THE GENESIS OF LOGARITHMS 165 



of the complement of the half-arc to the sine of the whole arc." 

 This is, of course, the Napierian equivalent of the trigono- 

 metrical theorem 



sin 2d = 2 sin 6 cos 8, 



which equation, written in the form 



. sin 20 



sin 6 = - — : — j- rt\, 



2 sin (90 - 6) 



enables us at once to compute the logarithm of sin 0, knowing 

 the logarithms of the sines of 2 and of (90 — ^) ; choosing so 

 that 2 0, to begin with, lies within the limits of the Radical 

 Table, the table may be gradually extended so as to include the 

 logarithms of the sines of all angles from 30 to o°. It is 

 further pointed out 1 that this method can be used for all angles 

 less than 45 ; so that the construction of the logarithms of the 

 sines of the angles between 45 and 30° is thereby rendered 

 much more simple, the use of the " difference theorem " and the 

 Radical Table being avoided. 



It is hoped that the preceding analysis will suffice to show 

 the uniqueness and originality of Napier's great discovery. 

 The publication of the Descriptio in 1614 was hailed with an 

 amount of enthusiasm and the full credit of Napier's work 

 awarded to him with a unanimity seldom paralleled in the 

 annals of mathematical discovery. After the fashion of the 

 times, the enthusiasm of Napier's contemporaries found vent in 

 a number of laudatory poems, of which one by Thomas Bretnor 

 possesses sufficient merit, apart from its somewhat too-fervid 

 patriotic spirit, to bear reproduction to the extent of a couple 

 of verses : 



"And bonnets vaile, you Germans! Rheticus, 

 Reignoldus, Oswald, and John Regiomont, 

 Lansbergius, Finckius and Copernicus, 

 And thou, Pitiscus, from whose clearer font 

 We sucked have the sweet from Hellespont. 

 For were your labours ne'er composed so well 

 Great Napier's worth they could not parallel. 



By thee great Lord we solve a tedious toyle, 

 In resolution of our trinall lines, 

 We need not now to carke, to care, or moile, 

 Sith from thy witty braine such splendor shines, 

 As dazels much the eyes of deepe divines. 

 Great the invention, greater is the praise, 

 Which thou unto thy nation hence doth raise." 



1 Constrnctio, Prop. LVIII. 



