262 PHILOSOPHICAL TRANSACTIONS. [aNNO 1787. 



With regard to the original object of these experiments, the discovery of the 

 relation which I thought might possibly subsist between the warmth of the sub- 

 stances in question, when made use of as clothing, and their powers of attract- 

 ing moisture from the atmosphere ; or, in other words, between the quantities 

 of water they contain, and their conducting powers with regard to heat ; I 

 could not find that these properties depended in any manner upon, or were in 

 any way connected with, each other. 



XXIIl. The Principles and Illustration of an Advantageous Method of Ar- 

 ranging the Differences of Logarithms^ on Lines Graduated for the Purpose 

 of Computation. By Mr. TV. Nicholson, p. 246. 



1. If two geometrical series of numbers, having the same common ratio, be 

 placed in order with the terms opposite each other ; the ratio, between any term 

 in one series and its opposite in the other, will be constant. 2. And likewise 

 the ratio of a term in one series to any term in the other, will be the same as 

 obtains between any other two terms having the same relative position and dis- 

 tance. 3. In all such pairs of geometrical series, as have the same common 

 ratio, the last-mentioned property obtains, though the first antecedent and con- 

 sequent be taken in one pair, and the second in any other pair. 



4. If the differences of the logarithms of numbers be laid in order on an ar- 

 rangement of equi-distant parallel right lines, in such a manner as that a right 

 line, drawn across the whole, shall intersect it at divisions which denote num- 

 bers in geometrical progression ; then, from the condition of the arrangement 

 and the property of this logarithmic line, it follows, first, that every right line, 

 so drawn, will, by its intersections, indicate a geometrical series of numbers ; 

 secondly, that such series, as are so indicated by parallel right lines, will have 

 the same common ratio; and, thirdly, that the series thus indicated by two pa- 

 rallel right lines, supposed to move laterally without changing either their mutual 

 distance, or parallelism to themselves, will have each the same common ratio; 

 and in all pairs of series indicated by such two lines, the ratio between an ante- 

 cedent on one parallel and the opposite term on the other, taken as a consequent, 

 will be constant. 



All these properties Mr. N. demonstrates or illustrates by examples. 5. Thus 

 far the logarithmic line has been considered as unlimited. If therefore an ante- 

 cedent and consequent be given, it will be possible to find both on the arrange- 

 ment, and to draw two parallel lines, one over each number: and if the lines be 

 then supposed to move, without changing either their distance or absolute direc- 

 tion, so that the line, which before marked an antecedent, may now mark a new 

 antecedent; the other (by art. 2 and 3) will mark a number, at the same relative 

 position and distance, which shall be the consequent to this last antecedent after 



