118 
CONIC SECTIONS 
areas, manipulation of simple equations between areas and, in 
particular, the application of areas ; in other words, plane 
problems were those which, if expressed algebraically, depend 
on equations of a degree not higher than the second. 
Problems, however, soon arose which did not yield to ‘ plane ’ 
methods. One of the first was that of the duplication of the 
cube, which was a problem of geometry in three dimensions or 
solid geometry. Consequently, when it was found that this 
problem could be solved by means of conics, and that no 
higher curves were necessary, it would be natural to speak of 
them as ‘ solid ’ loci, especially as they were in fact produced 
from sections of a solid figure, the cone. The propriety of the 
term would be only confirmed when it was found that, just as 
the duplication of the cube depended on the solution of a pure 
cubic equation, other problems such as the trisection of an 
angle, or the cutting of a sphere into two segments bearing 
a given ratio to one another, led to an equation between 
volumes in one form or another, i. e. a mixed cubic equation, 
and that this equation, which was also a solid problem, could 
likewise be solved by means of conics. 
Aristaeus’s Solid Loci. 
The Solid Loci of Aristaeus, then, presumably dealt with 
loci which proved to be conic sections. In particular, he must 
have discussed, however imperfectly, the locus with respect to 
three or four lines the synthesis of which Apollonius says that 
he found inadequately worked out in Euclid’s Conics. The 
theorems relating to this locus are enunciated by Pappus in 
this way: 
‘ If three straight lines be given in position and from one and 
the same point straight lines be drawn to meet the three 
straight lines at given angles, and if the ratio of the rectangle 
contained by two of the straight lines so drawn to the square 
on the remaining one be given, then the point will lie on a 
solid locus given in position, that is, on one of the three conic 
sections. And if straight lines be so drawn to meet, at given 
angles, four straight lines given in position, and the ratio of 
the rectangfe contained by two of the lines so drawn to the 
rectangle contained by the remaining two be given, then in