gives terrestrial bodies their weight and regulates the motion of
projectiles. [Because, while a stone drops 16 feet in a second, the
moon, which is 60 times as far from the centre of the earth, drops 16
feet in a minute.]
* * * * *
7. The moon is attracted not only by the earth, but by the sun also;
hence its orbit is perturbed, and Newton calculated out the chief of
these perturbations, viz.:--
(The equation of the centre, discovered by Hipparchus.)
(_a_) The evection, discovered by Hipparchus and Ptolemy.
(_b_) The variation, discovered by Tycho Brahe.
(_c_) The annual equation, discovered by Tycho Brahe.
(_d_) The retrogression of the nodes, then being observed at
Greenwich by Flamsteed.
(_e_) The variation of inclination, then being observed at
Greenwich by Flamsteed.
(_f_) The progression of the apses (with an error of one-half).
(_g_) The inequality of apogee, previously unknown.
(_h_) The inequality of nodes, previously unknown.
8. Each planet is attracted not only by the sun but by the other
planets, hence their orbits are slightly affected by each other. Newton
began the theory of planetary perturbations.
9. He recognized the comets as members of the solar system, obedient to
the same law of gravity and moving in very elongated ellipses; so their
return could be predicted (_e.g._ Halley's comet).
10. Applying the idea of centrifugal force to the earth considered as a
rotating body, he perceived that it could not be a true sphere, and
calculated its oblateness, obtaining 28 miles greater equatorial than
polar diameter.
11. Conversely, from the observed shape of Jupiter, or any planet, the
length of its day could be estimated.
12. The so-calculated shape of the earth, in combination with
centrifugal force, causes the weight of bodies to vary with latitude;
and Newton calculated the amount of this variation. 194 lbs. at pole
balance 195 lbs. at equator.
13. A homogeneous sphere attracts as if its mass were concentrated at
its centre. For any other figure, such as an oblate spheroid, this is
not exactly true. A hollow concentric spherical shell exerts no force on
small bodies inside it.
14. The earth's equatorial protuberance, being acted on by the
attraction of the sun and moon, must disturb its axis of rotation in a
calculated manner; and thus is produced the precession of the equinoxes.
[The attraction of the planets on the same protuberance causes a smaller
and rather different kind of precession.]
15. The waters of the ocean are attracted towards the sun and moon on
one side, and whirled a little further away than the solid earth on the
other side: hence Newton explained all the main phenomena of the tides.
16. The sun's mass being known, he calculated the height of the solar
tide.
17. From the observed heights of spring and neap tides he determined the
lunar tide, and thence made an estimate of the mass of the moon.
REFERENCE TABLE OF NUMERICAL DATA.
+---------+---------------+----------------------+-----------------+
| |Masses in Solar| Height dropped by a | Length of Day or|
| | System. |stone in first second.|time of rotation.|
+---------+---------------+----------------------+-----------------+
|Mercury | .065 | 7.0 feet | 24 hours |
|Venus | .885 | 15.8 " | 23-1/2 " |
|Earth | 1.000 | 16.1 " | 24 " |
|Mars | .108 | 6.2 " | 24-1/2 " |
|Jupiter | 300.8 | 45.0 " | 10 " |
|Saturn | 89.7 | 18.4 " | 10-1/2 " |
|The Sun | 316000. | 436.0 " | 608 " |
|The Moon | about .012 | 3.7 " | 702 " |
+---------+---------------+----------------------+-----------------+
The mass of the earth, taken above as unity, is 6,000 trillion tons.
_Observatories._--Uraniburg flourished from 1576 to 1597; the
Observatory of Paris was founded in 1667; Greenwich Observatory in 1675.
_Astronomers-Royal._--Flamsteed, Halley, Bradley, Bliss, Maskelyne,
Pond, Airy, Christie.
LECTURE IX
NEWTON'S "PRINCIPIA"
The law of gravitation, above enunciated, in conjunction with the laws
of motion rehearsed at the end of the preliminary notes of Lecture VII.,
now supersedes the laws of Kepler and includes them as special cases.
The more comprehensive law enables us to criticize Kepler's laws from a
higher standpoint, to see how far they are exact and how far they are
only approximations. They are, in fact, not precisely accurate, but the
reason for every discrepancy now becomes abundantly clear, and can be
worked out by the theory of gravitation.
We may treat Kepler's laws either as immediate consequences of the law
of gravitation, or as the known facts upon which that law was founded.
Historically, the latter is the more natural plan, and it is thus that
they are treated in the first three statements of the above notes; but
each proposition may be worked inversely, and we might state them
thus:--
1. The fact that the force acting on each planet is directed to the sun,
necessitates the equable description of areas.
2. The fact that the force varies as the inverse square of the distance,
necessitates motion in an ellipse, or some other conic section, with the
sun in one focus.
3. The fact that one attracting body acts on all the planets with an
inverse square law, causes the cubes of their mean distances to be
proportional to the squares of their periodic times.
Not only these but a multitude of other deductions follow rigorously
from the simple datum that every particle of matter attracts every other
particle with a force directly proportional to the mass of each and to
the inverse square of their mutual distance. Those dealt with in the
_Principia_ are summarized above, and it will be convenient to run over
them in order, with the object of giving some idea of the general
meaning of each, without attempting anything too intricate to be readily
intelligible.
[Illustration: FIG. 70.]
No. 1. Kepler's second law (equable description of areas) proves that
each planet is acted on by a force directed towards the sun as a centre
of force.
The equable description of areas about a centre of force has already
been fully, though briefly, established. (p. 175.) It is undoubtedly of
fundamental importance, and is the earliest instance of the serious
discussion of central forces, _i.e._ of forces directed always to a
fixed centre.
We may put it afresh thus:--OA has been the motion of a particle in a
unit of time; at A it receives a knock towards C, whereby in the next
unit it travels along AD instead of AB. Now the area of the triangle
CAD, swept out by the radius vector in unit time, is 1/2_bh_; _h_ being
the perpendicular height of the triangle from the base AC. (Fig. 70.)
Now the blow at A, being along the base, has no effect upon _h_; and
consequently the area remains just what it would have been without the
blow. A blow directed to any point other than C would at once alter the
area of the triangle.
One interesting deduction may at once be drawn. If gravity were a
radiant force emitted from the sun with a velocity like that of light,
the moving planet would encounter it at a certain apparent angle
(aberration), and the force experienced would come from a point a little
in advance of the sun. The rate of description of areas would thus tend
to increase; whereas in reality it is constant. Hence the force of
gravity, if it travel at all, does so with a speed far greater than that
of light. It appears to be practically instantaneous. (Cf. "Modern Views
of Electricity," Sec. 126, end of chap. xii.) Again, anything like a
retarding effect of the medium through which the planets move would
constitute a tangential force, entirely un-directed towards the sun.
Hence no such frictional or retarding force can appreciably exist. It
is, however, conceivable that both these effects might occur and just
neutralize each other. The neutralization is unlikely to be exact for
all the planets; and the fact is, that no trace of either effect has as
yet been discovered. (See also p. 176.)
The planets are, however, subject to forces not directed towards the
sun, viz. their attractions for each other; and these perturbing forces
do produce a slight discrepancy from Kepler's second law, but a
discrepancy which is completely subject to calculation.
No. 2. Kepler's first law proves that this central force diminishes in
the same proportion as the square of the distance increases.
To prove the connection between the inverse-square law of distance, and
the travelling in a conic section with the centre of force in one focus
(the other focus being empty), is not so simple. It obviously involves
some geometry, and must therefore be left to properly armed students.
But it may be useful to state that the inverse-square law of distance,
although the simplest possible law for force emanating from a point or
sphere, is not to be regarded as self-evident or as needing no
demonstration. The force of a magnetic pole on a magnetized steel scrap,
for instance, varies as the inverse cube of the distance; and the curve
described by such a particle would be quite different from a conic
section--it would be a definite class of spiral (called Cotes's spiral).
Again, on an iron filing the force of a single pole might vary more
nearly as the inverse fifth power; and so on. Even when the thing
concerned is radiant in straight lines, like light, the law of inverse
squares is not universally true. Its truth assumes, first, that the
source is a point or sphere; next, that there is no reflection or
refraction of any kind; and lastly, that the medium is perfectly
transparent. The law of inverse squares by no means holds from a prairie
fire for instance, or from a lighthouse, or from a street lamp in a fog.
Mutual perturbations, especially the pull of Jupiter, prevent the path
of a planet from being really and truly an ellipse, or indeed from being
any simple re-entrant curve. Moreover, when a planet possesses a
satellite, it is not the centre of the planet which ever attempts to
describe the Keplerian ellipse, but it is the common centre of gravity
of the two bodies. Thus, in the case of the earth and moon, the point
which really does describe a close attempt at an ellipse is a point
displaced about 3000 miles from the centre of the earth towards the
moon, and is therefore only 1000 miles beneath the surface.
No. 3. Kepler's third law proves that all the planets are acted on by
the same kind of force; of an intensity depending on the mass of the
sun.
The third law of Kepler, although it requires geometry to state and
establish it for elliptic motion (for which it holds just as well as it
does for circular motion), is very easy to establish for circular
motion, by any one who knows about centrifugal force. If _m_ is the mass
of a planet, _v_ its velocity, _r_ the radius of its orbit, and _T_ the
time of describing it; 2[pi]_r_ = _vT_, and the centripetal force
needed to hold it in its orbit is
mv^2 4[pi]^2_mr_
-------- or -----------
_r_ T^2
Now the force of gravitative attraction between the planet and the sun
is
_VmS_
-----,
r^2
where _v_ is a fixed quantity called the gravitation-constant, to be
determined if possible by experiment once for all. Now, expressing the
fact that the force of gravitation _is_ the force holding the planet in,
we write,
4[pi]^2_mr_ _VmS_
----------- = ---------,
T^2 r^2
whence, by the simplest algebra,
r^3 _VS_
------ = ---------.
T^2 4[pi]^2
The mass of the planet has been cancelled out; the mass of the sun
remains, multiplied by the gravitation-constant, and is seen to be
proportional to the cube of the distance divided by the square of the
periodic time: a ratio, which is therefore the same for all planets
controlled by the sun. Hence, knowing _r_ and _T_ for any single planet,
the value of _VS_ is known.
No. 4. So by knowing the length of year and distance of any planet from
the sun, the sun's mass can be calculated, in terms of that of the
earth.
No. 5. For the satellites, the force acting depends on the mass of
_their_ central body, a planet. Hence the mass of any planet possessing
a satellite becomes known.
The same argument holds for any other system controlled by a central
body--for instance, for the satellites of Jupiter; only instead of _S_
it will be natural to write _J_, as meaning the mass of Jupiter. Hence,
knowing _r_ and _T_ for any one satellite of Jupiter, the value of _VJ_
is known.
Apply the argument also to the case of moon and earth. Knowing the
distance and time of revolution of our moon, the value of _VE_ is at
once determined; _E_ being the mass of the earth. Hence, _S_ and _J_,
and in fact the mass of any central body possessing a visible satellite,
are now known in terms of _E_, the mass of the earth (or, what is
practically the same thing, in terms of _V_, the gravitation-constant).
Observe that so far none of these quantities are known absolutely. Their
relative values are known, and are tabulated at the end of the Notes
above, but the finding of their absolute values is another matter, which
we must defer.
But, it may be asked, if Kepler's third law only gives us the mass of a
_central_ body, how is the mass of a _satellite_ to be known? Well, it
is not easy; the mass of no satellite is known with much accuracy. Their
mutual perturbations give us some data in the case of the satellites of
Jupiter; but to our own moon this method is of course inapplicable. Our
moon perturbs at first sight nothing, and accordingly its mass is not
even yet known with exactness. The mass of comets, again, is quite
unknown. All that we can be sure of is that they are smaller than a
certain limit, else they would perturb the planets they pass near.
Nothing of this sort has ever been detected. They are themselves
perturbed plentifully, but they perturb nothing; hence we learn that
their mass is small. The mass of a comet may, indeed, be a few million
or even billion tons; but that is quite small in astronomy.
But now it may be asked, surely the moon perturbs the earth, swinging it
round their common centre of gravity, and really describing its own
orbit about this point instead of about the earth's centre? Yes, that is
so; and a more precise consideration of Kepler's third law enables us to
make a fair approximation to the position of this common centre of
gravity, and thus practically to "weigh the moon," i.e. to compare its
mass with that of the earth; for their masses will be inversely as their
respective distances from the common centre of gravity or balancing
point--on the simple steel-yard principle.
Hitherto we have not troubled ourselves about the precise point about
which the revolution occurs, but Kepler's third law is not precisely
accurate unless it is attended to. The bigger the revolving body the
greater is the discrepancy: and we see in the table preceding Lecture
III., on page 57, that Jupiter exhibits an error which, though very
slight, is greater than that of any of the other planets, when the sun
is considered the fixed centre.
Let the common centre of gravity of earth and moon be displaced a
distance _x_ from the centre of the earth, then the moon's distance
from the real centre of revolution is not _r_, but _r-x_; and the
equation of centrifugal force to gravitative-attraction is strictly
4[pi]^2 _VE_
--------- (_r-x_) = ------,
T^2 r^2
instead of what is in the text above; and this gives a slightly
modified "third law." From this equation, if we have any distinct
method of determining _VE_ (and the next section gives such a
method), we can calculate _x_ and thus roughly weigh the moon,
since
_r-x_ E
----- = -----,
_r_ E+M
but to get anything like a reasonable result the data must be very
precise.
No. 6. The force constraining the moon in her orbit is the same gravity
as gives terrestrial bodies their weight and regulates the motion of
projectiles.
Here we come to the Newtonian verification already several times
mentioned; but because of its importance I will repeat it in other
words. The hypothesis to be verified is that the force acting on the
moon is the same kind of force as acts on bodies we can handle and
weigh, and which gives them their weight. Now the weight of a mass _m_
is commonly written _mg_, where _g_ is the intensity of terrestrial
gravity, a thing easily measured; being, indeed, numerically equal to
twice the distance a stone drops in the first second of free fall. [See
table p. 205.] Hence, expressing that the weight of a body is due to
gravity, and remembering that the centre of the earth's attraction is
distant from us by one earth's radius (R), we can write
_Vm_E
_mg_ = ------,
R^2
or
_V_E = gR^2 = 95,522 cubic miles-per-second per second.
But we already know _v_E, in terms of the moon's motion, as
4[pi]^2r^3
-----------
T^2
approximately, [more accurately, see preceding note, this quantity is
_V_(E + M)]; hence we can easily see if the two determinations of this
quantity agree.[20]
All these deductions are fundamental, and may be considered as the
foundation of the _Principia_. It was these that flashed upon Newton
during that moment of excitement when he learned the real size of the
earth, and discovered his speculations to be true.
The next are elaborations and amplifications of the theory, such as in
ordinary times are left for subsequent generations of theorists to
discover and work out.
Newton did not work out these remoter consequences of his theory
completely by any means: the astronomical and mathematical world has
been working them out ever since; but he carried the theory a great way,
and here it is that his marvellous power is most conspicuous.
It is his treatment of No. 7, the perturbations of the moon, that
perhaps most especially has struck all future mathematicians with
amazement. No. 7, No. 14, No. 15, these are the most inspired of the
whole.
No. 7. The moon is attracted not only by the earth, but by the sun also;
hence its orbit is perturbed, and Newton calculated out the chief of
these perturbations.
Now running through the perturbations (p. 203) in order:--The first is
in parenthesis, because it is mere excentricity. It is not a true
perturbation at all, and more properly belongs to Kepler.
(_a_) The first true perturbation is what Ptolemy called "the evection,"
the principal part of which is a periodic change in the ellipticity or
excentricity of the moon's orbit, owing to the pull of the sun. It is a
complicated matter, and Newton only partially solved it. I shall not
attempt to give an account of it.
(_b_) The next, "the variation," is a much simpler affair. It is caused
by the fact that as the moon revolves round the earth it is half the
time nearer to the sun than the earth is, and so gets pulled more than
the average, while for the other fortnight it is further from the sun
than the earth is, and so gets pulled less. For the week during which
it is changing from a decreasing half to a new moon it is moving in the
direction of the extra pull, and hence becomes new sooner than would
have been expected. All next week it is moving against the same extra
pull, and so arrives at quadrature (half moon) somewhat late. For the
next fortnight it is in the region of too little pull, the earth gets
pulled more than it does; the effect of this is to hurry it up for the
third week, so that the full moon occurs a little early, and to retard
it for the fourth week, so that the decreasing half moon like the
increasing half occurs behind time again. Thus each syzygy (as new and
full are technically called) is too early; each quadrature is too late;
the maximum hurrying and slackening force being felt at the octants, or
intermediate 45 deg. points.
(_c_) The "annual equation" is a fluctuation introduced into the other
perturbations by reason of the varying distance of the disturbing body,
the sun, at different seasons of the year. Its magnitude plainly depends
simply on the excentricity of the earth's orbit.
Both these perturbations, (_b_) and (_c_), Newton worked out completely.
(_d_) and (_e_) Next come the retrogression of the nodes and the
variation of the inclination, which at the time were being observed at
Greenwich by Flamsteed, from whom Newton frequently, but vainly, begged
for data that he might complete their theory while he had his mind upon
it. Fortunately, Halley succeeded Flamsteed as Astronomer-Royal [see
list at end of notes above], and then Newton would have no difficulty in
gaining such information as the national Observatory could give.
The "inclination" meant is the angle between the plane of the moon's
orbit and that of the earth. The plane of the earth's orbit round the
sun is called the ecliptic; the plane of the moon's orbit round the
earth is inclined to it at a certain angle, which is slowly changing,
though in a periodic manner. Imagine a curtain ring bisected by a sheet
of paper, and tilted to a certain angle; it may be likened to the moon's
orbit, cutting the plane of the ecliptic. The two points at which the
plane is cut by the ring are called "nodes"; and these nodes are not
stationary, but are slowly regressing, _i.e._ travelling in a direction
opposite to that of the moon itself. Also the angle of tilt is varying
slowly, oscillating up and down in the course of centuries.
(_f_) The two points in the moon's elliptic orbit where it comes nearest
to or farthest from the earth, _i.e._ the points at the extremity of the
long axis of the ellipse, are called separately perigee and apogee, or
together "the apses." Now the pull of the sun causes the whole orbit to
slowly revolve in its own plane, and consequently these apses
"progress," so that the true path is not quite a closed curve, but a
sort of spiral with elliptic loops.
But here comes in a striking circumstance. Newton states with reference
to this perturbation that theory only accounts for 1-1/2 deg. per annum,
whereas observation gives 3 deg., or just twice as much.
This is published in the _Principia_ as a fact, without comment. It was
for long regarded as a very curious thing, and many great mathematicians
afterwards tried to find an error in the working. D'Alembert, Clairaut,
and others attacked the problem, but were led to just the same result.
It constituted the great outstanding difficulty in the way of accepting
the theory of gravitation. It was suggested that perhaps the inverse
square law was only a first approximation; that perhaps a more complete
expression, such as
A B
---- + -----,
r^2 r^4
must be given for it; and so on.
Ultimately, Clairaut took into account a whole series of neglected
terms, and it came out correct; thus verifying the theory.
But the strangest part of this tale is to come. For only a few years
ago, Prof. Adams, of Cambridge (Neptune Adams, as he is called), was
editing various old papers of Newton's, now in the possession of the
Duke of Portland, and he found manuscripts bearing on this very point,
and discovered that Newton had reworked out the calculations himself,
had found the cause of the error, had taken into account the terms
hitherto neglected, and so, fifty years before Clairaut, had completely,
though not publicly, solved this long outstanding problem of the
progression of the apses.
(_g_) and (_h_) Two other inequalities he calculated out and predicted,
viz. variation in the motions of the apses and the nodes. Neither of
these had then been observed, but they were afterwards detected and
verified.
A good many other minor irregularities are now known--some thirty, I
believe; and altogether the lunar theory, or problem of the moon's exact
motion, is one of the most complicated and difficult in astronomy; the
perturbations being so numerous and large, because of the enormous mass
of the perturbing body.
The disturbances experienced by the planets are much smaller, because
they are controlled by the sun and perturbed by each other. The moon is
controlled only by the earth, and perturbed by the sun. Planetary
perturbations can be treated as a series of disturbances with some
satisfaction: not so those of the moon. And yet it is the only way at
present known of dealing with the lunar theory.