84 REPORT — 1856. 



T'p, it will give the negative arc of the parabola Vp, corresponding to the 

 number V?^. Fractional numbers, or numbers between + I and 0, must 

 therefore be represented by the expression «i(sec ~ tan 0), since tanfl 

 changes its sign. 



When the number is 0, n coincides with V, and the angle NSQ in this 

 case is a right angle. Tlierefore the point T' will be the intersection of VT' 

 and SQ. Hence T' is at an infinite distance below the axis, and therefore 

 the logarithm of +0 is — go . 



Hence the tangential difference due to the amplittide 6, is the logarithm of 

 the number sec 6 + tan 6. 



Consequently it follows that negative numbers have no logarithms, at least 

 no real ones ; and imaginary ones can only be educed by the transformation 

 so often referred to, and this leads us to seek them among the properties of 

 the circle. For as always lies between and a right angle, or between 

 and the half of +7r, sec + tan 6 is alicays positive ; therelbre negative num- 

 bers can have no real ov parabolic logarithms, but they may have imaginary 

 or circular logarithms ; for in the expression 



log{cosS-+ V^sinS}=^V'^, (28) 



we may make ^= (2m + 1)^-, and we shall get log( — l)=(2w+l)7rv' — 1- 

 Hence also, as the length of the parabolic arc TP, without reference to 

 the sign, depends solely on the amplitude 0, it follows that the logarithm of 

 sec — tan is equal to the logarithm of sec ■\- tan 0. We may accordingly 

 infer that the logarithm of any number is equal to the logaritiim of its reci- 

 procal, with the sign changed, since (sec 0+ tan 0) (sec 0— tan 0)=1. 



When is very large, sec + tan 0=2 tan nearly. It follows, therefore, 

 if we represent a large number by an ordinate of a parabola whose focal 

 distance to the vertex is 1, the diiference between the corresponding arc and 

 its subtangent will represent its logarithm. 

 Since VT + TP > arc VP, therefore 



VT > arc VP-TP > log VN. 



Hence VT or tan is always greater than the logarithm of (sec + tan 0) in 

 the Napierian system of logarithms. This may be shown on other principles : 

 thus 



sin--+cos2i+2sin^cos— l + tan|. 



^ , , „ 1 + sin 0_ 2 2 2 2 2 



sec0+tan0= — — 



COS0 --«_sin=l l_tanl 



COS" — ^... -^- - ^ 



2 2 2 



a 



Let tan —=M. Then 

 2 



log(sec0+tan0)=log(i±H)=2(^«+^ + ^ + ^&c.), 



2tan — 

 and tan0= '^=2{u + u^ + w- + u^+ &c.). 



1-tan-- 



Hence tan >- 1 og (sec + tan 0), 



or ^~" is always greater than the logarithm of ?«. 



