scholium to Plato’s Charmides speaks of the problem “called by Archimedes the Cattle-Problem”. It is an extraordinarily difficult problem in indeterminate analysis, the solution of which involves enormous figures.

Of lost works of Archimedes the following can be identified:—

Investigations relating to polyhedra are referred to by Pappus, who, after speaking of the five regular solids, gives a description of thirteen other polyhedra discovered by Archimedes which are semi-regular, being contained by polygons equilateral and equiangular but not similar. One at least of these semi-regular solids was, however, already known to Plato.

A book of arithmetical content entitled Principles dealt, as we learn from Archimedes himself, with the naming of numbers, and

expounded a system of expressing large numbers which could not be written in the ordinary Greek notation. In setting out the same system in the Sandreckoner (see Chapter V. below), Archimedes explains that he does so for the benefit of those who had not seen the earlier work.

On Balances (or perhaps levers). Pappus says that in this work Archimedes proved that “greater circles overpower lesser circles when they rotate about the same centre”.

A book On Centres of Gravity is alluded to by Simplicius. It is not, however, certain that this and the last-mentioned work were separate treatises, Possibly Book I. On Plane Equilibriums may have been part of a larger work (called perhaps Elements of Mechanics), and On Balances may have been an alternative title. The

title On Centres of Gravity may be a loose way of referring to the same treatise.