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:-- 1. 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. 2. 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. 3. _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". 4. 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. 5. _Catoptrica_, an optical work from which Theon of Alexandria quotes a remark about refraction. 6. _On Sphere-making_, a mechanical work on the construction of a sphere to represent the motions of the heavenly bodies (cf. pp. 5-6 above). Arabian writers attribute yet further works to Archimedes, (1) On the circle, (2) On a heptagon in a circle, (3) On circles touching one another, (4) On parallel lines, (5) On triangles, (6) On the properties of right-angled triangles, (7) a book of _Data_; but we have no confirmation of these statements. CHAPTER IV. GEOMETRY IN ARCHIMEDES.