of Curves by their Curvature. 79 



as the curve becomes notably inclined to the axis the change of 

 reading on the axis between the two ends of a short arc even 

 may be more than appreciable, so that if the process were 

 carried ont exactly as described the needle-point would be on 

 the whole too late in making its move ; i. e. where the radius 

 is increasing it would be too small, and vice versa. To avoid 

 this it is convenient to choose such a small arc as shall change 

 the axial reading by any small number of divisions at a time 

 and to move the needle-point through a corresponding number 

 at a time also, but so that the needle-point is placed always 

 at that division which corresponds with the intermediate one of 

 the axial divisions. The curve can in this way be carried on very 

 rapidly by very small steps without ever even looking to see 

 where the pencil or pen is being taken, and it in no way suffers 

 from the small errors of each resetting of an ordinary compass. 

 It is not even necessary to draw the curve at all. The process 

 can be carried out as described, and the curve will be traced 

 as in imagination by the hole at oo . The perfection of the re- 

 sult which is obtainable, whether the compass or the reciprocal 

 rule method is employed, depends of course upon the fact that 

 whereas a very fair representation of a curve may be made by 

 means of a polygon with a very great number of very short 

 sides, each making very nearly an angle of two right with 

 its neighbour, a vastly more perfect result is produced by a 

 corresponding series of small arcs of circles, each having the 

 necessary radius of curvature and making no angular break 

 at each stage. The small discontinuity is one of curvature 

 only, not of direction. The gain in smoothness and accuracy 

 is still very great when the steps are made larger, so that a 

 polygon of the same number of sides would cease to fairly 

 represent the curve. Where the curvature is a maximum 

 or a minimum the contact of the circle of curvatures is of the 

 third instead of the second order, and at these parts a much 

 longer step may be taken without impairing the accuracy of 

 the curve. Similarly, if instead of the arc of curvature being 

 used for each small step some curve of curvature and change 

 of curvature could be employed, a still closer approximation 

 would result, or still larger steps could be safely employed. 



Going back now to the curve which has been chosen (a 

 nodoid), it will be found that the point on the axis gets 

 gradually farther away until the end of the rule is reached at 

 *05 in the case of the rule exhibited. The needle-point will 

 now be at division '4:5. For the next two or three steps the 

 strip will be rapidly approaching parallelism with the axis, 

 so that the axial reading is equally rapidly approaching ; 

 L e. at an infinite distance. During these steps the axial 



