A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

a
MEASUREMENT OF A CIRCLE 55
be
The series of values found by Archimedes are shown in the
following table
> 2339J 1007 (< \/(1007 2 + 66 2 ) 66
n a b c
265 306 153 1351 1560 780
571 > |V(571 2 +153 2 )]
153 1 2911 < 591|
< 3013JJ
1162|>[V{(1162|) 2 +153 2 }] 153 2 f5924j|
780]*
>1172|
1823 ( < v / (1823 2 +240 2 ) 240/
1
9
< 1838 TT
2334i>[v / {(2334£) 2 +153 2 }] 153 3 3661 T 9 T ... 240 f
4673| 153 4
2016|
( < 1009J
< V{(2016^) 2 + 66 2 } 66
< 2017f
and, bearing in mind that in the first case the final ratio
a :
4
c is the ratio A : AG = 2 OA : OH, and in the second case
the final ratio 6 4
: c is the ratio AB : BG, while GH in the first
figure and BG in the second are the sides of regular polygons
of 96 sides circumscribed and inscribed respectively, we have
finally
96X153 96x66
> 7T >
4673| 2017J
Archimedes simply infers from this that
As a matter of fact
3i >tt > 3if
96 x 153 667| 667
= 3 * _ i
and
"
4673J 4673|'
4672J
1
It is also to be observed that 3^£ = 3 -\ -, and it may
have been arrived at by a method equivalent to developing
the 6336
fraction in the form of a continued fraction.
2017j
It should be noted that, in the text as we have it, the values
of b lf b 2
, 63, 6 4
are simply stated in their final form without
the intermediate step containing the radical except in the first
* t Here the ratios of a to c are in the first instance reduced to lower
terms.
56 ARCHIMEDES
case of all, where we are told that 0D l :AD 2 > 349450 :
and then that OD.DA > 591j:153. At the points marked
* and f in the table Archimedes simplifies the ratio a :
2
c and
a :
z
c before calculating b 2
, b z
respectively, by multiplying each
term in the first case by 5 % and in the second case by JJ.
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
23409
that the method amounted to the use of the formula (a±b) 2
= a 2 + 2 ab + 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 Archimedes'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 rightangled
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 hyperboloid
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