A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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 : so do LH, HR,
and PQ, QM.
Since FC bisects the angle AFD and AC:CD = AK:KD
t
(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
'
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'
a circle
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.
Book VII. On the 'Treasury of Analysis'.
Any point F on
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
400 PAPPUS OF ALEXANDRIA
book, practically constitutes our only source of information.
The Book begins (p. 634) with a definition of analysis and
synthesis which, as being the most elaborate Greek utterance
on the subject, deserves to be quoted in full.
The so-called 'AvaXvo/xevos is, to put it shortly, a special
'
body of doctrine provided for the use of those who, after
finishing the ordinary Elements, are desirous of acquiring the
power of solving problems which may be set them involving
(the construction of) lines, and it is useful for this alone. It is
the work of three men, Euclid the author of the Elements,
Apollonius of Perga and Aristaeus the elder, and proceeds by
way of analysis and synthesis.'
Definition
of Analysis and Synthesis.
'
Analysis, then, takes that which is sought as if it were
admitted and passes from it through its successive consequences
to something which is admitted as the result of
synthesis : for in analysis we assume that which is sought
as if it were already done (yeyovos), and we inquire what it is
from which this results, and again what is the antecedent
cause of the latter, and so on, until by so retracing our steps
we come upon something already known or belonging to the
class of first principles, and such a method we call analysis
as being solution backwards {avaizaXiv Xvcriv).
1
But in synthesis, reversing the process, we take as already
done that which was last arrived at in the analysis and, by
arranging in their natural order as consequences what before
were antecedents, and successively connecting them one with
another, we arrive finally at the construction of what was
sought ; and this we call synthesis.
'
Now analysis is of two kinds, the one directed to searching
for the truth and called theoretical, the other directed to
finding what we are told to find and called 'problematical.
(1) In the theoretical kind we assume what is sought as if
it were existent and true, after which we pass through its
successive consequences, as if they too were true and established
by virtue of our hypothesis, to something admitted : then
(a), if that something admitted is true, that which is sought
will also be true and the proof will correspond in the reverse
order to the analysis, but (6), if we come upon something
admittedly false, that which is sought will also be false.
(2) In the problematical kind we assume that which is propounded
as if it were known, after which we pass through its