Pioneers of Science Part 18

(_f_) The progression of the apses (with an error of one-half).

(_g_) The inequality of apogee, previously unknown.

(_h_) The inequality of nodes, previously unknown.

8. Each planet is attracted not only by the sun but by the other planets, hence their orbits are slightly affected by each other. Newton began the theory of planetary perturbations.

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 (_e.g._ Halley's comet).

10. Applying the idea of centrifugal force to the earth considered as a rotating body, he perceived that it could not be a true sphere, and calculated its oblateness, obtaining 28 miles greater equatorial than polar diameter.

11. Conversely, from the observed shape of Jupiter, or any planet, the length of its day could be estimated.

12. The so-calculated shape of the earth, in combination with centrifugal force, causes the weight of bodies to vary with lat.i.tude; and Newton calculated the amount of this variation. 194 lbs. at pole balance 195 lbs. at equator.

13. A h.o.m.ogeneous sphere attracts as if its ma.s.s were concentrated at its centre. For any other figure, such as an oblate spheroid, this is not exactly true. A hollow concentric spherical sh.e.l.l exerts no force on small bodies inside it.

14. The earth's equatorial protuberance, being acted on by the attraction of the sun and moon, must disturb its axis of rotation in a calculated manner; and thus is produced the precession of the equinoxes.

[The attraction of the planets on the same protuberance causes a smaller and rather different kind of precession.]

15. The waters of the ocean are attracted towards the sun and moon on one side, and whirled a little further away than the solid earth on the other side: hence Newton explained all the main phenomena of the tides.

16. The sun's ma.s.s being known, he calculated the height of the solar tide.

17. From the observed heights of spring and neap tides he determined the lunar tide, and thence made an estimate of the ma.s.s of the moon.

REFERENCE TABLE OF NUMERICAL DATA.

+---------+---------------+----------------------+-----------------+ | |Ma.s.ses in Solar| Height dropped by a | Length of Day or| | | System. |stone in first second.|time of rotation.| +---------+---------------+----------------------+-----------------+ |Mercury | 065 | 70 feet | 24 hours | |Venus | 885 | 158 " | 23-1/2 " | |Earth | 1000 | 161 " | 24 " | |Mars | 108 | 62 " | 24-1/2 " | |Jupiter | 3008 | 450 " | 10 " | |Saturn | 897 | 184 " | 10-1/2 " | |The Sun | 316000 | 4360 " | 608 " | |The Moon | about 012 | 37 " | 702 " | +---------+---------------+----------------------+-----------------+

The ma.s.s of the earth, taken above as unity, is 6,000 trillion tons.

_Observatories._--Uraniburg flourished from 1576 to 1597; the Observatory of Paris was founded in 1667; Greenwich Observatory in 1675.

_Astronomers-Royal._--Flamsteed, Halley, Bradley, Bliss, Maskelyne, Pond, Airy, Christie.

LECTURE IX

NEWTON'S "PRINCIPIA"

The law of gravitation, above enunciated, in conjunction with the laws of motion rehea.r.s.ed at the end of the preliminary notes of Lecture VII., now supersedes the laws of Kepler and includes them as special cases.

The more comprehensive law enables us to criticize Kepler's laws from a higher standpoint, to see how far they are exact and how far they are only approximations. They are, in fact, not precisely accurate, but the reason for every discrepancy now becomes abundantly clear, and can be worked out by the theory of gravitation.

We may treat Kepler's laws either as immediate consequences of the law of gravitation, or as the known facts upon which that law was founded.

Historically, the latter is the more natural plan, and it is thus that they are treated in the first three statements of the above notes; but each proposition may be worked inversely, and we might state them thus:--

1. The fact that the force acting on each planet is directed to the sun, necessitates the equable description of areas.

2. The fact that the force varies as the inverse square of the distance, necessitates motion in an ellipse, or some other conic section, with the sun in one focus.

3. The fact that one attracting body acts on all the planets with an inverse square law, causes the cubes of their mean distances to be proportional to the squares of their periodic times.

Not only these but a mult.i.tude of other deductions follow rigorously from the simple datum that every particle of matter attracts every other particle with a force directly proportional to the ma.s.s of each and to the inverse square of their mutual distance. Those dealt with in the _Principia_ are summarized above, and it will be convenient to run over them in order, with the object of giving some idea of the general meaning of each, without attempting anything too intricate to be readily intelligible.

[Ill.u.s.tration: FIG. 70.]

No. 1. Kepler's second law (equable description of areas) proves that each planet is acted on by a force directed towards the sun as a centre of force.

The equable description of areas about a centre of force has already been fully, though briefly, established. (p. 175.) It is undoubtedly of fundamental importance, and is the earliest instance of the serious discussion of central forces, _i.e._ of forces directed always to a fixed centre.

We may put it afresh thus:--OA has been the motion of a particle in a unit of time; at A it receives a knock towards C, whereby in the next unit it travels along AD instead of AB. Now the area of the triangle CAD, swept out by the radius vector in unit time, is 1/2_bh_; _h_ being the perpendicular height of the triangle from the base AC. (Fig. 70.) Now the blow at A, being along the base, has no effect upon _h_; and consequently the area remains just what it would have been without the blow. A blow directed to any point other than C would at once alter the area of the triangle.

One interesting deduction may at once be drawn. If gravity were a radiant force emitted from the sun with a velocity like that of light, the moving planet would encounter it at a certain apparent angle (aberration), and the force experienced would come from a point a little in advance of the sun. The rate of description of areas would thus tend to increase; whereas in reality it is constant. Hence the force of gravity, if it travel at all, does so with a speed far greater than that of light. It appears to be practically instantaneous. (Cf. "Modern Views of Electricity," -- 126, end of chap. xii.) Again, anything like a r.e.t.a.r.ding effect of the medium through which the planets move would const.i.tute a tangential force, entirely un-directed towards the sun.

Hence no such frictional or r.e.t.a.r.ding force can appreciably exist. It is, however, conceivable that both these effects might occur and just neutralize each other. The neutralization is unlikely to be exact for all the planets; and the fact is, that no trace of either effect has as yet been discovered. (See also p. 176.)

The planets are, however, subject to forces not directed towards the sun, viz. their attractions for each other; and these perturbing forces do produce a slight discrepancy from Kepler's second law, but a discrepancy which is completely subject to calculation.

No. 2. Kepler's first law proves that this central force diminishes in the same proportion as the square of the distance increases.

To prove the connection between the inverse-square law of distance, and the travelling in a conic section with the centre of force in one focus (the other focus being empty), is not so simple. It obviously involves some geometry, and must therefore be left to properly armed students.

But it may be useful to state that the inverse-square law of distance, although the simplest possible law for force emanating from a point or sphere, is not to be regarded as self-evident or as needing no demonstration. The force of a magnetic pole on a magnetized steel sc.r.a.p, for instance, varies as the inverse cube of the distance; and the curve described by such a particle would be quite different from a conic section--it would be a definite cla.s.s of spiral (called Cotes's spiral).

Again, on an iron filing the force of a single pole might vary more nearly as the inverse fifth power; and so on. Even when the thing concerned is radiant in straight lines, like light, the law of inverse squares is not universally true. Its truth a.s.sumes, first, that the source is a point or sphere; next, that there is no reflection or refraction of any kind; and lastly, that the medium is perfectly transparent. The law of inverse squares by no means holds from a prairie fire for instance, or from a lighthouse, or from a street lamp in a fog.

Mutual perturbations, especially the pull of Jupiter, prevent the path of a planet from being really and truly an ellipse, or indeed from being any simple re-entrant curve. Moreover, when a planet possesses a satellite, it is not the centre of the planet which ever attempts to describe the Keplerian ellipse, but it is the common centre of gravity of the two bodies. Thus, in the case of the earth and moon, the point which really does describe a close attempt at an ellipse is a point displaced about 3000 miles from the centre of the earth towards the moon, and is therefore only 1000 miles beneath the surface.

No. 3. Kepler's third law proves that all the planets are acted on by the same kind of force; of an intensity depending on the ma.s.s of the sun.

The third law of Kepler, although it requires geometry to state and establish it for elliptic motion (for which it holds just as well as it does for circular motion), is very easy to establish for circular motion, by any one who knows about centrifugal force. If _m_ is the ma.s.s of a planet, _v_ its velocity, _r_ the radius of its...o...b..t, and _T_ the time of describing it; 2[pi]_r_ = _vT_, and the centripetal force needed to hold it in its...o...b..t is

mv^2 4[pi]^2_mr_ -------- or ----------- _r_ T^2

Now the force of gravitative attraction between the planet and the sun is

_VmS_ -----, r^2

where _v_ is a fixed quant.i.ty called the gravitation-constant, to be determined if possible by experiment once for all. Now, expressing the fact that the force of gravitation _is_ the force holding the planet in, we write,

4[pi]^2_mr_ _VmS_ ----------- = ---------, T^2 r^2

whence, by the simplest algebra,

r^3 _VS_ ------ = ---------.

T^2 4[pi]^2

The ma.s.s of the planet has been cancelled out; the ma.s.s of the sun remains, multiplied by the gravitation-constant, and is seen to be proportional to the cube of the distance divided by the square of the periodic time: a ratio, which is therefore the same for all planets controlled by the sun. Hence, knowing _r_ and _T_ for any single planet, the value of _VS_ is known.