392 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Let A be the first base, and B the second. Then B=A^. This is the exponential 
theorem. 
Let A be the Napierian base, then x=.l, and A=e. Hence B=e^. 
LXIV. Given the number to find its logarithm, may be exhibited by the following 
geometrical construction. 
Let GAP be a parabola. Through the focus O draw 
the perpendicular OQ to the axis AO. Through A let a 
tangent of indefinite length be drawn. On this tangent 
take the line AN to represent the given number. Join 
NO, and make the angle NOT always equal to the angle 
NOQ. Draw TP at right angles to TO. This line will 
touch the parabola in the point P, and the arc of the 
parabola AP— PT will be the logarithm of AN. 
When AN'=AO=the unit g, the angle N'OQ is equal 
to half a right angle. Hence the point T in this case wiil 
coincide with A. The parabolic arc therefore vanishes, 
or the logarithm of 1 is 0. When sec0+ tan0=l, 0=0. 
When the number is less than 1, the point N wfill fall 
below N' in the position n. Hence wOQ is greater than half a right angle. There- 
fore T will fall below the axis in the point T' ; and if we draw through T' a tangent T'/>, 
it will give the negative arc of the parabola T’p, corresponding to the number An. 
Fractional numbers, or numbers between +1 and 0, must therefore be represented 
by the expression g(secO— tan0), since tan0 changes its sign. 
When the number is 0, n coincides with A, and the angle NOQ in this case is a 
right angle. Therefore the point T' will be the intersection of AT' and OQ. Hence 
T' is at an infinite distance below the axis, and therefore the logarithm of -j-0 is — oo . 
Hence 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 0 always lies between 0 and 
a right angle, or between 0 and the half of sec0+tan0 is always positive ; hence 
negative numbers can have no real or parabolic logarithms, but they may have ima- 
ginary or circular logarithms ; for in the expression log (cosS^-|-\/ — 1 sin^)=3^v^ — 1, 
we may make ^=(2^-}- 1 ) t , and we shall get log (— l) = (2w-l- l)5rv^ — 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 sec0— tan0 is equal 
to the logarithm of sec0-|- tan0. As (sec0-l- tan0)(sec0— tan0) = l, we may hence 
infer, that the logarithm of any number is equal to the logarithm of its reciprocal, 
with the sign changed. 
When 0 is very large, sec0-l-tan0=2 tan0, nearly. Hence if we represent a large 
number by an ordinate of a parabola whose focal distance to the vertex is 1, the differ- 
ence between the corresponding arc and its protangent will represent its logarithm. 
Along the tangent to the vertex of the parabola, as in the preceding figure, draw. 
Fig. 23. 
