Vol. Ill, No, 7.] Notes on Indian Mathematics. 499 



the determination of the ratio of the circumference to the 

 diameter of the circle, the Arabian mathematician has copied 

 the Hindus. The result given by the Muhammadan may 



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be represented hy p = d X ^ , while the result as given in 



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the Lilavati is represented hy j[>=d x YoKh' "^^^ ^^^^ ^^^^ these 



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two results are identically equal forms the basis of Rosen's argu- 

 ment to prove that M. ibn Musa copied from the Hindus,^ not- 

 withstanding that M. ibn Musa lived three hundred years before 

 the Lilavati was composed. Now, as Chasles says, " La verite, 

 en Geometric, est la loi commune, elle est une, elle appartient a tons 

 les temps, k toutes les intelligences qui savent la comprendre ; et 

 sa presence sur plusieurs points, chez plusieurs peuples, n'est pas 

 une preuve de communications entre eux " (p. 430) ; but Rosen 

 thinks otherwise and says, *' It is extremely improbable that the 

 Arabs should, by mere accident, have the same proportion as the 

 Hindus ; particularly, if we bear in mind that the Arabs did not 

 seem to have troubled themselves about finding an exact method. " 

 He adds to this astonishing comment the following still more 

 astonishing foot-note ^ -. " This would appear from the very man- 

 ner in which our author (M, ibn Musa) introduces the several 

 methods ; but still more from the following marginal note of the 

 manuscx'ipt to the general passage. ' This is an approximation, 

 not the exact truth itself : nobody can ascertain the exact 

 truth of this, and find the real circumference, except the Omnis- 

 cient .... This is called an approximation, in the same manner 

 as it is said of the square-roots of irrational numbers, that they 

 are approximations, and not exact truths : for God alone knows 

 * what the exact truth is. 



The remarks of Rosen about it being mere accident, and the 

 Arabs not troubling themselves, are too absurd to be considered ; 

 and, as to the note he ridicules, I venture to state that in the 

 whole range of Hindu mathematical writings nothing approach- 

 ing such a clear exposition of a difficult point has been given.^ 

 The facts refen^ed to by Rosen further prove the ridiculousness 

 of his arguments. They are here given side by side, and a glance 

 will show that accurate results were given by the Muhammadan 

 some three centuries earlier than by the Hindus in question, and 

 that he could not have possibly borrowed his results from them. 

 It is not necessary here to take into account the value given by 

 Aryabhata as Brahmagupta, on whose work that of M. ibn Musa 



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1 It is to be borne in mind that it la only "Eopen'e arguments that are 

 here being confuted- Rosen knew nothing: of Aryabhata. 



* Of course such arguments are not, in themselves, worth repeating. 

 It is on such, however, that the current theories are based. 



3 Since writing the above it has atrack me that Rosen possibly believed 



that the circle could be exactly ' squared/ Such a belief was not uncommon 



in his day. 



