An interesting fact which we now know from Arabian sources is that the formula for the area of any triangle in terms of its sides which we write in the form

Δ = √{s (s − a) (s − b) (s − c) },

and which was supposed to be Heron’s because Heron gives the geometrical proof of it, was really due to Archimedes.

Archimedes is further credited with the authorship of the famous Cattle-Problem enunciated in a Greek epigram edited by Lessing in 1773. According to its heading the problem was communicated by Archimedes to the mathematicians at Alexandria in a letter to Eratosthenes; and a 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.