Vol. Ill, No. 7.] Notes on Indian 3Iathematic.i. 491 



when on division by nine the linear remainder is 1 or 8, 2 or 

 7, 3 or 6, 4 or 5 the square remainders are respectively 1, 4, 9, 

 IG (^^e., 7).^ (Of course it will be at once seen that this is 

 only a particular case of a more general proposition.) 



An interesting anecdote is related by Bayley in the following 

 quotation from jVIasandi, who visited India at the close of the 

 tenth century : " Un congres des sages renni par ordre du roi 

 composa le livre du Sind Hind . . • . lis inventerent aussi les 

 neuf chifEres qui forme le systeme numeriqne indien." Congresses 

 and councils are not often known to invent^ but it is quite con- 

 ceivable that at such a meeting the adojpHon of a new system . 

 (possibly foreign) might be considered. 



A similar anecdote is related about the Khalif Walid who 

 reigned from 705 to 715 A.D. . . . It is stated that he forbade, 

 by a special edict, the use of the Greek language in the public 

 accounts. He made, however, a special reservation in favour of 

 Greek letters as numeral signs, on the ground that the Arabic 

 language possessed no numerals of its own. Nmv the Arabic 

 ahjad is exactly the same as the Greek alphabetic notation and it 

 is ''undoubtedly ancient " as Bayley states, and therefore the 

 edict could 7iot refer to the Greek alphabetic notation. There are 

 only two possible conclusions, viz.^ (^) the edict referred to some 

 special notation of the Greeks ( ? the apices of the Neo-Pythago- 

 reans) ; or (n) the whole tale is false. 



Such evidence as is contained in this section, being more or 

 less legendary, does not carry very great weight. The points here 

 dealt with would not, in all probability, have been taken up in the 

 present argument if they had not been already used by the Indian- 

 ists. Their value here lies in the rather remarkable truth that 

 they help to prove just the opposite to the theory they were 

 intended to support. It is disappointing that so-called historical 

 evidence can avail so little in such an investigation as this. Even 

 when we come to the records of such a reliable investigator as 

 Albiruni, we find very little really pertinent to the question In 

 hand. When he visited India (in the eleventh century) the new 

 notation must have been fairly well established,* His language 

 is not always perfectly unambiguous, but what he says leads us to 

 conclude that the Hindus he came across 2cere ignorant of the 

 fundamental ^principles of matJiematics, " At first, " he writes, 

 *' [ stood to their astixjnomers in the relation of a pupil to his 

 master, being a stranger among them and not acquainted with 

 their peculiar national and traditional methods of science. On 

 having made some progress, I began to show them the elements 

 on which the science rests, to point to them some rules of logical 

 deduction and the scientific methods of all mathematics, and 

 then they flocked together round me from all parts and most 



1 A similar example is given by Theou of Smyrna, A.D. 130. 

 » Is the antiquity of the abjad so certain? What is the earliest epigra 

 phical instance ? 



