s, in Jupiter
and Mars, in Saturn and the earth, as well as in Venus and Mercury, the
cubes of their distances are as the squares of their periodic times;
and, therefore, the centripetal circum-solar force throughout all the
planetary regions decreases in the duplicate proportion of the distances
from the sun. Neglecting those little fractions which may have arisen
from insensible errors of observation, we shall always find the said
proportion to hold exactly; for the distances of Saturn, Jupiter, Mars,
the Earth, Venus, and Mercury from the sun, drawn from the observations
of astronomers, are (Kepler) as the numbers 951,000, 519,650, 152,350,
100,000, 70,000, 38,806; or (Bullialdus) as the numbers 954,198,
522,520, 152,350, 100,000, 72,398, 38,585; and from the periodic times
they come out 953,806, 520,116, 152,399, 100,000, 72,333, 38,710. Their
distances, according to Kepler and Bullialdus, scarcely differ by any
sensible quantity, and where they differ most the differences drawn from
the periodic times fall in between them.

_Earth as a Centre_

That the circum-terrestrial force likewise decreases in the duplicate
proportion of the distances, I infer thus:

The mean distance of the moon from the centre of the earth is, we may
assume, sixty semi-diameters of the earth; and its periodic time in
respect of the fixed stars 27 days 7 hr. 43 min. Now, it has been shown
in a previous book that a body revolved in our air, near the surface of
the earth supposed at rest, by means of a centripetal force which should
be to the same force at the distance of the moon in the reciprocal
duplicate proportion of the distances from the centre of the earth, that
is, as 3,600 to 1, would (secluding the resistance of the air) complete
a revolution in 1 hr. 24 min. 27 sec.

Suppose the circumference of the earth to be 123,249,600 Paris feet,
then the same body deprived of its circular motion and falling by the
impulse of the same centripetal force as before would in one second of
time describe 15-1/12 Paris feet. This we infer by a calculus formed
upon Prop. xxxvi. ("To determine the times of the descent of a body
falling from a given place"), and it agrees with the results of Mr.
Huyghens's experiments of pendulums, by which he demonstrated that
bodies falling by all the centripetal force with which (of whatever
nature it is) they are impelled near the surface of the earth do in one
second of time describe 15-1/12 Paris feet.

Bu