THE s OF DIOPHANTUS. 609 



" diese meine Vermuthung ist jetzt bestatigt worden durch Michaelis Pselli epistola 

 inedita, in Diophanti Op. ii. pp. 37, 3-6, 38, 22-39, 6 [ed. Tannery]." 



Before I conclude, I am bound to admit that there seems to me to be one serious defect 

 in my theory as I have stated it in the foregoing account, to wit, that " a heap," which 

 already conveys the notion of a solid, is not what we should expect as the starting-point 

 of a series where Svvafus and /cu'j3o9 are to follow consecutively. To remedy this defect, I 

 offer two suggestions, which are not alternatives, but may be both partially and comple- 

 mentarily true. The <rwp6? may convey the idea not necessarily of a heap, but merely 

 of a chain of links or units, and the passage then becomes natural and legitimate to the 

 square and cube, with their connotations of area and solid. Or again, if we dwell on 

 what has been said above, and especially on the passage I have quoted from Hultsch, the 

 possibility presents itself to us that the awpos may relate not so much to the " indefinite 

 number," the unknown quantity, the x itself, as to the method to be employed for its 

 discovery, the factors by whose substitution it was to be solved. The analogy of the heap 

 or chain of syllogisms, the acervus or a-wpelr^ of the logicians, suggests to me that the 

 word o-wpos here may not be simply a metaphor drawn from a heap of sand, with its 

 myriad grains taken in the gross, but may be a direct allusion to that resolved chain of 

 factors which is the distinctive feature of the Diophantine analysis. I find not a little sup- 

 port to this view in Aristotle's use of the word in the Metaphys., vii. 17, 10416, to '4k 



tivos crvvQerov ovtw? ware ev elvai to ttuv, aXXa fjnj 009 crwpos aXX' g5? f\ av\Xa/3>], where 



it is as clear as possible that there is, in contrast to vvWafiih a sense of disjunction in 

 crwpog, antagonizing and predominating over the merely collective notion of a heap. If 

 all this be so, it is as true of the hau or "heap-calculus" of the Egyptian as of the 

 <rmp6<s of him who wrote in Greek ; it applies to the fractions of Ahmes as to the factors 

 of Diophantus. In regard both to method and to nomenclature, to the processes both 

 of logical and of mathematical analysis, it may help us to link together the Alexandrine 

 with the Theban, and to discern Antiquity learning of the Antiquity before. Though it 

 is but a little matter in itself, it reminds us that a stream of thought flowed down out 

 of Ancient Egypt into the Western World, from the Eiver to the Sea ; it points to a 

 descent of traditional science and a continuity of intellectual activity, reaching down 

 to the Silver Age of Imperial Eome from the Scholar-Priesthood of the Shepherd Kings. 

 Hcbc JEgyptia quondam, nunc et sacra Romana sunt. 



