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

MENSURATION 319
Heiberg puts side by side with corresponding sections of the
Geometrica in parallel columns ; others he inserts in suitable
places ;
sections 78. 79 contain two important problems in
indeterminate analysis (= Geom. 24, 1-2, Heib.). Heiberg
adds, from the Constantinople manuscript containing the
Metrica, eleven more sections (chap. 24, 3-13) containing
indeterminate problems, and other sections (chap. 24, 14-30 and
37-51) giving the mensuration, mainly, of figures inscribed in or
circumscribed to others, e.g. squares or circles in triangles,
circles in squares, circles about triangles, and lastly of circles
and segments of circles.
The Stereometriea I has at the beginning the title Elcraycoyal
rcov arepeo/ieTpovfievcoy "Hpo&vos but, like the Geometrica,
seems to have been edited by Patricius. Chaps. 1-40 give the
mensuration of the geometrical solid figures, the sphere, the
cone, the frustum of a cone, the obelisk with circular base,
the cylinder* the 'pillar', the cube, the arfy-qvio-Kos (also called
6vv£), the fietovpov irpoeo-Kapupevfievov, pyramids, and frusta.
Some portions of this section of the book go back to Heron
thus in the measurement of the sphere chap. 1 = Metrica
II. 11, and both here and elsewhere the ordinary form of
fractions appears. Chaps. 41-54 measure the contents of certain
buildings or other constructions, e.g. a theatre, an amphitheatre,
a swimming-bath, a well, a ship, a wine-butt, and
the like.
The second collection, Stereometriea II, appears to be of
Byzantine origin and contains similar matter to Stereometriea I,
parts of which are here repeated. Chap. 31 (27, Heib.) gives
the problem of Thales, to find the height of a pillar or a tree
by the measurement of shadows ; the last sections measure
various pyramids, a prism, a /3co/j.i a 2 + b 2 ,
is the criterion determining which of the
and this
two propositions is
applicable. The method is directed to determining, first the
segments into which any side is divided by the perpendicular