A history of Greek mathematics - Wilbourhall.org

ON CONOIDS AND SPHEROIDS 61
correspond to it in the inscribed figure, and we should write
the ratio as (BD : zero).
Archimedes concludes, by means of a lemma in proportions
forming Prop. 1, that
(frustum BF) :
(inscribed figure)
= (BD + HN+ ...) :(TN + SM+ ...+ XO)
= n 2 k : (k + 2k + 3k + ... + n-lk),
where XO = k, so that BD = nk.
In like manner, he concludes that
(frustum BF) :
(circumscribed figure)
= M 2 Jc : (Jc 4- 2 h + 3 k + . . . + ?i&).
But, by the Lemma preceding Prop. 1,
/c + 2/s+3&+...+w— Ik < ^n z k < k+2k+3k+ ... +w&,
whence
(frustum i?jP) : (inscr. fig.) > 2 > (frustum BF) :
(circumscr. fig.).
This indicates the desired result, which is then confirmed by
the method of exhaustion, namely that
(frustum BF) = 2 (segment of paraboloid),
or, if V be the volume of the ' segment of a cone ', with vertex
A and base the same as that of the segment,
(volume of segment) = ^V.
Archimedes, it will be seen, proves in effect that, if k be
indefinitely diminished, and n indefinitely increased, while nk
remains equal to c,
then
limit of k{k+2k + 3k+ ... +(n— l)k} = \6\
that is,
in our notation,
Jo
JU(a/JU — ?> •
Prop. 23 proves that the volume is constant for a given
length of axis AD, whether the segment is cut off* by a plane
perpendicular or not perpendicular to the axis, and Prop. 24
shows that the volumes of two segments are as the squares on
their axes.