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