DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 395 
and if (k.a), (k.<p), (k.x) and are four corresponding parabolic arcs, 
(^.^) — (^.(p)-(^.;;^)-(yr.'4/) = A:tan(^-i-;^) tan(?!-‘--4/) tan(>^-i-^), . . (3.50.; 
which gives a simple relation between four conjugate parabolic arcs. 
Let, in the preceding formula, tp=x—'4'} we shall have 
{k.a) —3{k.(p) = ktaif{ip-^(p) = 8k tan®^ sec®(p (351.) 
We are thus enabled to assign the difference between an arc of a parabola and three 
times another arc, 6j={(p-^<p-^(p). 
If in (ct) (341.) we make <p=-)(,zzz-^, tan5=4 tanY+ tan?). 
Introduce into this expression, the imaginary transformation tan<p=y/ — 1 sin0, and 
we shall get sin30=— 4 sin®0+sin0, which is the known formula for the trisection of 
a circular arc. (351.) may therefore be taken as the formula which gives the trisec- 
tion of an arc of a parabola. 
When there are five parabolic arcs, whose normal angles <p, %, \p, v, ft are related 
as above, namely, 
we get the following relation, 
(/f.n) — (^.?)) — (^.;^) — (/f.^/) — (A-.u) = ^tan(?)-*-)/-L.u) tan(x-*-'4'-‘-v) tan(-4/-L-(p-i-y)^ (352.) 
a formula which connects five parabolic arcs, whose amplitudes are derived by the 
given law. 
We might pursue this subject very much further ; but enough has been done to 
show the analogy which exists between the trigonometry of the circle and that of the 
parabola. As the calculus of angular magnitude has always been referred to the 
circle as its type, so the calculus of logarithms may, in precisely the same way, be 
referred to the parabola as its type. 
The obscurities, which hitherto have hung over the geometrical theory of loga- 
rithms, have it is hoped been now removed. It is possible to represent logarithms, 
as elliptic integrals usually have been represented, by curves devised to exhibit some 
special property only ; and accordingly, such curves, while they place before us the 
properties they have been constructed to represent, fail generally to carry us any 
further. The close analogies which connect the theory of logarithms with the pro- 
perties of the circle will no longer appear inexplicable"^. 
* The views above developed, on the trigonometry of the parabola, throw much light on a controversy long 
parried on between Leibnitz and J. Bernoulli on the subject of the logarithms of negative numbers. Leib- 
nitz insisted they were imaginary, while Bernoulli argued they were real, and the same as the logarithms 
of equal positive numbers. Euler espoused the side of the former, while D’Alembert coincided with the views 
of Bernoulli. Indeed, if we derive the theory of logarithms from the properties of the hyperbola (as geo- 
meters always have done), it will not be easy satisfactorily to answer the argument of Bernoulli — that as an 
hyperbolic area represents the logarithm of a positive number, denoted by the positive abscissa -|-.r, so a negative 
number, according to conventional usage, being represented by the negative abscissa — ,r, the corresponding 
hyperbolic area should denote its logarithm also. All this obscurity is cleared up by the theory developed in. 
the text, which completely establishes the correctness of the views of Leibnitz and Euler, 
