WORKS OTHER THAN THE COLLECTION 357 
diameters at right angles and terminating at one point is 
equal to, but is not, a right angle. 1 (2) Pappus said that, 
in addition to the genuine axioms of Euclid, there were others 
on record about unequals added to 
equals and equals added to unequals. 
Others given by Pappus are (says 
Proclus) involved by the definitions, 
e.g. that ‘ all parts of the plane and of 
the straight line coincide with one 
another’, that ‘a point divides a line, 
a line a surface, and a surface a solid and that ‘ the infinite 
is (obtained) in magnitudes both by addition and diminution’. 2 
(3) Pappus gave a pretty proof of Eucl. I. 5, which modern 
editors have spoiled when introducing it into text-books. If 
AB, AC are the equal sides in an isosceles triangle, Pappus 
compares the triangles ABC and AGB (i.e. as if he were com 
paring the triangle ABC seen from the front with the same 
triangle seen from the back), and shows that they satisfy the 
conditions of I. 4, so that they are equal in all respects, whence 
the result follows. 3 
Marinus at the end of his commentary on Euclid’s Data 
refers to a commentary by Pappus on that book. 
Pappus’s commentary on Ptolemy’s Syntaxis has already 
been mentioned (p, 274); it seems to have extended to six 
Books, if not to the whole of Ptolemy’s work. The Fihrist 
says that he also wrote a commentary on Ptolemy’s Planl- 
sphae^ium, which was translated into Arabic by Thabit b. 
Qurra. Pappus himself alludes to his own commentary on 
the Analemma of Diodorus, in the course of which he used the 
conchoid of Nicomedes for the purpose of trisecting an angle. 
We come now to Pappus’s great work. 
The Synagoge or Collection. 
(a) Character of the work; 'wide range. 
Obviously written with the object of reviving the classical 
Greek geometry, it covers practically the whole field. It is, 
1 Proclus on Eucl. I, pp. 189-90. 2 lb., pp. 197. 6-198. 15. 
3 lb., pp. 249. 20-250. 12.