square on _g_C. But DP, CP, and PE are known; hence also one knows GC,
which is 98,779.
_Lemma which has been supposed_.
If a spheroid is touched by a straight line, and also by two or more
planes which are parallel to this line, though not parallel to one
another, all the points of contact of the line, as well as of the
planes, will be in one and the same ellipse made by a plane which
passes through the centre of the spheroid.
Let LED be the spheroid touched by the line BM at the point B, and
also by the planes parallel to this line at the points O and A. It is
required to demonstrate that the points B, O, and A are in one and the
same Ellipse made in the spheroid by a plane which passes through its
centre.
[Illustration]
Through the line BM, and through the points O and A, let there be
drawn planes parallel to one another, which, in cutting the spheroid
make the ellipses LBD, POP, QAQ; which will all be similar and
similarly disposed, and will have their centres K, N, R, in one and
the same diameter of the spheroid, which will also be the diameter of
the ellipse made by the section of the plane that passes through the
centre of the spheroid, and which cuts the planes of the three said
Ellipses at right angles: for all this is manifest by proposition 15
of the book of Conoids and Spheroids of Archimedes. Further, the two
latter planes, which are drawn through the points O and A, will also,
by cutting the planes which touch the spheroid in these same points,
generate straight lines, as OH and AS, which will, as is easy to see,
be parallel to BM; and all three, BM, OH, AS, will touch the Ellipses
LBD, POP, QAQ in these points, B, O, A; since they are in the planes
of these ellipses, and at the same time in the planes which touch the
spheroid. If now from these points B, O, A, there are drawn the
straight lines BK, ON, AR, through the centres of the same ellipses,
and if through these centres there are drawn also the diameters LD,
PP, QQ, parallel to the tangents BM, OH, AS; these will be conjugate
to the aforesaid BK, ON, AR. And because the three ellipses are
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