ON SPIRALS 65 
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then' two simple geometrical propositions, followed by pro 
positions (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 
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) : c. 
Archimedes takes D a straight line greater than 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 uevais 
whese 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 vergivg). 
Each of the propositions 5-9 depends on a i'evans of this kind, 
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 
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