60 REl'ORT— 1873. 



to ten places ; the left-hand pages contain the logarithms, and the right- 

 liand Images the proportional parts, which are given for every hundredth 

 of the differences. The change in the line is denoted by an asterisk ; and 

 the last figure is underlined -when it has been increased. 



The mode of using the tables is as follows : — If the first figures of the 

 number lie between 1000 and 1011, the logarithm can be taken out directly 

 from table 2 ; if not, a factor M is found from the auxiliary table, by which 

 the number must be multiidicd in order to make its initial figures lie between 

 these limits, and so bring it within the range of table 2. After performing 

 this multiplication the logarithm can be taken out ; and to neutralize the 



effect of the multiplication, as far as the result is concerned, log ( — J must 



be added ; this quantity is therefore given in an adjoining column to M in 

 the auxiliary table. A similar procedure gives the number answering to any 

 logarithm, only that another factor (approximately the reciprocal of M) is 

 given, so that in both cases multiplication is used. 



The laborious part of the work is the multiplication by the factor M ; 

 but this is compensated to a great extent by the ease with which, by the 

 proportional parts, the logarithm is taken out. Great pains have been taken 

 to choose the factors M (which are 300 in number) so as to minimize this 

 labour ; and of the 300 only 25 consist of three figures all diflPcrent and not 

 involving or 1. Whenever it was possible, factors containing two figures 

 alike or containing a 0, or of only one or two figures, have been found. The 

 process of taking out a logarithm is rather longer than if Ylacq or Yega 

 were used ; but, on the other hand, the size of this book (only about 80 pp. 

 8vo) is a great advantage, both of the former works being large folios. Also 

 both Vlacq and Vega are so scarce as to be very difficult to procure ; so that 

 Pineto's table will be often the only ten-figure table available for any one who 

 has not access to a good library ; and on this account it is unique. Though 

 the principle of multiplying by a factor, which is subsequently cancelled by 

 subtracting its logarithm, is frcquentl}'^ employed in the construction of tables, 

 this is, we believe, the first instance in which it forms part of the process of 

 iisvi{j the table. By taking the numbers to 12 instead of 10 places, in a 

 manner explained in the introduction, greater accuracy in the last place 

 is ensured than results from the use of Vlacq or Vega. It is not stated 

 whether the table is stereotyped ; so we pi'esume it is not. 



On the last page (p. 56) are given the first hundred multiples of the 

 modulus and its reciprocal to 10 places. (Notices and examples taken from 

 Pineto's tables will be found in the ' Quarterly Journal of Mathematics ' for 

 October 1871, and the ' Messenger of Mathematics ' for July 1872.) 



Sang, 1871. Ten-figure logarithms, from 1 to 1000, and seven-figure 

 logaritlims, from 20,000 to 200,000, with differences and multiples (not pro- 

 portional parts) of the differences throughout. 



The advantages arising from the table extending from 20,000 to 200,000, 

 instead of from 10,000 to 100,000, are, that whereas in the latter the dif- 

 ferences near the beginning of the table are so numerous that the propor- 

 tional parts must either be very crowded or some of them omitted, and even 

 if they are aU given the interpolation is inconvenient, in a table extending 

 from 20,000 to 200,000 the differences are halved in magnitude, while the 

 number of them in a page is quartered ; the space gained enables multiples 

 instead of proportional parts to be given. 



The table is printed without rules (except one dividing the logarithms 

 from the numbers) ; and the numbers are separated from the logarithms by 



