Christmas Day, 1642. His father, a small freehold farmer, also named
Isaac, died before his birth. His mother, _nee_ Hannah Ayscough, in two
years married a Mr. Smith, rector of North Witham, but was again left a
widow in 1656. His uncle, W. Ayscough, was rector of a near parish and a
graduate of Trinity College, Cambridge. At the age of fifteen Isaac was
removed from school at Grantham to be made a farmer of, but as it seemed
he would not make a good one his uncle arranged for him to return to
school and thence to Cambridge, where he entered Trinity College as a
sub-sizar in 1661. Studied Descartes's geometry. Found out a method of
infinite series in 1665, and began the invention of Fluxions. In the
same year and the next he was driven from Cambridge by the plague. In
1666, at Woolsthorpe, the apple fell. In 1667 he was elected a fellow of
his college, and in 1669 was specially noted as possessing an
unparalleled genius by Dr. Barrow, first Lucasian Professor of
Mathematics. The same year Dr. Barrow retired from his chair in favour
of Newton, who was thus elected at the age of twenty-six. He lectured
first on optics with great success. Early in 1672 he was elected a
Fellow of the Royal Society, and communicated his researches in optics,
his reflecting telescope, and his discovery of the compound nature of
white light. Annoying controversies arose; but he nevertheless
contributed a good many other most important papers in optics, including
observations in diffraction, and colours of thin plates. He also
invented the modern sextant. In 1672 a letter from Paris was read at the
Royal Society concerning a new and accurate determination of the size of
the earth by Picard. When Newton heard of it he began the _Principia_,
working in silence. In 1684 arose a discussion between Wren, Hooke, and
Halley concerning the law of inverse square as applied to gravity and
the path it would cause the planets to describe. Hooke asserted that he
had a solution, but he would not produce it. After waiting some time for
it Halley went to Cambridge to consult Newton on the subject, and thus
discovered the existence of the first part of the _Principia_, wherein
all this and much more was thoroughly worked out. On his representations
to the Royal Society the manuscript was asked for, and when complete was
printed and published in 1687 at Halley's expense. While it was being
completed Newton and seven others were sent to uphold the dignity of the
University, before the Court of High Commission and Judge Jeffreys,
against a high-handed action of James II. In 1682 he was sent to
Parliament, and was present at the coronation of William and Mary. Made
friends with Locke. In 1694 Montague, Lord Halifax, made him Warden, and
in 1697 Master, of the Mint. Whiston succeeded him as Lucasian
Professor. In 1693 the method of fluxions was published. In 1703 Newton
was made President of the Royal Society, and held the office to the end
of his life. In 1705 he was knighted by Anne. In 1713 Cotes helped him
to bring out a new edition of the _Principia_, completed as we now have
it. On the 20th of March 1727, he died: having lived from Charles I. to
George II.
THE LAWS OF MOTION, DISCOVERED BY GALILEO, STATED BY NEWTON.
_Law 1._--If no force acts on a body in motion, it continues to move
uniformly in a straight line.
_Law 2._--If force acts on a body, it produces a change of motion
proportional to the force and in the same direction.
_Law 3._--When one body exerts force on another, that other reacts with
equal force upon the one.
LECTURE VII
SIR ISAAC NEWTON
The little hamlet of Woolsthorpe lies close to the village of
Colsterworth, about six miles south of Grantham, in the county of
Lincoln. In the manor house of Woolsthorpe, on Christmas Day, 1642, was
born to a widowed mother a sickly infant who seemed not long for this
world. Two women who were sent to North Witham to get some medicine for
him scarcely expected to find him alive on their return. However, the
child lived, became fairly robust, and was named Isaac, after his
father. What sort of a man this father was we do not know. He was what
we may call a yeoman, that most wholesome and natural of all classes. He
owned the soil he tilled, and his little estate had already been in the
family for some hundred years. He was thirty-six when he died, and had
only been married a few months.
Of the mother, unfortunately, we know almost as little. We hear that she
was recommended by a parishioner to the Rev. Barnabas Smith, an old
bachelor in search of a wife, as "the widow Newton--an extraordinary
good woman:" and so I expect she was, a thoroughly sensible, practical,
homely, industrious, middle-class, Mill-on-the-Floss sort of woman.
However, on her second marriage she went to live at North Witham, and
her mother, old Mrs. Ayscough, came to superintend the farm at
Woolsthorpe, and take care of young Isaac.
By her second marriage his mother acquired another piece of land, which
she settled on her first son; so Isaac found himself heir to two little
properties, bringing in a rental of about L80 a year.
[Illustration: FIG. 56.--Manor-house of Woolsthorpe.]
He had been sent to a couple of village schools to acquire the ordinary
accomplishments taught at those places, and for three years to the
grammar school at Grantham, then conducted by an old gentleman named Mr.
Stokes. He had not been very industrious at school, nor did he feel
keenly the fascinations of the Latin Grammar, for he tells us that he
was the last boy in the lowest class but one. He used to pay much more
attention to the construction of kites and windmills and waterwheels,
all of which he made to work very well. He also used to tie paper
lanterns to the tail of his kite, so as to make the country folk fancy
they saw a comet, and in general to disport himself as a boy should.
It so happened, however, that he succeeded in thrashing, in fair fight,
a bigger boy who was higher in the school, and who had given him a
kick. His success awakened a spirit of emulation in other things than
boxing, and young Newton speedily rose to be top of the school.
Under these circumstances, at the age of fifteen, his mother, who had
now returned to Woolsthorpe, which had been rebuilt, thought it was time
to train him for the management of his land, and to make a farmer and
grazier of him. The boy was doubtless glad to get away from school, but
he did not take kindly to the farm--especially not to the marketing at
Grantham. He and an old servant were sent to Grantham every week to buy
and sell produce, but young Isaac used to leave his old mentor to do all
the business, and himself retire to an attic in the house he had lodged
in when at school, and there bury himself in books.
After a time he didn't even go through the farce of visiting Grantham at
all; but stopped on the road and sat under a hedge, reading or making
some model, until his companion returned.
We hear of him now in the great storm of 1658, the storm on the day
Cromwell died, measuring the force of the wind by seeing how far he
could jump with it and against it. He also made a water-clock and set it
up in the house at Grantham, where it kept fairly good time so long as
he was in the neighbourhood to look after it occasionally.
At his own home he made a couple of sundials on the side of the wall (he
began by marking the position of the sun by the shadow of a peg driven
into the wall, but this gradually developed into a regular dial) one of
which remained of use for some time; and was still to be seen in the
same place during the first half of the present century, only with the
gnomon gone. In 1844 the stone on which it was carved was carefully
extracted and presented to the Royal Society, who preserve it in their
library. The letters WTON roughly carved on it are barely visible.
All these pursuits must have been rather trying to his poor mother, and
she probably complained to her brother, the rector of Burton Coggles:
at any rate this gentleman found master Newton one morning under a hedge
when he ought to have been farming. But as he found him working away at
mathematics, like a wise man he persuaded his sister to send the boy
back to school for a short time, and then to Cambridge. On the day of
his finally leaving school old Mr. Stokes assembled the boys, made them
a speech in praise of Newton's character and ability, and then dismissed
him to Cambridge.
At Trinity College a new world opened out before the country-bred lad.
He knew his classics passably, but of mathematics and science he was
ignorant, except through the smatterings he had picked up for himself.
He devoured a book on logic, and another on Kepler's Optics, so fast
that his attendance at lectures on these subjects became unnecessary. He
also got hold of a Euclid and of Descartes's Geometry. The Euclid seemed
childishly easy, and was thrown aside, but the Descartes baffled him for
a time. However, he set to it again and again and before long mastered
it. He threw himself heart and soul into mathematics, and very soon made
some remarkable discoveries. First he discovered the binomial theorem:
familiar now to all who have done any algebra, unintelligible to others,
and therefore I say nothing about it. By the age of twenty-one or two he
had begun his great mathematical discovery of infinite series and
fluxions--now known by the name of the Differential Calculus. He wrote
these things out and must have been quite absorbed in them, but it never
seems to have occurred to him to publish them or tell any one about
them.
In 1664 he noticed some halos round the moon, and, as his manner was, he
measured their angles--the small ones 3 and 5 degrees each, the larger
one 22 deg..35. Later he gave their theory.
Small coloured halos round the moon are often seen, and are said to
be a sign of rain. They are produced by the action of minute
globules of water or cloud particles upon light, and are brightest
when the particles are nearly equal in size. They are not like the
rainbow, every part of which is due to light that has entered a
raindrop, and been refracted and reflected with prismatic
separation of colours; a halo is caused by particles so small as to
be almost comparable with the size of waves of light, in a way
which is explained in optics under the head "diffraction." It may
be easily imitated by dusting an ordinary piece of window-glass
over with lycopodium, placing a candle near it, and then looking at
the candle-flame through the dusty glass from a fair distance. Or
you may look at the image of a candle in a dusted looking-glass.
Lycopodium dust is specially suitable, for its granules are
remarkably equal in size. The large halo, more rarely seen, of
angular radius 22 deg..35, is due to another cause again, and is a
prismatic effect, although it exhibits hardly any colour. The angle
22-1/2 deg. is characteristic of refraction in crystals with angles of
60 deg. and refractive index about the same as water; in other words
this halo is caused by ice crystals in the higher regions of the
atmosphere.
He also the same year observed a comet, and sat up so late watching it
that he made himself ill. By the end of the year he was elected to a
scholarship and took his B.A. degree. The order of merit for that year
never existed or has not been kept. It would have been interesting, not
as a testimony to Newton, but to the sense or non-sense of the
examiners. The oldest Professorship of Mathematics at the University of
Cambridge, the Lucasian, had not then been long founded, and its first
occupant was Dr. Isaac Barrow, an eminent mathematician, and a kind old
man. With him Newton made good friends, and was helpful in preparing a
treatise on optics for the press. His help is acknowledged by Dr. Barrow
in the preface, which states that he had corrected several errors and
made some capital additions of his own. Thus we see that, although the
chief part of his time was devoted to mathematics, his attention was
already directed to both optics and astronomy. (Kepler, Descartes,
Galileo, all combined some optics with astronomy. Tycho and the old ones
combined alchemy; Newton dabbled in this also.)
Newton reached the age of twenty-three in 1665, the year of the Great
Plague. The plague broke out in Cambridge as well as in London, and the
whole college was sent down. Newton went back to Woolsthorpe, his mind
teeming with ideas, and spent the rest of this year and part of the next
in quiet pondering. Somehow or other he had got hold of the notion of
centrifugal force. It was six years before Huyghens discovered and
published the laws of centrifugal force, but in some quiet way of his
own Newton knew about it and applied the idea to the motion of the
planets.
We can almost follow the course of his thoughts as he brooded and
meditated on the great problem which had taxed so many previous
thinkers,--What makes the planets move round the sun? Kepler had
discovered how they moved, but why did they so move, what urged them?
Even the "how" took a long time--all the time of the Greeks, through
Ptolemy, the Arabs, Copernicus, Tycho: circular motion, epicycles, and
excentrics had been the prevailing theory. Kepler, with his marvellous
industry, had wrested from Tycho's observations the secret of their
orbits. They moved in ellipses with the sun in one focus. Their rate of
description of area, not their speed, was uniform and proportional to
time.
Yes, and a third law, a mysterious law of unintelligible import, had
also yielded itself to his penetrating industry--a law the discovery of
which had given him the keenest delight, and excited an outburst of
rapture--viz. that there was a relation between the distances and the
periodic times of the several planets. The cubes of the distances were
proportional to the squares of the times for the whole system. This law,
first found true for the six primary planets, he had also extended,
after Galileo's discovery, to the four secondary planets, or satellites
of Jupiter (p. 81).
But all this was working in the dark--it was only the first step--this
empirical discovery of facts; the facts were so, but how came they so?
What made the planets move in this particular way? Descartes's vortices
was an attempt, a poor and imperfect attempt, at an explanation. It had
been hailed and adopted throughout Europe for want of a better, but it
did not satisfy Newton. No, it proceeded on a wrong tack, and Kepler had
proceeded on a wrong tack in imagining spokes or rays sticking out from
the sun and driving the planets round like a piece of mechanism or mill
work. For, note that all these theories are based on a wrong idea--the
idea, viz., that some force is necessary to maintain a body in motion.
But this was contrary to the laws of motion as discovered by Galileo.
You know that during his last years of blind helplessness at Arcetri,
Galileo had pondered and written much on the laws of motion, the
foundation of mechanics. In his early youth, at Pisa, he had been
similarly occupied; he had discovered the pendulum, he had refuted the
Aristotelians by dropping weights from the leaning tower (which we must
rejoice that no earthquake has yet injured), and he had returned to
mechanics at intervals all his life; and now, when his eyes were useless
for astronomy, when the outer world has become to him only a prison to
be broken by death, he returns once more to the laws of motion, and
produces the most solid and substantial work of his life.
For this is Galileo's main glory--not his brilliant exposition of the
Copernican system, not his flashes of wit at the expense of a moribund
philosophy, not his experiments on floating bodies, not even his
telescope and astronomical discoveries--though these are the most taking
and dazzling at first sight. No; his main glory and title to immortality
consists in this, that he first laid the foundation of mechanics on a
firm and secure basis of experiment, reasoning, and observation. He
first discovered the true Laws of Motion.
I said little of this achievement in my lecture on him; for the work was
written towards the end of his life, and I had no time then. But I knew
I should have to return to it before we came to Newton, and here we are.
You may wonder how the work got published when so many of his
manuscripts were destroyed. Horrible to say, Galileo's own son destroyed
a great bundle of his father's manuscripts, thinking, no doubt, thereby
to save his own soul. This book on mechanics was not burnt, however. The
fact is it was rescued by one or other of his pupils, Toricelli or
Viviani, who were allowed to visit him in his last two or three years;
it was kept by them for some time, and then published surreptitiously in
Holland. Not that there is anything in it bearing in any visible way on
any theological controversy; but it is unlikely that the Inquisition
would have suffered it to pass notwithstanding.
I have appended to the summary preceding this lecture (p. 160) the three
axioms or laws of motion discovered by Galileo. They are stated by
Newton with unexampled clearness and accuracy, and are hence known as
Newton's laws, but they are based on Galileo's work. The first is the
simplest; though ignorance of it gave the ancients a deal of trouble. It
is simply a statement that force is needed to change the motion of a
body; _i.e._ that if no force act on a body it will continue to move
uniformly both in speed and direction--in other words, steadily, in a
straight line. The old idea had been that some force was needed to
maintain motion. On the contrary, the first law asserts, some force is
needed to destroy it. Leave a body alone, free from all friction or
other retarding forces, and it will go on for ever. The planetary motion
through empty space therefore wants no keeping up; it is not the motion
that demands a force to maintain it, it is the curvature of the path
that needs a force to produce it continually. The motion of a planet is
approximately uniform so far as speed is concerned, but it is not
constant in direction; it is nearly a circle. The real force needed is
not a propelling but a deflecting force.
The second law asserts that when a force acts, the motion changes,
either in speed or in direction, or both, at a pace proportional to the
magnitude of the force, and in the same direction as that in which the
force acts. Now since it is almost solely in direction that planetary
motion alters, a deflecting force only is needed; a force at right
angles to the direction of motion, a force normal to the path.
Considering the motion as circular, a force along the radius, a radial
or centripetal force, must be acting continually. Whirl a weight round
and round by a bit of elastic, the elastic is stretched; whirl it
faster, it is stretched more. The moving mass pulls at the elastic--that
is its centrifugal force; the hand at the centre pulls also--that is
centripetal force.
The third law asserts that these two forces are equal, and together
constitute the tension in the elastic. It is impossible to have one
force alone, there must be a pair. You can't push hard against a body
that offers no resistance. Whatever force you exert upon a body, with
that same force the body must react upon you. Action and reaction are
always equal and opposite.
Sometimes an absurd difficulty is felt with respect to this, even by
engineers. They say, "If the cart pulls against the horse with precisely
the same force as the horse pulls the cart, why should the cart move?"
Why on earth not? The cart moves because the horse pulls it, and because
nothing else is pulling it back. "Yes," they say, "the cart is pulling
back." But what is it pulling back? Not itself, surely? "No, the horse."
Yes, certainly the cart is pulling at the horse; if the cart offered no
resistance what would be the good of the horse? That is what he is for,
to overcome the pull-back of the cart; but nothing is pulling the cart
back (except, of course, a little friction), and the horse is pulling it
forward, hence it goes forward. There is no puzzle at all when once you
realise that there are two bodies and two forces acting, and that one
force acts on each body.[16]
If, indeed, two balanced forces acted on one body that would be in
equilibrium, but the two equal forces contemplated in the third law act
on two different bodies, and neither is in equilibrium.
So much for the third law, which is extremely simple, though it has
extraordinarily far-reaching consequences, and when combined with a
denial of "action at a distance," is precisely the principle of the
Conservation of Energy. Attempts at perpetual motion may all be regarded
as attempts to get round this "third law."
[Illustration: FIG. 57.]
On the subject of the _second_ law a great deal more has to be said
before it can be in any proper sense even partially appreciated,
but a complete discussion of it would involve a treatise on
mechanics. It is _the_ law of mechanics. One aspect of it we must
attend to now in order to deal with the motion of the planets, and
that is the fact that the change of motion of a body depends solely
and simply on the force acting, and not at all upon what the body
happens to be doing at the time it acts. It may be stationary, or
it may be moving in any direction; that makes no difference.
Thus, referring back to the summary preceding Lecture IV, it is
there stated that a dropped body falls 16 feet in the first second,
that in two seconds it falls 64 feet, and so on, in proportion to
the square of the time. So also will it be the case with a thrown
body, but the drop must be reckoned from its line of motion--the
straight line which, but for gravity, it would describe.
Thus a stone thrown from _O_ with the velocity _OA_ would in one
second find itself at _A_, in two seconds at _B_, in three seconds
at _C_, and so on, in accordance with the first law of motion, if
no force acted. But if gravity acts it will have fallen 16 feet by
the time it would have got to _A_, and so will find itself at _P_.
In two seconds it will be at _Q_, having fallen a vertical height
of 64 feet; in three seconds it will be at _R_, 144 feet below _C_;
and so on. Its actual path will be a curve, which in this case is a
parabola. (Fig. 57.)
If a cannon is pointed horizontally over a level plain, the cannon
ball will be just as much affected by gravity as if it were
dropped, and so will strike the plain at the same instant as
another which was simply dropped where it started. One ball may
have gone a mile and the other only dropped a hundred feet or so,
but the time needed by both for the vertical drop will be the same.
The horizontal motion of one is an extra, and is due to the powder.
As a matter of fact the path of a projectile in vacuo is only
approximately a parabola. It is instructive to remember that it is
really an ellipse with one focus very distant, but not at infinity.
One of its foci is the centre of the earth. A projectile is really
a minute satellite of the earth's, and in vacuo it accurately obeys
all Kepler's laws. It happens not to be able to complete its orbit,
because it was started inconveniently close to the earth, whose
bulk gets in its way; but in that respect the earth is to be
reckoned as a gratuitous obstruction, like a target, but a target
that differs from most targets in being hard to miss.
[Illustration: FIG. 58.]
Now consider circular motion in the same way, say a ball whirled
round by a string. (Fig. 58.)
Attending to the body at _O_, it is for an instant moving towards
_A_, and if no force acted it would get to _A_ in a time which for
brevity we may call a second. But a force, the pull of the string,
is continually drawing it towards _S_, and so it really finds
itself at _P_, having described the circular arc _OP_, which may
be considered to be compounded of, and analyzable into the
rectilinear motion _OA_ and the drop _AP_. At _P_ it is for an
instant moving towards _B_, and the same process therefore carries
it to _Q_; in the third second it gets to _R_; and so on: always
falling, so to speak, from its natural rectilinear path, towards
the centre, but never getting any nearer to the centre.
The force with which it has thus to be constantly pulled in towards
the centre, or, which is the same thing, the force with which it is
tugging at whatever constraint it is that holds it in, is
_mv^2/r_; where _m_ is the mass of the particle, _v_ its
velocity, and _r_ the radius of its circle of movement. This is the
formula first given by Huyghens for centrifugal force.
We shall find it convenient to express it in terms of the time of
one revolution, say _T_. It is easily done, since plainly T =
circumference/speed = _2[pi]r/v_; so the above expression for
centrifugal force becomes _4[pi]^2mr/T^2_.
As to the fall of the body towards the centre every microscopic
unit of time, it is easily reckoned. For by Euclid III. 36, and
Fig. 58, _AP.AA' = AO^2_. Take _A_ very near _O_, then _OA = vt_,
and _AA' = 2r_; so _AP = v^2t^2/2r = 2[pi]^2r
t^2/T^2_; or the fall per second is _2[pi]^2r/T^2_,
_r_ being its distance from the centre, and _T_ its time of going
once round.
In the case of the moon for instance, _r_ is 60 earth radii; more
exactly 60.2; and _T_ is a lunar month, or more precisely 27 days,
7 hours, 43 minutes, and 11-1/2 seconds. Hence the moon's
deflection from the tangential or rectilinear path every minute
comes out as very closely 16 feet (the true size of the earth being
used).
Returning now to the case of a small body revolving round a big one, and
assuming a force directly proportional to the mass of both bodies, and
inversely proportional to the square of the distance between them:
_i.e._ assuming the known force of gravity, it is
_V Mm/r^2_
where _V_ is a constant, called the gravitation constant, to be
determined by experiment.
If this is the centripetal force pulling a planet or satellite in, it
must be equal to the centrifugal force of this latter, viz. (see above).
_4[pi]^2mr/T^2
Equate the two together, and at once we get
_r^3/T^2 = V/4[pi]^2M;_
or, in words, the cube of the distance divided by the square of the
periodic time for every planet or satellite of the system under
consideration, will be constant and proportional to the mass of the
central body.