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

comparatively
MENSURATION 317
does not use concrete measures, but simple numbers or units
which may then in particular cases be taken to be feet, cubits,
or any other unit of measurement. Up to 1896, when a
manuscript of it was discovered by R. Schone at Constantinople,
it was only known by an allusion to it in Eutocius
(on Archimedes's Measurement of a Circle), who states that
the way to obtain an approximation to the square root of
a non-square number is shown by Heron in his Metrica, as
well as by Pappus, Theon, and others who had commented on
the Syntaxis of Ptolemy. 1 Tannery 2 had already in 1894
discovered a fragment of Heron's Metrica giving the particular
rule in a Paris manuscript of the thirteenth century containing
Prolegomena to the Syntaxis compiled presumably from
the commentaries of Pappus and Theon. Another interesting
difference between the Metrica and the other works is that in
the former the Greek way of writing fractions (which is our
method) largely preponderates, the Egyptian form (which
expresses a fraction as the sum of
being used .
case in the other works.
diminishing submultiples)
rarely, whereas the reverse is the
In view of the greater authority of the Metrica, we shall
take it as the basis of our account of the mensuration, while
keeping the other works in view. It is desirable at the
outset to compare broadly the contents of
the various collections.
Book I of the Metrica contains the mensuration of
squares, rectangles and triangles (chaps. 1-9), parallel-trapezia,
rhombi, rhomboids and quadrilaterals with one angle right
(10-16), regular polygons from the equilateral triangle to the
regular dodecagon (17-25), a ring between two concentric
circles (26), segments of circles (27-33), an ellipse (34), a parabolic
segment (35), the surfaces of a cylinder (36), an isosceles
cone (37), a sphere (38) and a segment of a sphere (39).
Book II gives the mensuration of certain solids, the solid
content of a cone (chap. 1), a cylinder (2), rectilinear solid
figures, a parallelepiped, a prism, a pyramid and a frustum,
&c. (3-8), a frustum of a cone (9, 10), a sphere and a segment
of a sphere (11, 12), a spire or tore (13), the section of a
cylinder measured in Archimedes's Method (14), and the solid
1
2
Archimedes, vol. iii, p. 232. 13-17.
Tannery, Memoires scientifiques, ii, 1912, pp. 447-54.
318 HERON OF ALEXANDRIA
formed by the intersection of two cylinders with axes at right
angles inscribed in a cube, also measured in the Method (15),
the five regular solids (16-19). Book III deals with the division
of figures into parts having given ratios to one another,
first plane figures (1-19), then solids, a pyramid, a cone and a
frustum, a sphere (20-3),
The Geometrla or Geometrumena is a collection based upon
Heron, but not his work in its present form. The addition of
a theorem due to Patricius 1 and a reference to him in the
Stereometrica (I.
22) suggest that Patricius edited both works,
but the date of Patricius is uncertain. Tannery identifies
him with a mathematical professor of the tenth century,
Nicephorus Patricius ; if this is correct, he would be contemporary
with the Byzantine writer (erroneously called Heron)
who is known to have edited genuine works of Heron, and
indeed Patricius and the anonymous Byzantine might be one
and the same person. The mensuration in the Geometry has
reference almost entirely to the same figures as those
measured in Book I of the Metrica, the difference being that
in the Geometry (1) the rules are not explained but merely
applied to examples, (2)
a large number of numerical illustrations
are given for each figure, (3) the Egyptian way of
writing fractions as the sum of submultiples is followed,
(4) lengths and areas are given in terms of particular
measures, and the calculations are lengthened by a considerable
amount of conversion from one measure into another.
The first chapters (1-4) are of the nature of a general introduction,
including certain definitions and ending with a table
of measures. Chaps. 5-99, Hultsch ( = 5-20, 14, Heib.), though
for the most part corresponding in content to Metrica I,
seem to have been based on a different collection, because
chaps. 100-3 and 105 ( = 21, 1-25, 22, 3-24, Heib.) are clearly
modelled on the Metrica, and 101 is headed 'A definition
(really measurement ' ') of a "circle in another book of Heron \
Heiberg transfers to the Geometrica U considerable amount of
the content of the so-called Liber Geeponicus, a badly ordered
collection consisting to a large extent of extracts from the
other works. Thus it begins with 41 definitions identical
with the same number of the Definitiones. Some sections
1
Geometrica, 21 26 (vol. iv, p. 386. 23).