A history of Greek mathematics - Wilbourhall.org

OTHER LOST WORKS 193
of plane and solid angles, and his attempts to prove the axioms ;
it must also have included the three definitions (13-15) in
Euclid's Data which, according to a scholium, were due to
Apollonius and must therefore
have been interpolated (they
are definitions of KarrjyfjLevr), dvrjy/ievr], and the elliptical
phrase irapa Oeaei, which means 'parallel to a straight line
given in position ').
Probably the same work also contained
Apollonius's alternative constructions for the problems of
Eucl. I. 10, 11 and 23 given by Proclus. Pappus speaks
of a mention by Apollonius ' before his own elements ' of the
class of locus called e0e/cri/coy, and it may be that the treatise
now in question is referred to rather than the Plane Loci
itself.
(i) The work On the Cochlias was on the cylindrical helix.
It included the theoretical generation of the curve on the
surface of the cylinder, and the proof that the curve is
homoeomeric or uniform, i.e. such that any part will fit upon
or coincide with any other.
(k) A work on Unordered Irrationals is mentioned by
Proclus, and a scholium on Eucl. X. 1 extracted from Pappus's
commentary remarks that Euclid did not deal with ' all
rationals and irrationals, but only with the simplest kinds by
the combination of which an infinite number of irrationals
are formed, of which latter Apollonius also gave some '.
To a like effect is a passage of the fragment of Pappus's
commentary on Eucl. X discovered in an Arabic translation
c
by Woepcke : it was Apollonius who, besides the ordered
irrational magnitudes, showed the existence of the unordered,
and by accurate methods set forth a great number of them '.
The hints given by the author of the commentary seem to imply
that Apollonius's extensions of the theory of irrationals took
two directions, (1) generalizing the medial straight line of
Euclid, on the basis that, between two lines commensurable in
square (only), we may take not only one sole medial line
but
three or four, and so on ad infinitum, since we can take,
between any two given straight lines, as many lines as
we please in continued proportion, (2) forming compound
irrationals by the addition and subtraction of more than two
terms of the sort composing the binomials, apotomes, &c.
1523.2 O