234
THE SQUARING OF THE CIRCLE
1 Archimedes, ed. Heib., vol. iii, pp. 258-9.
There is evidence of a still closer calculation than Ptolemy’s,
due to some Greek whose name we do not know. The Indian
mathematician Aryabhatta (born A. D. 476) says in his Lessons
in Calculation:
‘To 100 add 4; multiply the sum by 8; add (52000 more
and thus (we have), for a diameter of 2 myriads, the approxi
mate length of the circumference of the circle ’;
that is, he gives or 3-1416 as the value of ?r. But the
way in which he expresses it points indubitably to a Greek
source, ‘ for the Greeks alone of all peoples made the myriad
the unit of the second order ’ (Rodet).
This brings us to the notice at the end of Eutocius’s com
mentary on the Measurement of a Circle of Archimedes, which
records 1 that other mathematicians made similar approxima
tions, though it does not give their results.
‘ It is to be observed that Apollonius of Perga solved the
same problem in his ’LIkvtoklov (“ means of quick delivery ”),
using other numbers and making the approximation closer
[than that of Archimedes]. While Apollonius’s figures seem
to be more accurate, they do not serve the purpose which
Archimedes had in view: for, as we said, his object in this
book was to find an approximate figure suitable for use in
daily life. Hence we cannot regard as appropriate the censure
of Sporus of Nicaea, who seems to charge Archimedes with
having failed to determine with accuracy (the length of) the
straight line which is equal to the circumference of £he circle,
to judge by the passage in his Keria where Sporus' observes
that his own teacher, meaning Phil on of Gadara, reduced (the
matter) to more exact numerical expression than Archimedes
did, I mean in his i and ; in fact people seem, one after the
other, to have failed to appreciate Archimedes’s object. They
have also used multiplications and divisions of myriads, a
method not easy to follow for any one who has not gone
through a course of Magnus’s Logistica.’
It is possible that, as Apollonius used myriads, ‘ second
myriads’, ‘third myriads’, &c., as orders of integral numbers,
he may have worked with the fractions ----- > , &c.;
J 10000 10000 2