used by different Nations. 323 



of the astronomer of Alexandria, where degrees, minutes, and 

 seconds are indicated, the value is evidently expressed by the 

 position of the figures. They follow one another in different 

 layers, and thus, they express an absolute and a relative value. 

 But, as in the last mentioned scale, every layer is composed of 

 two figures (for want of n— 1 or 59 figures) ; the value expressed 

 by position does here not procure the advantages accruing from 

 the Indian figures. When the three hundred and sixty parts 

 of a circle are considered as so many entires, the minutes are 

 sixtieths of them, the seconds are sixtieths of the minutes, &c. 

 Considering them as fractions, Ptolomaeus distinguished them 

 by the sign of fraction, a line placed above, and, in order to ex- 

 press their descending progression, by which every layer of two 

 figures has sixty times less value than the preceding, the frac- 

 tion-signs were increased in number from layer to layer. 

 According to these principles, the minutes are indicated by one 

 line, the common designation of fractions (the numerator of 

 which is the unit), the seconds by two such lines, the terces by 

 three ; but, the degrees themselves, as being entires, were not 

 distinguished by a line, but, perhaps, by nought (o^^g'v), or a 

 cypher* — I say, perhaps, for in the writings of Ptolomaeus 

 and Theon, the figure of the cypher is not yet used to indicate 

 degrees. 



The simple enumeration of the methods used by those 

 nations who did not know the Indian system of position, in ex- 

 pressing the multipla of the fundamental groups, shews, in my 

 opinion at least, the way in which the Indian system probably 

 has been invented by degrees. When the number 3568 is 

 written either in a perpendicular or in a horizontal direction, by 



3 5 6 8 



means of indicators: M C X I, it is evident that the figures of 

 the groups M C, &c., may be omitted. For our Indian figures 

 are only the multiplicators of the different groups. This mode 



* On the use of the cypher, see Leslie, p. 12, 135; Germanen und Griechen 

 Hist., V. ii., p. 2 — 33 ; Ducange, Glossar. Mediae Graecitatis, t. ii., p. 572. Mannert 

 de Nummorum quos Arabicos vocant Origine Pythagor. p 17. In the Greek 

 arithmetic M,, signifies the unity; f/.cms, as a delta (A), when a cypher (properly 

 omicron) is placed over it, signifies tetartos. Bast, Gregor., Corinth, p. 85 1 . Thus, 

 we find in Diophantus, M''*«:=:21. The Indian grammatical sign, anuswara, has 

 indeed the figure of the Indian cypher. It indicates, however, nothing but a modi- 

 fication in the accentuation of the vowel placed nearest to it, and is ia no way 

 connected with the sunya. 



