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-cla.s.s, 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 80 a year.
[Ill.u.s.tration: 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 cla.s.s 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 circ.u.mstances, 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 a.s.sembled 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 cla.s.sics pa.s.sably, 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 2235. 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-gla.s.s over with lycopodium, placing a candle near it, and then looking at the candle-flame through the dusty gla.s.s from a fair distance. Or you may look at the image of a candle in a dusted looking-gla.s.s.
Lycopodium dust is specially suitable, for its granules are remarkably equal in size. The large halo, more rarely seen, of angular radius 2235, is due to another cause again, and is a prismatic effect, although it exhibits hardly any colour. The angle 22-1/2 is characteristic of refraction in crystals with angles of 60 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 t.i.tle 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 ma.n.u.scripts were destroyed. Horrible to say, Galileo"s own son destroyed a great bundle of his father"s ma.n.u.scripts, 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 surrept.i.tiously 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 pa.s.s 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 a.s.serts, some force is needed to destroy it. Leave a body alone, free from all friction or other r.e.t.a.r.ding forces, and it will go on for ever. The planetary motion through empty s.p.a.ce 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 a.s.serts 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 ma.s.s pulls at the elastic--that is its centrifugal force; the hand at the centre pulls also--that is centripetal force.
The third law a.s.serts that these two forces are equal, and together const.i.tute 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."
[Ill.u.s.tration: 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...o...b..t, 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.
[Ill.u.s.tration: 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 a.n.a.lyzable 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 ma.s.s 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 = circ.u.mference/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 602; 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 a.s.suming a force directly proportional to the ma.s.s of both bodies, and inversely proportional to the square of the distance between them: _i.e._ a.s.suming the known force of gravity, it is
_V Mm/r^2_