Part 19 (1/2)

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.