240 
THE TRISECTION OF ANY ANGLE 
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of his treatise has come down to us except the construction 
of the ‘ first ’ conchoid, its fundamental property, and the fact 
that the curve has the ruler or base as an asymptote in 
each direction. The distinction, however, drawn by Pappus 
between the ‘ first ‘ second ’, ‘ third ’ and ‘ fourth ’ conchoids 
may well have been taken from the original treatise, directly 
or indirectly. We are not told the nature of the conchoids 
other than the ‘ first but it is probable that they were three 
other curves produced by varying the conditions in the figure. 
Let a be the distance or fixed intercept between the curve and 
the base, h the distance of the pole from the base. Then 
clearly, if along each radius vector drawn through the pole 
we measure a backwards from the base towards the pole, 
we get a conchoidal figure on the side of the base towards 
the pole. This curve takes three forms according as a is 
greater than, equal to, or less than h. Each of them has 
the base for asymptote, but in the first of the three cases 
the curve has a loop as shown in the figure, in the second 
case it has a cusp at the pole, in the third it has no double 
point. The most probable hypothesis seems to be that the 
other three cochloidal curves mentioned by Pappus are these 
three varieties. 
(8) Another reduction to a vevcns (Archimedes). 
A proposition leading to the reduction of the trisection 
of an angle to another v ever is is included in the collection of 
Lemmas {Liber Assumptorum) which has come to us under