A history of Greek mathematics - Wilbourhall.org

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.