A history of Greek mathematics - Wilbourhall.org

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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. /*
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
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.
m
\
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.