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sections being established, we can at once, by the help of this 

 discovery, enunciate its dual. In this theory, having assumed 

 a point winch may be termed the origin, let us conceive a plane 

 drawn any where in space, and a perpendicular let fall upon it 

 from the origin : a point taken on tins perpendicular, whose 

 distance from the origin is inversely proportional to the length 

 of this perpendicular, is defined as the pole of the plane ; and 

 the plane as the polar plane of tins point. Again, let there 

 be a right line anywhere in space ; through tins right line con- 

 ceive two planes drawn, and then poles taken by the last 

 construction, the right bne which joins these points is called 

 the polar line of the given right line. It follows then, that a 

 point and a plane are connected as pole and polar, and that 

 the polar of a right line is also a right Une. These right lines 

 are termed "conjugate polars." As the results of this 

 theory have been deduced by means of graphical methods 

 only ; it occurred to me, some years ago, that there must be 

 an analytical method equally general and co-extensive with the 

 Cartesian method of co-ordinates. The fundamental idea of 

 this theory may be developed briefly, without entering at 

 length into mathematical details, or using algebraical symbols. 



" In the Cartesian method, three planes are drawn through a 

 point — co-ordinate planes they are called — for simplicity we 

 shall suppose them at right angles to each other. On those 

 planes perpendiculars are let fall from a point, and if we assume 

 a relation as existing between the perpendiculars, whether 

 algebraical or transcendental — the point will in general always 

 be found on a curved surface, whose nature and properties 

 may be deduced from the assumed relation. 



" The perpendiculars are usually termed the co-ordinates of 

 the point, projective co-ordinates we shall call them, and are 

 denoted by the symbols x, y, z ; and the constant relation 

 between them by some such as F (x, y, .rj=0. 



" In the method which I have discovered, a tangent plane 

 is conceived to be drawn to a curved surface, meeting the 



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