446 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



(M) m =-''"■ ■■••(') 



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



a^ b ""^ *^o -^ \a" hWa^ b) 



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



«=^'{M}^ then.,=/'{:--+f}, and .+«,=/'{^4'}; 

 we have now manifestly, 



/'(M)-/'(H')=/1(M)-(H')}- 



But, in any system of logarithms, 



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

 as the logarithms of the quantities - +7, - +f', - + r- 



We have now this important property of the hyperbola ; 

 Let X and y he the co-ordinates of a, a sector of a hyjyerhola ivliose transverse and 

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



''a 



• 



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

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

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

 solved by MERCATORf 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 labour of computing tables of logarithms had been accomplished. Its dis- 

 covery induced the 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. X 



* Gregorii a S. Vincentio Vera Quadratura Circuli et Hyperbolce. Antwerp, 1647. 



+ Mercator. Logarithmotechma, &:c. London, 1668. 



X PhilosopJdcal Transaclions (No. 27), vol. i., Lowthorpe's Abridgment. 



