UNIVERSAL GRAVITATION.
“Nature and Nature’s laws lay hid in night;
God said ‘Let Newton be!’ and all was light.”
POPE.
164. Newton’s life may be conveniently divided into three portions.
First came 22 years (1643-1665) of boyhood and undergraduate life;
then followed his great productive period, of almost exactly the same
length, culminating in the publication of the _Principia_ in 1687;
while the rest of his life (1687-1727), which lasted nearly as long as
the other two periods put together, was largely occupied with official
work and studies of a non-scientific character, and was marked by no
discoveries ranking with those made in his middle period, though some
of his earlier work received important developments and several new
results of decided interest were obtained.
165. Isaac Newton was born at Woolsthorpe, near Grantham, in
Lincolnshire, on January 4th, 1643;[100] this was very nearly a year
after the death of Galilei, and a few months after the beginning of
our Civil Wars. His taste for study does not appear to have developed
very early in life, but ultimately became so marked that, after some
unsuccessful attempts to turn him into a farmer, he was entered at
Trinity College, Cambridge, in 1661.
Although probably at first rather more backward than most
undergraduates, he made extremely rapid progress in mathematics and
allied subjects, and evidently gave his teachers some trouble by
the rapidity with which he absorbed what little they knew. He met
with Euclid’s _Elements of Geometry_ for the first time while an
undergraduate, but is reported to have soon abandoned it as being “a
trifling book,” in favour of more advanced reading. In January 1665
graduated in the ordinary course as Bachelor of Arts.
166. The external events of Newton’s life during the next 22 years may
be very briefly dismissed. He was elected a Fellow in 1667, became
M.A. in due course in the following year, and was appointed Lucasian
Professor of Mathematics, in succession to his friend Isaac Barrow, in
1669. Three years later he was elected a Fellow of the recently founded
Royal Society. With the exception of some visits to his Lincolnshire
home, he appears to have spent almost the whole period in quiet study
at Cambridge, and the history of his life is almost exclusively the
history of his successive discoveries.
167. His scientific work falls into three main groups, astronomy
(including dynamics), optics, and pure mathematics. He also spent a
good deal of time on experimental work in chemistry, as well as on
heat and other branches of physics, and in the latter half of his life
devoted much attention to questions of chronology and theology; in none
of these subjects, however, did he produce results of much importance.
168. In forming an estimate of Newton’s genius it is of course
important to bear in mind the range of subjects with which he dealt;
from our present point of view, however, his mathematics only presents
itself as a tool to be used in astronomical work; and only those of
his optical discoveries which are of astronomical importance need be
mentioned here. In 1668 he constructed a =reflecting telescope=, that
is, a telescope in which the rays of light from the object viewed
are concentrated by means of a curved mirror instead of by a lens,
as in the =refracting telescopes= of Galilei and Kepler. Telescopes
on this principle, differing however in some important particulars
from Newton’s, had already been described in 1663 by _James Gregory_
(1638-1675), with whose ideas Newton was acquainted, but it does
not appear that Gregory had actually made an instrument. Owing to
mechanical difficulties in construction, half a century elapsed before
reflecting telescopes were made which could compete with the best
refractors of the time, and no important astronomical discoveries were
made with them before the time of William Herschel (chapter XII.), more
than a century after the original invention.
Newton’s discovery of the effect of a prism in resolving a beam of
white light into different colours is in a sense the basis of the
method of spectrum analysis (chapter XIII., § 299), to which so many
astronomical discoveries of the last 40 years are due.
169. The ideas by which Newton is best known in each of his three great
subjects—gravitation, his theory of colours, and fluxions—seem to have
occurred to him and to have been partly thought out within less than
two years after he took his degree, that is before he was 24. His own
account—written many years afterwards—gives a vivid picture of his
extraordinary mental activity at this time:—
“In the beginning of the year 1665 I found the method of
approximating Series and the Rule for reducing any dignity of any
Binomial into such a series. The same year in May I found the method
of tangents of Gregory and Slusius, and in November had the direct
method of Fluxions, and the next year in January had the Theory of
Colours, and in May following I had entrance into the inverse method
of Fluxions. And the same year I began to think of gravity extending
to the orb of the Moon, and having found out how to estimate the force
with which [a] globe revolving within a sphere presses the surface of
the sphere, from Kepler’s Rule of the periodical times of the Planets
being in a sesquialterate proportion of their distances from the
centers of their orbs I deduced that the forces which keep the Planets
in their orbs must [be] reciprocally as the squares of their distances
from the centers about which they revolve: and thereby compared the
force requisite to keep the Moon in her orb with the force of gravity
at the surface of the earth, and found them answer pretty nearly. All
this was in the two plague years of 1665 and 1666, for in those days I
was in the prime of my age for invention, and minded Mathematicks and
Philosophy more than at any time since.”[101]
170. He spent a considerable part of this time (1665-1666) at
Woolsthorpe, on account of the prevalence of the plague.
The well-known story, that he was set meditating on gravity by the
fall of an apple in the orchard, is based on good authority, and is
perfectly credible in the sense that the apple may have reminded him at
that particular time of certain problems connected with gravity. That
the apple seriously suggested to him the existence of the problems or
any key to their solution is wildly improbable.
Several astronomers had already speculated on the “cause” of the known
motions of the planets and satellites; that is they had attempted to
exhibit these motions as consequences of some more fundamental and
more general laws. Kepler, as we have seen (chapter VII., § 150), had
pointed out that the motions in question should not be considered as
due to the influence of mere geometrical points, such as the centres
of the old epicycles, but to that of other bodies; and in particular
made some attempt to explain the motion of the planets as due to a
special kind of influence emanating from the sun. He went, however,
entirely wrong by looking for a force to keep up the motion of the
planets and as it were push them along. Galilei’s discovery that the
motion of a body goes on indefinitely unless there is some cause at
work to alter or stop it, at once put a new aspect on this as on other
mechanical problems; but he himself did not develop his idea in this
particular direction. _Giovanni Alfonso Borelli_ (1608-1679), in a book
on Jupiter’s satellites published in 1666, and therefore about the
time of Newton’s first work on the subject, pointed out that a body
revolving in a circle (or similar curve) had a tendency to recede from
the centre, and that in the case of the planets this might be supposed
to be counteracted by some kind of attraction towards the sun. We have
then here the idea— in a very indistinct form certainly—that the
motion of a planet is to be explained, not by a force acting in the
direction in which it is moving, but by a force directed towards the
sun, that is about at right angles to the direction of the planet’s
motion. Huygens carried this idea much further—though without special
reference to astronomy—and obtained (chapter VIII., § 158) a numerical
measure for the tendency of a body moving in a circle to recede from
the centre, a tendency which had in some way to be counteracted if the
body was not to fly away. Huygens published his work in 1673, some
years after Newton had obtained his corresponding result, but before
he had published anything; and there can be no doubt that the two men
worked quite independently.
[Illustration: FIG. 70.—Motion in a circle.]
171. Viewed as a purely general question, apart from its astronomical
applications, the problem may be said to be to examine under what
conditions a body can revolve with uniform speed in a circle.
Let A represent the position at a certain instant of a body which is
revolving with uniform speed in a circle of centre O. Then at this
instant the body is moving in the direction of the tangent A _a_ to the
circle. Consequently by Galilei’s First Law (chapter VI., §§ 130, 133),
if left to itself and uninfluenced by any other body, it would continue
to move with the same speed and in the same direction, _i.e._ along the
line A _a_, and consequently would be found after some time at such a
point as _a_. But actually it is found to be at B on the circle. Hence
some influence must have been at work to bring it to B instead of to
_a_. But B is nearer to the centre of the circle than _a_ is; hence
some influence must be at work tending constantly to draw the body
towards O, or counteracting the tendency which it has, in virtue of the
First Law of Motion, to get farther and farther away from O. To express
either of these tendencies numerically we want a more complex idea
than that of velocity or rate of motion, namely =acceleration= or rate
of change of velocity, an idea which Galilei added to science in his
discussion of the law of falling bodies (chapter VI., §§ 116, 133). A
falling body, for example, is moving after one second with the velocity
of about 32 feet per second, after two seconds with the velocity of
64, after three seconds with the velocity of 96, and so on; thus in
every second it gains a downward velocity of 32 feet per second; and
this may be expressed otherwise by saying that the body has a downward
acceleration of 32 feet per second per second. A further investigation
of the motion in a circle shews that the motion is completely explained
if the moving body has, in addition to its original velocity, an
acceleration of a certain magnitude _directed towards the centre of
the circle_. It can be shewn further that the acceleration may be
numerically expressed by taking the square of the velocity of the
moving body (expressed, say, in feet per second), and dividing this by
the radius of the circle in feet. If, for example, the body is moving
in a circle having a radius of four feet, at the rate of ten feet a
second, then the acceleration towards the centre is (10 × 10)∕4 = 25
feet per second per second.
These results, with others of a similar character, were first published
by Huygens—not of course precisely in this form—in his book on the
_Pendulum Clock_ (chapter VIII., § 158); and discovered independently
by Newton in 1666.
If then a body is seen to move in a circle, its motion becomes
intelligible if some other body can be discovered which produces this
acceleration. In a common case, such as when a stone is tied to a
string and whirled round, this acceleration is produced by the string
which pulls the stone; in a spinning-top the acceleration of the outer
parts is produced by the forces binding them on to the inner part, and
so on.
172. In the most important cases of this kind which occur in astronomy,
a planet is known to revolve round the sun in a path which does not
differ much from a circle. If we assume for the present that the path
is actually a circle, the planet must have an acceleration towards the
centre, and it is possible to attribute this to the influence of the
central body, the sun. In this way arises the idea of attributing to
the sun the power of influencing in some way a planet which revolves
round it, so as to give it an acceleration towards the sun; and the
question at once arises of how this “influence” differs at different
distances. To answer this question Newton made use of Kepler’s Third
Law (chapter VII., § 144). We have seen that, according to this
law, the squares of the times of revolution of any two planets are
proportional to the cubes of their distances from the sun; but the
velocity of the planet may be found by dividing the length of the
path it travels in its revolution round the sun by the time of the
revolution, and this length is again proportional to the distance of
the planet from the sun. Hence the velocities of the two planets are
proportional to their distances from the sun, divided by the times
of revolution, and consequently the squares of the velocities are
proportional to the squares of the distances from the sun divided
by the squares of the times of revolution. Hence, by Kepler’s law,
the squares of the velocities are proportional to the squares of the
distances divided by the cubes of the distances, that is the squares
of the velocities are _inversely_ proportional to the distances, the
more distant planet having the less velocity and _vice versa_. Now by
the formula of Huygens the acceleration is measured by the square of
the velocity divided by the radius of the circle (which in this case
is the distance of the planet from the sun). The accelerations of the
two planets towards the sun are therefore inversely proportional to the
distances each multiplied by itself, that is are inversely proportional
to the squares of the distances. Newton’s first result therefore is:
that the motions of the planets—regarded as moving in circles, and in
strict accordance with Kepler’s Third Law—can be explained as due to
the action of the sun, if the sun is supposed capable of producing on a
planet an acceleration towards the sun itself which is proportional to
the inverse square of its distance from the sun; _i.e._ at twice the
distance it is 1∕4 as great, at three times the distance 1∕9 as great,
at ten times the distance 1∕100 as great, and so on.
The argument may perhaps be made clearer by a numerical example. In
round numbers Jupiter’s distance from the sun is five times as great as
that of the earth, and Jupiter takes 12 years to perform a revolution
round the sun, whereas the earth takes one. Hence Jupiter goes in
12 years five times as far as the earth goes in one, and Jupiter’s
velocity is therefore about 5∕12 that of the earth’s, or the two
velocities are in the ratio of 5 to 12; the squares of the velocities
are therefore as 5 × 5 to 12 × 12, or as 25 to 144. The accelerations
of Jupiter and of the earth towards the sun are therefore as 25 ÷ 5
to 144, or as 5 to 144; hence Jupiter’s acceleration towards the sun
is about 1∕28 earth, and if we had taken more accurate figures this
fraction would have come out more nearly 1∕25. Hence at five times the
distance the acceleration is 25 times less.
This =law of the inverse square=, as it may be called, is also the law
according to which the light emitted from the sun or any other bright
body varies, and would on this account also be not unlikely to suggest
itself in connection with any kind of influence emitted from the sun.
173. The next step in Newton’s investigation was to see whether the
motion of the moon round the earth could be explained in some similar
way. By the same argument as before, the moon could be shewn to have
an acceleration towards the earth. Now a stone if let drop falls
downwards, that is in the direction of the centre of the earth, and, as
Galilei had shewn (chapter VI., § 133), this motion is one of uniform
acceleration; if, in accordance with the opinion generally held at
that time, the motion is regarded as being due to the earth, we may
say that the earth has the power of giving an acceleration towards
its own centre to bodies near its surface. Newton noticed that this
power extended at any rate to the tops of mountains, and it occurred
to him that it might possibly extend as far as the moon and so give
rise to the required acceleration. Although, however, the acceleration
of falling bodies, as far as was known at the time, was the same
for terrestrial bodies wherever situated, it was probable that at
such a distance as that of the moon the acceleration caused by the
earth would be much less. Newton assumed as a working hypothesis that
the acceleration diminished according to the same law which he had
previously arrived at in the case of the sun’s action on the planets,
that is that the acceleration produced by the earth on any body is
inversely proportional to the square of the distance of the body from
the centre of the earth.
It may be noticed that a difficulty arises here which did not present
itself in the corresponding case of the planets. The distances of the
planets from the sun being large compared with the size of the sun, it
makes little difference whether the planetary distances are measured
from the centre of the sun or from any other point in it. The same is
true of the moon and earth; but when we are comparing the action of the
earth on the moon with that on a stone situated on or near the ground,
it is clearly of the utmost importance to decide whether the distance
of the stone is to be measured from the nearest point of the earth, a
few feet off, from the centre of the earth, 4000 miles off, or from
some other point. Provisionally at any rate Newton decided on measuring
from the centre of the earth.
It remained to verify his conjecture in the case of the moon by a
numerical calculation; this could easily be done if certain things were
known, _viz._ the acceleration of a falling body on the earth, the
distance of the surface of the earth from its centre, the distance of
the moon, and the time taken by the moon to perform a revolution round
the earth. The first of these was possibly known with fair accuracy;
the last was well known; and it was also known that the moon’s distance
was about 60 times the radius of the earth. How accurately Newton at
this time knew the size of the earth is uncertain. Taking moderately
accurate figures, the calculation is easily performed. In a month of
about 27 days the moon moves about 60 times as far as the distance
round the earth; that is she moves about 60 × 24,000 miles in 27 days,
which is equivalent to about 3,300 feet per second. The acceleration
of the moon is therefore measured by the square of this, divided by
the distance of the moon (which is 60 times the radius of the earth,
or 20,000,000 feet); that is, it is (3,300 × 3,300)∕(60 × 20,000,000),
which reduces to about 1∕110. Consequently, if the law of the inverse
square holds, the acceleration of a falling body at the surface of the
earth, which is 60 times nearer to the centre than the moon is, should
be (60 × 60)∕110, or between 32 and 33; but the actual acceleration
of falling bodies is rather more than 32. The argument is therefore
satisfactory, and Newton’s hypothesis is so far verified.
The analogy thus indicated between the motion of the moon round the
earth and the motion of a falling stone may be illustrated by a
comparison, due to Newton, of the moon to a bullet shot horizontally
out of a gun from a high place on the earth. Let the bullet start from
B in fig. 71, then moving at first horizontally it will describe a
curved path and reach the ground at a point such as C, at some distance
from the point A, vertically underneath its starting-point. If it were
shot out with a greater velocity, its path at first would be flatter
and it would reach the ground at a point C′ beyond C; if the velocity
were greater still, it would reach the ground at C″ or at C‴; and
it requires only a slight effort of the imagination to conceive that,
with a still greater velocity to begin with, it would miss the earth
altogether and describe a circuit round it, such as B D E. This is
exactly what the moon does, the only difference being that the moon is
at a much greater distance than we have supposed the bullet to be, and
that her motion has not been produced by anything analogous to the gun;
but the motion being once there it is immaterial how it was produced
or whether it was ever produced in the past. We may in fact say of the
moon “that she is a falling body, only she is going so fast and is so
far off that she falls quite round to the other side of the earth,
instead of hitting it; and so goes on for ever.”[102]
[Illustration: FIG. 71.—The moon as a projectile.]
In the memorandum already quoted (§ 169) Newton speaks of the
hypothesis as fitting the facts “pretty nearly”; but in a letter of
earlier date (June 20th, 1686) he refers to the calculation as not
having been made accurately enough. It is probable that he used a
seriously inaccurate value of the size of the earth, having overlooked
the measurements of Snell and Norwood (chapter VIII., § 159); it is
known that even at a later stage he was unable to deal satisfactorily
with the difficulty above mentioned, as to whether the earth might
for the purposes of the problem be identified with its centre; and he
was of course aware that the moon’s path differed considerably from a
circle. The view, said to have been derived from Newton’s conversation
many years afterwards, that he was so dissatisfied with his results
as to regard his hypothesis as substantially defective, is possible,
but by no means certain; whatever the cause may have been, he laid the
subject aside for some years without publishing anything on it, and
devoted himself chiefly to optics and mathematics.
174. Meanwhile the problem of the planetary motions was one of the
numerous subjects of discussion among the remarkable group of men who
were the leading spirits of the Royal Society, founded in 1662. _Robert
Hooke_ (1635-1703), who claimed credit for most of the scientific
discoveries of the time, suggested with some distinctness, not later
than 1674, that the motions of the planets might be accounted for
by attraction between them and the sun, and referred also to the
possibility of the earth’s attraction on bodies varying according
to the law of the inverse square. _Christopher Wren_ (1632-1723),
better known as an architect than as a man of science, discussed some
questions of this sort with Newton in 1677, and appears also to have
thought of a law of attraction of this kind. A letter of Hooke’s to
Newton, written at the end of 1679, dealing amongst other things
with the curve which a falling body would describe, the rotation of
the earth being taken into account, stimulated Newton, who professed
that at this time his “affection to philosophy” was “worn out,” to go
on with his study of the celestial motions. Picard’s more accurate
measurement of the earth (chapter VIII., § 159) was now well known,
and Newton repeated his former calculation of the moon’s motion, using
Picard’s improved measurement, and found the result more satisfactory
than before.
175. At the same time (1679) Newton made a further discovery of the
utmost importance by overcoming some of the difficulties connected with
motion in a path other than a circle.
He shewed that if a body moved round a central body, in such a way
that the line joining the two bodies sweeps out equal areas in equal
times, as in Kepler’s Second Law of planetary motion (chapter VII.,
§ 141), then the moving body is acted on by an attraction directed
exactly towards the central body; and further that if the path is an
ellipse, with the central body in one focus, as in Kepler’s First
Law of planetary motion, then this attraction must vary in different
parts of the path as the inverse square of the distance between the
two bodies. Kepler’s laws of planetary motion were in fact shewn to
lead necessarily to the conclusions that the sun exerts on a planet an
attraction inversely proportional to the square of the distance of the
planet from the sun, and that such an attraction affords a sufficient
explanation of the motion of the planet.
Once more, however, Newton published nothing and “threw his
calculations by, being upon other studies.”
176. Nearly five years later the matter was again brought to his
notice, on this occasion by _Edmund Halley_ (chapter X., §§ 199-205),
whose friendship played henceforward an important part in Newton’s
life, and whose unselfish devotion to the great astronomer forms a
pleasant contrast to the quarrels and jealousies prevalent at that
time between so many scientific men. Halley, not knowing of Newton’s
work in 1666, rediscovered, early in 1684, the law of the inverse
square, as a consequence of Kepler’s Third Law, and shortly afterwards
discussed with Wren and Hooke what was the curve in which a body would
move if acted on by an attraction varying according to this law; but
none of them could answer the question.[103] Later in the year Halley
visited Newton at Cambridge and learnt from him the answer. Newton had,
characteristically enough, lost his previous calculation, but was able
to work it out again and sent it to Halley a few months afterwards.
This time fortunately his attention was not diverted to other topics;
he worked out at once a number of other problems of motion, and devoted
his usual autumn course of University lectures to the subject. Perhaps
the most interesting of the new results was that Kepler’s Third Law,
from which the law of the inverse square had been deduced in 1666,
only on the supposition that the planets moved in circles, was equally
consistent with Newton’s law when the paths of the planets were taken
to be ellipses.
177. At the end of the year 1684 Halley went to Cambridge again and
urged Newton to publish his results. In accordance with this request
Newton wrote out, and sent to the Royal Society, a tract called
_Propositiones de Motu_, the 11 propositions of which contained the
results already mentioned and some others relating to the motion
of bodies under attraction to a centre. Although the propositions
were given in an abstract form, it was pointed out that certain
of them applied to the case of the planets. Further pressure from
Halley persuaded Newton to give his results a more permanent form by
embodying them in a larger book. As might have been expected, the
subject grew under his hands, and the great treatise which resulted
contained an immense quantity of material not contained in the _De
Motu_. By the middle of 1686 the rough draft was finished, and some
of it was ready for press. Halley not only undertook to pay the
expenses, but superintended the printing and helped Newton to collect
the astronomical data which were necessary. After some delay in the
press, the book finally appeared early in July 1687, under the title
_Philosophiae Naturalis Principia Mathematica_.
178. The _Principia_, as it is commonly called, consists of three books
in addition to introductory matter: the first book deals generally
with problems of the motion of bodies, solved for the most part in
an abstract form without special reference to astronomy; the second
book deals with the motion of bodies through media which resist
their motion, such as ordinary fluids, and is of comparatively small
astronomical importance, except that in it some glaring inconsistencies
in the Cartesian theory of vortices are pointed out; the third book
applies to the circumstances of the actual solar system the results
already obtained, and is in fact an explanation of the motions of the
celestial bodies on Newton’s mechanical principles.
179. The introductory portion, consisting of “Definitions” and “Axioms,
or Laws of Motion,” forms a very notable contribution to dynamics,
being in fact the first coherent statement of the fundamental laws
according to which the motions of bodies are produced or changed.
Newton himself does not appear to have regarded this part of his book
as of very great importance, and the chief results embodied in it,
being overshadowed as it were by the more striking discoveries in other
parts of the book, attracted comparatively little attention. Much of it
must be passed over here, but certain results of special astronomical
importance require to be mentioned.
Galilei, as we have seen (chapter VI., §§ 130, 133), was the first to
enunciate the law that a body when once in motion continues to move in
the same direction and at the same speed unless some cause is at work
to make it change its motion. This law is given by Newton in the form
already quoted in § 130, as the first of three fundamental laws, and is
now commonly known as the First Law of Motion.
Galilei also discovered that a falling body moves with continually
changing velocity, but with a uniform acceleration (chapter VI., §
133), and that this acceleration is the same for all bodies (chapter
VI., § 116). The tendency of a body to fall having been generally
recognised as due to the earth, Galilei’s discovery involved the
recognition that one effect of one body on another may be an
acceleration produced in its motion. Newton extended this idea by
shewing that the earth produced an acceleration in the motion of the
moon, and the sun in the motion of the planets, and was led to the
general idea of acceleration in a body’s motion, which might be due
in a variety of ways to the action of other bodies, and which could
conveniently be taken as a measure of the effect produced by one body
on another.
180. To these ideas Newton added the very important and difficult
conception of =mass=.
If we are comparing two different bodies of the same material but
of different sizes, we are accustomed to think of the larger one as
heavier than the other. In the same way we readily think of a ball
of lead as being heavier than a ball of wood of the same size. The
most prominent idea connected with “heaviness” and “lightness” is
that of the muscular effort required to support or to lift the body
in question; a greater effort, for example, is required to hold the
leaden ball than the wooden one. Again, the leaden ball if supported by
an elastic string stretches it farther than does the wooden ball; or
again, if they are placed in the scales of a balance, the lead sinks
and the wood rises. All these effects we attribute to the “weight” of
the two bodies, and the weight we are mostly accustomed to attribute
in some way to the action of the earth on the bodies. The ordinary
process of weighing a body in a balance shews, further, that we are
accustomed to think of weight as a measurable quantity. On the other
hand, we know from Galilei’s result, which Newton tested very carefully
by a series of pendulum experiments, that the leaden and the wooden
ball, if allowed to drop, fall with the same acceleration. If therefore
we measure the effect which the earth produces on the two balls by
their acceleration, then the earth affects them equally; but if we
measure it by the power which they have of stretching strings, or by
the power which one has of supporting the other in a balance, then the
effect which the earth produces on the leaden ball is greater than that
produced on the wooden ball. Taken in this way, the action of the earth
on either ball may be spoken of as weight, and the weight of a body can
be measured by comparing it in a balance with standard bodies.
The difference between two such bodies as the leaden and wooden ball
may, however, be recognised in quite a different way. We can easily
see, for example, that a greater effort is needed to set the one in
motion than the other; or that if each is tied to the end of a string
of given kind and whirled round at a given rate, the one string is
more tightly stretched than the other. In these cases the attraction
of the earth is of no importance, and we recognise a distinction
between the two bodies which is independent of the attraction of the
earth. This distinction Newton regarded as due to a difference in the
quantity of matter or material in the two bodies, and to this quantity
he gave the name of mass. It may fairly be doubted whether anything is
gained by this particular definition of mass, but the really important
step was the distinct recognition of mass as a property of bodies,
of fundamental importance in dynamical questions, and capable of
measurement.
Newton, developing Galilei’s idea, gave as one measurement of the
action exerted by one body on another the product of the mass by the
acceleration produced—a quantity for which he used different names,
now replaced by force. The _weight_ of a body was thus identified with
the force exerted on it by the earth. Since the earth produces the
same acceleration in all bodies at the same place, it follows that the
masses of bodies at the same place are proportional to their weights;
thus if two bodies are compared at the same place, and the weight of
one (as shewn, for example, by a pair of scales) is found to be ten
times that of the other, then its mass is also ten times as great.
But such experiments as those of Richer at Cayenne (chapter VIII., §
161) shewed that the acceleration of falling bodies was less at the
equator than in higher latitudes; so that if a body is carried from
London or Paris to Cayenne, its weight is altered but its mass remains
the same as before. Newton’s conception of the earth’s gravitation as
extending as far as the moon gave further importance to the distinction
between mass and weight; for if a body were removed from the earth to
the moon, then its mass would be unchanged, but the acceleration due
to the earth’s attraction would be 60 × 60 times less, and its weight
diminished in the same proportion.
Rules are also given for the effect produced on a body’s motion by the
simultaneous action of two or more forces.[104]
A further principle of great importance, of which only very indistinct
traces are to be found before Newton’s time, was given by him as the
Third Law of Motion in the form: “To every action there is always an
equal and contrary reaction; or, the mutual actions of any two bodies
are always equal and oppositely directed.” Here action and reaction are
to be interpreted primarily in the sense of force. If a stone rests on
the hand, the force with which the stone presses the hand downwards is
equal to that with which the hand presses the stone upwards; if the
earth attracts a stone downwards with a certain force, then the stone
attracts the earth upwards with the same force, and so on. It is to
be carefully noted that if, as in the last example, two bodies are
acting on one another, the _accelerations_ produced are not the same,
but since force is measured by the product of mass and acceleration,
the body with the larger mass receives the lesser acceleration. In the
case of a stone and the earth, the mass of the latter being enormously
greater,[105] its acceleration is enormously less than that of the
stone, and is therefore (in accordance with our experience) quite
insensible.
181. When Newton began to write the _Principia_ he had probably
satisfied himself (§ 173) that the attracting power of the earth
extended as far as the moon, and that the acceleration thereby produced
in any body—whether the moon, or whether a body close to the earth—is
inversely proportional to the square of the distance from the centre
of the earth. With the ideas of force and mass this result may be
stated in the form: _the earth attracts any body with a force inversely
proportional to the square of the distance on the earth’s centre, and
also proportional to the mass of the body_.
In the same way Newton had established that the motions of the planets
could be explained by an attraction towards the sun producing an
acceleration inversely proportional to the square of the distance from
the sun’s centre, not only in the _same_ planet in different parts of
its path, but also in _different_ planets. Again, it follows from this
that the sun attracts any planet with a force inversely proportional
to the square of the distance of the planet from the sun’s centre, and
also proportional to the mass of the planet.
But by the Third Law of Motion a body experiencing an attraction
towards the earth must in turn exert an equal attraction on the earth;
similarly a body experiencing an attraction towards the sun must exert
an equal attraction on the sun. If, for example, the mass of Venus is
seven times that of Mars, then the force with which the sun attracts
Venus is seven times as great as that with which it would attract
Mars if placed at the same distance; and therefore also the force
with which Venus attracts the sun is seven times as great as that
with which Mars would attract the sun if at an equal distance from
it. Hence, in all the cases of attraction hitherto considered and in
which the comparison is possible, the force is proportional not only
to the mass of the attracted body, but also to that of the attracting
body, as well as being inversely proportional to the square of the
distance. Gravitation thus appears no longer as a property peculiar to
the central body of a revolving system, but as belonging to a planet in
just the same way as to the sun, and to the moon or to a stone in just
the same way as to the earth.
Moreover, the fact that separate bodies on the surface of the earth
are attracted by the earth, and therefore in turn attract it, suggests
that this power of attracting other bodies which the celestial bodies
are shewn to possess does not belong to each celestial body as a whole,
but to the separate particles making it up, so that, for example, the
force with which Jupiter and the sun mutually attract one another is
the result of compounding the forces with which the separate particles
making up Jupiter attract the separate particles making up the sun.
Thus is suggested finally the law of gravitation in its most general
form: _every particle of matter attracts every other particle with a
force proportional to the mass of each, and inversely proportional to
the square of the distance between them_.[106]
182. In all the astronomical cases already referred to the attractions
between the various celestial bodies have been treated as if they were
accurately directed towards their centres, and the distance between
the bodies has been taken to be the distance between their centres.
Newton’s doubts on this point, in the case of the earth’s attraction
of bodies, have been already referred to (§ 173); but early in 1685
he succeeded in justifying this assumption. By a singularly beautiful
and simple course of reasoning he shewed (_Principia_, Book I.,
propositions 70, 71) that, if a body is spherical in form and equally
dense throughout, it attracts any particle external to it exactly as
if its whole mass were concentrated at its centre. He shewed, further,
that the same is true for a sphere of variable density, provided it can
be regarded as made up of a series of spherical shells, having a common
centre, each of uniform density throughout, different shells being,
however, of different densities. For example, the result is true for
a hollow indiarubber ball as well as for a solid one, but is not true
for a sphere made up of a hemisphere of wood and a hemisphere of iron
fastened together.
183. The law of gravitation being thus provisionally established,
the great task which lay before Newton, and to which he devotes the
greater part of the first and third books of the _Principia_, was
that of deducing from it and the “laws of motion” the motions of the
various members of the solar system, and of shewing, if possible, that
the motions so calculated agreed with those observed. If this were
successfully done, it would afford a verification of the most delicate
and rigorous character of Newton’s principles.
The conception of the solar system as a mechanism, each member of
which influences the motion of every other member in accordance with
one universal law of attraction, although extremely simple in itself,
is easily seen to give rise to very serious difficulties when it is
proposed actually to calculate the various motions. If in dealing
with the motion of a planet such as Mars it were possible to regard
Mars as acted on only by the attraction of the sun, and to ignore the
effects of the other planets, then the problem would be completely
solved by the propositions which Newton established in 1679 (§ 175);
and by their means the position of Mars at any time could be calculated
with any required degree of accuracy. But in the case which actually
exists the motion of Mars is affected by the forces with which all the
other planets (as well as the satellites) attract it, and these forces
in turn depend on the position of Mars (as well as upon that of the
other planets) and hence upon the motion of Mars. A problem of this
kind in all its generality is quite beyond the powers of any existing
mathematical methods. Fortunately, however, the mass of even the
largest of the planets is so very much less than that of the sun, that
the motion of any one planet is only slightly affected by the others;
and it may be regarded as moving very nearly as it would move if the
other planets did not exist, the effect of these being afterwards
allowed for as producing disturbances or =perturbations= in its path.
Although even in this simplified form the problem of the motion of
the planets is one of extreme difficulty (cf. chapter XI., § 228), and
Newton was unable to solve it with anything like completeness, yet he
was able to point out certain general effects which must result from
the mutual action of the planets, the most interesting being the slow
forward motion of the apses of the earth’s orbit, which had long ago
been noticed by observing astronomers (chapter III., § 59). Newton also
pointed out that Jupiter, on account of its great mass, must produce a
considerable perturbation in the motion of its neighbour Saturn, and
thus gave some explanation of an irregularity first noted by Horrocks
(chapter VIII., § 156).
184. The motion of the moon presents special difficulties, but
Newton, who was evidently much interested in the problems of lunar
theory, succeeded in overcoming them much more completely than the
corresponding ones connected with the planets.
The moon’s motion round the earth is primarily due to the attraction
of the earth; the perturbations due to the other planets are
insignificant; but the sun, which though at a very great distance has
an enormously greater mass than the earth, produces a very sensible
disturbing effect on the moon’s motion. Certain irregularities, as we
have seen (chapter II., §§ 40, 48; chapter V., § 111), had already been
discovered by observation. Newton was able to shew that the disturbing
action of the sun would necessarily produce perturbations of the same
general character as those thus recognised, and in the case of the
motion of the moon’s nodes and of her apogee he was able to get a very
fairly accurate numerical result;[107] and he also discovered a number
of other irregularities, for the most part very small, which had not
hitherto been noticed. He indicated also the existence of certain
irregularities in the motions of Jupiter’s and Saturn’s moons analogous
to those which occur in the case of our moon.
185. One group of results of an entirely novel character resulted from
Newton’s theory of gravitation. It became for the first time possible
to estimate the _masses_ of some of the celestial bodies, by comparing
the attractions exerted by them on other bodies with that exerted by
the earth.
The case of Jupiter may be given as an illustration. The time of
revolution of Jupiter’s outermost satellite is known to be about 16
days 16 hours, and its distance from Jupiter was estimated by Newton
(not very correctly) at about four times the distance of the moon from
the earth. A calculation exactly like that of § 172 or § 173 shews
that the acceleration of the satellite due to Jupiter’s attraction
is about ten times as great as the acceleration of the moon towards
the earth, and that therefore, the distance being four times as
great, Jupiter attracts a body with a force 10 × 4 × 4 times as great
as that with which the earth attracts a body at the same distance;
consequently Jupiter’s mass is 160 times that of the earth. This
process of reasoning applies also to Saturn, and in a very similar
way a comparison of the motion of a planet, Venus for example, round
the sun with the motion of the moon round the earth gives a relation
between the masses of the sun and earth. In this way Newton found the
mass of the sun to be 1067, 3021, and 169282 times greater than that
of Jupiter, Saturn, and the earth, respectively. The corresponding
figures now accepted are not far from 1047, 3530, 324439. The large
error in the last number is due to the use of an erroneous value of the
distance of the sun—then not at all accurately known—upon which depend
the other distances in the solar system, except those connected with
the earth-moon system. As it was necessary for the employment of this
method to be able to observe the motion of some other body attracted
by the planet in question, it could not be applied to the other three
planets (Mars, Venus, and Mercury), of which no satellites were known.
186. From the equality of action and reaction it follows that, since
the sun attracts the planets, they also attract the sun, and the sun
consequently is in motion, though—owing to the comparative smallness of
the planets—only to a very small extent. It follows that Kepler’s Third
Law is not strictly accurate, deviations from it becoming sensible in
the case of the large planets Jupiter and Saturn (cf. chapter VII., §
144). It was, however, proved by Newton that in any system of bodies,
such as the solar system, moving about in any way under the influence
of their mutual attractions, there is a particular point, called the
=centre of gravity=, which can always be treated as at rest; the sun
moves relatively to this point, but so little that the distance between
the centre of the sun and the centre of gravity can never be much more
than the diameter of the sun.
It is perhaps rather curious that this result was not seized upon by
some of the supporters of the Church in the condemnation of Galilei,
now rather more than half a century old; for if it was far from
supporting the view that the earth is at the centre of the world, it at
any rate negatived that part of the doctrine of Coppernicus and Galilei
which asserted the sun to be _at rest_ in the centre of the world.
Probably no one who was capable of understanding Newton’s book was a
serious supporter of any anti-Coppernican system, though some still
professed themselves obedient to the papal decrees on the subject.[108]
187. The variation of the time of oscillation of a pendulum in
different parts of the earth, discovered by Richer in 1672 (chapter
VIII., § 161), indicated that the earth was probably not a sphere.
Newton pointed out that this departure from the spherical form was
a consequence of the mutual gravitation of the particles making up
the earth and of the earth’s rotation. He supposed a canal of water
to pass from the pole to the centre of the earth, and then from the
centre to a point on the equator (B O _a_ A in fig. 72), and then
found the condition that these two columns of water O B, O A, each
being attracted towards the centre of the earth, should balance. This
method involved certain assumptions as to the inside of the earth, of
which little can be said to be known even now, and consequently, though
Newton’s general result, that the earth is flattened at the poles and
bulges out at the equator, was right, the actual numerical expression
which he found was not very accurate. If, in the figure, the dotted
line is a circle the radius of which is equal to the distance of the
pole B from the centre of the earth O, then the actual surface of the
earth extends at the equator beyond this circle as far as A, where,
according to Newton, _a_ A is about 1∕230 of O B or O A, and according
to modern estimates, based on actual measurement of the earth as well
as upon theory (chapter X., § 221), it is about 1∕293 of O A. Both
Newton’s fraction and the modern one are so small that the resulting
flattening cannot be made sensible in a figure; in fig. 72 the length
_a_ A is made, for the sake of distinctness, nearly 30 times as great
as it should be.
[Illustration: FIG. 72.—The spheroidal form of the earth.]
Newton discovered also in a similar way the flattening of Jupiter,
which, owing to its more rapid rotation, is considerably more flattened
than the earth; this was also detected telescopically by Domenico
Cassini four years after the publication of the _Principia_.
188. The discovery of the form of the earth led to an explanation of
the precession of the equinoxes, a phenomenon which had been discovered
1,800 years before (chapter II., § 42), but had remained a complete
mystery ever since.
If the earth is a perfect sphere, then its attraction on any other body
is exactly the same as if its mass were all concentrated at its centre
(§ 182), and so also the attraction on it of any other body such as
the sun or moon is equivalent to a single force passing through the
centre O of the earth; but this is no longer true if the earth is not
spherical. In fact the action of the sun or moon on the spherical part
of the earth, inside the dotted circle in fig. 72, is equivalent to
a force through O, and has no tendency to turn the earth in any way
about its centre; but the attraction on the remaining portion is of a
different character, and Newton shewed that from it resulted a motion
of the axis of the earth of the same general character as precession.
The amount of the precession as calculated by Newton did as a matter of
fact agree pretty closely with the observed amount, but this was due to
the accidental compensation of two errors, arising from his imperfect
knowledge of the form and construction of the earth, as well as from
erroneous estimates of the distance of the sun and of the mass of the
moon, neither of which quantities Newton was able to measure with any
accuracy.[109] It was further pointed out that the motion in question
was necessarily not quite uniform, but that, owing to the unequal
effects of the sun in different positions, the earth’s axis would
oscillate to and fro every six months, though to a very minute extent.
189. Newton also gave a general explanation of the tides as due to
the disturbing action of the moon and sun, the former being the more
important. If the earth be regarded as made of a solid spherical
nucleus, covered by the ocean, then the moon attracts different
parts unequally, and in particular the attraction, measured by the
acceleration produced, on the water nearest to the moon is greater
than that on the solid earth, and that on the water farthest from
the moon is less. Consequently the water moves on the surface of the
earth, the general character of the motion being the same as if the
portion of the ocean on the side towards the moon were attracted and
that on the opposite side repelled. Owing to the rotation of the earth
and the moon’s motion, the moon returns to nearly the same position
with respect to any place on the earth in a period which exceeds a
day by (on the average) about 50 minutes, and consequently Newton’s
argument shewed that low tides (or high tides) due to the moon would
follow one another at any given place at intervals equal to about half
this period; or, in other words, that two tides would in general occur
daily, but that on each day any particular phase of the tides would
occur on the average about 50 minutes later than on the preceding day,
a result agreeing with observation. Similar but smaller tides were
shewn by the same argument to arise from the action of the sun, and the
actual tide to be due to the combination of the two. It was shewn that
at new and full moon the lunar and solar tides would be added together,
whereas at the half moon they would tend to counteract one another, so
that the observed fact of greater tides every fortnight received an
explanation. A number of other peculiarities of the tides were also
shewn to result from the same principles.
Newton ingeniously used observations of the height of the tide when
the sun and moon acted together and when they acted in opposite ways
to compare the tide-raising powers of the sun and moon, and hence
to estimate the mass of the moon in terms of that of the sun, and
consequently in terms of that of the earth (§ 185). The resulting mass
of the moon was about twice what it ought to be according to modern
knowledge, but as before Newton’s time no one knew of any method of
measuring the moon’s mass even in the roughest way, and this result
had to be disentangled from the innumerable complications connected
with both the theory and with observation of the tides, it cannot
but be regarded as a remarkable achievement. Newton’s theory of the
tides was based on certain hypotheses which had to be made in order to
render the problem at all manageable, but which were certainly not
true, and consequently, as he was well aware, important modifications
would necessarily have to be made, in order to bring his results into
agreement with actual facts. The mere presence of land not covered
by water is, for example, sufficient by itself to produce important
alterations in tidal effects at different places. Thus Newton’s theory
was by no means equal to such a task as that of predicting the times
of high tide at any required place, or the height of any required
tide, though it gave a satisfactory explanation of many of the general
characteristics of tides.
190. As we have seen (chapter V., § 103; chapter VII., § 146), comets
until quite recently had been commonly regarded as terrestrial objects
produced in the higher regions of our atmosphere, and even the
more enlightened astronomers who, like Tycho, Kepler, and Galilei,
recognised them as belonging to the celestial bodies, were unable to
give an explanation of their motions and of their apparently quite
irregular appearances and disappearances. Newton was led to consider
whether a comet’s motion could not be explained, like that of a planet,
by gravitation towards the sun. If so then, as he had proved near the
beginning of the _Principia_, its path must be either an ellipse or
one of two other allied curves, the =parabola= and =hyperbola=. If a
comet moved in an ellipse which only differed slightly from a circle,
then it would never recede to any very great distance from the centre
of the solar system, and would therefore be regularly visible, a result
which was contrary to observation. If, however, the ellipse was very
elongated, as shewn in fig. 73, then the period of revolution might
easily be very great, and, during the greater part of it, the comet
would be so far from the sun and consequently also from the earth as
to be invisible. If so the comet would be seen for a short time and
become invisible, only to reappear after a very long time, when it
would naturally be regarded as a new comet. If again the path of the
comet were a parabola (which may be regarded as an ellipse indefinitely
elongated), the comet would not return at all, but would merely be seen
once when in that part of its path which is near the sun. But if a
comet moved in a parabola, with the sun in a focus, then its positions
when not very far from the sun would be almost the same as if it moved
in an elongated ellipse (see fig. 73), and consequently it would hardly
be possible to distinguish the two cases. Newton accordingly worked
out the case of motion in a parabola, which is mathematically the
simpler, and found that, in the case of a comet which had attracted
much attention in the winter 1680-1, a parabolic path could be found,
the calculated places of the comet in which agreed closely with those
observed. In the later editions of the _Principia_ the motions of a
number of other comets were investigated with a similar result. It was
thus established that in many cases a comet’s path is either a parabola
or an elongated ellipse, and that a similar result was to be expected
in other cases. This reduction to rule of the apparently arbitrary
motions of comets, and their inclusion with the planets in the same
class of bodies moving round the sun under the action of gravitation,
may fairly be regarded as one of the most striking of the innumerable
discoveries contained in the _Principia_.
[Illustration: FIG. 73.—An elongated ellipse and a parabola.]
In the same section Newton discussed also at some length the nature of
comets and in particular the structure of their tails, arriving at the
conclusion, which is in general agreement with modern theories (chapter
XIII., § 304), that the tail is formed by a stream of finely divided
matter of the nature of smoke, rising up from the body of the comet,
and so illuminated by the light of the sun when tolerably near it as to
become visible.
191. The _Principia_ was published, as we have seen, in 1687. Only a
small edition seems to have been printed, and this was exhausted in
three or four years. Newton’s earlier discoveries, and the presentation
to the Royal Society of the tract _De Motu_ (§ 177), had prepared the
scientific world to look for important new results in the _Principia_,
and the book appears to have been read by the leading Continental
mathematicians and astronomers, and to have been very warmly received
in England. The Cartesian philosophy had, however, too firm a hold to
be easily shaken; and Newton’s fundamental principle, involving as it
did the idea of an action between two bodies separated by an interval
of empty space, seemed impossible of acceptance to thinkers who had
not yet fully grasped the notion of judging a scientific theory by the
extent to which its consequences agree with observed facts. Hence even
so able a man as Huygens (chapter VIII., §§ 154, 157, 158), regarded
the idea of gravitation as “absurd,” and expressed his surprise that
Newton should have taken the trouble to make such a number of laborious
calculations with no foundation but this principle, a remark which
shewed Huygens to have had no conception that the agreement of the
results of these calculations with actual facts was proof of the
soundness of the principle. Personal reasons also contributed to the
Continental neglect of Newton’s work, as the famous quarrel between
Newton and Leibniz as to their respective claims to the invention of
what Newton called fluxions and Leibniz the differential method (out of
which the differential and integral calculus have developed) grew in
intensity and fresh combatants were drawn into it on both sides. Half
a century in fact elapsed before Newton’s views made any substantial
progress on the Continent (cf. chapter XI., § 229). In our country the
case was different; not only was the _Principia_ read with admiration
by the few who were capable of understanding it, but scholars like
Bentley, philosophers like Locke, and courtiers like Halifax all made
attempts to grasp Newton’s general ideas, even though the details
of his mathematics were out of their range. It was moreover soon
discovered that his scientific ideas could be used with advantage as
theological arguments.
192. One unfortunate result of the great success of the _Principia_
was that Newton was changed from a quiet Cambridge professor, with
abundant leisure and a slender income, into a public character, with a
continually increasing portion of his time devoted to public business
of one sort or another.
Just before the publication of the _Principia_ he had been appointed
one of the representatives of his University to defend its rights
against the encroachments of James II., and two years later he sat
as member for the University in the Convention Parliament, though he
retired after its dissolution.
Notwithstanding these and many other distractions, he continued to
work at the theory of gravitation, paying particular attention to
the lunar theory, a difficult subject with his treatment of which he
was never quite satisfied.[110] He was fortunately able to obtain
from time to time first-rate observations of the moon (as well as of
other bodies) from the Astronomer Royal Flamsteed (chapter X., §§
197-8), though Newton’s continual requests and Flamsteed’s occasional
refusals led to strained relations at intervals. It is possible that
about this time Newton contemplated writing a new treatise, with more
detailed treatment of various points discussed in the _Principia_;
and in 1691 there was already some talk of a new edition of the
_Principia_, possibly to be edited by some younger mathematician. In
any case nothing serious in this direction was done for some years,
perhaps owing to a serious illness, apparently some nervous disorder,
which attacked Newton in 1692 and lasted about two years. During this
illness, as he himself said, “he had not his usual consistency of
mind,” and it is by no means certain that he ever recovered his full
mental activity and power.
[Illustration: NEWTON.]
Soon after recovering from this illness he made some preparations
for a new edition of the _Principia_, besides going on with the lunar
theory, but the work was again interrupted in 1695, when he received
the valuable appointment of Warden to the Mint, from which he was
promoted to the Mastership four years later. He had, in consequence, to
move to London (1696), and much of his time was henceforward occupied
by official duties. In 1701 he resigned his professorship at Cambridge,
and in the same year was for the second time elected the Parliamentary
representative of the University. In 1703 he was chosen President of
the Royal Society, an office which he held till his death, and in 1705
he was knighted on the occasion of a royal visit to Cambridge.
During this time he published (1704) his treatise on _Optics_, the
bulk of which was probably written long before, and in 1709 he finally
abandoned the idea of editing the _Principia_ himself, and arranged
for the work to be done by _Roger Cotes_ (1682-1716), the brilliant
young mathematician whose untimely death a few years later called from
Newton the famous eulogy, “If Mr. Cotes had lived we might have known
something.” The alterations to be made were discussed in a long and
active correspondence between the editor and author, the most important
changes being improvements and additions to the lunar theory, and to
the discussions of precession and of comets, though there were also a
very large number of minor changes; and the new edition appeared in
1713. A third edition, edited by Pemberton, was published in 1726,
but this time Newton, who was over 80, took much less part, and the
alterations were of no great importance. This was Newton’s last piece
of scientific work, and his death occurred in the following year (March
3rd, 1727).
193. It is impossible to give an adequate idea of the immense magnitude
of Newton’s scientific discoveries except by a free use of the
mathematical technicalities in which the bulk of them were expressed.
The criticism passed on him by his personal enemy Leibniz that, “Taking
mathematics from the beginning of the world to the time when Newton
lived, what he had done was much the better half,” and the remark of
his great successor Lagrange (chapter XI., § 237), “Newton was the
greatest genius that ever existed, and the most fortunate, for we
cannot find more than once a system of the world to establish,” shew
the immense respect for his work felt by those who were most competent
to judge it.
With these magnificent eulogies it is pleasant to compare Newton’s
own grateful recognition of his predecessors, “If I have seen further
than other men, it is because I have stood upon the shoulders of the
giants,” and his modest estimate of his own performances:—
“I do not know what I may appear to the world; but to myself I seem
to have been only like a boy playing on the seashore, and diverting
myself in now and then finding a smoother pebble or a prettier shell
than ordinary, whilst the great ocean of truth lay all undiscovered
before me.”
194. It is sometimes said, in explanation of the difference between
Newton’s achievements and those of earlier astronomers, that whereas
they discovered _how_ the celestial bodies moved, he shewed _why_ the
motions were as they were, or, in other words, that they _described_
motions while he _explained_ them or ascertained their cause. It
is, however, doubtful whether this distinction between How and Why,
though undoubtedly to some extent convenient, has any real validity.
Ptolemy, for example, represented the motion of a planet by a certain
combination of epicycles; his scheme was equivalent to a particular
method of describing the motion; but if any one had asked him why the
planet would be in a particular position at a particular time, he
might legitimately have answered that it was so because the planet was
connected with this particular system of epicycles, and its place could
be deduced from them by a rigorous process of calculation. But if any
one had gone further and asked why the planet’s epicycles were as they
were, Ptolemy could have given no answer. Moreover, as the system of
epicycles differed in some important respects from planet to planet,
Ptolemy’s system left unanswered a number of questions which obviously
presented themselves. Then Coppernicus gave a partial answer to some
of these questions. To the question why certain of the planetary
motions, corresponding to certain epicycles, existed, he would have
replied that it was because of certain motions of the earth, from
which these (apparent) planetary motions could be deduced as necessary
consequences. But the same information could also have been given as a
mere descriptive statement that the earth moves in certain ways and the
planets move in certain other ways. But again, if Coppernicus had been
asked why the earth rotated on its axis, or why the planets revolved
round the sun, he could have given no answer; still less could he have
said why the planets had certain irregularities in their motions,
represented by his epicycles.
Kepler again described the same motions very much more simply and
shortly by means of his three laws of planetary motion; but if any
one had asked why a planet’s motion varied in certain ways, he might
have replied that it was because all planets moved in ellipses so as
to sweep out equal areas in equal times. _Why_ this was so Kepler was
unable to say, though he spent much time in speculating on the subject.
This question was, however, answered by Newton, who shewed that the
planetary motions were necessary consequences of his law of gravitation
and his laws of motion. Moreover from these same laws, which were
extremely simple in statement and few in number, followed as necessary
consequences the motion of the moon and many other astronomical
phenomena, and also certain familiar terrestrial phenomena, such as
the behaviour of falling bodies; so that a large number of groups of
observed facts, which had hitherto been disconnected from one another,
were here brought into connection as necessary consequences of certain
fundamental laws. But again Newton’s view of the solar system might
equally well be put as a mere descriptive statement that the planets,
etc., move with accelerations of certain magnitudes towards one
another. As, however, the actual position or rate of motion of a planet
at any time can only be deduced by an extremely elaborate calculation
from Newton’s laws, they are not at all obviously equivalent to the
observed celestial motions, and we do not therefore at all easily think
of them as being merely a description.
Again Newton’s laws at once suggest the question why bodies attract one
another in this particular way; and this question, which Newton fully
recognised as legitimate, he was unable to answer. Or again we might
ask why the planets are of certain sizes, at certain distances from
the sun, etc., and to these questions again Newton could give no answer.
But whereas the questions left unanswered by Ptolemy, Coppernicus,
and Kepler were in whole or in part answered by their successors,
that is, their unexplained facts or laws were shewn to be necessary
consequences of other simpler and more general laws, it happens that up
to the present day no one has been able to answer, in any satisfactory
way, these questions which Newton left unanswered. In this particular
direction, therefore, Newton’s laws mark the boundary of our present
knowledge. But if any one were to succeed this year or next in shewing
gravitation to be a consequence of some still more general law, this
new law would still bring with it a new Why.
If, however, Newton’s laws cannot be regarded as an ultimate
explanation of the phenomena of the solar system, except in the
historic sense that they have not yet been shewn to depend on other
more fundamental laws, their success in “explaining,” with fair
accuracy, such an immense mass of observed results in all parts of the
solar system, and their universal character, gave a powerful impetus
to the idea of accounting for observed facts in other departments
of science, such as chemistry and physics, in some similar way as
the consequence of forces acting between bodies, and hence to the
conception of the material universe as made up of a certain number
of bodies, each acting on one another with definite forces in such a
way that all the changes which can be observed to go on are necessary
consequences of these forces, and are capable of prediction by any one
who has sufficient knowledge of the forces and sufficient mathematical
skill to develop their consequences.
Whether this conception of the material universe is adequate or not,
it has undoubtedly exercised a very important influence on scientific
discovery as well as on philosophical thought, and although it was
never formulated by Newton, and parts of it would probably have been
repudiated by him, there are indications that some such ideas were in
his head, and those who held the conception most firmly undoubtedly
derived their ideas directly or indirectly from him.
195. Newton’s scientific method did not differ essentially from that
followed by Galilei (chapter VI., § 134), which has been variously
described as =complete induction= or as the =inverse deductive method=,
the difference in name corresponding to a difference in the stress laid
upon different parts of the same general process. Facts are obtained
by observation or experiment; a hypothesis or provisional theory is
devised to account for them; from this theory are obtained, if possible
by a rigorous process of deductive reasoning, certain consequences
capable of being compared with actual facts, and the comparison is then
made. In some cases the first process may appear as the more important,
but in Newton’s work the really convincing part of the proof of his
results lay in the verification involved in the two last processes.
This has perhaps been somewhat obscured by his famous remark,
_Hypotheses non fingo_ (I do not invent hypotheses), dissociated from
its context. The words occur in the conclusion of the _Principia_,
after he has been speaking of universal gravitation:—
“I have not yet been able to deduce (_deducere_) from phenomena
the reason of these properties of gravitation, and I do not invent
hypotheses. For any thing which cannot be deduced from phenomena
should be called a hypothesis.”
Newton probably had in his mind such speculations as the Cartesian
vortices, which could not be deduced directly from observations, and
the consequences of which either could not be worked out and compared
with actual facts or were inconsistent with them. Newton in fact
rejected hypotheses which were unverifiable, but he constantly made
hypotheses, suggested by observed facts, and verified by the agreement
of their consequences with fresh observed facts. The extension of
gravity to the moon (§ 173) is a good example: he was acquainted with
certain facts as to the motion of falling bodies and the motion of the
moon; it occurred to him that the earth’s attraction might extend as
far as the moon, and certain other facts connected with Kepler’s Third
Law suggested the law of the inverse square. If this were right, the
moon’s acceleration towards the earth ought to have a certain value,
which could be obtained by calculation. The calculation was made and
found to agree roughly with the actual motion of the moon.
Moreover it may be fairly urged, in illustration of the great
importance of the process of verification, that Newton’s fundamental
laws were not rigorously established by him, but that the deficiencies
in his proofs have been to a great extent filled up by the elaborate
process of verification that has gone on since. For the motions of
the solar system, as deduced by Newton from gravitation and the laws
of motion, only agreed roughly with observation; many outstanding
discrepancies were left; and though there was a strong presumption that
these were due to the necessary imperfections of Newton’s processes
of calculation, an immense expenditure of labour and ingenuity on the
part of a series of mathematicians has been required to remove these
discrepancies one by one, and as a matter of fact there remain even
to-day a few small ones which are unexplained (chapter XIII., § 290).