Archimedes's works are all original, and are perfect models of mathematical exposition; their wide range will be seen from the list of those which survive: _On the Sphere and Cylinder_ I, II, _Measurement of a Circle_, _On Conoids and Spheroids_, _On Spirals_, _On Plane Equilibriums_ I, II, the _Sandreckoner_, _Quadrature of the Parabola_, _On Floating Bodies_ I, II, and lastly the _Method_ (only discovered in 1906). The difficult Cattle-Problem is also attributed to him, and a _Liber Assumptorum_ which has reached us through the Arabic, but which cannot be his in its present form, although some of the propositions in it (notably that about the 'Salinon', salt-cellar, and others about circles inscribed in the αρβηλος {arbêlos}, shoemaker's knife) are quite likely to be of Archimedean origin. Among lost works were the _Catoptrica_, _On Sphere-making_, and investigations into polyhedra, including thirteen semi-regular solids, the discovery of which is attributed by Pappus to Archimedes. Speaking generally, the geometrical works are directed to the measurement of curvilinear areas and volumes; and Archimedes employs a method which is a development of Eudoxus's method of exhaustion. Eudoxus apparently approached the figure to be measured from below only, i. e. by means of figures successively inscribed to it. Archimedes approaches it from both sides by successively inscribing figures and circumscribing others also, thereby compressing them, as it were, until they coincide as nearly as we please with the figure to be measured. In many cases his procedure is, when the analytical equivalents are set down, seen to amount to real _integration_; this is so with his investigation of the areas of a parabolic segment and a spiral, the surface and volume of a sphere, and the volume of any segments of the conoids and spheroids. The newly-discovered _Method_ is especially interesting as showing how Archimedes originally obtained his results; this was by a clever mechanical method of (theoretically) _weighing_ infinitesimal elements of the figure to be measured against elements of another figure the area or content of which (as the case may be) is known; it amounts to an _avoidance_ of integration. Archimedes, however, would only admit that the mechanical method is useful for finding results; he did not consider them proved until they were established geometrically.