for circular motion only, so as to avoid geometrical difficulties, but
even so it is very instructive. The reason of the proportion between
_r^3_ and _T^2_ is at once manifest; and as soon as the constant _V_
became known, _the mass of the central body_, the sun in the case of a
planet, the earth in the case of the moon, Jupiter in the case of his
satellites, was at once determined.
Newton's reasoning at this time might, however, be better displayed
perhaps by altering the order of the steps a little, as thus:--
The centrifugal force of a body is proportional to _r^3/T^2_, but by
Kepler's third law _r^3/T^2_ is constant for all the planets,
reckoning _r_ from the sun. Hence the centripetal force needed to hold
in all the planets will be a single force emanating from the sun and
varying inversely with the square of the distance from that body.
Such a force is at once necessary and sufficient. Such a force would
explain the motion of the planets.
But then all this proceeds on a wrong assumption--that the planetary
motion is circular. Will it hold for elliptic orbits? Will an inverse
square law of force keep a body moving in an elliptic orbit about the
sun in one focus? This is a far more difficult question. Newton solved
it, but I do not believe that even he could have solved it, except that
he had at his disposal two mathematical engines of great power--the
Cartesian method of treating geometry, and his own method of Fluxions.
One can explain the elliptic motion now mathematically, but hardly
otherwise; and I must be content to state that the double fact is
true--viz., that an inverse square law will move the body in an ellipse
or other conic section with the sun in one focus, and that if a body so
moves it _must_ be acted on by an inverse square law.
[Illustration: FIG. 59.]
This then is the meaning of the first and third laws of Kepler. What
about the second? What is the meaning of the equable description of
areas? Well, that rigorously proves that a planet is acted on by a force
directed to the centre about which the rate of description of areas is
equable. It proves, in fact, that the sun is the attracting body, and
that no other force acts.
For first of all if the first law of motion is obeyed, _i.e._ if no
force acts, and if the path be equally subdivided to represent
equal times, and straight lines be drawn from the divisions to any
point whatever, all these areas thus enclosed will be equal,
because they are triangles on equal base and of the same height
(Euclid, I). See Fig. 59; _S_ being any point whatever, and _A_,
_B_, _C_, successive positions of a body.
Now at each of the successive instants let the body receive a
sudden blow in the direction of that same point _S_, sufficient to
carry it from _A_ to _D_ in the same time as it would have got to
_B_ if left alone. The result will be that there will be a
compromise, and it will really arrive at _P_, travelling along the
diagonal of the parallelogram _AP_. The area its radius vector
sweeps out is therefore _SAP_, instead of what it would have been,
_SAB_. But then these two areas are equal, because they are
triangles on the same base _AS_, and between the same parallels
_BP_, _AS_; for by the parallelogram law _BP_ is parallel to _AD_.
Hence the area that would have been described is described, and as
all the areas were equal in the case of no force, they remain equal
when the body receives a blow at the end of every equal interval of
time, _provided_ that every blow is actually directed to _S_, the
point to which radii vectores are drawn.
[Illustration: FIG. 60.]
[Illustration: FIG. 61.]
It is instructive to see that it does not hold if the blow is any
otherwise directed; for instance, as in Fig. 61, when the blow is
along _AE_, the body finds itself at _P_ at the end of the second
interval, but the area _SAP_ is by no means equal to _SAB_, and
therefore not equal to _SOA_, the area swept out in the first
interval.
In order to modify Fig. 60 so as to represent continuous motion and
steady forces, we have to take the sides of the polygon _OAPQ_,
&c., very numerous and very small; in the limit, infinitely
numerous and infinitely small. The path then becomes a curve, and
the series of blows becomes a steady force directed towards _S_.
About whatever point therefore the rate of description of areas is
uniform, that point and no other must be the centre of all the
force there is. If there be no force, as in Fig. 59, well and good,
but if there be any force however small not directed towards _S_,
then the rate of description of areas about _S_ cannot be uniform.
Kepler, however, says that the rate of description of areas of each
planet about the sun is, by Tycho's observations, uniform; hence
the sun is the centre of all the force that acts on them, and there
is no other force, not even friction. That is the moral of Kepler's
second law.
We may also see from it that gravity does not travel like light, so
as to take time on its journey from sun to planet; for, if it did,
there would be a sort of aberration, and the force on its arrival
could no longer be accurately directed to the centre of the sun.
(See _Nature_, vol. xlvi., p. 497.) It is a matter for accuracy of
observation, therefore, to decide whether the minutest trace of
such deviation can be detected, _i.e._ within what limits of
accuracy Kepler's second law is now known to be obeyed.
I will content myself by saying that the limits are extremely
narrow. [Reference may be made also to p. 208.]
Thus then it became clear to Newton that the whole solar system depended
on a central force emanating from the sun, and varying inversely with
the square of the distance from him: for by that hypothesis all the laws
of Kepler concerning these motions were completely accounted for; and,
in fact, the laws necessitated the hypothesis and established it as a
theory.
Similarly the satellites of Jupiter were controlled by a force emanating
from Jupiter and varying according to the same law. And again our moon
must be controlled by a force from the earth, decreasing with the
distance according to the same law.
Grant this hypothetical attracting force pulling the planets towards
the sun, pulling the moon towards the earth, and the whole mechanism of
the solar system is beautifully explained.
If only one could be sure there was such a force! It was one thing to
calculate out what the effects of such a force would be: it was another
to be able to put one's finger upon it and say, this is the force that
actually exists and is known to exist. We must picture him meditating in
his garden on this want--an attractive force towards the earth.
If only such an attractive force pulling down bodies to the earth
existed. An apple falls from a tree. Why, it does exist! There is
gravitation, common gravity that makes bodies fall and gives them their
weight.
Wanted, a force tending towards the centre of the earth. It is to hand!
It is common old gravity that had been known so long, that was perfectly
familiar to Galileo, and probably to Archimedes. Gravity that regulates
the motion of projectiles. Why should it only pull stones and apples?
Why should it not reach as high as the moon? Why should it not be the
gravitation of the sun that is the central force acting on all the
planets?
Surely the secret of the universe is discovered! But, wait a bit; is it
discovered? Is this force of gravity sufficient for the purpose? It must
vary inversely with the square of the distance from the centre of the
earth. How far is the moon away? Sixty earth's radii. Hence the force of
gravity at the moon's distance can only be 1/3600 of what it is on the
earth's surface. So, instead of pulling it 16 ft. per second, it should
pull it 16/3600 ft. per second, or 16 ft. a minute.[17] How can one
decide whether such a force is able to pull the moon the actual amount
required? To Newton this would seem only like a sum in arithmetic. Out
with a pencil and paper and reckon how much the moon falls toward the
earth in every second of its motion. Is it 16/3600? That is what it
ought to be: but is it? The size of the earth comes into the
calculation. Sixty miles make a degree, 360 degrees a circumference.
This gives as the earth's diameter 6,873 miles; work it out.
The answer is not 16 feet a minute, it is 13.9 feet.
Surely a mistake of calculation?
No, it is no mistake: there is something wrong in the theory, gravity is
too strong.
Instead of falling toward the earth 5-1/3 hundredths of an inch every
second, as it would under gravity, the moon only falls 4-2/3 hundredths
of an inch per second.
With such a discovery in his grasp at the age of twenty-three he is
disappointed--the figures do not agree, and he cannot make them agree.
Either gravity is not the force in action, or else something interferes
with it. Possibly, gravity does part of the work, and the vortices of
Descartes interfere with it.
He must abandon the fascinating idea for the time. In his own words, "he
laid aside at that time any further thought of the matter."
So far as is known, he never mentioned his disappointment to a soul. He
might, perhaps, if he had been at Cambridge, but he was a shy and
solitary youth, and just as likely he might not. Up in Lincolnshire, in
the seventeenth century, who was there for him to consult?
True, he might have rushed into premature publication, after our
nineteenth century fashion, but that was not his method. Publication
never seemed to have occurred to him.
His reticence now is noteworthy, but later on it is perfectly
astonishing. He is so absorbed in making discoveries that he actually
has to be reminded to tell any one about them, and some one else always
has to see to the printing and publishing for him.
I have entered thus fully into what I conjecture to be the stages of
this early discovery of the law of gravitation, as applicable to the
heavenly bodies, because it is frequently and commonly misunderstood. It
is sometimes thought that he discovered the force of gravity; I hope I
have made it clear that he did no such thing. Every educated man long
before his time, if asked why bodies fell, would reply just as glibly as
they do now, "Because the earth attracts them," or "because of the force
of gravity."
His discovery was that the motions of the solar system were due to the
action of a central force, directed to the body at the centre of the
system, and varying inversely with the square of the distance from it.
This discovery was based upon Kepler's laws, and was clear and certain.
It might have been published had he so chosen.
But he did not like hypothetical and unknown forces; he tried to see
whether the known force of gravity would serve. This discovery at that
time he failed to make, owing to a wrong numerical datum. The size of
the earth he only knew from the common doctrine of sailors that 60 miles
make a degree; and that threw him out. Instead of falling 16 feet a
minute, as it ought under gravity, it only fell 13.9 feet, so he
abandoned the idea. We do not find that he returned to it for sixteen
years.
LECTURE VIII
NEWTON AND THE LAW OF GRAVITATION
We left Newton at the age of twenty-three on the verge of discovering
the mechanism of the solar system, deterred therefrom only by an error
in the then imagined size of the earth. He had proved from Kepler's laws
that a centripetal force directed to the sun, and varying as the inverse
square of the distance from that body, would account for the observed
planetary motions, and that a similar force directed to the earth would
account for the lunar motion; and it had struck him that this force
might be the very same as the familiar force of gravitation which gave
to bodies their weight: but in attempting a numerical verification of
this idea in the case of the moon he was led by the then received notion
that sixty miles made a degree on the earth's surface into an erroneous
estimate of the size of the moon's orbit. Being thus baffled in
obtaining such verification, he laid the matter aside for a time.
The anecdote of the apple we learn from Voltaire, who had it from
Newton's favourite niece, who with her husband lived and kept house for
him all his later life. It is very like one of those anecdotes which are
easily invented and believed in, and very often turn out on scrutiny to
have no foundation. Fortunately this anecdote is well authenticated, and
moreover is intrinsically probable; I say fortunately, because it is
always painful to have to give up these child-learnt anecdotes, like
Alfred and the cakes and so on. This anecdote of the apple we need not
resign. The tree was blown down in 1820 and part of its wood is
preserved.
I have mentioned Voltaire in connection with Newton's philosophy. This
acute critic at a later stage did a good deal to popularise it
throughout Europe and to overturn that of his own countryman Descartes.
Cambridge rapidly became Newtonian, but Oxford remained Cartesian for
fifty years or more. It is curious what little hold science and
mathematics have ever secured in the older and more ecclesiastical
University. The pride of possessing Newton has however no doubt been the
main stimulus to the special pursuits of Cambridge.
He now began to turn his attention to optics, and, as was usual with
him, his whole mind became absorbed in this subject as if nothing else
had ever occupied him. His cash-book for this time has been discovered,
and the entries show that he is buying prisms and lenses and polishing
powder at the beginning of 1667. He was anxious to improve telescopes by
making more perfect lenses than had ever been used before. Accordingly
he calculated out their proper curves, just as Descartes had also done,
and then proceeded to grind them as near as he could to those figures.
But the images did not please him; they were always blurred and rather
indistinct.
At length, it struck him that perhaps it was not the lenses but the
light which was at fault. Perhaps light was so composed that it _could_
not be focused accurately to a sharp and definite point. Perhaps the law
of refraction was not quite accurate, but only an approximation. So he
bought a prism to try the law. He let in sunlight through a small round
hole in a window shutter, inserted the prism in the light, and received
the deflected beam on a white screen; turning the prism about till it
was deviated as little as possible. The patch on the screen was not a
round disk, as it would have been without the prism, but was an
elongated oval and was coloured at its extremities. Evidently
refraction was not a simple geometrical deflection of a ray, there was a
spreading out as well.
[Illustration: FIG. 63.--A prism not only _deviates_ a beam of sunlight,
but also spreads it out or _disperses_ it.]
Why did the image thus spread out? If it were due to irregularities in
the glass a second prism should rather increase them, but a second prism
when held in appropriate position was able to neutralise the dispersion
and to reproduce the simple round white spot without deviation.
Evidently the spreading out of the beam was connected in some definite
way with its refraction. Could it be that the light particles after
passing through the prism travelled in variously curved lines, as
spinning racquet balls do? To examine this he measured the length of the
oval patch when the screen was at different distances from the prism,
and found that the two things were directly proportional to each other.
Doubling the distance of the screen doubled the length of the patch.
Hence the rays travelled in straight lines from the prism, and the
spreading out was due to something that occurred within its substance.
Could it be that white light was compound, was a mixture of several
constituents, and that its different constituents were differently bent?
No sooner thought than tried. Pierce the screen to let one of the
constituents through and interpose a second prism in its path. If the
spreading out depended on the prism only it should spread out just as
much as before, but if it depended on the complex character of white
light, this isolated simple constituent should be able to spread out no
more. It did not spread out any more: a prism had no more dispersive
power over it; it was deflected by the appropriate amount, but it was
not analysed into constituents. It differed from sunlight in being
simple. With many ingenious and beautifully simple experiments, which
are quoted in full in several books on optics, he clinched the argument
and established his discovery. White light was not simple but compound.
It could be sorted out by a prism into an infinite number of constituent
parts which were differently refracted, and the most striking of which
Newton named violet, indigo, blue, green, yellow, orange, and red.
[Illustration: FIG. 64.--A single constituent of white light, obtained
by the use of perforated screens is capable of no more dispersion.]
At once the true nature of colour became manifest. Colour resided not in
the coloured object as had till now been thought, but in the light which
illuminated it. Red glass for instance adds nothing to sunlight. The
light does not get dyed red by passing through the glass; all that the
red glass does is to stop and absorb a large part of the sunlight; it is
opaque to the larger portion, but it is transparent to that particular
portion which affects our eyes with the sensation of red. The prism acts
like a sieve sorting out the different kinds of light. Coloured media
act like filters, stopping certain kinds but allowing the rest to go
through. Leonardo's and all the ancient doctrines of colour had been
singularly wrong; colour is not in the object but in the light.
Goethe, in his _Farbenlehre_, endeavoured to controvert Newton, and to
reinstate something more like the old views; but his failure was
complete.
Refraction analysed out the various constituents of white light and
displayed them in the form of a series of overlapping images of the
aperture, each of a different colour; this series of images we call a
spectrum, and the operation we now call spectrum analysis. The reason of
the defect of lenses was now plain: it was not so much a defect of the
lens as a defect of light. A lens acts by refraction and brings rays to
a focus. If light be simple it acts well, but if ordinary white light
fall upon a lens, its different constituents have different foci; every
bright object is fringed with colour, and nothing like a clear image can
be obtained.
[Illustration: FIG. 65.--Showing the boundary rays of a parallel beam
passing through a lens.]
A parallel beam passing through a lens becomes conical; but instead of a
single cone it is a sheaf or nest of cones, all having the edge of the
lens as base, but each having a different vertex. The violet cone is
innermost, near the lens, the red cone outermost, while the others lie
between. Beyond the crossing point or focus the order of cones is
reversed, as the above figure shows. Only the two marginal rays of the
beam are depicted.
If a screen be held anywhere nearer the lens than the place marked 1
there will be a whitish centre to the patch of light and a red and
orange fringe or border. Held anywhere beyond the region 2, the border
of the patch will be blue and violet. Held about 3 the colour will be
less marked than elsewhere, but nowhere can it be got rid of. Each point
of an object will be represented in the image not by a point but by a
coloured patch: a fact which amply explains the observed blurring and
indistinctness.
Newton measured and calculated the distance between the violet and red
foci--VR in the diagram--and showed that it was 1/50th the diameter of
the lens. To overcome this difficulty (called chromatic aberration)
telescope glasses were made small and of very long focus: some of them
so long that they had no tube, all of them egregiously cumbrous. Yet it
was with such instruments that all the early discoveries were made. With
such an instrument, for instance, Huyghens discovered the real shape of
Saturn's ring.
The defects of refractors seemed irremediable, being founded in the
nature of light itself. So he gave up his "glass works"; and proceeded
to think of reflexion from metal specula. A concave mirror forms an
image just as a lens does, but since it does so without refraction or
transmission through any substance, there is no accompanying dispersion
or chromatic aberration.
The first reflecting telescope he made was 1 in. diameter and 6 in.
long, and magnified forty times. It acted as well as a three or four
feet refractor of that day, and showed Jupiter's moons. So he made a
larger one, now in the library of the Royal Society, London, with an
inscription:
"The first reflecting telescope, invented by Sir Isaac Newton, and made
with his own hands."
This has been the parent of most of the gigantic telescopes of the
present day. Fifty years elapsed before it was much improved on, and
then, first by Hadley and afterwards by Herschel and others, large and
good reflectors were constructed.
The largest telescope ever made, that of Lord Rosse, is a Newtonian
reflector, fifty feet long, six feet diameter, with a mirror weighing
four tons. The sextant, as used by navigators, was also invented by
Newton.
The year after the plague, in 1667, Newton returned to Trinity College,
and there continued his experiments on optics. It is specially to be
noted that at this time, at the age of twenty-four, Newton had laid the
foundations of all his greatest discoveries:--
[Illustration: FIG. 66.--Newton's telescope.]
The Theory of Fluxions; or, the Differential Calculus.
The Law of Gravitation; or, the complete theory of astronomy.
The compound nature of white light; or, the beginning of Spectrum
Analysis.
[Illustration: FIG. 67.--The sextant, as now made.]
His later life was to be occupied in working these incipient discoveries
out. But the most remarkable thing is that no one knew about any one of
them. However, he was known as an accomplished young mathematician, and
was made a fellow of his college. You remember that he had a friend
there in the person of Dr. Isaac Barrow, first Lucasian Professor of
Mathematics in the University. It happened, about 1669, that a
mathematical discovery of some interest was being much discussed, and
Dr. Barrow happened to mention it to Newton, who said yes, he had worked
out that and a few other similar things some time ago. He accordingly
went and fetched some papers to Dr. Barrow, who forwarded them to other
distinguished mathematicians, and it thus appeared that Newton had
discovered theorems much more general than this special case that was
exciting so much interest. Dr. Barrow, being anxious to devote his time
more particularly to theology, resigned his chair the same year in
favour of Newton, who was accordingly elected to the Lucasian
Professorship, which he held for thirty years. This chair is now the
most famous in the University, and it is commonly referred to as the
chair of Newton.
Still, however, his method of fluxions was unknown, and still he did not
publish it. He lectured first on optics, giving an account of his
experiments. His lectures were afterwards published both in Latin and
English, and are highly valued to this day.
The fame of his mathematical genius came to the ears of the Royal
Society, and a motion was made to get him elected a fellow of that body.
The Royal Society, the oldest and most famous of all scientific
societies with a continuous existence, took its origin in some private
meetings, got up in London by the Hon. Robert Boyle and a few scientific
friends, during all the trouble of the Commonwealth.