56 
ARCHIMEDES 
case of all, where we are told that OD 2 : A D 2 > 349450:23409 
and then that 0D:DA > 591^:153. At the points marked 
* and f in the table Archimedes simplifies the ratio <x 2 : c and 
a 3 : c before calculating b 2 , b s respectively, by multiplying each 
term in the first case by T 4j and in the second case by |q . 
He gives no explanation of the exact figure taken as the 
approximation to the square root in each case, or of the 
method by which he obtained it. We may, however, be sure 
that the method amounted to the use of the formula (a±b) 2 
= a 2 ±2ab + b 2 , much as our method of extracting the square 
root also depends upon it. 
We have already seen (vol. i, p. 232) that, according to 
Heron, Archimedes made a still closer approximation to the 
value of 77. 
On Conoids and Spheroids. 
The main problems attacked in this treatise are, in Archi 
medes’s manner, stated in his preface addressed to Dositheus, 
which also sets out the premisses with regard to the solid 
figures in question. These premisses consist of definitions and 
obvious inferences from them. The figures are (1) the right- 
angled conoid (paraboloid of revolution), (2) the obtuse-angled 
conoid (hyperboloid of revolution), and (3) the spheroids 
(a) the oblong, described by the revolution of an ellipse about 
its ‘ greater diameter ’ (major axis), (b) the flat, described by 
the revolution of an ellipse about its ‘ lesser diameter ’ (minor 
axis). Other definitions are those of the vertex and axis of the 
figures or segments thereof, the vertex of a segment being 
the point of contact of the tangent plane to the solid which 
is parallel to the base of the segment. The centre is only 
recognized in the case of the spheroid; what corresponds to 
the centre in the case of the hyperboloid is the ‘ vertex of 
the enveloping cone’ (described by the revolution of what 
Archimedes calls the ‘nearest lines to the section of the 
obtuse-angled cone’, i.e. the asymptotes of the hyperbola), 
and the line between this point and the vertex of the hyper 
boloid or segment is called, not the axis or diameter, but (the 
line) ‘adjacent to the axis’. The axis of the segment is in 
the case of the paraboloid the line through the vertex of the 
segment parallel to the axis of the paraboloid, in the case