446 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



(^f)e-f) ='■"'• ■ ■ • « 



Let X and Y be the co-ordinates of a third sector equal to a + a.,, then we have 



Now, considering « as a function of -+^, let us put 



.=/•{%!}; then.,=/-{M'}, and .+.,=/{^4'}, 

 we have now manifestly, 



^■(M)*/-(M)=/1(M)-(^f)} 



But, in any system of logarithms, 



'»«(M)->"s(t-^)='«s{(M)(M')}- 



Hence it appears that the sectors a, «.„ « + «, are related among themselves exactly 



as the logarithms of the quantities - +f , - + r- - + r- 



^ a h a b a 



We have now this important property of the h3ri3erbola ; 

 Let X andy be the co-ordinates of a, a sector of a hyperhola whose transverse and 

 conjugate semi-axes are a and b ; then c being some given number, c a is the logarithm 



•'a b 



This theorem, at least one deducible from it, was first discovered by Gbegory 

 of St Vincent* and was a most important step in the theory of logarithms, for it 

 identified their construction witli the quadrature of tlie hyperbola, a problem re- 

 solved bj^ MercatorI and Brounker. This beautiful analogy between loga- 

 rithms and hyperbolic sectors led to gi-eat improvements in their computation ; 

 these, however, came too late to be of practical use ; for before it was found, the 

 great labom- of computing tables of logarithms had been accomplished. Its dis- 

 covery induced tlie geometers of that day to regard logarithms as a geometrical 

 theory ; but Dr Halley shewed that although the theory of logarithms had some 

 relations with geometry, yet it was properly a purely arithmetical theory, and as 

 such he treated it. :j: 



• Gkegorii a 5. Vincentio Vera. Quadratura Circtdi el H//perbolce. Antwerp, 1647- 



+ Mercator. Lot/arit/imo/echma, &t. London, 1668. 



t Pkihsophical Transuctious (No. 27), vol. i., LowthorPe's Abridc/ment. 



