244 New Protractor for Trisecting Angles. 



in the conditions prescribed for the research ; and the failure is to 

 be ascribed (if we may continue the metaphor) more to the fas- 

 tidiousness of the suitors than the caprice of the object. The 

 severe synthetic geometry of the ancients admitted only fixed 

 points and constructions upon lines drawn from or to such points. 

 Hence motion, except around a fixed point (as for instance in the 

 description of a circle) although neither unknown nor uninvesti- 

 gated in that early science, found no place in the majestic stability 

 which the school of Plato sought to uphold. It was only in 

 despair (or perhaps he meant it to show what he could do if he 

 would) that Plato himself resorted to a mechanical solution in 

 effecting the duplication of the cube. The quadrat rix of Dinos- 

 tratus, therefore, as well as other methods that might have been 

 proposed, involving sections of the cone, for the trisection of an 

 angle, was classed in the interdicted caste of mechanical curves 

 (as such figures were denominated) ; and so failed, in the verdict 

 of antique severity, to be admitted among the triumphs of pure 

 geometry. Modern times, less prepossessed or more just, have 

 nevertheless made for it a niche in the history of the progress of 

 human intellect ; and modern science, less strict and classical but 

 more fecund, has seized on the idea and held it as a thread to 

 guide among the intricacies of a more profound and wide-reach- 

 ing analysis than that which wove it first. Abandoning the 

 former inadequate distinctions of plane and solid and hypersolid, 

 it has imparted a wider generality to the whole theory of curves; 

 those that were before excluded as mechanical, it now treats as 

 transcendant ; and for the present application, shews how, by the 

 revolution of a cubic equation, the condition for trisection may 

 be met. 



But the elaboration and refinements of these methods of cal- 

 culation, unfit them for practical use, as in the majority of cases 

 where the result alone is wanted ; and I have thought, therefore, 

 that it might be of some interest to present a mechanical method 

 for effecting the purpose with as much accuracy as is desirable or 

 necessary in most instances. This is done in the present descrip- 

 tion of the accompanying figure (p. 245) of a New Protractor for 

 Trisecting Angles. New, I believe it is ; for it will hardly be found 

 elsewhere ; but it is to be observed that its device does not require 

 much ingenuity; and its merit, if it should be allowed to have 

 any, consists in its appropriateness and simplicity. An explana- 

 tion of the principle on which its device reposes, will be at once 

 the best introduction and commentary upon its mode of working 

 and use. 



Referring to the lines and letters in the figure which gives the 

 projection of the instrument, if we have any angle ACD requir- 

 ed to be trisected ; and with any radius AC, be described the 

 arc AED with its centre at j and with the same radius but with 





