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later, while there are hundreds intervening with examples of the old non-place value system. The references in mediæval works to India do not necessarily indicate India proper but often simply refer to 'the East' and the use of the term with regard to numbers has been further confused by the misreading by Wœpcke and others of the Arabic term hindasi (geometrical, having to do with numeration, etc.) which has nothing to do with India. Again, it has been assumed that the use of the abacus "has been universal in India from time immemorial," but this assumption is not based upon fact, there being actually no evidence of its use in India until quite modern times. Further, there is evidence that indicates that the notation was introduced into India, as it was into Europe, from a right to left script.

23. In paragraph 7 above certain attempts at squaring the circle are briefly described and it has been pointed out (in § 10) that Āryabhata gives an extremely accurate value of $\scriptstyle {\pi }$. The topic is perhaps of sufficient interest to deserve some special mention. The Indian values given and used are not altogether consistent and the subject is wrapped in some mystery. Briefly put—the Indians record an extremely accurate value at a very early date but seldom or never actually use it. The following table roughly exhibits how the matter stands:—

Date Circa.	Authority.	Value of $\scriptstyle {\pi }$
B.C. 1700	Ahmes the Egyptian	$\scriptstyle {({\frac {16}{9}})^{2}}$	$\scriptstyle {=3\cdot 160.}$
„  250	Archimedes	$\scriptstyle {<3{\frac {1}{7}}{\text{ and }}>3{\frac {10}{71}}}$
„    ?	The S'ulvasūtras	$\scriptstyle {({\frac {26}{15}})^{2}}$	$\scriptstyle {=3\cdot 004.}$
	"	$\scriptstyle {({\frac {9785}{5568}})^{2}}$	$\scriptstyle {=3\cdot 097.}$
„  230	Apollonius	$\scriptstyle {3{\frac {17}{120}}}$	$\scriptstyle {=3\cdot 14166.}$
„  120	Heron	$\scriptstyle {3}$	$\scriptstyle {=3\cdot }$
	"	$\scriptstyle {3{\frac {1}{7}}}$	$\scriptstyle {=3\cdot 14286.}$