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XVII. — TJie $' of Diophantus. By Prof. D'Arcy Wentworth Thompson. 



(Read 18th May 1896.) 



The Arithmetical, or, as we should say, Algebraic, Problems of Diophantus involve 

 a single unknown quantity, symbolised by / or ? o/ . The number for which this symbol 

 stands is described as TrXrjOos fxovdSoov aXoyov* and is spoken of as 6 aopio-ros apidi*6$, the 

 undefined number, or simply as 6 apiO/mos, the number, par excellence, of the problem 

 in question. 



By all commentators, with but one exception so far as I know, the s' of Diophantus 

 is taken as an abbreviation of apiOfxog, and I find the same statement made unquestion- 

 ingly in the standard Palaeography of Gardthausen. The mathematicians, while 

 expressing no doubt as to the fact, show, however, in some cases, a clear enough 

 perception of the difficulty of accounting for such an abbreviation, and are not all agreed 

 as to its more precise origin. Nesselmann, Cantor, and most others, treat the ? as 

 the final letter of api6/u.6$ : Heath, who devotes many pages to a discussion of the 

 subject, puts forward, as an alternative and preferable hypothesis, a suggestion that the 

 ? may be a corrupt cursive abbreviation of the two first letters ap of the same word.t 



Objections to these two theories, or modifications of a theory, are easy to find, and 

 it is not necessary to discuss all the difficulties involved in the acceptance of either. 

 The first, viz., that ?' represents the last letter of aptO/mos, is demolished over and over 

 again, for not only is improbability written on the face of it, but it is incompatible 

 with the occurrence of such inflected forms as ?°°, s ' (cf. r' = tov), ?? ot ', ?? w ", &c. The 

 other, viz., that 9 = ap, leaves unexplained the alternative form ? '. 



In my opinion, the ?' has no connection with apidixos, but is the initial letter of 

 trwjoo'?, a heap. 



This view is countenanced by the following circumstances : — 



1. We are now dealing with an initial letter, as in the contractions used by the 

 same writer and by others for such words as Svva/m.1? and Kiifios. 



* I am not quite content with Mr Heath's rendering {Dioph. of Alex., p. 57) of TrXrjOos [jt.ova.8wv aXoyov 

 as " a number of units of which no account is given, or undefined," but I rather think that dAoyos is here 

 used in a more technical sense. The especial significance of aAoyos is irrational, usually in the sense of 

 " surd," i.e., a number having no determinable " ratio" : I take it here to refer, in a similar way, to a number 

 whose "ratio" is not yet determined. The equivalent phrase 6 dopioros dpifyios is precisely comparable to 

 the Aristotelian ddpio-ros Suds, fj aopio-ros Suds is, in our nomenclature, 2re or 2x; and the statement 

 (Arist., Met., xii. 7, 1081a) that 6 dpifyids eo-nv e« tov Iv6s ko.1 t^s SudSos ttJs aopio-Tov, simply means that any 

 number can be expressed either as 2n or 2n + 1. In like manner 6 aopio-ros dpifyio's is n or x. 



t The one writer who declines to commit himself as to the derivation of the s from dptfyxos is Gow 

 (Hist, of GJc. Mathem., p. 108, footnote) ; the symbol, he says, " may be Egyptian, or Indian, or Babylonian, 

 and may reveal an entirely unknown chapter in the history of mathematics." He is inclined to look for the 

 origin of the s (and also of the Diophantine /p) in some pictorial hieroglyph. 



VOL. XXXVIII. PART III. (NO. 17). 4 P 



