GEOMETRICAL INSTRUMENTS AND MODELS. 45 



C. MODELS OF FIGURES IN SPACE. 



Models of certain geometrical solids, for example, of the so- 

 called three round bodies the sphere, the right cone, and the 

 right cylinder; of the five regular solids the tetrahedron, the 

 cube, the octahedron, the dodecahedron, and the icosahedron ; 

 and of the various forms of prisms, pyramids, and parallelepipeds, 

 have been long in use as educational appliances. * All these 

 geometrical forms have been well known to mathematicians from 

 very ancient times ; surprising as it may appear that researches 

 into the nature of such solids as the dodecahedron and icosahedron 

 should have preceded other inquiries of a more elementary and 

 at the same time of a more important character. 



In very recent times the great development of geometrical specu- 

 lation has led to the effort to reproduce in a material shape a much 

 greater variety of geometrical conceptions. We proceed to enu- 

 merate some of the more important classes of objects of this kind. 



(i.) Models of surfaces of the sedond order. 



Among these, two of the central surfaces, the ellipsoid and the 

 hyperboloid of one sheet, have occupied a great place in the recent 

 history of geometry. We may mention a few important theoretical 

 considerations which models of these surfaces have served to 

 illustrate. 



The curvature of a surface at any point is either such that the 

 surface at that point is entirely concave on one side and convex 

 on the other, or else it is such that the surface on each side alike 

 is partly convex and partly concave. A sphere is an example of a 

 surface of the first kind ; the upper surface of a saddle may serve 

 as an instance of curvature of the second kind. But the ellipsoid 

 and the hyperboloid furnish perfect typical examples of these two 

 kinds of curvature. 



At any point of a surface there are always two directions at 



* A complete collection of such models, made by Stronkoff, is exhibited 

 by the Committee of the Russian Pedagogical Museum. 



