ON THE TRIGONOMETRY OF THE PARABOLA. 89 



in the ratio of log 10: 1. Hence, as is well known, the modulus m' of the 

 decimal system is tiie reciprocal of the Napierian logarithm of 10. 



It is therefore obvious, that as any number of systems of logarithms may be 

 represented by the differences between the similar arcs and their subtangents 

 of as many confocal parabolas, the logarithms of the same number in these 

 different systems will be to one another simply as the magnitudes of the para- 

 bolas whose arcs represent them, that is, as the parameters of these parabolas. 

 Accordingly the moduli of these several systems are represented by the halves 

 of the semiparameters of the several parabolas. 



The Napierian parabola differs from the decimal and other parabolas in 

 this, that the focal distance of its vertex is taken as the arithmetical unit, and 

 that the scalar line on which the numbers are set off is a tangent to it at its 

 vertex. 



Hence if m, the vertical focal distance of the Napierian parabola, be taken 

 as 1, the vertical focal distance m' of the decimal parabola is . 4342 &c., or 

 if m=l, m'— .4342 &e. 



XXIX. In every system of logarithms whatever, the logarithm of 1 is 0. 



For when the point T coincides with V, the corresponding point r will coin- 

 cide with V, whatever be the magnitude of its modulus m'. It is obvious that 

 the circle whose radius is unity is analogous to the parabola whose vertical 

 focal distance is unity, and that the Napierian logarithms have the same 

 analogy to trigonometrical lines computed from a radius equal to unity, which 

 any other system of logarithms has to trigonometrical lines computed from a 

 radius r. As we may represent different systems of trigonometry by a series 

 of concentric circles whose radii are I, r, r' &c., so we may in like manner 

 exhibit as many systems of logarithms by a series of confocal parabolas 

 whose focal distances or moduli are 1, m', m" &c. The modulus in the 

 trigonometry of the parabola corresponds with the radius in the trigonometry 

 of the circle. But while the base in the trigonometry of the parabola is real, 

 in the circle it is imaginary. In the parabola, the angle of the base is given 

 by the equation sec0 + tan0=e. In the circle, cos 04- a/— 1 sin0=e®V^^; 

 and making 0=1, we get 



cos(l)+A/^risin(l)=eV^ (35) 



Hence, while e^ is the parabolic base, e'^~ is the circular base. Or as 

 [sec e + tan e] is the Napierian base, [cos(l)+ a/^ sin(l)] is the circular 

 or imaginary base. Thus 



[cos (1 ) + V^ sin (l)]a=cos Sr+ ^T-i sin ^. 



We may therefore infer, speaking more precisely, that imaginary numbers 

 have real logarithms, but an imaginary base. We may always pass from the 

 real logarithms of the para bola to the imaginary logarithms of the circle by 

 changing tan into V —\ sin ^, sec into cos ^, and e^ into e^~^ 



As in the parabola the angle is non-periodic, its limit being \k, while in 

 the circle ^ has no limit, it follows that while a number can have only one 

 real or parabolic logarithm, it may have innumerable imaginary or circular 

 logarithms. 



Along the scalar, which is a tangent to the Napierian parabola at its vertex, 

 as in the preceding figure, draw, measured from the vertex, a series of lines 

 in geometrical progression, 



OT(sec + tan 0), ;w(sec + tan 0)^ ?w(sec + tan 0)^ m{sec + tan 0)». 



Join N, the general representative of the extremities of fhese right lines, with 

 the focus S. Erect the perpendicular SQ, and make the angle NST always 



