114 ASTRONOMY. 
The Primary Causes of the Elliptical Orbits of the Planets ; Kepler’s 
Laws. 
32. In the explanation of the yearly course of the earth about the sun (by 
means of the figure on p/. 8), it was mentioned that the paths of the planets, 
and consequently those of the earth and moon, are ellipses, which are pro- 
duced by the co-operation of two special forces. How that takes place will 
be explained hereafter. Let us suppose that any body (as m, pl. 10, fig. 2) 
once set in motion is impelled by two forces; let the line mP represent the 
direction and intensity of the one force, the line mV the direction and 
intensity of the other. It is evident that the body m will not move towards 
S, the sun alone, nor towards z alone. It must rather (see what is said, 
section 26, about compound motion) follow the direction mo, and pass 
to the point +; the force represented by mP is the attractive force of the 
sun at S, and the force represented by mV, is the tangential force produced by 
an impulse. As the ever-varying central force, namely the attraction of 
the sun, is constantly acting upon this tangential force, this must also vary. 
The curvature m ~, ma, of a planet’s orbit, produced by the co-operation of 
these two forces in the first, and in all following moments, must manifestly 
depend upon their relative proportion. The central force again depends 
upon the distance of the planet from the sun. Should the original velocity 
of the planet in the first second be exactly equal to the planet's fall 
towards the sun in the same second, then the ratio will be 1:1, and the 
orbit will be a circle. It is perfectly plain, however, and much more pro- 
bable, that the original velocity may have been a little greater or less than 
what would be necessary to the production of a circular orbit. Then the 
planet would move in an ellipse (pl. 10, fig. 2), in which the point @, where 
the planet started, would be the perihelion if the projectile force had been 
the greatest, for it would recede from the sun from the very beginning of 
its motion. But if the planet had started in the point =, the projectile force 
must have been the lesser of the two, and the point = would be the aphelion. 
The higher mechanics shows that ellipses arise when, beginning at the 
perihelion, the original velocity amounts to from 51,2, to '73;3, English geo- | 
graphical miles in a second, and that ellipses likewise arise when, beginning 
in the aphelion, the original velocity amounts to from 74; to 51,%3% geo- 
graphical miles in a second. It has actually been found that the planets 
(and their moons), whether starting in their perihelion or aphelion, must 
have had initial velocities falling within the above limits, and conse- 
quently must describe elliptic orbits. 
If the tangential force operate on the point m, in the direction Vm, and 
the attractive force of the sun in the direction Pm, the point m will move in 
the direction m= tox. At = the tangential force operates in the direction 
Bx, and the central force in the direction p-, therefore the point ~ now 
moves in the direction =m to m. At m the tangential force acts afresh 
in the direction Dm, and the central force in the direction km, conse- 
quently the point ™ now moves in the direction ms to 9, &c. It is hence» 
evident that the planet must describe an ellipse, not, however, the broken 
114 
