THE ALEXANDRIAN PERIOD 75 



quantities whose difference may be less than any 

 given quantity. 



The work On Spirals contains the study of the curve 

 to which Archimedes has given his name, and which is 

 described by a radius vector r increasing uniformly with 

 the vectorial angle 6 : r = cd, c being a constant. 



The writing entitled On the Equilibrium of Planes 

 ox The Centres of Gravity of Planes is composed of two 

 books. The first one begins by establishing the theory 

 of the equilibrium of the lever, then enunciates and 

 demonstrates various theorems relating to the centres 

 of gravity of the parallelogram, triangle, and recti- 

 linear trapezium. The second treats of the deter- 

 mination of the centre of gravity of a parabolic seg- 

 ment. The Arenarius is one of the most valuable 

 documents we possess on the astronomy and system 

 of numeration of the Greeks ; amongst other things 

 it contains a description of the heliocentric system of 

 Aristarchus of Samos (Heiberg edition, II, p. 244, 12). 

 To calculate numbers of any desired magnitude, 

 Archimedes makes use of progressions, one arithmetical, 

 the other geometrical, the former being used to find 

 any term of the latter. On the Quadrature of the 

 Parabola estimates the area of the parabola, first 

 by means of pure geometry (method of exhaustion), 

 then by considerations of equilibrium (infinitesimal 

 mechanical method). The treatise On Floating Bodies 

 establishes the fundamental laws of hydrostatics ; the 

 state of equilibrium of a liquid ; the position of equi- 

 librium of a solid immersed in a liquid according to 

 the ratio of its density to that of this liquid. Accord- 

 ing to a legend related by Vitruvius (Bk. IX, 215, 10) 

 Archimedes discovered the laws of hydrostatics whilst 

 in his bath, thinking of the crown adulterated by the 

 goldsmith of King Hiero. 



The treatise On the Method relating to Mechanical 

 Theorems has been recently discovered on a palimpsest 



