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 Constanti 
nople, 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 contain 
ing 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 diminishing submultiples) 
being used comparatively rarely, whereas the reverse is the 
case in the other works. 
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 collec 
tions. 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 para 
bolic 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 Archimedes, vol. iii, p. 232. 13-17. 
2 Tannery, Mémoires scientifiques, ii, 1912, pp. 447—54.