200 SCIENCE PROGRESS 



triangular diagram of numbers to facilitate the calculation of 

 the higher powers. The extraction of square and cube roots 

 and of higher roots compounded of these is clearly expounded ; 

 and when a number will not "extract" perfectly Napier shows 

 how to find limits between which the incommensurable value lies. 

 Perhaps, however, the most interesting feature is his recog- 

 nition of the imaginary quantity, namely, the square root of a 

 negative number. In particular the imaginary ■/— 9 is intro- 

 duced with the warning that the radicle and the sign must not 

 be transposed. Prof. Steggall infers that something of value 

 has been lost, to which Napier refers when he speaks of a 

 11 great algebraic secret, from which (although it has, as far as I 

 know, not been revealed by any one) it will afterwards appear 

 how great advantage will follow to this art and to the rest of 

 mathematics." 



We know on the authority of Robert Napier that the manu- 

 script of De Arte Logistica was by a wonderful chance saved 

 from destruction by a fire in which many of John Napier's 

 papers perished. Among these there may well have been 

 a fuller discussion of the imaginary. There is no doubt that 

 Napier's reference to imaginaries is the first on record. To 

 quote from Dr. George Philip : " Historians of algebra usu- 

 ally credit Girard with being the first to use imaginary roots of 

 equations, but in view of the above the Flemish mathematician 

 must waive his claim in favour of Napier. As Girard's most 

 important work was published in 1629, there is no question of 

 Napier having got the idea from him, and it is superfluous to 

 remark that Girard could not have borrowed from Napier." 



It is not possible within the limits of a short article to do 

 more than touch upon the many points of history which were 

 dealt with by the Congress. It may be of interest, however, 

 to refer more particularly to one line of discussion which 

 received attention at the Tercentenary Celebration, namely, 

 the development of logarithmic and trigonometrical tables since 

 the days of Napier. 



Thus questions were raised and discussed as to the improve- 

 ment of mathematical tables in general, the mode of arrangement, 

 the economising of space, the advisability of publishing new 

 logarithmic tables to 12 or 15 significant figures, and the like. 



In connection with this last question of extended logarithmic 

 tables, one of the most interesting exhibits at the Celebration 



