matured into a system by Lord Bacon, a celebrated English philosopher,
early in the seventeenth century,--indeed, during the life of Galileo.
Previous to that time, the inquirers into Nature did not open their eyes
to see how the facts really _are_; but, by metaphysical processes, in
imitation of Aristotle, determined how they _ought to be_, and hastily
concluded that they were so. Thus, they did not study into the laws of
motion, by observing how motion actually takes place, under various
circumstances, but first, in their closets, constructed a definition of
motion, and thence inferred all its properties. The system of reasoning
respecting the phenomena of Nature, introduced by Lord Bacon, was this:
in the first place, to examine all the facts of the case, and then from
these to determine the laws of Nature. To derive general conclusions
from the comparison of a great number of individual instances
constitutes the peculiarity of the Baconian philosophy. It is called the
_inductive_ system, because its conclusions were built on the induction,
or comparison, of a great many single facts. Previous to the time of
Lord Bacon, hardly any insight had been gained into the causes of
natural phenomena, and hardly one of the laws of Nature had been clearly
established, because all the inquirers into Nature were upon a wrong
road, groping their way through the labyrinth of error. Bacon pointed
out to them the true path, and held before them the torch-light of
experiment and observation, under whose guidance all successful students
of Nature have since walked, and by whose illumination they have gained
so wonderful an insight into the mysteries of the natural world.
It is a remarkable fact, that two such characters as Bacon and Galileo
should appear on the stage at the same time, who, without any
communication with each other, or perhaps without any personal knowledge
of each other's existence, should have each developed the true method of
investigating the laws of Nature. Galileo practised what Bacon only
taught; and some, therefore, with much reason, consider Galileo as a
greater philosopher than Bacon. "Bacon," says Hume, "pointed out, at a
great distance, the road to philosophy; Galileo both pointed it out to
others, and made, himself, considerable advances in it. The Englishman
was ignorant of geometry; the Florentine revived that science, excelled
in it, and was the first who applied it, together with experiment, to
natural philosophy. The former rejected, with the most positive disdain,
the system of Copernicus; the latter fortified it with new proofs,
derived both from reason and the senses."
When we reflect that geometry is a science built upon self-evident
truths, and that all its conclusions are the result of pure
demonstration, and can admit of no controversy; when we further reflect,
that experimental evidence rests on the testimony of the senses, and we
infer a thing to be true because we actually see it to be so; it shows
us the extreme bigotry, the darkness visible, that beclouded the human
intellect, when it not only refused to admit conclusions first
established by pure geometrical reasoning, and afterwards confirmed by
experiments exhibited in the light of day, but instituted the most cruel
persecutions against the great philosopher who first proclaimed these
truths. Galileo was hated and persecuted by two distinct bodies of men,
both possessing great influence in their respective spheres,--the one
consisting of the learned doctors of philosophy, who did nothing more,
from age to age, than reiterate the doctrines of Aristotle, and were
consequently alarmed at the promulgation of principles subversive of
those doctrines; the other consisting of the Romish priesthood,
comprising the terrible Inquisition, who denounced the truths taught by
Galileo, as inconsistent with certain declarations of the Holy
Scriptures. We shall see, as we advance, what a fearful warfare he had
to wage against these combined powers of darkness.
Aristotle had asserted, that, if two different weights of the same
material were let fall from the same height, the heavier one would reach
the ground sooner than the other, in proportion as it was more weighty.
For example: if a ten-pound leaden weight and a one-pound were let fall
from a given height at the same instant, the former would reach the
ground ten times as soon as the latter. No one thought of making the
trial, but it was deemed sufficient that Aristotle had said so; and
accordingly this assertion had long been received as an axiom in the
science of motion. Galileo ventured to appeal from the authority of
Aristotle to that of his own senses, and maintained, that both weights
would fall in the same time. The learned doctors ridiculed the idea.
Galileo tried the experiment in their presence, by letting fall, at the
same instant, large and small weights from the top of the celebrated
leaning tower of Pisa. Yet, with the sound of the two weights clicking
upon the pavement at the same moment, they still maintained that the
ten-pound weight would reach the ground in one tenth part of the time of
the other, because they could quote the chapter and verse of Aristotle
where the fact was asserted. Wearied and disgusted with the malice and
folly of these Aristotelian philosophers, Galileo, at the age of
twenty-eight, resigned his situation in the university of Pisa, and
removed to Padua, in the university of which place he was elected
professor of mathematics. Up to this period, Galileo had devoted himself
chiefly to the studies of the laws of motion, and the other branches of
mechanical philosophy. Soon afterwards, he began to publish his
writings, in rapid succession, and became at once among the most
conspicuous of his age,--a rank which he afterwards well sustained and
greatly exalted, by the invention of the telescope, and by his numerous
astronomical discoveries. I will reserve an account of these great
achievements until we come to that part of astronomy to which they were
more immediately related, and proceed, now, to explain to you the
leading principles of _terrestrial gravity_, as exemplified in falling
bodies.
First, _all bodies near the earth's surface fall in straight lines
towards the centre of the earth_. We are not to infer from this fact,
that there resides at the centre any peculiar force, as a great
loadstone, for example, which attracts bodies towards itself; but bodies
fall towards the centre of the sphere, because the combined attractions
of all the particles of matter in the earth, each exerting its proper
force upon the body, would carry it towards the centre. This may be
easily illustrated by a diagram. Let B, Fig. 29, page 140, be the
centre of the earth, and A a body without it. Every portion of matter in
the earth exerts some force on A, to draw it down to the earth. But
since there is just as much matter on one side of the line A B, as on
the other side, each half exerts an equal force to draw the body towards
itself; therefore it falls in the direction of the diagonal between the
two forces. Thus, if we compare the effects of any two particles of
matter at equal distances from the line A B, but on opposite sides of
it, as _a_, _b_, while the force of the particle at _a_ would tend to
draw A in the direction of A _a_, that of _b_ would draw it in the
direction of A _b_, and it would fall in the line A B, half way between
the two. The same would hold true of any other two corresponding
particles of matter on different sides of the earth, in respect to a
body situated in any place without it.
[Illustration Fig. 29.]
Secondly, _all bodies fall towards the earth, from the same height, with
equal velocities_. A musket-ball, and the finest particle of down, if
let fall from a certain height towards the earth, tend to descend
towards it at the same rate, and would proceed with equal speed, were it
not for the resistance of the air, which retards the down more than it
does the ball, and finally stops it. If, however, the air be removed out
of the way, as it may be by means of the air-pump, the two bodies keep
side by side in falling from the greatest height at which we can try the
experiment.
Thirdly, _bodies, in falling towards the earth, have their rate of
motion continually accelerated_. Suppose we let fall a musket-ball from
the top of a high tower, and watch its progress, disregarding the
resistance of the air: the first second, it will pass over sixteen feet
and one inch, but its speed will be constantly increased, being all the
while urged onward by the same force, and retaining all that it has
already acquired; so that the longer it is in falling, the swifter its
motion becomes. Consequently, when bodies fall from a great height, they
acquire an immense velocity before they reach the earth. Thus, a man
falling from a balloon, or from the mast-head of a ship, is broken in
pieces; and those meteoric stones, which sometimes fall from the sky,
bury themselves deep in the earth. On measuring the spaces through which
a body falls, it is found, that it will fall four times as far in two
seconds as in one, and one hundred times as far in ten seconds as in
one; and universally, the space described by a falling body is
proportioned to the time multiplied into itself; that is, to the square
of the time.
Fourthly, _gravity is proportioned to the quantity of matter_. A body
which has twice as much matter as another exerts a force of attraction
twice as great, and also receives twice as much from the same body as it
would do, if it were only just as heavy as that body. Thus the earth,
containing, as it does, forty times as much matter as the moon, exerts
upon the moon forty times as much force as it would do, were its mass
the same with that of the moon; but it is also capable of _receiving_
forty times as much gravity from the moon as it would do, were its mass
the same as the moon's; so that the power of attracting and that of
being attracted are reciprocal; and it is therefore correct to say, that
the moon attracts the earth _just as much_ as the earth attracts the
moon; and the same may be said of any two bodies, however different in
quantity of matter.
Fifthly, _gravity, when acting at a distance from the earth, is not as
intense as it is near the earth_. At such a distance as we are
accustomed to ascend above the general level of the earth, no great
difference is observed. On the tops of high mountains, we find bodies
falling towards the earth, with nearly the same speed as they do from
the smallest elevations. It is found, nevertheless, that there is a real
difference; so that, in fact, the weight of a body (which is nothing
more than the measure of its force of gravity) is not quite so great on
the tops of high mountains as at the general level of the sea. Thus, a
thousand pounds' weight, on the top of a mountain half a mile high,
would weigh a quarter of a pound less than at the level of the sea; and
if elevated four thousand miles above the earth,--that is, _twice_ as
far from the centre of the earth as the surface is from the centre,--it
would weigh only one fourth as much as before; if _three times_ as far,
it would weigh only one ninth as much. So that the force of gravity
decreases, as we recede from the earth, in the same proportion as the
square of the distance increases. This fact is generalized by saying,
that _the force of gravity, at different distances from the earth, is
inversely as the square of the distance_.
Were a body to fall from a great distance,--suppose a thousand times
that of the radius of the earth,--the force of gravity being one million
times less than that at the surface of the earth, the motion of the body
would be exceedingly slow, carrying it over only the sixth part of an
inch in a day. It would be a long time, therefore, in making any
sensible approaches towards the earth; but at length, as it drew near to
the earth it would acquire a very great velocity, and would finally rush
towards it with prodigious violence. Falling so far, and being
continually accelerated on the way, we might suppose that it would at
length attain a velocity infinitely great; but it can be demonstrated,
that, if a body were to fall from an infinite distance, attracted to the
earth only by gravity, it could never acquire a velocity greater than
about seven miles per second. This, however, is a speed inconceivably
great, being about eighteen times the greatest velocity that can be
given to a cannon-ball, and more than twenty-five thousand miles per
hour.
But the phenomena of falling bodies must have long been observed, and
their laws had been fully investigated by Galileo and others, before the
cause of their falling was understood, or any such principle as
gravity, inherent in the earth and in all bodies, was applied to them.
The developement of this great principle was the work of Sir Isaac
Newton; and I will give you, in my next Letter, some particulars
respecting the life and discoveries of this wonderful man.
LETTER XIV.
SIR ISAAC NEWTON.--UNIVERSAL GRAVITATION.--FIGURE OF THE EARTH'S
ORBIT.--PRECESSION OF THE EQUINOXES.
"The heavens are all his own; from the wild rule
Of whirling vortices, and circling spheres,
To their first great simplicity restored.
The schools astonished stood; but found it vain
To combat long with demonstration clear,
And, unawakened, dream beneath the blaze
Of truth. At once their pleasing visions fled,
With the light shadows of the morning mixed,
When Newton rose, our philosophic sun."--_Thomson's Elegy._
SIR ISAAC NEWTON was born in Lincolnshire, England, in 1642, just one
year after the death of Galileo. His father died before he was born, and
he was a helpless infant, of a diminutive size, and so feeble a frame,
that his attendants hardly expected his life for a single hour. The
family dwelling was of humble architecture, situated in a retired but
beautiful valley, and was surrounded by a small farm, which afforded but
a scanty living to the widowed mother and her precious charge. The cut
on page 144, Fig 30, represents the modest mansion, and the emblems of
rustic life that first met the eyes of this pride of the British nation,
and ornament of human nature. It will probably be found, that genius has
oftener emanated from the cottage than from the palace.
[Illustration Fig. 30.]
The boyhood of Newton was distinguished chiefly for his ingenious
mechanical contrivances. Among other pieces of mechanism, he constructed
a windmill so curious and complete in its workmanship, as to excite
universal admiration. After carrying it a while by the force of the
wind, he resolved to substitute animal power, and for this purpose he
inclosed in it a mouse, which he called the miller, and which kept the
mill a-going by acting on a tread-wheel. The power of the mouse was
brought into action by unavailing attempts to reach a portion of corn
placed above the wheel. A water-clock, a four-wheeled carriage propelled
by the rider himself, and kites of superior workmanship, were among the
productions of the mechanical genius of this gifted boy. At a little
later period, he began to turn his attention to the motions of the
heavenly bodies, and constructed several sun-dials on the walls of the
house where he lived. All this was before he had reached his fifteenth
year. At this age, he was sent by his mother, in company with an old
family servant, to a neighboring market-town, to dispose of products of
their farm, and to buy articles of merchandise for their family use; but
the young philosopher left all these negotiations to his worthy partner,
occupying himself, mean-while, with a collection of old books, which he
had found in a garret. At other times, he stopped on the road, and took
shelter with his book under a hedge, until the servant returned. They
endeavored to educate him as a farmer; but the perusal of a book, the
construction of a water-mill, or some other mechanical or scientific
amusement, absorbed all his thoughts, when the sheep were going astray,
and the cattle were devouring or treading down the corn. One of his
uncles having found him one day under a hedge, with a book in his hand,
and entirely absorbed in meditation, took it from him, and found that it
was a mathematical problem which so engrossed his attention. His
friends, therefore, wisely resolved to favor the bent of his genius, and
removed him from the farm to the school, to prepare for the university.
In the eighteenth year of his age, Newton was admitted into Trinity
College, Cambridge. He made rapid and extraordinary advances in the
mathematics, and soon afforded unequivocal presages of that greatness
which afterwards placed him at the head of the human intellect. In 1669,
at the age of twenty-seven, he became professor of mathematics at
Cambridge, a post which he occupied for many years afterwards. During
the four or five years previous to this he had, in fact, made most of
those great discoveries which have immortalized his name. We are at
present chiefly interested in one of these, namely, that of _universal
gravitation_; and let us see by what steps he was conducted to this
greatest of scientific discoveries.
In the year 1666, when Newton was about twenty-four years of age, the
plague was prevailing at Cambridge, and he retired into the country. One
day, while he sat in a garden, musing on the phenomena of Nature around
him, an apple chanced to fall to the ground. Reflecting on the
mysterious power that makes all bodies near the earth fall towards its
centre, and considering that this power remains unimpaired at
considerable heights above the earth, as on the tops of trees and
mountains, he asked himself,--"May not the same force extend its
influence to a great distance from the earth, even as far as the moon?
Indeed, may not this be the very reason, why the moon is drawn away
continually from the straight line in which every body tends to move,
and is thus made to circulate around the earth?" You will recollect that
it was mentioned, in my Letter which contained an account of the first
law of motion, that if a body is put in motion by any force, it will
always move forward in a straight line, unless some other force compels
it to turn aside from such a direction; and that, when we see a body
moving in a curve, as a circular orbit, we are authorized to conclude
that there is some force existing within the circle, which continually
draws the body away from the direction in which it tends to move.
Accordingly, it was a very natural suggestion, to one so well acquainted
with the laws of motion as Newton, that the moon should constantly bend
towards the earth, from a tendency to fall towards it, as any other
heavy body would do, if carried to such a distance from the earth.
Newton had already proved, that if such a power as gravity extends from
the earth to distant bodies, it must decrease, as the square of the
distance from the centre of the earth increases; that is, at double the
distance, it would be four times less; at ten times the distance, one
hundred times less; and so on. Now, it was known that the moon is about
sixty times as far from the centre of the earth as the surface of the
earth is from the centre, and consequently, the force of attraction at
the moon must be the square of sixty, or thirty-six hundred times less
than it is at the earth; so that a body at the distance of the moon
would fall towards the earth very slowly, only one thirty-six hundredth
part as far in a given time, as at the earth. Does the moon actually
fall towards the earth at this rate; or, what is the same thing, does
she depart at this rate continually from the straight line in which she
tends to move, and in which she would move, if no external force
diverted her from it? On making the calculation, such was found to be
the fact. Hence gravity, and no other force than gravity, acts upon the
moon, and compels her to revolve around the earth. By reasonings equally
conclusive, it was afterwards proved, that a similar force compels all
the planets to circulate around the sun; and now, we may ascend from the
contemplation of this force, as we have seen it exemplified in falling
bodies, to that of a universal power whose influence extends to all the
material creation. It is in this sense that we recognise the principle
of universal gravitation, the law of which may be thus enunciated; _all
bodies in the universe, whether great or small, attract each other, with
forces proportioned to their respective quantities of matter, and
inversely as the squares of their distances from each other_.
This law asserts, first, that attraction reigns throughout the material
world, affecting alike the smallest particle of matter and the greatest
body; secondly, that it acts upon every mass of matter, precisely in
proportion to its quantity; and, thirdly, that its intensity is
diminished as the square of the distance is increased.
Observation has fully confirmed the prevalence of this law throughout
the solar system; and recent discoveries among the fixed stars, to be
more fully detailed hereafter, indicate that the same law prevails
there. The law of universal gravitation is therefore held to be the
grand principle which governs all the celestial motions. Not only is it
consistent with all the observed motions of the heavenly bodies, even
the most irregular of those motions, but, when followed out into all its
consequences, it would be competent to assert that such irregularities
must take place, even if they had never been observed.
Newton first published the doctrine of universal gravitation in the
'Principia,' in 1687. The name implies that the work contains the
fundamental principles of natural philosophy and astronomy. Being
founded upon the immutable basis of mathematics, its conclusions must of
course be true and unalterable, and thenceforth we may regard the great
laws of the universe as traced to their remotest principle. The greatest
astronomers and mathematicians have since occupied themselves in
following out the plan which Newton began, by applying the principles of
universal gravitation to all the subordinate as well as to the grand
movements of the spheres. This great labor has been especially achieved
by La Place, a French mathematician of the highest eminence, in his
profound work, the 'Mecanique Celeste.' Of this work, our distinguished
countryman, Dr. Bowditch, has given a magnificent translation, and
accompanied it with a commentary, which both illustrates the original,
and adds a great amount of matter hardly less profound than that.
[Illustration Fig. 31.]
We have thus far taken the earth's orbit around the sun as a great
circle, such being its projection on the sphere constituting the
celestial ecliptic. The real path of the earth around the sun is
learned, as I before explained to you, by the apparent path of the sun
around the earth once a year. Now, when a body revolves about the earth
at a great distance from us, as is the case with the sun and moon, we
cannot certainly infer that it moves in a circle because it appears to
describe a circle on the face of the sky, for such might be the
appearance of its orbit, were it ever so irregular a curve. Thus, if E,
Fig. 31, represents the earth, and ACB, the irregular path of a body
revolving about it, since we should refer the body continually to some
place on the celestial sphere, XYZ, determined by lines drawn from the
eye to the concave sphere through the body, the body, while moving from
A to B through C, would appear to move from X to Z, through Y. Hence, we
must determine from other circumstances than the actual appearance, what
is the true figure of the orbit.
[Illustration Fig. 32.]
Were the earth's path a circle, having the sun in the centre, the sun
would always appear to be at the same distance from us; that is, the
radius of the orbit, or _radius vector_, (the name given to a line drawn
from the centre of the sun to the orbit of any planet,) would always be
of the same length. But the earth's distance from the sun is constantly
varying, which shows that its orbit is not a circle. We learn the true
figure of the orbit, by ascertaining the _relative distances_ of the
earth from the sun, at various periods of the year. These distances all
being laid down in a diagram, according to their respective lengths, the
extremities, on being connected, give us our first idea of the shape of
the orbit, which appears of an oval form, and at least resembles an
ellipse; and, on further trial, we find that it has the properties of an
ellipse. Thus, let E, Fig. 32, be the place of the earth, and _a_, _b_,
_c_, &c., successive positions of the sun; the _relative_ lengths of the
lines E _a_, E _b_, &c., being known, on connecting the points _a_,
_b_, _c_, &c., the resulting figure indicates the true figure of the
earth's orbit.