GEOMETRICAL INSTRUMENTS AND MODELS. 43 



arrangement is possible, which renders this curve of so much use 

 in mechanics. 



But notwithstanding these prerogatives of the straight line and 

 circle, the tracing of other curves is occasionally indispensable both 

 in theoretical and practical geometry. It is by no means an easy 

 matter to invent a good method of tracing a curve. Even when 

 the theory of a curve is pretty well known, that theory may fail to 

 suggest any mode of describing it mechanically; and not every 

 mode which theory suggests can be made to work accurately in 

 practice. Of all curves, after the circle, the ellipse would seem to 

 be the simplest and easiest to draw, but some authorities on the 

 subject recommend the draughtsman not to attempt a true ellipse, 

 but to put together an imitation semi-ellipse, by means of six or 

 seven arcs of circles with centres and radii appropriately chosen. 

 It is said that such an imitation will impose even on a well-trained 

 eye, although it is plain that, whereas the curvature of an ellipse 

 changes continuously, the curvature of the imitation curve changes 

 abruptly at the points of junction of the circular arcs. 



The discovery of M. Peaucellier that a linkage can be con- 

 structed capable of describing a straight line may perhaps hereafter 

 revolutionise this part of geometry. Already it is known that any 

 conic section, and several of the more important curves of the 

 third and fourth orders can be described by linkage movements, or 

 compound compasses, as M. Peaucellier has called them, riot too 

 complicated to work steadily. Theoretically, the results obtained 

 are of a far wider scope, and Professor Sylvester has shown good 

 reason for believing that every geometrical curve is capable of 

 being described by a link-movement 



We may briefly mention some other mechanical arrangements 

 which exist for describing certain plane curves. 



(i.) The ellipse can hardly be described with great accuracy by 

 means of a thread attached to its two foci and stretched by a pencil, 

 because of the extensibility of the thread. It can be described as 

 an hypocycloid, as in the apparatus of Mr. A. E. Donkin ; or 



