Pioneers of Science Part 19

No. 4. So by knowing the length of year and distance of any planet from the sun, the sun's ma.s.s can be calculated, in terms of that of the earth.

No. 5. For the satellites, the force acting depends on the ma.s.s of _their_ central body, a planet. Hence the ma.s.s of any planet possessing a satellite becomes known.

The same argument holds for any other system controlled by a central body--for instance, for the satellites of Jupiter; only instead of _S_ it will be natural to write _J_, as meaning the ma.s.s of Jupiter. Hence, knowing _r_ and _T_ for any one satellite of Jupiter, the value of _VJ_ is known.

Apply the argument also to the case of moon and earth. Knowing the distance and time of revolution of our moon, the value of _VE_ is at once determined; _E_ being the ma.s.s of the earth. Hence, _S_ and _J_, and in fact the ma.s.s of any central body possessing a visible satellite, are now known in terms of _E_, the ma.s.s of the earth (or, what is practically the same thing, in terms of _V_, the gravitation-constant).

Observe that so far none of these quant.i.ties are known absolutely. Their relative values are known, and are tabulated at the end of the Notes above, but the finding of their absolute values is another matter, which we must defer.

But, it may be asked, if Kepler's third law only gives us the ma.s.s of a _central_ body, how is the ma.s.s of a _satellite_ to be known? Well, it is not easy; the ma.s.s of no satellite is known with much accuracy. Their mutual perturbations give us some data in the case of the satellites of Jupiter; but to our own moon this method is of course inapplicable. Our moon perturbs at first sight nothing, and accordingly its ma.s.s is not even yet known with exactness. The ma.s.s of comets, again, is quite unknown. All that we can be sure of is that they are smaller than a certain limit, else they would perturb the planets they pa.s.s near.

Nothing of this sort has ever been detected. They are themselves perturbed plentifully, but they perturb nothing; hence we learn that their ma.s.s is small. The ma.s.s of a comet may, indeed, be a few million or even billion tons; but that is quite small in astronomy.

But now it may be asked, surely the moon perturbs the earth, swinging it round their common centre of gravity, and really describing its own orbit about this point instead of about the earth's centre? Yes, that is so; and a more precise consideration of Kepler's third law enables us to make a fair approximation to the position of this common centre of gravity, and thus practically to "weigh the moon," i.e. to compare its ma.s.s with that of the earth; for their ma.s.ses will be inversely as their respective distances from the common centre of gravity or balancing point--on the simple steel-yard principle.

Hitherto we have not troubled ourselves about the precise point about which the revolution occurs, but Kepler's third law is not precisely accurate unless it is attended to. The bigger the revolving body the greater is the discrepancy: and we see in the table preceding Lecture III., on page 57, that Jupiter exhibits an error which, though very slight, is greater than that of any of the other planets, when the sun is considered the fixed centre.

Let the common centre of gravity of earth and moon be displaced a distance _x_ from the centre of the earth, then the moon's distance from the real centre of revolution is not _r_, but _r-x_; and the equation of centrifugal force to gravitative-attraction is strictly

4[pi]^2 _VE_ --------- (_r-x_) = ------, T^2 r^2

instead of what is in the text above; and this gives a slightly modified "third law." From this equation, if we have any distinct method of determining _VE_ (and the next section gives such a method), we can calculate _x_ and thus roughly weigh the moon, since

_r-x_ E ----- = -----, _r_ E+M

but to get anything like a reasonable result the data must be very precise.

No. 6. The force constraining the moon in her orbit is the same gravity as gives terrestrial bodies their weight and regulates the motion of projectiles.

Here we come to the Newtonian verification already several times mentioned; but because of its importance I will repeat it in other words. The hypothesis to be verified is that the force acting on the moon is the same kind of force as acts on bodies we can handle and weigh, and which gives them their weight. Now the weight of a ma.s.s _m_ is commonly written _mg_, where _g_ is the intensity of terrestrial gravity, a thing easily measured; being, indeed, numerically equal to twice the distance a stone drops in the first second of free fall. [See table p. 205.] Hence, expressing that the weight of a body is due to gravity, and remembering that the centre of the earth's attraction is distant from us by one earth's radius (R), we can write

_Vm_E _mg_ = ------, R^2

or

_V_E = gR^2 = 95,522 cubic miles-per-second per second.

But we already know _v_E, in terms of the moon's motion, as

4[pi]^2r^3 ----------- T^2

approximately, [more accurately, see preceding note, this quant.i.ty is _V_(E + M)]; hence we can easily see if the two determinations of this quant.i.ty agree.[20]

All these deductions are fundamental, and may be considered as the foundation of the _Principia_. It was these that flashed upon Newton during that moment of excitement when he learned the real size of the earth, and discovered his speculations to be true.

The next are elaborations and amplifications of the theory, such as in ordinary times are left for subsequent generations of theorists to discover and work out.

Newton did not work out these remoter consequences of his theory completely by any means: the astronomical and mathematical world has been working them out ever since; but he carried the theory a great way, and here it is that his marvellous power is most conspicuous.

It is his treatment of No. 7, the perturbations of the moon, that perhaps most especially has struck all future mathematicians with amazement. No. 7, No. 14, No. 15, these are the most inspired of the whole.

No. 7. The moon is attracted not only by the earth, but by the sun also; hence its...o...b..t is perturbed, and Newton calculated out the chief of these perturbations.

Now running through the perturbations (p. 203) in order:--The first is in parenthesis, because it is mere excentricity. It is not a true perturbation at all, and more properly belongs to Kepler.

(_a_) The first true perturbation is what Ptolemy called "the evection,"

the princ.i.p.al part of which is a periodic change in the ellipticity or excentricity of the moon's...o...b..t, owing to the pull of the sun. It is a complicated matter, and Newton only partially solved it. I shall not attempt to give an account of it.

(_b_) The next, "the variation," is a much simpler affair. It is caused by the fact that as the moon revolves round the earth it is half the time nearer to the sun than the earth is, and so gets pulled more than the average, while for the other fortnight it is further from the sun than the earth is, and so gets pulled less. For the week during which it is changing from a decreasing half to a new moon it is moving in the direction of the extra pull, and hence becomes new sooner than would have been expected. All next week it is moving against the same extra pull, and so arrives at quadrature (half moon) somewhat late. For the next fortnight it is in the region of too little pull, the earth gets pulled more than it does; the effect of this is to hurry it up for the third week, so that the full moon occurs a little early, and to r.e.t.a.r.d it for the fourth week, so that the decreasing half moon like the increasing half occurs behind time again. Thus each syzygy (as new and full are technically called) is too early; each quadrature is too late; the maximum hurrying and slackening force being felt at the octants, or intermediate 45 points.

(_c_) The "annual equation" is a fluctuation introduced into the other perturbations by reason of the varying distance of the disturbing body, the sun, at different seasons of the year. Its magnitude plainly depends simply on the excentricity of the earth's...o...b..t.

Both these perturbations, (_b_) and (_c_), Newton worked out completely.

(_d_) and (_e_) Next come the retrogression of the nodes and the variation of the inclination, which at the time were being observed at Greenwich by Flamsteed, from whom Newton frequently, but vainly, begged for data that he might complete their theory while he had his mind upon it. Fortunately, Halley succeeded Flamsteed as Astronomer-Royal [see list at end of notes above], and then Newton would have no difficulty in gaining such information as the national Observatory could give.

The "inclination" meant is the angle between the plane of the moon's...o...b..t and that of the earth. The plane of the earth's...o...b..t round the sun is called the ecliptic; the plane of the moon's...o...b..t round the earth is inclined to it at a certain angle, which is slowly changing, though in a periodic manner. Imagine a curtain ring bisected by a sheet of paper, and tilted to a certain angle; it may be likened to the moon's...o...b..t, cutting the plane of the ecliptic. The two points at which the plane is cut by the ring are called "nodes"; and these nodes are not stationary, but are slowly regressing, _i.e._ travelling in a direction opposite to that of the moon itself. Also the angle of tilt is varying slowly, oscillating up and down in the course of centuries.

(_f_) The two points in the moon's elliptic orbit where it comes nearest to or farthest from the earth, _i.e._ the points at the extremity of the long axis of the ellipse, are called separately perigee and apogee, or together "the apses." Now the pull of the sun causes the whole orbit to slowly revolve in its own plane, and consequently these apses "progress," so that the true path is not quite a closed curve, but a sort of spiral with elliptic loops.

But here comes in a striking circ.u.mstance. Newton states with reference to this perturbation that theory only accounts for 1-1/2 per annum, whereas observation gives 3, or just twice as much.

This is published in the _Principia_ as a fact, without comment. It was for long regarded as a very curious thing, and many great mathematicians afterwards tried to find an error in the working. D'Alembert, Clairaut, and others attacked the problem, but were led to just the same result.

It const.i.tuted the great outstanding difficulty in the way of accepting the theory of gravitation. It was suggested that perhaps the inverse square law was only a first approximation; that perhaps a more complete expression, such as

A B ---- + -----, r^2 r^4

must be given for it; and so on.

Ultimately, Clairaut took into account a whole series of neglected terms, and it came out correct; thus verifying the theory.

But the strangest part of this tale is to come. For only a few years ago, Prof. Adams, of Cambridge (Neptune Adams, as he is called), was editing various old papers of Newton's, now in the possession of the Duke of Portland, and he found ma.n.u.scripts bearing on this very point, and discovered that Newton had reworked out the calculations himself, had found the cause of the error, had taken into account the terms. .h.i.therto neglected, and so, fifty years before Clairaut, had completely, though not publicly, solved this long outstanding problem of the progression of the apses.

(_g_) and (_h_) Two other inequalities he calculated out and predicted, viz. variation in the motions of the apses and the nodes. Neither of these had then been observed, but they were afterwards detected and verified.

A good many other minor irregularities are now known--some thirty, I believe; and altogether the lunar theory, or problem of the moon's exact motion, is one of the most complicated and difficult in astronomy; the perturbations being so numerous and large, because of the enormous ma.s.s of the perturbing body.

The disturbances experienced by the planets are much smaller, because they are controlled by the sun and perturbed by each other. The moon is controlled only by the earth, and perturbed by the sun. Planetary perturbations can be treated as a series of disturbances with some satisfaction: not so those of the moon. And yet it is the only way at present known of dealing with the lunar theory.

To deal with it satisfactorily would demand the solution of such a problem as this:--Given three rigid spherical ma.s.ses thrown into empty s.p.a.ce with any initial motions whatever, and abandoned to gravity: to determine their subsequent motions. With two ma.s.ses the problem is simple enough, being pretty well summed up in Kepler's laws; but with three ma.s.ses, 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.