A history of Greek mathematics - Wilbourhall.org

GEMINUS 225
was applied by the early Pythagoreans more particularly
to geometry and arithmetic, sciences which deal with the pure,
the eternal and the unchangeable, but was extended by later
writers to cover what we call ' mixed ' or applied mathematics,
which, though theoretical, has to do with sensible objects, e.g.
astronomy and optics. Other extracts from Geminus are found
in
extant manuscripts in connexion with Damianus's treatise
on optics (published by R. Schone, Berlin, 1897). The definitions
of logistic and geometry also appear, but with decided
differences, in the scholia to Plato's Charmides 165 e.
isolated extracts appear in Eutocius, (1)
Lastly,
a remark reproduced
in the commentary on Archimedes's Plane Equilibriums to
the effect that Archimedes in that work gave the name of
postulates to what are really axioms, (2) the statement that
before Apollonius's time the conies were produced by cutting
different cones (right-angled, acute-angled, and obtuse-angled)
by sections perpendicular in each case to a generator. 1
The object of Geminus's work was evidently the examination
of the first principles, the logical building up of mathematics
on the basis of those admitted principles, and the
defence of the whole structure against the criticisms of
the enemies of the science, the Epicureans and Sceptics, some
of
whom questioned the unproved principles, and others the
logical validity of the deductions from them. Thus in
geometry Geminus dealt first with the principles or hypotheses
(dp^ai, vTroBecreis) and then with the logical deductions, the
theorems and problems (rot jiera ras dp\d?). The distinction
is between the things which must be taken for granted but
are incapable of proof and the things which must not be
assumed but are matter for demonstration. The principles
consisting of definitions, postulates, and axioms, Geminus
subjected them severally to a critical examination from this
point of view, distinguishing carefully between postulates and
axioms, and discussing the legitimacy or otherwise of those
formulated by Euclid in each class.
In his notes on the definitions
Geminus treated them historically, giving the various
alternative definitions which had been suggested for each
fundamental concept such as (
'
line ', surface ', ' figure ', 'body',
'
angle ', &c, and frequently adding instructive classifications
1
Eutocius, Comm. on Apollonius's Conies, ad init,
1523.2
Q