344 On the Curves of Trisection. 
pire and the Logistic Curve and Spiral, which are 
d nly by points ; and stand on the same footing 
with the Geaideil of Nicomedes, the Ellipse and the Hy- 
perbola, while they are superior to the Cye/oid, described 
by the motion of a wheel, and to the Parabola, which can 
be deseribed only, I believe, by points, and in a continued 
line by means of a threa 
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 that is 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’ aR Archi- 
medes, I make use of the © oi r the construction of 
problems.” _ With: these remarks in ica the claim of the 
wil he ee sing little snfeNOr i in sip iep 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 
the 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. pe that this equa- 
tion has three rea] roots: sand. that th ie problem cannot be 
constructed, except by the intersection vat a right line with a 
pester the thind: =i or by the intersection of two curves 
second degree ; the analysis, which they have given 
ef this problem 4 is aicuaiei and has for a long time left 
nothing to desite.” With these impressions the Academy 
resolved ia 1775, that they would not examine any new SO- 
lution of the: problem of the trisection of an angle. Geome- 
tricians must decide, whether this deiextostie is to be 
