A history of Greek mathematics - Wilbourhall.org

THE COLLECTION. BOOKS VI, VII 399
The test of apparent equality is of course that the two straight
lines should subtend equal angles at F.
The main points in the proof are these.
The plane through
CF, CK is perpendicular to the planes BFE, PFM and LFR
;
hence CF is
perpendicular to BE, QF to PM and HF to LR,
whence BC and CE subtend equal angles at F :
and PQ, QM.
Since FC bisects the angle AFD
t
so do LH, HR,
and AC:CD = AK:KD
(by the polar property), Z CFK is a right angle. And CF is
the intersection of two planes at right angles, namely AFK
and BFE, in the former of which FK lies; therefore KF is
perpendicular to the plane BFE, and therefore to FN. Since
therefore (by the polar property) LN : NP = IjK : KP, it
follows that the angle LFP is bisected by FN] hence LN, NP
are apparently equal.
Again
LC:CM = LN:NP = LF: FP = LF: FM.
Therefore the angles LFC, CFM are equal, and LC, CM
are apparently equal.
Lastly
LR:PM=LK:KP=LN:NP=LF:FP; therefore
the isosceles triangles FLR, FPM are equiangular; therefore
the angles PFM, LFR, and consequently PFQ, LFH, are
equal. Hence LP, RM will appear to be parallel to AD.
We have, based on this proposition, an easy method of
a circle
'
solving Pappus's final problem (Prop. 54). Given
ABBE and any point within it, to find outside the plane of
the circle a point from which the circle will have the appearance
of an ellipse with centre C'
We have only to produce the diameter AD through C to the
pole K of the chord BE perpendicular to
AD and then, in
the plane through AK perpendicular to the plane of the circle,
to describe a semicircle on CK as diameter.
this semicircle satisfies the condition.
Any point F on
Book VII. On the 'Treasury of Analysis'.
Book VII is of much greater importance, since it gives an
account of the books forming what was called the Treasury of
Analysis (dvaXvouevo? tottos) and, as regards those of the books
which are now lost, Pappus's account, with the hints derivable
from the large collection of lemmas supplied by him to each