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the successive steps of a proof with facility and conviction. This 

 may to some extent also explain why the method has hitherto received 

 so little countenance as not to be admitted into any elementary work 

 on the application of the principles and notation of algebra to the 

 investigation and discussion of the properties of space. But the ad- 

 dition of a new method of investigation to those already in use, the 

 development of its principles, with illustrations of the mode of its 

 application, are surely not of less value to a philosophical apprecia- 

 tion of what that is in which mathematical knowledge truly consists, 

 than the giving of pr jblems, which, while they embody no general 

 principle, are yet often difficult to solve ; and when solved, frequently 

 afford no clue by which the solution may be rendered available in 

 other cases. 



The radical vice in mathematical instruction in this country and 

 in our time would seem to be, that knowledge of principles and 

 familiarity with methods of investigation are subordinated to nimble 

 dexterity in the manipulation of symbols, and to cramming the me- 

 mory with long formulae and tabular expressions. 



Again, it often happens that an investigation, which, if pursued by 

 one method, would prove barren of results or altogether impracticable, 

 when followed out from a different point of view and by the help of 

 another method, not unfrequently leads by a few easy steps to the dis- 

 covery of important truths, or to the consideration of others under a 

 novel aspect. Hence the multiplication of methods of investigation 

 tends widely to enlarge the boundaries of science. 



My object in the following paper will be to show that problems of 

 great difficulty, some of which have not hitherto been solved, while 

 others by the ordinary methods admit only of complicated and tedious 

 modes of proof, may by this method be treated with singular brevity 

 and remarkable simplicity. I will first premise a few simple principles. 



When two figures in the same plane, or more generally in space, 

 are so related that one is the reciprocal polar of the other, then to 

 every point in the one corresponds a plane in the other ; to every 

 right line in the one a right line also in the other ; to any number of 

 points in the same right line in the one, as many planes all intersect- 

 ing in the same right line in the other ; to any number of points in 

 the same plane in the one, as many planes all meeting in the same 

 point in the other. I might easily proceed to any length with this 



