ON MATHEMATICAL TABLES. .^J 



On Byrge's claim see De Morgan's careful resume (article " Tables," under 

 Justus Byrgius, 1620, in the 'Eng. Cyclop.,' where references are given), 

 and Mr, Mark Napier's ' Memoirs of John Napier of Merchiston,' Edin- 

 burgh, 1834 (where the question how far Napier received any assistance 

 from his predecessors in the discovery is fully discussed). We have also seen 

 ' Justus Byrg als Mathematiker und dessen Eiuleitung in seine Logarith- 

 men,' by Dr. Gieswald, Dantzig, 1856, 4to (pp. 36). Napier's ' Canonis 

 Logarithmorum Mirifici Descriptio ' (which contained the first announcement 

 and the first table of logarithms) was published in 1614 ; and in 1619 (two 

 years after his death, which occurred on April 4, 1617) appeared the ' Mirifici 

 logarithmoiTim Canonis Constructio,' edited by his son Robert, in which the 

 method of constructing the canon is explained. The various reprints and 

 translations of the ' Descriptio ' and ' Constructio ' are described under 

 Napier, 1614 and 1619 ; and the relations between Napier and Briggs with 

 regard to the invention of decimal logarithms are noticed in § 3, art. 13. 

 The most elaborate canon of Napierian logarithms is Ursinus (1624-1625), 

 described below. 



The diflference between the logarithms introduced Napier and hyperbolic 

 logarithms is explained under Napier (1614). We have paid considerable 

 attention to the early logarithmic tables, and have examined all of them that 

 were accessible to us ; and it is with some regret that we omit to notice them 

 in detail here : the accounts of the smaller tables that immediately suc- 

 ceeded Napier would be of only bibliographical or historical interest ; and to 

 describe them with sufiicient detaU to render the accounts of value would 

 occupy too much space. However, as the works of this period are very rare, 

 it is worth while remarking that there is a copy of Napier's * Constructio ' 

 in the Cambridge University Library (there is none in the British Museum 

 or Royal Society's Library), where also are to be found Ursinus's ' Cursus ' of 

 1618, Speidell 1619, and Kepler 1624: we have generally, in describing 

 works of this date, mentioned the library containing the copy we have seen. 

 We have found De Morgan to be very accurate (except where he has had to 

 form his opinions from secondhand or imperfect evidence) ; and he has 

 'devoted much care to the early logarithmic tables, so that we feel the less 

 reluctance in omitting to notice them further here. 



Napier, 1614. The book consists of 57 pp. explaining the nature of 

 logarithms &c., and 90 pp. of tabular matter, giving natural sines and their 

 Napierian logarithms to every minute of the quadrant (seraiquadrantaUy 

 arranged) to seven or eight figures (seven decimals). Logarithmic tangents 

 are also given under the heading differenUce (they are the differences between 

 the sine and cosine, which, though the latter name is not used, are both on 

 the same line, as a consequence of the semiquadrantal arrangement of the 

 table). 



The logarithms introduced by Napier were not hyperbolic or Napierian 

 logarithms as we now understand these terms, viz. logarithms to the base e 

 (2-71828 . . ), but somewhat difierent ; the relation between the two being 



L^ 



e' = lO^.e 10^ or L = 10^ log, 10^ - 10^ I, 



I being the logarithm to base e, and L the Napierian logarithm j the relation 

 between N (a sine) and L, its Napierian logarithm is therefore 



L 



N = 10,000,000 <5 10,000,000; 



