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

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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. /*
j
Others given by Pappus are (says /
Proclus) involved by the definitions, I
e.g. that 'all parts of the plane and of \
the straight line coincide with one n^
y
another', that 'a point divides a line,
j
^ ^
\f
N^
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 ACB (i.e. as if he were comparing
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 Syntdxis has already
been mentioned (p. 274); it seems to have extended to six
Books, if not to the whole of Ptolemy's work. The Flhrld
says that he also wrote a commentary on Ptolemy's Planisphaermm,
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; ivicle range.
Obviously written with the object of reviving the classical
Greek geometry, it covers practically the whole field. It is,
1 2
Proclus on Eucl. I, pp. 189-90. lb., pp. 197. 6-198. 15.
3
lb., pp. 249. 20-250. 12.
m
\
358 PAPPUS OF ALEXANDRIA
however, a handbook or guide to Greek geometry rather than
an encyclopaedia ; it was intended, that is, to be read with the
original works (where still extant) rather than to enable them
to be dispensed with. Thus in the case of the treatises
included in the Treasury of Analysis there is
a general introduction,
followed by a general account of the contents, w r ith
lemmas, &c, designed to facilitate the reading of the treatises
themselves. On the other hand, where the history of a subject
is given, e.g. that of the problem of the duplication of the
cube or the finding of the two mean proportionals, the various
solutions themselves are reproduced, presumably because they
were not easily accessible, but had to be collected from various
sources. Even when it is some accessible classic which is
being described, the opportunity is taken to give alternative
methods, or to make improvements in proofs, extensions, and
so on. Without pretending to great originality, the whole
work shows, on the part of the author, a thorough grasp of
all the subjects treated, independence of judgement, mastery
of technique ; the style is terse and clear ; in short, Pappus
stands out as an accomplished and versatile mathematician,
a worthy representative of the classical Greek geometry.
(j8) List of authors mentioned.
The immense range of the Collection can be gathered from
a mere enumeration of the names of the various mathematicians
quoted or referred to in the course of it. The greatest of
them, Euclid, Archimedes and Apollonius, are of course continually
cited, others are mentioned for some particular
achievement, and in a few cases the mention of a name by
Pappus is the whole of the information we possess about the
person mentioned. In giving the list of the names occurring
in the book, it will, I think, be convenient and may economize
future references if I note in brackets the particular occasion
of the reference to the writers who are mentioned for one
achievement or as the authors of a particular book or investigation.
The list in alphabetical order is : Apollonius of Perga,
Archimedes, Aristaeus the elder (author of a treatise in five
Books on the Elements of Conies or of ' five Books on Solid
Loci connected with the conies '),
Aristarchus of Samos (On the