386 
PAPPUS OF ALEXANDRIA 
means of a hyperbola which has to be constructed from certain 
data, namely the asymptotes and a certain point through 
which the curve must pass (this easy construction is given in 
Prop. 33, chap. 41-2). Then the problem is directly solved 
(chaps. 43, 44) by means of a hyperbola in two ways prac 
tically equivalent, the hyperbola being determined in the one 
case by the ordinary Apollonian property, but in the other by 
means of the focus-directrix property. Solutions follow of 
the problem of dividing any angle in a given ratio by means 
(1) of the quadratrix, (2) of the spiral of Archimedes (chaps. 
45, 46). All these solutions have been sufficiently described 
above (vol. i, pp. 235-7, 241-3, 225-7). 
Some problems follow (chaps. 47-51) depending on these 
results, namely those of constructing an isosceles triangle in 
which either of the base angles has a given ratio to the vertical 
angle (Prop. 37), inscribing in a circle a regular polygon of 
any number of sides (Prop. 38), drawing a circle the circum 
ference of which shall be equal to a given straight line (Prop. 
39), constructing on a given straight line AB a segment of 
a circle such that the arc of the segment may have a given 
ratio to the base (Prop, 40), and constructing an angle incom 
mensurable with a given angle (Prop. 41). 
Section (5). Solution of the vevens of Archimedes, ‘ On Spirals', 
Prop. 8, hy means of conics. 
Book IV concludes with the solution of the v ever is which, 
according to Pappus, Archimedes unnecessarily assumed in 
On Spirals, Prop. 8. Archimedes’s assumption is this. Given 
a circle, a chord (BC) in it less than the diameter, and a point 
A on the circle the perpendicular from which to BC cuts BC 
in a point D such that BD > DC and meets the circle again 
in E, it is possible to draw through A a straight line ARP 
cutting BC in R and the circle in P in such a way that RP 
shall be equal to DE (or, in the phraseology of revaeis, to 
place between the straight line BC and the circumference 
of the circle a straight line equal to DE and verging 
towards A). 
Pappus makes the problem rather more general by not 
requiring PR to be equal to DE, but making it of any given