316
HERON OF ALEXANDRIA
or any angle), multiples, Defs, 119-21 ; proportion in magni
tudes, 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, converts ndo, alternando, in-
vertendo 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 alter
native 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 Apol
lonius, 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 collec
tion 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, Geoinetrica, 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 not a mere application of rules to
particular examples. It is also more akin to theory in that it
