Baron of Merch iston. 177 



(san-ching-fang .*) In continuing to the end, we may deduce 

 the figures of 30 multiplications, and even more. Such is 

 the excellence of this method. Here we stop at the figure 

 of 5 multiplications. By imitating it, we may obtain all 

 the degrees of co-efficients sought. 



A note placed at the side of the triangle indicates, that 

 this table was already given in a more ancient work (the 

 collection of Ou- Chi), but without an explanation of the 

 method of forming it. 



The notation of the powers is made, as we see, by a. 

 regular method, which points out how many times the root 

 is multiplied by itself. 



The power 1 is called the root. 



,, 2 ,, the equal figure, the figure for one 

 multiplication , that is to say, as the 

 text expresses it, the figure where 

 we multiply the root by itself. 



,, 3 ,, the right or solid figure, that is, 

 the figure for two multiplications, 

 or where we multiply the root 

 twice by itself. 



,, 4 ,, the figure for three multiplications, 

 or where we multiply the root 

 three times by itself. 



,, 5 ,, the figure for four multiplications. 



Thus, the power of the order m will be the figure for 

 m — 1 multiplications. 



The author explains, that the two first powers alone 

 are represented by figures, and that the others are only the 

 result of numerical operations. 



In employing the co-efficients given by the table, several 

 square and cube roots are extracted in the same work. 

 There is even an example of the extraction of a root of the 

 4th power, which is effected by employing the co-efficients 

 4, 6, 4, given by the table. In relation to this last case, 

 the author adds, that it is more simple to extract first the 



• Such is the orthography of the Chinese second and third numerals, (li, sail) 

 in the French version. The common pronunciation, even hy the best educated 

 Chinese I have met with is, ye.e (two,) samm (three). Perhaps, in the first, the 

 liquid // might convey the sound of y. — Edit. 

 VOL. HI. 



