problem as this:--Given three rigid spherical masses thrown into empty
space with any initial motions whatever, and abandoned to gravity: to
determine their subsequent motions. With two masses the problem is
simple enough, being pretty well summed up in Kepler's laws; but with
three masses, strange to say, it is so complicated as to be beyond the
reach of even modern mathematics. It is a famous problem, known as that
of "the three bodies," but it has not yet been solved. Even when it is
solved it will be only a close approximation to the case of earth, moon,
and sun, for these bodies are not spherical, and are not rigid. One may
imagine how absurdly and hopelessly complicated a complete treatment of
the motions of the entire solar system would be.
No. 8. Each planet is attracted not only by the sun but by the other
planets, hence their orbits are slightly affected by each other.
The subject of planetary perturbation was only just begun by Newton.
Gradually (by Laplace and others) the theory became highly developed;
and, as everybody knows, in 1846 Neptune was discovered by means of it.
No. 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.
It was a long time before Newton recognized the comets as real members
of the solar system, and subject to gravity like the rest. He at first
thought they moved in straight lines. It was only in the second edition
of the _Principia_ that the theory of comets was introduced.
Halley observed a fine comet in 1682, and calculated its orbit on
Newtonian principles. He also calculated when it ought to have been seen
in past times; and he found the year 1607, when one was seen by Kepler;
also the year 1531, when one was seen by Appian; again, he reckoned
1456, 1380, 1305. All these appearances were the same comet, in all
probability, returning every seventy-five or seventy-six years. The
period was easily allowed to be not exact, because of perturbing
planets. He then predicted its return for 1758, or perhaps 1759, a date
he could not himself hope to see. He lived to a great age, but he died
sixteen years before this date.
As the time drew nigh, three-quarters of a century afterwards,
astronomers were greatly interested in this first cometary prediction,
and kept an eager look-out for "Halley's comet." Clairaut, a most
eminent mathematician and student of Newton, proceeded to calculate out
more exactly the perturbing influence of Jupiter, near which it had
passed. After immense labour (for the difficulty of the calculation was
extreme, and the mass of mere figures something portentous), he
predicted its return on the 13th of April, 1759, but he considered that
he might have made a possible error of a month. It returned on the 13th
of March, 1759, and established beyond all doubt the rule of the
Newtonian theory over comets.
[Illustration: FIG. 71.--Well-known model exhibiting the oblate
spheroidal form as a consequence of spinning about a central axis. The
brass strip _a_ looks like a transparent globe when whirled, and bulges
out equatorially.]
No. 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.
Here we return to one of the more simple deductions. A spinning body of
any kind tends to swell at its circumference (or equator), and shrink
along its axis (or poles). If the body is of yielding material, its
shape must alter under the influence of centrifugal force; and if a
globe of yielding substance subject to known forces rotates at a
definite pace, its shape can be calculated. Thus a plastic sphere the
size of the earth, held together by its own gravity, and rotating once a
day, can be shown to have its equatorial diameter twenty-eight miles
greater than its polar diameter: the two diameters being 8,000 and 8,028
respectively. Now we have no guarantee that the earth is of yielding
material: for all Newton could tell it might be extremely rigid. As a
matter of fact it is now very nearly rigid. But he argued thus. The
water on it is certainly yielding, and although the solid earth might
decline to bulge at the equator in deference to the diurnal rotation,
that would not prevent the ocean from flowing from the poles to the
equator and piling itself up as an equatorial ocean fourteen miles deep,
leaving dry land everywhere near either pole. Nothing of this sort is
observed: the distribution of land and water is not thus regulated.
Hence, whatever the earth may be now, it must once have been plastic
enough to accommodate itself perfectly to the centrifugal forces, and to
take the shape appropriate to a perfectly plastic body. In all
probability it was once molten, and for long afterwards pasty.
Thus, then, the shape of the earth can be calculated from the length of
its day and the intensity of its gravity. The calculation is not
difficult: it consists in imagining a couple of holes bored to the
centre of the earth, one from a pole and one from the equator; filling
these both with water, and calculating how much higher the water will
stand in one leg of the gigantic V tube so formed than in the other. The
answer comes out about fourteen miles.
The shape of the earth can now be observed geodetically, and it accords
with calculation, but the observations are extremely delicate; in
Newton's time the _size_ was only barely known, the _shape_ was not
observed till long after; but on the principles of mechanics, combined
with a little common-sense reasoning, it could be calculated with
certainty and accuracy.
No. 11. From the observed shape of Jupiter or any planet the length of
its day could be estimated.
Jupiter is much more oblate than the earth. Its two diameters are to one
another as 17 is to 16; the ellipticity of its disk is manifest to
simple inspection. Hence we perceive that its whirling action must be
more violent--it must rotate quicker. As a matter of fact its day is ten
[Illustration: FIG. 72.--Jupiter.]
hours long--five hours daylight and five hours night. The times of
rotation of other bodies in the solar system are recorded in a table
above.
No. 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.
But following from the calculated shape of the earth follow several
interesting consequences. First of all, the intensity of gravity will
not be the same everywhere; for at the equator a stone is further from
the average bulk of the earth (say the centre) than it is at the poles,
and owing to this fact a mass of 590 pounds at the pole; would suffice
to balance 591 pounds at the equator, if the two could be placed in the
pans of a gigantic balance whose beam straddled along an earth's
quadrant. This is a _true_ variation of gravity due to the shape of the
earth. But besides this there is a still larger _apparent_ variation due
to centrifugal force, which affects all bodies at the equator but not
those at the poles. From this cause, even if the earth were a true
sphere, yet if it were spinning at its actual pace, 288 pounds at the
pole could balance 289 pounds at the equator; because at the equator the
true weight of the mass would not be fully appreciated, centrifugal
force would virtually diminish it by 1/289th of its amount.
In actual fact both causes co-exist, and accordingly the total variation
of gravity observed is compounded of the real and the apparent effects;
the result is that 194 pounds at a pole weighs as much as 195 pounds at
the equator.
No. 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.
A sphere composed of uniform material, or of materials arranged in
concentric strata, can be shown to attract external bodies as if its
mass were concentrated at its centre. A hollow sphere, similarly
composed, does the same, but on internal bodies it exerts no force at
all.
Hence, at all distances above the surface of the earth, gravity
decreases in inverse proportion as the square of the distance from the
centre of the earth increases; but, if you descend a mine, gravity
decreases in this case also as you leave the surface, though not at the
same rate as when you went up. For as you penetrate the crust you get
inside a concentric shell, which is thus powerless to act upon you, and
the earth you are now outside is a smaller one. At what rate the force
decreases depends on the distribution of density; if the density were
uniform all through, the law of variation would be the direct distance,
otherwise it would be more complicated. Anyhow, the intensity of gravity
is a maximum at the surface of the earth, and decreases as you travel
from the surface either up or down.
No. 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.
Here we come to a truly awful piece of reasoning. A sphere attracts as
if its mass were concentrated at its centre (No. 12), but a spheroid
does not. The earth is a spheroid, and hence it pulls and is pulled by
the moon with a slightly uncentric attraction. In other words, the line
of pull does not pass through its precise centre. Now when we have a
spinning body, say a top, overloaded on one side so that gravity acts on
it unsymmetrically, what happens? The axis of rotation begins to rotate
cone-wise, at a pace which depends on the rate of spin, and on the shape
and mass of the top, as well as on the amount and leverage of the
overloading.
Newton calculated out the rapidity of this conical motion of the axis of
the earth, produced by the slightly unsymmetrical pull of the moon, and
found that it would complete a revolution in 26,000 years--precisely
what was wanted to explain the precession of the equinoxes. In fact he
had discovered the physical cause of that precession.
Observe that there were three stages in this discovery of precession:--
First, the observation by Hipparchus, that the nodes, or intersections
of the earth's orbit (the sun's apparent orbit) with the plane of the
equator, were not stationary, but slowly moved.
Second, the description of this motion by Copernicus, by the statement
that it was due to a conical motion of the earth's axis of rotation
about its centre as a fixed point.
Third, the explanation of this motion by Newton as due to the pull of
the moon on the equatorial protuberance of the earth.
The explanation _could_ not have been previously suspected, for the
shape of the earth, on which the whole theory depends, was entirely
unknown till Newton calculated it.
Another and smaller motion of a somewhat similar kind has been worked
out since: it is due to the unsymmetrical attraction of the other
planets for this same equatorial protuberance. It shows itself as a
periodic change in the obliquity of the ecliptic, or so-called recession
of the apses, rather than as a motion of the nodes.[21]
No. 15. The waters of the ocean are attracted towards the sun and moon
on one side, and whirled a little farther away than the solid earth on
the other side: hence Newton explained all the main phenomena of the
tides.
And now comes another tremendous generalization. The tides had long been
an utter mystery. Kepler likens the earth to an animal, and the tides to
his breathings and inbreathings, and says they follow the moon.
Galileo chaffs him for this, and says that it is mere superstition to
connect the moon with the tides.
Descartes said the moon pressed down upon the waters by the centrifugal
force of its vortex, and so produced a low tide under it.
Everything was fog and darkness on the subject. The legend goes that an
astronomer threw himself into the sea in despair of ever being able to
explain the flux and reflux of its waters.
Newton now with consummate skill applied his theory to the effect of
the moon upon the ocean, and all the main details of tidal action
gradually revealed themselves to him.
He treated the water, rotating with the earth once a day, somewhat as if
it were a satellite acted on by perturbing forces. The moon as it
revolves round the earth is perturbed by the sun. The ocean as it
revolves round the earth (being held on by gravitation just as the moon
is) is perturbed by both sun and moon.
The perturbing effect of a body varies directly as its mass, and
inversely as the cube of its distance. (The simple law of inverse square
does not apply, because a perturbation is a differential effect: the
satellite or ocean when nearer to the perturbing body than the rest of
the earth, is attracted more, and when further off it is attracted less
than is the main body of the earth; and it is these differences alone
which constitute the perturbation.) The moon is the more powerful of the
two perturbing bodies, hence the main tides are due to the moon; and its
chief action is to cause a pair of low waves or oceanic humps, of
gigantic area, to travel round the earth once in a lunar day, _i.e._ in
about 24 hours and 50 minutes. The sun makes a similar but still lower
pair of low elevations to travel round once in a solar day of 24 hours.
And the combination of the two pairs of humps, thus periodically
overtaking each other, accounts for the well-known spring and neap
tides,--spring tides when their maxima agree, neap tides when the
maximum of one coincides with the minimum of the other: each of which
events happens regularly once a fortnight.
These are the main effects, but besides these there are the effects of
varying distances and obliquity to be taken into account; and so we have
a whole series of minor disturbances, very like those discussed in No.
7, under the lunar theory, but more complex still, because there are two
perturbing bodies instead of only one.
The subject of the tides is, therefore, very recondite; and though one
may give some elementary account of its main features, it will be best
to defer this to a separate lecture (Lecture XVII).
I had better, however, here say that Newton did not limit himself to the
consideration of the primary oceanic humps: he pursued the subject into
geographical detail. He pointed out that, although the rise and fall of
the tide at mid-ocean islands would be but small, yet on stretches of
coast the wave would fling itself, and by its momentum would propel the
waters, to a much greater height--for instance, 20 or 30 feet;
especially in some funnel-shaped openings like the Bristol Channel and
the Bay of Fundy, where the concentrated impetus of the water is
enormous.
He also showed how the tidal waves reached different stations in
successive regular order each day; and how some places might be fed with
tide by two distinct channels; and that if the time of these channels
happened to differ by six hours, a high tide might be arriving by one
channel and a low tide by the other, so that the place would only feel
the difference, and so have a very small observed rise and fall;
instancing a port in China (in the Gulf of Tonquin) where that
approximately occurs.
In fact, although his theory was not, as we now know, complete or final,
yet it satisfactorily explained a mass of intricate detail as well as
the main features of the tides.
No. 16. The sun's mass being known, he calculated the height of the
solar tide.
No. 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.
Knowing the sun's mass and distance, it was not difficult for Newton to
calculate the height of the protuberance caused by it in a pasty ocean
covering the whole earth. I say pasty, because, if there was any
tendency for impulses to accumulate, as timely pushes given to a
pendulum accumulate, the amount of disturbance might become excessive,
and its calculation would involve a multitude of data. The Newtonian
tide ignored this, thus practically treating the motion as either
dead-beat, or else the impulses as very inadequately timed. With this
reservation the mid-ocean tide due to the action of the sun alone comes
out about one foot, or let us say one foot for simplicity. Now the
actual tide observed in mid-Atlantic is at the springs about four feet,
at the neaps about two. The spring tide is lunar plus solar; the neap
tide is lunar minus solar. Hence it appears that the tide caused by the
moon alone must be about three feet, when unaffected by momentum. From
this datum Newton made the first attempt to approximately estimate the
mass of the moon. I said that the masses of satellites must be
estimated, if at all, by the perturbation they are able to cause. The
lunar tide is a perturbation in the diurnal motion of the sea, and its
amount is therefore a legitimate mode of calculating the moon's mass.
The available data were not at all good, however; nor are they even now
very perfect; and so the estimate was a good way out. It is now
considered that the mass of the moon is about one-eightieth that of the
earth.
* * * * *
Such are some of the gems extracted from their setting in the
_Principia_, and presented as clearly as I am able before you.
Do you realize the tremendous stride in knowledge--not a stride, as
Whewell says, nor yet a leap, but a flight--which has occurred between
the dim gropings of Kepler, the elementary truths of Galileo, the
fascinating but wild speculations of Descartes, and this magnificent and
comprehensive system of ordered knowledge. To some his genius seemed
almost divine. "Does Mr. Newton eat, drink, sleep, like other men?" said
the Marquis de l'Hopital, a French mathematician of no mean eminence; "I
picture him to myself as a celestial genius, entirely removed from the
restrictions of ordinary matter." To many it seemed as if there was
nothing more to be discovered, as if the universe were now explored, and
only a few fragments of truth remained for the gleaner. This is the
attitude of mind expressed in Pope's famous epigram:--
"Nature and Nature's laws lay hid in Night,
God said, Let Newton be, and all was light."
This feeling of hopelessness and impotence was very natural after the
advent of so overpowering a genius, and it prevailed in England for
fully a century. It was very natural, but it was very mischievous; for,
as a consequence, nothing of great moment was done by England in
science, and no Englishman of the first magnitude appeared, till some
who are either living now or who have lived within the present century.
It appeared to his contemporaries as if he had almost exhausted the
possibility of discovery; but did it so appear to Newton? Did it seem to
him as if he had seen far and deep into the truths of this great and
infinite universe? It did not. When quite an old man, full of honour and
renown, venerated, almost worshipped, by his contemporaries, these were
his words:--
"I know not what the world will think of my labours, but to myself it
seems that I have been but as a child playing on the sea-shore; now
finding some pebble rather more polished, and now some shell rather more
agreeably variegated than another, while the immense ocean of truth
extended itself unexplored before me."
And so it must ever seem to the wisest and greatest of men when brought
into contact with the great things of God--that which they know is as
nothing, and less than nothing, to the infinitude of which they are
ignorant.
Newton's words sound like a simple and pleasing echo of the words of
that great unknown poet, the writer of the book of Job:--
"Lo, these are parts of His ways,
But how little a portion is heard of Him;
The thunder of His power, who can understand?"
END OF PART I.
PART II
_A COUPLE OF CENTURIES' PROGRESS._
NOTES TO LECTURE X
_Science during the century after Newton_
The _Principia_ published, 1687
Roemer 1644-1710
James Bradley 1692-1762
Clairaut 1713-1765
Euler 1707-1783
D'Alembert 1717-1783
Lagrange 1736-1813
Laplace 1749-1827
William Herschel 1738-1822
_Olaus Roemer_ was born in Jutland, and studied at Copenhagen. Assisted
Picard in 1671 to determine the exact position of Tycho's observatory on
Huen. Accompanied Picard to Paris, and in 1675 read before the Academy
his paper "On Successive Propagation of Light as revealed by a certain
inequality in the motion of Jupiter's First Satellite." In 1681 he
returned to Copenhagen as Professor of Mathematics and Astronomy, and
died in 1710. He invented the transit instrument, mural circle,
equatorial mounting for telescopes, and most of the other principal
instruments now in use in observatories. He made as many observations as
Tycho Brahe, but the records of all but the work of three days were
destroyed by a great fire in 1728.
_Bradley_, Professor of Astronomy at Oxford, discovered the aberration
of light in 1729, while examining stars for parallax, and the nutation
of the earth's axis in 1748. Was appointed Astronomer-Royal in 1742.
LECTURE X
ROEMER AND BRADLEY AND THE VELOCITY OF LIGHT
At Newton's death England stood pre-eminent among the nations of Europe
in the sphere of science. But the pre-eminence did not last long. Two
great discoveries were made very soon after his decease, both by
Professor Bradley, of Oxford, and then there came a gap. A moderately
great man often leaves behind him a school of disciples able to work
according to their master's methods, and with a healthy spirit of
rivalry which stimulates and encourages them. Newton left, indeed, a
school of disciples, but his methods of work were largely unknown to
them, and such as were known were too ponderous to be used by ordinary
men. Only one fresh result, and that a small one, has ever been attained
by other men working according to the methods of the _Principia_. The
methods were studied and commented on in England to the exclusion of all
others for nigh a century, and as a consequence no really important work
was done.
On the Continent, however, no such system of slavish imitation
prevailed. Those methods of Newton's which had been simultaneously
discovered by Leibnitz were more thoroughly grasped, modified, extended,
and improved. There arose a great school of French and German
mathematicians, and the laurels of scientific discovery passed to France
and Germany--more especially, perhaps, at this time to France. England
has never wholly recovered them. During the present century this country
has been favoured with some giants who, as they become distant enough
for their true magnitude to be perceived, may possibly stand out as
great as any who have ever lived; but for the mass and bulk of
scientific work at the present day we have to look to Germany, with its
enlightened Government and extensive intellectual development. England,
however, is waking up, and what its Government does not do, private
enterprise is beginning to accomplish. The establishment of centres of
scientific and literary activity in the great towns of England, though
at present they are partially encumbered with the supply of education of
an exceedingly rudimentary type, is a movement that in the course of
another century or so will be seen to be one of the most important and
fruitful steps ever taken by this country. On the Continent such centres
have long existed; almost every large town is the seat of a University,
and they are now liberally endowed. The University of Bologna (where,
you may remember, Copernicus learnt mathematics) has recently celebrated
its 800th anniversary.
The scientific history of the century after Newton, summarized in the
above table of dates, embraces the labours of the great mathematicians
Clairaut, Euler, D'Alembert, and especially of Lagrange and Laplace.