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

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.
Lastly,
isolated extracts appear in Eutocius, (1) 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
225 SUCCESSORS OF THE GREAT GEOMETERS
•of the different species of the thing defined. Thus in the
case of ' lines ' (which include curves) he distinguishes, first,
the composite (e.g. a broken line
forming an angle) and the
incomposite. The incomposite are subdivided into those
'
forming a figure '
(o"x^ fiaTonoiovo-ai) or determinate (e.g.
circle, ellipse, cissoid) and those not forming a figure, indeterminate
and extending without limit (e. g. straight line,
parabola, hyperbola, conchoid). In a second classification
incomposite lines are divided into (1) ' simple ', namely the circle
and straight line, the one ' making a figure ', the other extending
without limit, and (2) 'mixed'. 'Mixed' lines again are
divided into (a) 'lines in planes', one kind being a line meeting
itself (e.g. the cissoid) and another a line extending
without limit, and (b) ' lines on solids ', subdivided into lines
formed by sections (e.g. conic sections, spiric curves) and
'lines round solids'
(e.g. a helix round a cylinder, sphere, or
cone, the first of which is uniform, homoeomeric, alike in all
its parts, while the others are non-uniform). Geminus gave
a corresponding division of surfaces into simple and mixed,
the former being plane surfaces and spheres, while examples
of the latter are the tore or anchor-ring (though formed by
the revolution of a circle
about an axis) and the conicoids of
revolution (the right-angled conoid, the obtuse-angled conoid,
and the two spheroids, formed by the revolution of a parabola,
a hyperbola, and an ellipse respectively about their
axes). He observes that, while there are three homoeomeric
or uniform ' lines ' (the straight line, the circle, and the
cylindrical helix), there are only two homoeomeric surfaces,
the plane and the sphere. Other classifications are those of
'
angles ' (according to the nature of the two lines or curves
which form them) and of figures and plane figures.
When Proclus gives definitions, &c, by Posidonius, it is
Such
evident that he obtained them from Geminus's work.
are Posidonius's definitions of figure ' ' and parallels ' ', and his
division of quadrilaterals into seven kinds. We may assume
further that, even where Geminus did not mention the name
of Posidonius, he was, at all events so far as the philosophy of
mathematics was concerned, expressing views which were
mainly those of his master.
1
Eutocius, Comm. on Apollonius's Conies, ad init,
1523.2
Q