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 ma.s.s of the central body.

This is Kepler's third law, with a notable addition. It is stated above for circular motion only, so as to avoid geometrical difficulties, but even so it is very instructive. The reason of the proportion between _r^3_ and _T^2_ is at once manifest; and as soon as the constant _V_ became known, _the ma.s.s of the central body_, the sun in the case of a planet, the earth in the case of the moon, Jupiter in the case of his satellites, was at once determined.

Newton's reasoning at this time might, however, be better displayed perhaps by altering the order of the steps a little, as thus:--

The centrifugal force of a body is proportional to _r^3/T^2_, but by Kepler's third law _r^3/T^2_ is constant for all the planets, reckoning _r_ from the sun. Hence the centripetal force needed to hold in all the planets will be a single force emanating from the sun and varying inversely with the square of the distance from that body.

Such a force is at once necessary and sufficient. Such a force would explain the motion of the planets.

But then all this proceeds on a wrong a.s.sumption--that the planetary motion is circular. Will it hold for elliptic orbits? Will an inverse square law of force keep a body moving in an elliptic orbit about the sun in one focus? This is a far more difficult question. Newton solved it, but I do not believe that even he could have solved it, except that he had at his disposal two mathematical engines of great power--the Cartesian method of treating geometry, and his own method of Fluxions.

One can explain the elliptic motion now mathematically, but hardly otherwise; and I must be content to state that the double fact is true--viz., that an inverse square law will move the body in an ellipse or other conic section with the sun in one focus, and that if a body so moves it _must_ be acted on by an inverse square law.

[Ill.u.s.tration: FIG. 59.]

This then is the meaning of the first and third laws of Kepler. What about the second? What is the meaning of the equable description of areas? Well, that rigorously proves that a planet is acted on by a force directed to the centre about which the rate of description of areas is equable. It proves, in fact, that the sun is the attracting body, and that no other force acts.

For first of all if the first law of motion is obeyed, _i.e._ if no force acts, and if the path be equally subdivided to represent equal times, and straight lines be drawn from the divisions to any point whatever, all these areas thus enclosed will be equal, because they are triangles on equal base and of the same height (Euclid, I). See Fig. 59; _S_ being any point whatever, and _A_, _B_, _C_, successive positions of a body.

Now at each of the successive instants let the body receive a sudden blow in the direction of that same point _S_, sufficient to carry it from _A_ to _D_ in the same time as it would have got to _B_ if left alone. The result will be that there will be a compromise, and it will really arrive at _P_, travelling along the diagonal of the parallelogram _AP_. The area its radius vector sweeps out is therefore _SAP_, instead of what it would have been, _SAB_. But then these two areas are equal, because they are triangles on the same base _AS_, and between the same parallels _BP_, _AS_; for by the parallelogram law _BP_ is parallel to _AD_.

Hence the area that would have been described is described, and as all the areas were equal in the case of no force, they remain equal when the body receives a blow at the end of every equal interval of time, _provided_ that every blow is actually directed to _S_, the point to which radii vectores are drawn.

[Ill.u.s.tration: FIG. 60.]

[Ill.u.s.tration: FIG. 61.]

It is instructive to see that it does not hold if the blow is any otherwise directed; for instance, as in Fig. 61, when the blow is along _AE_, the body finds itself at _P_ at the end of the second interval, but the area _SAP_ is by no means equal to _SAB_, and therefore not equal to _SOA_, the area swept out in the first interval.

In order to modify Fig. 60 so as to represent continuous motion and steady forces, we have to take the sides of the polygon _OAPQ_, &c., very numerous and very small; in the limit, infinitely numerous and infinitely small. The path then becomes a curve, and the series of blows becomes a steady force directed towards _S_.

About whatever point therefore the rate of description of areas is uniform, that point and no other must be the centre of all the force there is. If there be no force, as in Fig. 59, well and good, but if there be any force however small not directed towards _S_, then the rate of description of areas about _S_ cannot be uniform.

Kepler, however, says that the rate of description of areas of each planet about the sun is, by Tycho's observations, uniform; hence the sun is the centre of all the force that acts on them, and there is no other force, not even friction. That is the moral of Kepler's second law.

We may also see from it that gravity does not travel like light, so as to take time on its journey from sun to planet; for, if it did, there would be a sort of aberration, and the force on its arrival could no longer be accurately directed to the centre of the sun.

(See _Nature_, vol. xlvi., p. 497.) It is a matter for accuracy of observation, therefore, to decide whether the minutest trace of such deviation can be detected, _i.e._ within what limits of accuracy Kepler's second law is now known to be obeyed.

I will content myself by saying that the limits are extremely narrow. [Reference may be made also to p. 208.]

Thus then it became clear to Newton that the whole solar system depended on a central force emanating from the sun, and varying inversely with the square of the distance from him: for by that hypothesis all the laws of Kepler concerning these motions were completely accounted for; and, in fact, the laws necessitated the hypothesis and established it as a theory.

Similarly the satellites of Jupiter were controlled by a force emanating from Jupiter and varying according to the same law. And again our moon must be controlled by a force from the earth, decreasing with the distance according to the same law.

Grant this hypothetical attracting force pulling the planets towards the sun, pulling the moon towards the earth, and the whole mechanism of the solar system is beautifully explained.

If only one could be sure there was such a force! It was one thing to calculate out what the effects of such a force would be: it was another to be able to put one's finger upon it and say, this is the force that actually exists and is known to exist. We must picture him meditating in his garden on this want--an attractive force towards the earth.

If only such an attractive force pulling down bodies to the earth existed. An apple falls from a tree. Why, it does exist! There is gravitation, common gravity that makes bodies fall and gives them their weight.

Wanted, a force tending towards the centre of the earth. It is to hand!

It is common old gravity that had been known so long, that was perfectly familiar to Galileo, and probably to Archimedes. Gravity that regulates the motion of projectiles. Why should it only pull stones and apples?

Why should it not reach as high as the moon? Why should it not be the gravitation of the sun that is the central force acting on all the planets?

Surely the secret of the universe is discovered! But, wait a bit; is it discovered? Is this force of gravity sufficient for the purpose? It must vary inversely with the square of the distance from the centre of the earth. How far is the moon away? Sixty earth's radii. Hence the force of gravity at the moon's distance can only be 1/3600 of what it is on the earth's surface. So, instead of pulling it 16 ft. per second, it should pull it 16/3600 ft. per second, or 16 ft. a minute.[17] How can one decide whether such a force is able to pull the moon the actual amount required? To Newton this would seem only like a sum in arithmetic. Out with a pencil and paper and reckon how much the moon falls toward the earth in every second of its motion. Is it 16/3600? That is what it ought to be: but is it? The size of the earth comes into the calculation. Sixty miles make a degree, 360 degrees a circ.u.mference.

This gives as the earth's diameter 6,873 miles; work it out.

The answer is not 16 feet a minute, it is 139 feet.

Surely a mistake of calculation?

No, it is no mistake: there is something wrong in the theory, gravity is too strong.

Instead of falling toward the earth 5-1/3 hundredths of an inch every second, as it would under gravity, the moon only falls 4-2/3 hundredths of an inch per second.

With such a discovery in his grasp at the age of twenty-three he is disappointed--the figures do not agree, and he cannot make them agree.

Either gravity is not the force in action, or else something interferes with it. Possibly, gravity does part of the work, and the vortices of Descartes interfere with it.

He must abandon the fascinating idea for the time. In his own words, "he laid aside at that time any further thought of the matter."

So far as is known, he never mentioned his disappointment to a soul. He might, perhaps, if he had been at Cambridge, but he was a shy and solitary youth, and just as likely he might not. Up in Lincolnshire, in the seventeenth century, who was there for him to consult?

True, he might have rushed into premature publication, after our nineteenth century fashion, but that was not his method. Publication never seemed to have occurred to him.

His reticence now is noteworthy, but later on it is perfectly astonishing. He is so absorbed in making discoveries that he actually has to be reminded to tell any one about them, and some one else always has to see to the printing and publishing for him.

I have entered thus fully into what I conjecture to be the stages of this early discovery of the law of gravitation, as applicable to the heavenly bodies, because it is frequently and commonly misunderstood. It is sometimes thought that he discovered the force of gravity; I hope I have made it clear that he did no such thing. Every educated man long before his time, if asked why bodies fell, would reply just as glibly as they do now, "Because the earth attracts them," or "because of the force of gravity."

His discovery was that the motions of the solar system were due to the action of a central force, directed to the body at the centre of the system, and varying inversely with the square of the distance from it.

This discovery was based upon Kepler's laws, and was clear and certain.

It might have been published had he so chosen.

But he did not like hypothetical and unknown forces; he tried to see whether the known force of gravity would serve. This discovery at that time he failed to make, owing to a wrong numerical datum. The size of the earth he only knew from the common doctrine of sailors that 60 miles make a degree; and that threw him out. Instead of falling 16 feet a minute, as it ought under gravity, it only fell 139 feet, so he abandoned the idea. We do not find that he returned to it for sixteen years.

LECTURE VIII

NEWTON AND THE LAW OF GRAVITATION