344 On the Curves of Trisection. 
Archimedes, and the Logistic Curve and Spiral, which are 
described, only by points ; and stand on the same footing 
with the Conchoid of Nicomedes, the Ellipse and the Hy- 
perbola, while they are superior to the Cyclotd, described 
by the motion of a wheel, and to the Parabola, which can 
be described only, I believe, by points, and in a continued 
line by means of a thread. 
Newton has said,——‘‘ We ought either to exclude all lines, 
beisde the circle and right line, out of geometry, or admit 
them according to the simplicity of their descriptions, in 
which case the Conchoid yields to none except the circle.” 
“That is arithmetically more simple, which is determined 
by the more simple equations, but thatis geometrically more 
simple, which is determined by the more simple drawing of 
lines; and in geometry that ought to be reckoned _ best, 
which is geometrically most simple ; wherefore I ought not 
to be blamed, if with that prince of mathematicians, Archi- 
medes, I make use of the Conchoid for the construction of 
problems.” With these remarks in view, the claim of the 
Curves, which I have discovered, to be regarded as geomet- 
rical curves will probably not be controverted ; andjpossibly 
the description of one of them by the instrument invented 
will be thought little inferior in simplicity to that of the Con- 
choid of Nicomedes by means of the instrument which he 
invented, and for the invention of which he felt an extreme 
elevation of mind. 
It seems, that the Greek geometricians, although they 
could not trisect an angle by a right line and a circle, yet 
were able to solve the problem by means of the Conic sec- 
tions and the Conchoid. ‘‘The moderns,” as is stated in 
ihe History of the Royal Academy of Sciences in France, 
‘“ have demonstrated, that this problem depends on the res- 
olution of an equation of the third degree ; that this equa- 
tion has three real roots ;-—and that the problem cannot be 
constructed, except by the intersection of a right line with a 
curve of the third degree, or by the intersection of two curves 
of the second degree ; the analysis, which they have given 
of this problem is complete, and has for a long time left 
nothing to desire.” With these impressions the Academy 
resolved in 1775, that they would not examine any new so- 
lution of the problem of the trisection of an angle. Geome- 
triclans must decide, whether this determination is to be 
