178 Memoir of John Napier, Baron of Merchiston. 



square root of the number given, and then the square root 

 of the number found. 



From what precedes, we see that the formation of co- 

 efficients of the different powers of a binomial expressed in 

 whole numbers, was known to the Chinese, at least, in 1593. 

 At this period, the first Jesuits, as missionaries, arrived in 

 China ; but the arithmetical triangle of Pascal was not 

 published in Europe till 1665. And, besides, if we examine 

 the nature of the questions treated of in the work cited, 

 and the methods there given for resolving them ; if we 

 consider the squares and magic circles which it contains, 

 we cannot believe that any European had assisted in digest- 

 ing it. 



The theorem of the formation of the co-efficients of a 

 binomial existed with the Arabs in 1430, and, perhaps, was 

 unknown in China, till after the conquest of the Mongols, 

 who called some Arabian philosophers to their court. We 

 find, even in the last section of the Chinese work, the mode 

 of multiplication, by triangular network, adopted for a long 

 time by the Arabs ; and again, in the classification of dif- 

 ferent numerical units, the units of a very high order 

 are distinguished by the name of ' sands of the river Ganges,' 

 which indicates some prior relations of the Chinese with 

 the Hindoos, who had also, as we know, notions of alge- 

 braical science and geometry. 



From the same Chinese work, we see that the Chinese, 

 at this period, knew the theory of similar triangles, the 

 exact measure of the pyramid and cone, as well as that of 

 the mass of the cone and pyramid, and the proportion 2 T 2 of 

 the circumference to the diameter, as they generally used 

 in these calculations the proportion of 3 to 1. Besides the 

 summation of series which we have noticed, they under- 

 stood the resolution of equations of the second degree with 

 one unknown number ; they even resolved by groping, 

 numerical equations of the third degree with one unknown 

 number, of which, it is true, they only extracted one root. 

 But, neither in this work, nor in those of the Arabs and 

 Hindoos, do we find notation by letters employed symboli- 

 cally to express numerical quantities as we do at present. 

 This invention, which constitutes truly the strength of 

 algebra, is altogether European, and due to Vieta. 



Edouard Biot. 



