42 SCIENTIFIC APPARATUS 



to describe any figure, the other will describe a similar and 

 similarly situated figure, the centre of similitude of the two figures 

 being the fixed point. In the older pantograph the similarity is 

 direct, in the Milan pantograph it is inverse ; i.e. in the first case 

 the figures are on the same side of their centres of similitude, in 

 the second they are on opposite sides. 



B. INSTRUMENTS FOR TRACING SPECIAL CURVES. 



Geometrical drawings consist very mainly, but not exclusively, 

 of straight lines and circles. And the same limitation is observable 

 in all the ordinary constructions of theoretical geometry. It has 

 been ascertained that every problem which admits of one solution, 

 and one only, can, if the data of it are given graphically, be solved 

 with the ruler only, i.e. by drawing straight lines only, and with- 

 out using the compasses ; and, again, that every problem which is 

 quadratic, i.e. which admits two solutions, but not more, can be 

 solved with the ruler and compasses. Indeed, it is further known 

 that, for this purpose, one circle, traced once for all, would, as a 

 matter of theory, be sufficient. 



These considerations are perhaps sufficient to account for the 

 great preponderance of importance which attaches, in theoretical 

 geometry, to the straight line and circle. On the other hand, the 

 straight line and circle possess in common a property which is 

 peculiar to them among all plane curves, and which is invaluable 

 in all the practical applications of geometry. They are the only 

 plane (or untwisted) lines any part of which can be applied exactly 

 to any other. In very many mechanical arrangements this pro- 

 perty is indispensable, and it is advantageous in all cases where 

 accuracy of form is required, because it offers a simple means of 

 verifying that accuracy has been obtained. There is but one 

 twisted curve which has the same property, viz., the helix, or 

 screw-curve, and it is precisely because any part of a screw-curve 

 can be superposed upon any other that the screw and nut 



