TRANSACTIONS OF THE SECTIONS. 17 



in the British Museum. On the titlep.age of the 'Disme ' are the words "Preniiere- 

 ment descripte en Flameng, & maintenant conuertie en Francois, par Sinidn Stevin 

 de Bruges." These words, appearing also in Albert Girard's collected edition of 

 Stevinus's works (1634), no doubt gave rise to De Morgan's inference that " the 

 method of decimal fractions was announced before 158o in Dutch." The Cambridge 

 University Library possesses a 1585 copy entitled " De Thiende . . . Beschreven 



door Simon Stevin van Brugghe Tot Leyden, By ChristoHel Plautijn, 



M.D.LXXXV " (privilege dated December 20, 1.584) ^ and there seems every reason 

 to believe, in the absence of any evidence to the contrary, that this was the first 

 edition of this celebrated tract. Peacock's statement that " it was first published 

 in Fleniish about the year 1590, and afterwards translated into Ijarbarous French 

 by Simon of Bruges," is also, I suspect, founded on no other evidence than the 

 sentence on the titlepage of the 'Disme,' which appears also in Girard. De 

 Morgan rightly remarks that Simon of Bruges is Stevinus himself, but he cannot 

 tell whence Peacock derived the date 1-590. It is probable that it was merely a 

 rough estimate obtained bj^ considering tlie dates of the other works of Stevinus. 

 Stevinus's method involved the use of his cumbrous exponents : thus he wrote 



27-847 as ^1 8 (T) 4 (V) 7 (¥) , and read it 27 commencements, 8 primes, 



4 seconds, 7 thirds ; and the question chiefly noticed in tliis abstract is the conside- 

 ration of who first saw that, by a simple notation, the exponents might be omitted, 

 and introduced this abbreviation into arithmetic. 



Napier's ' liabdologia ' was translated into several languages soon after its ap- 

 pearance ; and I have taken some pains to examine the different ways in wliich the 

 translators treated the example which Peacock regarded as the first use of the 

 decimal point, as we can thereby infer something with regard to the state of 

 decimal arithmetic in the different countries. Napier (1617) wrote 199.3,273 in 

 the work and 1993,2'7"3"' in the text. In Locatello's translation (Verona, 1623) 

 this is just reversed, via. there is 1993.2'7"3"' in the work and 1993,273 in the text. 



The Lyons edition (1626) has 199-3,273 in the work and 199-3,2 (T) 7 (s) 3 (s) 



in the text, while De Decker's edition (Gouda, 1626) has 1993,273 in the work, 



and in the text 1993 (o^ 2 (T) 7 (2^ 3 (S^ , the last being exactly as Stevinus 



would have written it. Ursinus's ' Rhabdologia Neperiana,' Berlin, 1623, is not an 

 exact translation ; and the example in question does not occur there. 



Some Sur/c/esthns towards the Formation of an extended Table of Logarithms. 



By G. 0. Hanlon. 



On the Theory of Differentvd Resolvents. 

 By the Rev, Robert Haklet, F.It.S. 



In the earlier development of the theory of differential resolvents attention was 

 confined almost exclusively to certain trinomial forms of algebraic equations, and 

 the resolvents were calculated for these forms. A connexion not before noticed 

 was found to exist between algebraic and differential equations ; and results re- 

 markahle for their simplicity and elegance were obtained. Some of these results 

 have been laid before the Section at former Meetings (see Reports of the Asso- 

 ciation, ' Transactions of Sections,' 1862, pp. 4, 5 ; 1865, p. 6 ; 1866, pp. 2, 3). 



Every differential resolvent may he regarded under two distinct aspects: it 

 may he considered either (first) as giving in its complete integration the solution 

 of the algebraic equation from which it has been derived, or (.secondly) as itself 

 solvable by means of that equation. The two equations, the algebraic and the 

 differential, are in fact coresolvents. The subject was first considered in the former 

 aspect by Sir .lames Cockle, the originator of the theory, and bv Mr. Harley : and 

 their researches will be found emhodied in various papers published in the ' Phi- 



1873. 2 



