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Proposition LX. Theorem XXIII.\hfill\break
\strut\hskip\parindent
\emph{If two bodies $S$ and $P$, attracting each
other with forces reciprocally proportional to the squares of their
distance, revolve about their common centre of gravity; I say, that
the principal axis of the ellipsis which either of the bodies, as $P$,
describes by this motion about the other $S$, will be to the principal
axis of the ellipsis, which the same body $P$ may describe in
the same periodical time about the other body $S$ quiescent, as the
sum of the two bodies $S+P$ to the first of the two mean
proportionals between that sum and the other body $S$.\hfill\break}
\strut\hskip\parindent
For if the ellipses described were equal to each other, their
periodic times by the last Theorem would be in a subduplicate ratio
of the body $S$ to the sum of the bodies $S+P$. Let the periodic
time in the latter ellipsis be diminished in that ratio, and the
periodic times will become equal; but, by Prop. XV, the principal axis
of the ellipsis will be diminished in a ratio sesquiplicate to the
former ratio; that is, in a ratio to which the ratio of $S$ to $S+P$
is triplicate; and therefore that axis will be to the principal axis
of the other ellipsis as the first of two mean proportionals between
$S+P$ and $S$ to $S+P$. And inversely the principal axis of the
ellipsis described about the movable body will be to the principal
axis of that described round the immovable as $S+P$ to the first of two
mean proportionals between $S+P$ and $S$. Q.E.D.\hfill\break
\strut\hskip\parindent
(Newton, \emph{The mathematical principles of natural philosophy},
translated by Andrew Motte, 1848.)} etex;
