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, 
but their bearing is not obvious : Props. 4 and 12 are remark 
able 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. 
Section (2). On circles inscribed in the dp(3rjXos 
(‘ shoemakers knife ’). 
The next section (pp. 208-32), directed towards the demon 
stration of a theorem about the relative sizes of successive 
circles inscribed in the dpfirjXos (shoemaker’s knife), is ex 
tremely interesting and clever, and I wish that I had space 
to reproduce it completely. The dpfirjXos, which we have 
already met with in Archimedes’s ‘ Book of Lemmas ’, is 
formed thus. BG is the diameter of a semicircle BGG and 
BG is divided into two parts (in general unequal) at H; 
semicircles are described on BD, DC as diameters on the same 
side of BG as BGG is; the figure included between the three 
semicircles is the dp(3r]Xos. 
b b 2