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

' '
concave ' and
THE DEFINITIONS 315
convex ', lune, garland (these last two are
composite of homogeneous parts) and axe (ireXeKvs), bounded byfour
circular arcs, two concave and two convex, Defs. 2 7-38.
Rectilineal figures follow, the various kinds of triangles and
of quadrilaterals, the gnomon in a parallelogram, and the
gnomon in the more general sense of the figure which added
to a given figure makes the whole into a similar figure,
polygons, the parts of figures (side, diagonal, height of a
triangle), perpendicular, parallels, the three figures which will
fill up the space round a point, Defs. 39-73. Solid figures are
next classified according to the surfaces bounding them, and
lines on surfaces are divided into (1) simple and circular,
(2) mixed, like the conic and spiric curves, Defs. 74, 75. The
sphere is then defined, with its parts, and stated to be
the figure which, of all figures having the same surface, is the
greatest in content, Defs. 76-82. Next the cone, its different
species and its parts are taken up, with the distinction
between the three conies, the section of the acute-angled cone
(' by some also called ellipse ') and the sections of the rightangled
and obtuse-angled cones (also called 'parabola and
hyperbola), Defs. 83-94; the cylinder, a section in general,
the spire or tore in its three varieties, open, continuous (or
just closed) and ' crossing-itself ', which respectively have
sections possessing special properties, ' square rings ' which
are cut out of cylinders (i. e. presumably rings the cross-section
of which through the centre is two squares), and various other
figures cut out of spheres or mixed surfaces, Defs. 95-7
rectilineal solid figures, pyramids, the five regular solids, the
semi-regular solids of Archimedes two of which (each with
fourteen faces) were known to Plato, Defs. 98-104; prisms
of different kinds, parallelepipeds, with the special varieties,
the cube, the beam, Bokos (length longer than breadth and
depth, which may be equal), the brick, ttXivOis (length less
than breadth and depth), the o-cprjvicrKos or /3co/jll(tkos with
length, breadth and depth unequal, Defs. 105-14.
Lastly come definitions of relations, equality of lines, surfaces,
and solids respectively, similarity of figures, reciprocal
'
figures', Defs. 115-18; indefinite increase in magnitude,
parts
(which must be homogeneous with the wholes, so that
e. g. the horn-like angle is not a part or submultiple of a right
316 HERON OF ALEXANDRIA
or any angle), multiples, Dels. 119-21
;
proportion in magnitudes,
what magnitudes can have a ratio to one another,
magnitudes in the same ratio or magnitudes in proportion,
definition of greater ratio, Defs. 122-5; transformation of
ratios (componendo, separando, convertendo, altemando, invertendo
and ex aequali), Defs. 126-7 ;
commensurable and
incommensurable magnitudes and straight lines, Defs. 128,
129. There follow two tables of measures, Defs. 130—2.
The Definitions are very valuable from the point of view of
the historian of mathematics, for they give the different alternative
definitions of the fundamental conceptions; thus we
'
find the Archimedean definition ' of a straight line, other
definitions which we know from Proclus to be due to Apollonius,
others from Posidonius, and so on. No doubt the
collection may have been recast by some editor or editors
after Heron's time, but it seems, at least in substance, to go
back to Heron or earlier still. So far as it contains original
definitions of Posidonius, it cannot have been compiled earlier
than the first century B.C.; but its content seems to belong in
the main to the period before the Christian era. Heiberg
adds to his edition of the Definitions extracts from Heron's
Geometry, postulates and axioms from Euclid, extracts from
Geminus on the classification of mathematics, the principles
of geometry, &c, extracts from Proclus or some early collection
of scholia on Euclid, and extracts from Anatolius and
Theon of Smyrna, which followed the actual definitions in the
manuscripts. These various additions were apparently collected
by some Byzantine editor, perhaps of the eleventh century.
Mensuration.
The Metrica, Geometrica, Stereometrica, Geodaesia,
Mensurae.
We now come to the mensuration of Heron. Of the
different works under this head the Metrica is the most
important from our point of view because it seems, more than
any of the others, to have preserved its original form. It is
also more fundamental in that it gives the theoretical basis of
the formulae used, and is
particular examples.
not a mere application of rules to
It is also more akin to theory in that it