88 



REPORT — 1856. 



their subtangents in the two parabolas will give the logarithms in the two 

 system.s, of the scmte number drawn upon the scalar ; for as all parabolas, like 

 circles, are similar figures, and these are confocal and similarly placed, any 

 line drawn through their common focus will cut the curves in the same angle, 

 and cut off proportional segments. Hence the two triangles SPT and Srarr 

 are similar, and the tangential differences PV — PT and zsv—wt are propor- 

 tional to 4m and 4//*', the parameters of the parabolas. 



Fig. 6. 



Let log denote the Napierian logarithm, and Log the decimal logarithm of 

 the same number. 



Draw the line ST, making the angle e with the axis such that sec e + tan e^e. 

 Then as PV— PT : wr— or : : m : m', and PV— PT=m=l, since e is the 

 base of the Napierian system; and ■urv—mT=^L,og e on the decimal parabola, 

 therefore 



m : Log e ::tn : m', or m'= Log <?. 



We may therefore conclude that the modulus of the decimal system is the 

 decimal logarithm of the Napierian base e. 



Draw the line ST'makingwith the axis an angle ^,suchthatsec2 + tanS= 10. 

 Now 



P'V-PT' : ctV-c/V : : m : m': 

 but 



P'V— PT'=mIoglO, hence ot'z>— ■zzrV=m'log 10. 



Now in order that 10 may be a base, or in other words, in order that its loga- 

 rithm may be unity, we must have ra-'y- •zEr'r'=m' log 10=»i ; or if m=l, we 



I 

 must have m'log 10^1, or m'=-, r— • ; that is, the parameter of the Deci- 

 mal parabola must be reduced compax'ed with that of the Napierian parabola 



