III. MODELS. 17 



handled, the strings are apt to become entangled or break. They should never 

 be used except by a person who understands them, and they should not be 

 shifted without some good reason. 



FIG. 1. FIG. 2. 



Fig. 1 is an example of the first, and Fig. 2 of the second. In both cases, 

 the curve, although not actually drawn, is indicated with sufficient approxi- 

 mation for most practical purposes. Models Nos. 10 and 30 also afford illus- 

 trations of the principle exhibited in Fig. 2. 



Geometrical drawings of most of the surfaces represented by these models 

 are contained in BRADLEY'S Practical Geometry (2 vols., oblong folio, pub- 

 lished by Chapman and Hall). Many of them will also be found iu the 

 French treatises on practical and descriptive geometry, such as LEROY, 

 ADHEMAR, LEFEBURE DE FOURCY, DE LA GOURNERIE, and in their treatises 

 on Stereotomy and Stone-cutting (coupe des pierres}. Many of them are also 

 given in SONNET'S Dictionnaire des Mathematiques Appliquees. A catalogue 

 of this collection of models, with an appendix containing an account of the 

 application of analysis to their investigation and classification, was prepared 

 for the South Kensington Museum in 1872, by Mr. C. W. Merrifield, F.R.S. 

 The following descriptions are extracted from this catalogue : 



77. Hyperbolic Paraboloid generated by a single system 

 of right lines. 



Two bars, each pierced with holes, equally spaced. One bar is fixed, 

 the other swings round an axis, which, moreover, can be inclined at different 

 angles to the fixed bar. 



When the bars are parallel the strings indicate a plane. When they are 

 inclined to one another, but still in the same plane, the strings still indicate a 

 plane ; but when the bars are not in the same plane, the surface is the hyper- 

 bolic paraboloid. 



The surface is sometimes called the tivisted plane. But it must not be 

 supposed that it can be made by bending a plane. On the contrary, when 

 the surface is twisted, no two of the strings lie in the same plane, and, there- 

 fore, no part of the surface is plane. It can neither be flattened nor made 

 from a plane, without stretching or contraction. 



The hyperbolic paraboloid is the natural surface proper for a ploughshare. 



78. Hyperbolic Paraboloid. 



Two bars, pierced with- holes at equal distances, the holes being connected 

 by two different systems of strings. The surface, as well as the arrangement, 

 is very nearly the same as in No. 1, only that there are two paraboloids in- 

 stead of one. As the movable bar swings round, the paraboloid opens out 

 while the other closes up. If the bars are swung so as to be in the same 

 39508. B 



