GEOMETRICAL AND ENGINEERING DEA WING. 83 



demonstrate the early problems of that work. These 

 boards represent the horizontal and vertical planes, and 

 the strings represent the lines in space. Then there are 

 lines drawn on the boards themselves, which are the 

 projections of these lines ; so that with a very little 

 trouble the student can determine at once the meaning of 

 the traces of planes, the traces and projections of lines. 

 There is here also a very convenient little apparatus 

 intended to explain plan and elevation, and also a means 

 of obtaining a second elevation. All these are obtained at 

 small cost, and certainly serve as a good commencement to 

 this study. I should mention that De Fourcy's work has 

 been partly translated in Weale's series, and also by Hall ; 

 I think it is the very best elementary work on descriptive 

 geometry. 



There is another point with respect to descriptive 

 geometry to which I must advert, and that is that 

 the student of analytical geometry is greatly helped by 

 a knowledge of it. For instance, if you are given the 

 equation of lines or planes in space, you can always draw 

 them by laying down co-ordinate planes and a ground 

 line. 



But as time presses I must hurry on 'to this beautiful 

 series of models made by M. Fabre de Lagrange, from 

 M. Ollivier's lectures, which belong to the South Kensington 

 Museum. These models, although numerous individually, 

 are not very numerous in class. They are all what are 

 called ruled surfaces. I should perhaps recall to your 

 minds what the ruled surface is. It is the term applied 

 to every surface, whether twisted or developable, which 

 is described by the movement of a right line in space, so 

 that at every point in such a surface a ruler may be so 

 placed as to be on the surface itself in all its length. In 

 general, there are two systems of construction; so that 

 not only can you lay the ruler on the surface in one direc- 

 tion, but in two. These models contain a little more than 

 the ordinary ruled surfaces, because there are some inter- 

 sections as well. I will take a few of the examples. The 

 first one is what is generally termed a twisted plane. It 

 is the hyperbolic paraboloid, and is engendered by a line 

 sliding along two other lines, not in the same plane, and 

 remaining constantly parallel to a given plane. These 



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