A history of Greek mathematics - Wilbourhall.org

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64 ARCHIMEDES
The conclusion, confirmed as usual by the method of exhaustion,
is that
(frustum BF) :
(segment
of spheroid) = (c + b) : { c + b - (%v + J b)
= (c + 6):(|c + #6) 5
whence (volume of segment) : (volume of cone ABB')
=:(|c + 26):(c + 6)
= (3GA-AD):(2GA-AD), since CA = ±c + b.
As a particular case (Props. 27, 28), half the spheroid is
double of the corresponding cone.
Props. 31, 32, concluding the treatise, deduce the similar
formula for the volume of the greater segment, namely, in our
figure,
(greater segmt.) :
(cone or segmt.of cone with same base and axis)
= (CA + AD):AD.
On Spirals.
The treatise On Spirals begins with a preface addressed to
Dositheus in which Archimedes mentions the death of Conon
as a grievous loss to mathematics, and then summarizes the
main results of the treatises On the Sphere and Cylinder and
On Conoids and Spheroids, observing that the last two propositions
of Book II of the former treatise took the place
of two which, as originally enunciated to Dositheus, were
wrong; lastly, he states the main results of the treatise
On Spirals, premising the definition of a spiral which is as
follows:
'
If a straight line one extremity of which remains fixed be
made to revolve at a uniform rate in a plane until it returns
to the position from which it started, and if, at the same time
as the straight line is revolving, a point move at a uniform
rate along the straight line, starting from the fixed extremity,
the point will describe a spiral in the plane.'
As usual, we have a series of propositions preliminary to
the main subject, first two propositions about uniform motion,