ON MATHEMATICAL TABLES. 69 



those now called h/jjei-holic (viz. to base e) and very frequently also Naj^ierian 

 logarithms. It is also to be noticed that JSTapier calculated no logarithms of 

 numbers. Jonx Speidell, 1019 (see below), first published logarithms to 

 base e bothof numbers and sines. The most complete table of hj'perbolic 

 logarithms is Base's, described below, which could be used, though not so 

 convenieutly, as an ordinary seven-figure Briggian table extending from 1000 

 to 105,000. It would sometimes be useful to have also a complete seven- 

 place table of hyperbolic logarithms of numbers from 1000 to 100,000, ex- 

 actly similar to the corresponding' Briggian tables, as in some cases it is con- 

 venient to perform calculations in duplicate, first by Briggian, and then by 

 hyperbolic logarithms ; and such a table would be of use in multiplying- five 

 figures by five figures : but hyperbolic logarithms cannot be rendered conve- 

 nient for general purposes. 



The most elaborate hyperbolic logarithmic table is "Wolfkam's, which prac- 

 tically gives the hyperbolic logarithms of all numbers from unity to 10,000 

 ioforty-eiglit decimal places. It first appeared, we believe, in Schulze (§ 4), 

 and was reprinted in Vega, folio, 1794 (§ 4). 



Wolfram was a Dutch Ueutenant of artillery ; and his table represents six 

 years of very laborious work. Just before its completion he was attacked by 

 a serious illness ; and a few logarithms were in consequence omitted in Schuize 

 (see Introduction, last page but two, to vol. i. of Schulze). The omissions 

 were supplied in Vega's ' Thesaurus,' 1794. De Morgan speaks of Wolfram's 

 table as one of the most striking additions that have been made in the sub- 

 ject of logarithms in modern times. 



Montucla (' Histoire,' vol. iii. p. 360) states that in 1781 Alexander Jom- 

 bert proposed to publish by subscription new tables of hyperbolic logarithms 

 to 21 places for all prime numbers to 100,000, with a table of all odd numbers 

 of two factors to the same limit. The author was Dom Vallej're, advised by 

 Dom Robe, benedictine of St. Maur. Only two hundred subscribers were re- 

 quired before the commencement of the printing, and nothing was asked in 

 advance; but the project fell through, no doubt for want of subscribers. 

 We infer from this account that the table was calculated. 



The Catalogue of the Royal Society's Library contains, under the name of 

 Prony, the title, " Formules pour calculer I'efl'et d'une machine a vapeur a 



detente et a un seul cylindre Tables de logarithmes hyperboliques calcu- 



lees de 100^ en 100" d'unite, fol. lithog.," but without any reference to the 

 place where the book is to be found in the library, so that we have not seen it. 



Speidell, 1619. Logarithmic sines, tangents, and secants, semiquadi-antally 

 arranged, to every minute, to five places. The logarithms are hyperbolic (viz. 

 to base e), and the first of the kind ever published. When the characteristic 

 is negative SpeideU adds 10 to it, and does not separate the characteristic so 

 increased from the rest of the figures by any space or mark. Thus he prints 

 the logarithm of the sine of 21° 30' as 899625, its true value being 2-99625 ; 

 but the logarithm of the cotangent is given as 93163 ; it would now bo 

 written -93163. The Royal Society has " the 5-impression, 1623," with the 

 " Breefe Treatise of Sphaericall Triangles " prefixed, and also some ordinary 

 hj-perbolic logarithms of numbers (the first published) &c. On this see De 

 Morgan's long account of Speidell's works, who, however, had never seen the 

 edition of 1619, in which the canon occurs by itself without the logarithms 

 of numbers. We cannot enter into the question of Speidell's fairness here. 

 The 1619 copy we have seen (Cambridge Univ. Lib.) has an obUteration 

 where, in the 1623 copy, there occur the words " the S-impression." 



