THEORETICAL ASTRONOMY. 115 
line m=, m2, For we may suppose the parallelogram of forces to be con- 
structed anew every successive moment, and consequently infinitely small, 
so that instead of the broken line, a continually curved one, namely an 
ellipse, will be produced, since both forces, the central and tangential, 
operate incessantly upon the planet. It is only for the more intelligible 
illustration of the subject that the single parallelograms in fig. 2 are repre- 
sented on so large a scale; the eccentricity, CS, is, for the same reason, 
assumed tolerably great, as the real eccentricity of the planetary orbits is 
much less. 
The proposition that the planets describe ellipses, the centre of the sun 
being in one of their foci, is the first of the three celebrated laws of Kepler; 
upon which the whole theory of the planetary motions depends. The 
second law (discovered, however, first) is, that any two areas (sectors), 
SQq and SQ’q', described by the radii vectores, SQ and Sq, SQ’ and Sq’, are 
proportional to the times (pl. 6, fig. 4). This law may also be expressed 
in this manner: The different velocities of a planet are as the squares of 
its different distances from the sun. Suppose the planet to describe the 
distances Qq, and Q’g’ of its orbit in equal times, then the elliptical 
sectors, SQq and SQ’q’, will have equivalent areas. Hence it follows, that 
if the areas of the sectors at the perihelion B and at the aphelion A are to 
be equivalent, the arc described by the planet at the perihelion must be 
greater than that described in equal time at the aphelion. Thus the planet 
must move with the greatest velocity at the perihelion, and the least at the 
aphelion, and these two different velocities are as the squares of the dis- 
tances SA and SB. In general, the velocity of a planet’s motion must be 
greater as it approaches the sun, and less as it recedes from it. 
The first law of Kepler expresses the character of the curve described by 
the planets; the second, the varying velocities of the planetary motion ; 
while the third law is a bond of union connecting the different planets 
together. This third law is expressed as follows: the squares of the times 
of revolution of two planets are as the cubes of their mean distances from 
the sun. The great value of this law consists in its presenting a geometrical 
proportion, so that knowing three of the four elements, the mean distances 
from the sun, and the periods of revolution, the fourth can always be 
obtained. In conclusion, the laws of Kepler are true laws of nature, since 
Newton has demonstrated that they are only consequences of that single 
and supreme law discovered by him—the law of universal gravitation. 
The Moons of Jupiter, Saturn, and Uranus. 
33. The planet Jupiter is accompanied by four moons in his journey ot 
113 years around the sun. Immediately after the discovery of the 
telescope, Simon Marius (at Ansbach), in November, 1609, observed four 
small stars very near to Jupiter, which, almost always in a straight line with 
him, appeared sometimes to the right, sometimes to the left, never separating 
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