
Newton discovered how to calculate the orbit of a body moving under
a central force, and showed that if the force varied as the inverse
square of the distance, the orbit would be an ellipse with the
centre of force in one focus. The great discovery, which made the
writing of his "Philosophiæ Naturalis Principia Mathematica"
possible, was that the attraction between two spheres is the same
as it would be if we supposed each sphere condensed to a point at
its centre. The book was published as a whole in 1687. Of its
author it was said by Lagrange that not only was he the greatest
genius that ever existed, but also the most fortunate, "for we
cannot find more than once a system of the world to establish."
Newton died on March 20, 1727.
Our design (writes Newton in his preface) not respecting arts but
philosophy, and our subject not manual but natural powers, we consider
those things which relate to gravity, levity, elastic force, the
resistance of fluids and the like forces, whether attractive or
impulsive; and, therefore, we offer this work as the mathematical
principles 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 from these forces to demonstrate the other phenomena, and
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 the
explication of 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.
Upon this subject I had (he says) composed the third book in a popular
method, that it might be read by many, but afterward, considering that
such as had not sufficiently entered into the principles could not
easily discern the strength of the consequences, nor lay aside the
prejudices to which they had been many years accustomed, therefore, to
prevent the disputes which might be raised upon such accounts, I chose
to reduce the substance of this book into the form of Propositions (in
the mathematical way). So that this third book is composed both "in
popular method" and in the form of mathematical propositions.
_Books I and II_
The principle of universal gravitation, namely, "That every particle of
matter is attracted by or gravitates to every other particle of matter
with a force inversely proportional to the squares of their distances,"
is the discovery which characterises the "Principia." This principle the
author deduced from the motion of the moon and the three laws of Kepler;
and these laws in turn Newton, by his greater law, demonstrated to be
true.
From the first law of Kepler, namely, the proportionality of the areas
to the times of their description, Newton inferred that the force which
retained the planet in its orbit was always directed to the sun. From
the second, namely, that every planet moves in an ellipse with the sun
as one of foci, he drew the more general inference that the force by
which the planet moves round that focus varies inversely as the square
of its distance therefrom. He demonstrated that a planet acted upon by
such a force could not move in any other curve than a conic section; and
he showed when the moving body would describe a circular, an elliptical,
a parabolic, or hyperbolic orbit. He demonstrated, too, that this force
or attracting, gravitating power resided in even the least particle; but
that in spherical masses it operates as if confined to their centres, so
that one sphere or body will act upon another sphere or body with a
force directly proportional to the quantity of matter and inversely as
the square of the distance between their centres, and that their
velocities of mutual approach will be in the inverse ratio of their
quantities of matter. Thus he outlined the universal law.
_The System of the World_
It was the ancient opinion of not a few (writes Newton in Book III.) in
the earliest ages of philosophy that the fixed stars stood immovable in
the highest parts of the world; that under the fixed stars the planets
were carried about the sun; that the earth, as one of the planets,
described an annual course about the sun, while, by a diurnal motion, it
was in the meantime revolved about its own axis; and that the sun, as
the common fire which served to warm the whole, was fixed in the centre
of the universe. It was from the Egyptians that the Greeks derived their
first, as well as their soundest notions of philosophy. It is not to be
denied that Anaxagoras, Democritus and others would have it that the
earth possessed the centre of the world, but it was agreed on both sides
that the motions of the celestial bodies were performed in spaces
altogether free and void of resistance. The whim of solid orbs was[1] of
later date, introduced by Endoxus, Calippus and Aristotle, when the
ancient philosophy began to decline.
As it was the unavoidable consequence of the hypothesis of solid orbs
while it prevailed that the comets must be thrust down below the moon,
so no sooner had the late observations of astronomers restored the
comets to their ancient places in the higher heavens than these
celestial spaces were at once cleared of the encumbrance of solid orbs,
which by these observations were broken to pieces and discarded for
ever.
Whence it was that the planets came to be retained within any certain
bounds in these free spaces, and to be drawn off from the rectilinear
courses, which, left to themselves, they should have pursued, into
regular revolutions in curvilinear orbits, are questions which we do not
know how the ancients explained; and probably it was to give some sort
of satisfaction to this difficulty that solid orbs were introduced.
The later philosophers pretend to account for it either by the action of
certain vortices, as Kepler and Descartes, or by some other principle of
impulse or attraction, for it is most certain that these effects must
proceed from the action of some force or other. This we will call by the
general name of a centripetal force, as it is a force which is directed
to some centre; and, as it regards more particularly a body in that
centre, we call it circum-solar, circum-terrestrial, circum-jovial.
_Centre-Seeking Forces_
That by means of centripetal forces the planets may be retained in
certain orbits we may easily understand if we consider the motions of
projectiles, for a stone projected is by the pressure of its own weight
forced out of the rectilinear path, which, by the projection alone, it
should have pursued, and made to describe a curve line in the air; and
through that crooked way is at last brought down to the ground, and the
greater the velocity is with which it is projected the further it goes
before it falls to earth. We can, therefore, suppose the velocity to be
so increased that it would describe an arc of 1, 2, 5, 10, 100, 1,000
miles before it arrived at the earth, till, at last, exceeding the
limits of the earth, it should pass quite by it without touching it.
And because the celestial motions are scarcely retarded by the little or
no resistance of the spaces in which they are performed, to keep up the
parity of cases, let us suppose either that there is no air about the
earth or, at least, that it is endowed with little or no power of
resisting.
And since the areas which by this motion it describes by a radius drawn
to the centre of the earth have previously been shown to be proportional
to the times in which they are described, its velocity when it returns
to the point from which it started will be no less than at first; and,
retaining the same velocity, it will describe the same curve over and
over by the same law.
But if we now imagine bodies to be projected in the directions of lines
parallel to the horizon from greater heights, as from 5, 10, 100, 1,000
or more miles, or, rather, as many semi-diameters of the earth, those
bodies, according to their different velocity and the different force of
gravity in different heights, will describe arcs either concentric with
the earth or variously eccentric, and go on revolving through the
heavens in those trajectories just as the planets do in their orbs.
As when a stone is projected obliquely, the perpetual deflection thereof
towards the earth is a proof of its gravitation to the earth no less
certain than its direct descent when suffered to fall freely from rest,
so the deviation of bodies moving in free spaces from rectilinear paths
and perpetual deflection therefrom towards any place, is a sure
indication of the existence of some force which from all quarters impels
those bodies towards that place.
That there are centripetal forces actually directed to the bodies of
the sun, of the earth, and other planets, I thus infer.
The moon revolves about our earth, and by radii drawn to its centre
describes areas nearly proportional to the times in which they are
described, as is evident from its velocity compared with its apparent
diameter; for its motion is slower when its diameter is less (and
therefore its distance greater), and its motion is swifter when its
diameter is greater.
The revolutions of the satellites of Jupiter about the planet are more
regular; for they describe circles concentric with Jupiter by equable
motions, as exactly as our senses can distinguish.
And so the satellites of Saturn are revolved about this planet with
motions nearly circular and equable, scarcely disturbed by any
eccentricity hitherto observed.
That Venus and Mercury are revolved about the sun is demonstrable from
their moon-like appearances. And Venus, with a motion almost uniform,
describes an orb nearly circular and concentric with the sun. But
Mercury, with a more eccentric motion, makes remarkable approaches to
the sun and goes off again by turns; but it is always swifter as it is
near to the sun, and therefore by a radius drawn to the sun still
describes areas proportional to the times.
Lastly, that the earth describes about the sun, or the sun about the
earth, by a radius from one to the other, areas exactly proportional to
the times is demonstrable from the apparent diameter of the sun compared
with its apparent motion.
These are astronomical experiments; from which it follows that there are
centripetal forces actually directed to the centres of the earth, of
Jupiter, of Saturn, and of the sun.[2]
That these forces decrease in the duplicate proportion of the distances
from the centre of every planet appears by Cor. vi., Prop. iv., Book
I.[3] for the periodic times of the satellites of Jupiter are one to
another in the sesquiplicate proportion of their distances from the
centre of this planet. Cassini assures us that the same proportion is
observed in the circum-Saturnal planets. In the circum-solar planets
Mercury and Venus, the same proportional holds with great accuracy.
That Mars is revolved about the sun is demonstrated from the phases
which it shows and the proportion of its apparent diameters; for from
its appearing full near conjunction with the sun and gibbous in its
quadratures,[4] it is certain that it travels round the sun. And since
its diameter appears about five times greater when in opposition to the
sun than when in conjunction therewith, and its distance from the earth
is reciprocally as its apparent diameter, that distance will be about
five times less when in opposition to than when in conjunction with the
sun; but in both cases its distance from the sun will be nearly about
the same with the distance which is inferred from its gibbous appearance
in the quadratures. And as it encompasses the sun at almost equal
distances, but in respect of the earth is very unequally distant, so by
radii drawn to the sun it describes areas nearly uniform; but by radii
drawn to the earth it is sometimes swift, sometimes stationary, and
sometimes retrograde.
That Jupiter in a higher orbit than Mars is likewise revolved about the
sun with a motion nearly equable as well in distance as in the areas
described, I infer from Mr. Flamsted's observations of the eclipses of
the innermost satellite; and the same thing may be concluded of Saturn
from his satellite by the observations of Mr. Huyghens and Mr. Halley.
If Jupiter was viewed from the sun it would never appear retrograde or
stationary, as it is seen sometimes from the earth, but always to go
forward with a motion nearly uniform. And from the very great inequality
of its apparent geocentric motion we infer--as it has been previously
shown that we may infer--that the force by which Jupiter is turned out
of a rectilinear course and made to revolve in an orbit is not directed
to the centre of the earth. And the same argument holds good in Mars and
in Saturn. Another centre of these forces is, therefore, to be looked
for, about which the areas described by radii intervening may be
equable; and that this is the sun, we have proved already in Mars and
Saturn nearly, but accurately enough in Jupiter.
The distances of the planets from the sun come out the same whether,
with Tycho, we place the earth in the centre of the system, or the sun
with Copernicus; and we have already proved that, these distances are
true in Jupiter. Kepler and Bullialdus have with great care determined
the distances of the planets from the sun, and hence it is that their
tables agree best with the heavens. And in all the planets, in Jupiter
and Mars, in Saturn and the earth, as well as in Venus and Mercury, the
cubes of their distances are as the squares of their periodic times;
and, therefore, the centripetal circum-solar force throughout all the
planetary regions decreases in the duplicate proportion of the distances
from the sun. Neglecting those little fractions which may have arisen
from insensible errors of observation, we shall always find the said
proportion to hold exactly; for the distances of Saturn, Jupiter, Mars,
the Earth, Venus, and Mercury from the sun, drawn from the observations
of astronomers, are (Kepler) as the numbers 951,000, 519,650, 152,350,
100,000, 70,000, 38,806; or (Bullialdus) as the numbers 954,198,
522,520, 152,350, 100,000, 72,398, 38,585; and from the periodic times
they come out 953,806, 520,116, 152,399, 100,000, 72,333, 38,710. Their
distances, according to Kepler and Bullialdus, scarcely differ by any
sensible quantity, and where they differ most the differences drawn from
the periodic times fall in between them.
_Earth as a Centre_
That the circum-terrestrial force likewise decreases in the duplicate
proportion of the distances, I infer thus:
The mean distance of the moon from the centre of the earth is, we may
assume, sixty semi-diameters of the earth; and its periodic time in
respect of the fixed stars 27 days 7 hr. 43 min. Now, it has been shown
in a previous book that a body revolved in our air, near the surface of
the earth supposed at rest, by means of a centripetal force which should
be to the same force at the distance of the moon in the reciprocal
duplicate proportion of the distances from the centre of the earth, that
is, as 3,600 to 1, would (secluding the resistance of the air) complete
a revolution in 1 hr. 24 min. 27 sec.
Suppose the circumference of the earth to be 123,249,600 Paris feet,
then the same body deprived of its circular motion and falling by the
impulse of the same centripetal force as before would in one second of
time describe 15-1/12 Paris feet. This we infer by a calculus formed
upon Prop. xxxvi. ("To determine the times of the descent of a body
falling from a given place"), and it agrees with the results of Mr.
Huyghens's experiments of pendulums, by which he demonstrated that
bodies falling by all the centripetal force with which (of whatever
nature it is) they are impelled near the surface of the earth do in one
second of time describe 15-1/12 Paris feet.
But if the earth is supposed to move, the earth and moon together will
be revolved about their common centre of gravity. And the moon (by Prop,
lx.) will in the same periodic time, 27 days 7 hr. 43 min., with the
same circum-terrestrial force diminished in the duplicate proportion of
the distance, describe an orbit whose semi-diameter is to the
semi-diameter of the former orbit, that is, to the sixty semi-diameters
of the earth, as the sum of both the bodies of the earth and moon to the
first of two mean proportionals between this sum and the body of the
earth; that is, if we suppose the moon (on account of its mean apparent
diameter 31-1/2 min.) to be about 1/42 of the earth, as 43 to (42 +
42^2)^1/3 or as about 128 to 127. And, therefore, the semi-diameter of
the orbit--that is, the distance of the centres of the moon and
earth--will in this case be 60-1/2 semi-diameters of the earth, almost
the same with that assigned by Copernicus; and, therefore, the duplicate
proportion of the decrement of the force holds good in this distance.
(The action of the sun is here disregarded as inconsiderable.)
This proportion of the decrement of the forces is confirmed from the
eccentricity of the planets, and the very slow motion of their apsides;
for in no other proportion, it has been established, could the
circum-solar planets once in every revolution descend to their least,
and once ascend to their greatest distance from the sun, and the places
of those distances remain immovable. A small error from the duplicate
proportion would produce a motion of the apsides considerable in every
revolution, but in many enormous.
_The Tides_
While the planets are thus revolved in orbits about remote centres, in
the meantime they make their several rotations about their proper axes:
the sun in 26 days, Jupiter in 9 hr. 56 min., Mars in 24-2/3 hr., Venus
in 23 hr., and in like manner is the moon revolved about its axis in 27
days 7 hr. 43 min.; so that this diurnal motion is equal to the mean
motion of the moon in its orbit; upon which account the same face of the
moon always respects the centre about which this mean motion is
performed--that is, the exterior focus of the moon's orbit nearly.
By reason of the diurnal revolutions of the planets the matter which
they contain endeavours to recede from the axis of this motion; and
hence the fluid parts, rising higher towards the equator than about the
poles, would lay the solid parts about the equator under water if those
parts did not rise also; upon which account the planets are something
thicker about the equator than about the poles.
And from the diurnal motion and the attractions of the sun and moon our
sea ought twice to rise and twice to fall every day, as well lunar as
solar. But the two motions which the two luminaries raise will not
appear distinguished but will make a certain mixed motion. In the
conjunction or opposition of the luminaries their forces will be
conjoined and bring on the greatest flood and ebb. In the quadratures
the sun will raise the waters which the moon depresseth and depress the
waters which the moon raiseth; and from the difference of their forces
the smallest of all tides will follow.
But the effects of the lumniaries depend upon their distances from the
earth, for when they are less distant their effects are greater and when
more distant their effects are less, and that in the triplicate
proportion of their apparent diameters. Therefore it is that the sun in
winter time, being then in its perigee, has a greater effect, whether
added to or subtracted from that of the moon, than in the summer season,
and every month the moon, while in the perigee raiseth higher tides than
at the distance of fifteen days before or after when it is in its
apogee.
The fixed stars being at such vast distances from one another, can
neither attract each other sensibly nor be attracted by our sun.
_Comets_
There are three hypotheses about comets. For some will have it that they
are generated and perish as often as they appear and vanish; others that
they come from the regions of the fixed stars, and are near by us in
their passage through the sytem of our planets; and, lastly, others that
they are bodies perpetually revolving about the sun in very eccentric
orbits.
In the first case, the comets, according to their different velocities,
will move in conic sections of all sorts; in the second they will
describe hyperbolas; and in either of the two will frequent
indifferently all quarters of the heavens, as well those about the poles
as those towards the ecliptic; in the third their motions will be
performed in eclipses very eccentric and very nearly approaching to
parabolas. But (if the law of the planets is observed) their orbits will
not much decline from the plane of the ecliptic; and, so far as I could
hitherto observe, the third case obtains; for the comets do indeed
chiefly frequent the zodiac, and scarcely ever attain to a heliocentric
latitude of 40 degrees. And that they move in orbits very nearly
parabolical, I infer from their velocity; for the velocity with which a
parabola is described is everywhere to the velocity with which a comet
or planet may be revolved about the sun in a circle at the same
distance in the subduplicate ratio of 2 to 1; and, by my computation,
the velocity of comets is found to be much about the same. I examined
the thing by inferring nearly the velocities from the distances, and the
distances both from the parallaxes and the phenomena of the tails, and
never found the errors of excess or defect in the velocities greater
than what might have arisen from the errors in the distances collected
after that manner.