A history of Greek mathematics - Wilbourhall.org

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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
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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 conies, 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 conies.
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 Conies. 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
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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 rectangle contained by two of the lines so drawn to the
rectangle contained by the remaining two be given, then in