153. A double projectile force will always balance a quadruple power of
gravity. Let the Planet at _B_ have twice as great an impulse from
thence towards _X_, as it had before: that is, in the same length of
time that it was projected from _B_ to _b_, as in the last example, let
it now be projected from _B_ to _c_; and it will require four times as
much gravity to retain it in it’s Orbit: that is, it must fall as far as
from _B_ to 4 in the time that the projectile force would carry it from
_B_ to _c_; otherwise it could not describe the curve _BD_, as is
evident by the Figure. But, in as much time as the Planet moves from _B_
to _C_ in the higher part of it’s Orbit, it moves from _I_ to _K_ or
from _K_ to _L_ in the lower part thereof; because, from the joint
action of these two forces, it must always describe equal areas in equal
times, throughout it’s annual course. These Areas are represented by the
triangles _BSC_, _CSD_, _DSE_, _ESF_, &c. whose contents are equal to
one another, quite round the Figure.
[Sidenote: A difficulty removed.]
154. As the Planets approach nearer the Sun, and recede farther from
him, in every Revolution; there may be some difficulty in conceiving the
reason why the power of gravity, when it once gets the better of the
projectile force, does not bring the Planets nearer and nearer the Sun
in every Revolution, till they fall upon and unite with him. Or why the
projectile force, when it once gets the better of gravity, does not
carry the Planets farther and farther from the Sun, till it removes them
quite out of the sphere of his attraction, and causes them to go on in
straight lines for ever afterward. But by considering the effects of
these powers as described in the two last Articles, this difficulty will
be removed. Suppose a Planet at _B_ to be carried by the projectile
force as far as from _B_ to _b_, in the time that gravity would have
brought it down from _B_ to 1: by these two forces it will describe the
curve _BC_. When the Planet comes down to _K_, it will be but half as
far from the Sun _S_ as it was at _B_; and therefore, by gravitating
four times as strongly towards him, it would fall from _K_ to _V_ in the
same length of time that it would have fallen from _B_ to 1 in the
higher part of it’s Orbit, that is, through four times as much space;
but it’s projectile force is then so much increased at _K_, as would
carry it from _K_ to _k_ in the same time; being double of what it was
at _B_, and is therefore too strong for the tendency of the gravitating
power, either to draw the Planet to the Sun, or cause it to go round him
in the circle _Klmn_, &c. which would require it’s falling from _K_ to
_w_, through a greater space than gravity can draw it whilst the
projectile force is such as would carry it from _K_ to _k_: and
therefore the Planet ascends in it’s Orbit _KLMN_, decreasing in it’s
velocity for the cause already assigned in § 152.
[Sidenote: The Planetary Orbits elliptical.