used by different Nations. 313 



nations of the west as (ibax or tabula logistica, are used in the 

 same manner, only that the strings indicate groups ot second, 

 third, and fourth order upwards and downwards. The kquas, 

 ^V^ich sire niore ancient than the characters used at present by 

 the Chinese, and even the magic drawings (ralm) of central 



sja and Mexico, which exhibit knotty parallels, often broken 



*, almost in the manner of musical notes, seem, piuy'j^p be 

 graphic projections of these calculation and reflection-strings *. 



In the Asiatic suanpan, and in the abacus, which was much 

 mpj^f used by the Romans, on account of the inconvenient 

 figures adopted by them, than by the Greeks, who had been 

 much more successful in their mode of writing numbers f, the 

 quinary rows are preserved together with the denary ones, 

 forming geometrical progressions upwards and downwards. 

 Outside of every calculation-string, indicating a group or 

 order, (n, n«, n',) a shorter string was placed, on which every 

 ball expressed the amount of five balls of the longer string. 

 By this contrivance the number of the unities was diminished 

 to such a degree, that the principal string needed only to con- 

 tain four balls, and the accessary only one J. 



It seems that among the Chinese, from the most ancient 

 times, the custom had prevailed to consider arbitrarily any 

 one of the parallel strings as containing the units, and that 

 thus they obtained, upwards and downwards, decimal frac- 

 tions, entire numbers, and powers of ten §. How late (in the 



* In the East, rfl/»e is called the negromantic art of the sand. Uninterrupted 

 lines, and others broken off, present the elements of it, and direct the nej^romant. 

 (Richardson and Wilkins, Diction. Persian and Arabic. 180G. t. i., p. 482.) In 

 Dresden a remarkable manuscript is preserved. It was brought from Mexico, and 

 exhibits nothing but figures like musical notes. I have published it in my 

 Monumens Americ. PI. xUv. When at Paris, I was visited by a learned Persian, 

 who, at the first view, recognized in it an oriental ralm, Quite similar, and tndy 

 American kouas and linear drawings, I discovered after that time in some 

 Aztetic hieroglyphic manuscript;!, and on the sculptures of Palenque, in the repub- 

 lic of Guatimala. In the ancient Chinese numerical figures, the sign of the 

 croup of ten is a pearl on a string, and evidently an imitative drawing of the 

 quippu. 



>:f Nicomachus, in Ast. Theologiimena Arith. 1817, p. 96. In the iinaBcial 

 system of the middle age, the account table (abax) became the exchequer. 



, t So the lloman abacus was contrived. In China five balls were placed on the 

 first, and two balls on the last. The balls, which were not employed in a calcula- 

 tion, were pushed aside. , ^ 

 "'l^ On the first attempts to establish the decimal system, made by Michael 

 Sti/e/ius of Kslingen, Stevenus of 3nigge; and BombdU pf Bologna, ae^ Ji^eslie, 

 Philos. of Anthmetic, p, 134, 



