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ARCHIMEDES 
The conclusion, confirmed as usual by the method of ex 
haustion, is that 
(frustum BF) : (segment of spheroid) = (c + 6):{c + 6-(^c + ^6)} 
= (c + 6):(£c + |6), 
whence (volume of segment) : (volume of cone ABB') 
= (|c + 2h) : (c + b) 
= (3GA — AD) :{2GA—AD), since GA — ^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 scgmt.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 pro 
positions 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,