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

THE COLLECTION. BOOK IV 371
we may say that the algebraic sum of the three parallelograms
is equal to zero.
Though Pappus only takes one case, as was the Greek habit,
I see no reason to doubt that he was aware of the results
in the other possible cases.
Props. 2, 3 are noteworthy in that they use the method and
phraseology of Eucl. X, proving that a certain line in one
figure is the irrational called minor (see Eucl. X. 76), and
a certain line in another figure is ' the excess by which the
binomial exceeds the straight line which produces with a
rational area a medial whole ' (Eucl. X. 77). The propositions
4-7 and 11-12 are quite interesting as geometrical exercises,
bat their bearing is not obvious : Props. 4 and 12 are remarkable
in
that they are cases of analysis followed by synthesis
applied to the proof of theorems. Props. 8-10 belong to the
subject of tangencies, being the sort of propositions that would
come as particular cases in a book such as that of Apollonius
On Contacts ; Prop. 8 shows that, if there are two equal
circles and a given point outside both, the diameter of the
circle passing through the point and touching both circles
is ' given '
; the proof is in many places obscure and assumes
lemmas of the same kind as those given later a propos of
Apollonius's treatise; Prop. 10 purports to show how, given
three unequal circles touching one another two and two, to
find the diameter of the circle including them and touching
all three.
372 PAPPUS OF ALEXANDRIA
There is, says Pappus, on record an ancient proposition to
the following effect. Let successive circles be inscribed in the
dpftrjXos touching the semicircles and one another as shown
in the figure on p. 376, their centres being A, P, ... . Then, if
Pi* Vv Vz
••• be the perpendiculars from the centres A, P, ...
on BG and d lf
c£ 2
, d 3
... the diameters of the corresponding
circles,
p1
= d 1
, p
2
=2d 2
, p
3
= Bd B
....
He begins by some lemmas, the course of which I shall
reproduce as shortly as I can.
I. If (Fig. 1) two circles with centres A, C of which the
former is the greater touch externally at B, and another circle
with centre G touches the two circles at K, L respectively,
then KL produced cuts the circle BL again in D and meets
AC produced in a point E such that AB :BG = AE : EG.
This is easily proved, because the circular segments DL, LK
are similar, and CD is parallel to AG. Therefore
AB:BC = AK:GD = AE: EC.
Also KE.EL = EB 2 .
For AE:EC=AB:BC = AB:CF= (AE- AB) :
= BE:EF.
(EC- CF)
Section (2). On circles inscribed in the dpfirjXos
(' shoemakers knife ').
The next section
(pp. 208-32), directed towards the demonstration
of a theorem about the relative sizes of successive
circles inscribed in the apfi-qXos (shoemaker's knife), is extremely
interesting and clever, and I wish that I had space
to reproduce it completely. The dpf3r)Xos, which we have
already met with in Archimedes's ' Book of Lemmas ', is
formed thus. BC is the diameter of a semicircle BGC and
BC is divided into two parts (in general unequal) at B;
semicircles are described on BD, DC as diameters on the same
side of BC as BGC is ; the figure included between the three
semicircles is the apftrjXos.
Bb2
But AE:EC= KE : ED
Therefore KE .
EL
:
;
EL
.
Fig 1.
therefore KE:ED = BE: EF.
ED
= BE* :
BE
.
EF.
And EL. ED = BE. EF; therefore KE. EL = EB 2 .