16 SEC. 2. GEOMETRY. 



zoiital slab, whose upper surface has been shaped, as hereafter described, in 

 accordance with numerical tables, calculated from the desired function, the 

 height of its surface at each point being the tabular value corresponding to 

 the two entries severally represented by the distance of that point from the 

 front and from one side of the slab. The plate that carries the two traces is 

 placed horizontally on a frame that travels in front of the slab. Two slides 

 move at right angles to this plate, and have microscopes attached to them, that 

 traverse the paper along ordinates having the same abscissa. One of these 

 slides is rigidly connected with a frame on which the slab is able to move from 

 side to side ; when this slide is pushed, the frame and the slab together are 

 pushed with it. The other slide gives a sidelong movement of the slab on the 

 frame. Thus the particular point of the slab that corresponds to the values 

 of the two ordinates is brought vertically below a descending rod, and this is 

 caused to drop gently on the surface of the slab by touching a treadle. The 

 vertical space through which the rod descends is consequently the function 

 required. The rod carries a horizontal pricker, with which it makes a dot on 

 a plate held vertically in the same stage that carries the plate on which the 

 two traces are drawn. The slabs can readily be fashioned by instrument- 

 makers who possess the necessary apparatus according to any required tables. 

 They are drilled to the requisite depth at various points, and are afterwards 

 smoothed down. 



76b. Rule, with joint, which serves to curve an elastic plate 

 into an arc of the circle, of any radius. 



Professor Tchebichef, University of St. Petersburg. 



III. MODELS OF FIGURES IN SPACE. 



COLLECTION OF MODELS OP RULED SURFACES, CONSTRUCTED 

 BY M. FABRE DE LAGRANGE, IN 1872, FOR THE SOUTH 

 KENSINGTON MUSEUM. 



This collection illustrates the principal types of the class of surfaces which 

 can be traced out in space by the motion of a straight line. 



These surfaces, on account of the facility with which they can be con- 

 structed and represented, and of the ease with which their intersections can 

 be determined, are of more consequence than any others in the geometry of 

 the Industrial Arts. It is only in small work, which can be put into the 

 lathe, that the class of surfaces of revolution approaches them in respect of 

 general utility. The most important surfaces of all, the plane, the right 

 cylinder, the right cone, and the common screw, belong to both classes. 



The representation of the surfaces by means of silk threads is of course 

 only approximate ; an approximation of the same character as the representa- 

 tion of a curve by a dotted or chain line, or by a series of right lines touching 

 the actual curve. 



The models are constructed with especial reference to the possibility of 

 changing their shape, by moving some of the supports of the strings, by altering 

 the lengths or positions of certain parts, or by converting upright forms into 

 oblique. This possibility of deformation, as the process is technically called, 

 greatly enhances the value of the models, by allowing them to represent a 

 much greater variety of surfaces than if they were fixed. They are, how- 

 ever, too delicate to be much pulled about, and, unless they are very cautiously 



