TRANSACTIONS OF THE SECTIONS. 13 



On the Negative Minima of the Gamma function. 

 By J. W. L. GiAisHER, B.A. 



The definition of the gamma function usually adopted is in effect, that between the 

 values and 1 of a; it is defined by the equation r(.r+l) = f'°v''e~''dv, and for 

 all other values of x by the equation T(x+l) = xT(x). 



The curve »/=r(a;) has a minimum corresponding to a;=l'4616321 . . . , as is well 

 known ; but as r{x) is infinite whenever a; is a negative integer, there are minima 

 values of r(a:) between x=—l and —2, —2 and —3, &c. The author had deter- 

 mined the positions of the first ten of these minima (or, algebraically considered, 

 minima and maxima alternately) to four places of decimals, and also their values, 

 the chief object being to obtain data to form a moderately accurate drawing of the 

 curve. The abscissae of the minima were found by the aid of the table of ^(a;) 

 in Gauss's Gottingen memoir of 1812 and Oakes's Table of Reciprocals, as fol- 

 lows. Writing, with Gauss, n(a?) for r(ii? + l) and logf a; (Gauss's •*•(«)) for 

 n'{x)-^n(x), the first minimum corresponds to the abscissa — 1+the root of 



logf a? = - + 1 ; 



° X x — 1 ' 



the second to the abscissa, — 2 + the root of 

 logix = - + - — r + 



"°' " X x—1 x—2 ■ 

 the third to — 3 + the root of 



logfa: = - +^3^+^32 + —3 &c. 



On the Introduction of the Decimal Point into Arithmetic. 

 By J. W. L. GiAisHEE, B.A. 



The following is an extract from Peacock's excellent history of Arithmetic in 

 the ' Encyclopaedia Metropolitana,' which forms the standard (not to say the only) 

 work on the subject. Speaking of Stevinus's 'Arithmetique,' Peacock wi-ites: — 

 " We find no traces, however, of decimal arithmetic in this work ; and the first 

 notice of decimal, properly so called, is to be found in a short tract which is put 

 at the end of his ' Arithmetique ' in the collection of his works by Albert Girard, 

 entitled 'La Disme.' It was first published in Flemish, about the year 1590, and 

 afterwards translated into barbarous French by Simon of Bruges .... Whatever 

 advantages, however, this admirable invention, combined as it still was with the 

 addition of the exponents, possessed above the ordinary methods of calculation in 

 the case of abstract or concrete fractions, it does not appear that they were readily 



perceived or adopted by his contemporaries The last and final' improvement 



m this decimal Arithmetic, of assimilating the notation of integers and decimal 

 fractions, by placing a. point or comma between them, and omitting the exponents 

 altogether, is unquestionably due to the illustrious Napier, and is not one of the 

 least of the many precious benefits which he conferred upon the science of cal- 

 culation. No notice whatever is taken of them in the * Mirifici Logarithmorum 

 Canonis Descriptio,' nor in its accompanying tables, which was published in 1614. 

 In a short abstract, however, of the theory of these logarithms, with a short 

 table of the logarithms of natural mmibers, which was published by Wright, 

 1616, we find a few examples of decimals expressed with reference to the deci- 

 mal point ; but they are first distinctly noticed in the ' Rabdologia,' which was 

 published in 1617. In an ' Admonitio pro decimal! Arithmetica 'he mentions in 

 terms of the highest praise the invention of StcAdnus, and explains his notation ; 

 and, without noticing his own simplification of it, he exhibits it in the follow- 

 ing example, in which it is required to divide 861094 by 432 The quotient is 



1993,273 or 1993,27"3"', the form under which he afterwards writes it, in partial 



