A history of Greek mathematics - Wilbourhall.org

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ON SPIRALS 65
then two simple geometrical propositions, followed by propositions
(5-9) which are all of one type.
Prop. 5 states that,
given a circle with centre 0, a tangent to it at A, and c, the
Fig. 1.
circumference of any circle whatever, it is possible to draw
a straight line OPF meeting the circle in P and the tangent
in F such that
FP:OP < (arc AP) :
Archimedes takes D a straight line greater than c,
c.
draws
OH parallel to the tangent at A and then says ' let PH be
placed equal to D verging (vevovcra) towards A '. This is the
usual phraseology of the type of problem known as vevcris
where a straight line of given length has to be placed between
two lines or curves in such a position that, if produced, it
passes through a given point (this is the meaning of verging)
Each of the propositions 5-9 depends on a vevcris of this kind,
Fig. 2.
which Archimedes assumes as * possible ' without showing how
it is effected. Except in the case of Prop. 5, the theoretical
solution cannot be effected by means of the straight line and
circle ; it depends in general on the solution of an equation
of the fourth degree, which can be solved by means of the
1523.2 F