used by different Nations, ' 319 



and the htin5i*eds, and in this manner the numerical value of 

 the Greek alphabetical figures approaches a little nearer to the 

 Semitic aboudjed *. M. Bockh, in his learned observations on 

 the digamma, has shewn that bau is the wau of the Semites, 

 (the F of the Latins, J koppa, the Semitic koph (9) , and sanpi 

 the Semitic shiri'f. The row of the unities beginning with 

 alpha, and ending with thefa, forms in the Greek system the 

 6eot-numbers (tirt/d/txEver) , and Apollonius had invented a con- 

 trivance f , by the help of which they were reduced, in the last 

 results, to the corresponding members of the second and third 

 h)W (the analogues). 



Second Method. Multiplication or diminution of the 

 value by signs placed over or under the figures. In the fourth 

 row of the Greek notations, the pythmenes return by analogy, 

 but increased a thousand times by a line placed under the 

 figures. In this way the Greeks arrived, in their numerical 

 system, at a myriad, — they wrote every number up to 9999. 

 Had they adopted this notation with a line for all the groups, 

 and suppressed all the figures after theta (9), the letter /S with 

 one, two or three lines would have expressed 20, 200, and 2000, 

 and thus the Greek system would have approached, as we shall 

 see afterwards, the system of the Arabic Gobar figures, which 

 is very little known, and at the same time the system expressing 

 value by position. But, unhappily, the Greeks did not adopt 

 this notation for the tenths and hundreds, applied it only for 

 the thousands, and did not try to employ it in higher groups. 



As a line added under the figures increases their value a 

 thousand times, thus a vertical line, in the Greek system, added 

 over the figure, indicates a fraction, whose numerator is the 

 unit, and whose denominator is expressed by the figure itself. 

 Thus in Diophantus, y'=i, ^'=i; but if the numerator is 

 greater than the unity, it is expressed by the principal figure, 

 and the denominator is written like an exponent 7'= J§. In 



♦ Hervas, Aritmetica delle Nazioni, p. 78. On the ancient order of the letters 

 in the Semitic alphabet see Description de I'Egypte mQderne,t. ii. P. ii. p. 208. 



■f Staatshaushaltung der Athener. B. II. p. 385. 



I Delambre, Hist, de T Astronomie Ancienne, t. ii. p. 10. 



§ Delambre, t. ii. p. 11 . The line added over the alphabetical letters to indicate 

 that they are used as numerical figures, ought not to be confounded with the sign 

 of fractions. The first is also never vertical in the oldest manuscripts of mathe- 



