By A. S. Smogorzhevski

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**Extra resources for Acerca de la Geometría de Lobachevski**

**Example text**

E. the plane figure in which (i. e. the elements are lines t) contains an infinite number of flat pencils. In the sheaf either the planes, or the straight lines or rays, may be regarded as the elements. If we take the planes as elements, the rays of the sheaf are the axes of so many axial pencils if, on the other hand, the rays are considered as the elements, the planes of the sheaf are so many flat ; pencils. The sheaf contains therefore an infinite number of axial * One of these ranges has all its points at an infinite distance ; each of the others has only one point at infinity.

O- is therefore proved. 17. Conversely, triangles and A^B^ two If A 2 B C2 2 , same plane, are such that the sides B l Cl and lying in the Fig. B2 C 3- and C^! and C2 A 2 A l B A 2 B2 cut one another in 2 l , , pairs in three collinear points A Q B Q CQ then the straight lines A^4 2 B^Bo, C^C2 which join corresponding angular points, will pass through one and the same point 0. (Fig. ) , , Through the straight line A Q B C draw another plane, project, from an arbitrary centre S19 the triangle A 1 B 1 C 1 and upon B { , , , C-i this plane.

By any the same are cut a- thus be called corresponding lines. , B C , project It follows may the straight lines correspond . . and that ... [10 to all which pass through a given point A of the plane acorrespond straight lines which pass through the corresponding- straight lines point A of the plane a . A describe a curve in the plane the A will in describe another curve the corresponding point said to correspond to the first curve. which may be plane 10. If the point <r, </, Tangents to the two curves at corresponding points are clearly corresponding straight lines ; and two curves are cut again, the by corresponding straight lines in corresponding points.