664 SCIENTIFIC THOUGHT. 



26. By the law of continuity he showed how in pure 

 continuity, geometry it became necessary to introduce the considera- 

 tion of points and lines which vanish into infinity or 

 which become imaginary, establishing by their invisible 

 elements the continuous transition from one geometric 

 form to another ; just as in algebra these conceptions 

 had forced themselves on the attention of analysts. 

 Ideal elements were thus made use of to lead to the dis- 

 covery of real properties. 



The consideration of lines and points which vanish 

 or lie at infinity was familiar to students of perspective 

 from the conception of the " vanishing line " ; but the 

 inclusion of ideal points and lines was, as Hankel says, 

 a gift which pure geometry received from analysis, 

 where imaginary (i.e., ideal or complex) quantities behave 



27. in the same way as real ones. "Without the inclusion of 



Ideal _ "^ 



elements, tlicsc ideal or invisible elements the generality or con- 

 tinuity of purely geometrical reasoning was impossible. 

 The geometrical reasoning of Monge, Carnot, and 

 Poncelet was thus largely admixed with algebraical or 

 analytic elements. It is true that Monge's descriptive 

 geometry was a purely graphical method, and that 



gested to Poncelet by the prop- 

 erty, known already to De la 

 Hire ("Sectiones ConicEc," 1685), 

 that in the plane of a conic 

 section every point corresponds 

 to a straight line called its 

 "polar," that to everj^ straight 

 line corresponds a point called 

 its " pole," that the " polars " 

 corresponding to all the points of 

 a straight line meet in one and 

 the same point, and vice versa 

 that the "poles" corresponding to 

 all lines going through one and 

 the same point lie on a straight 



line ; the line and point in ques- 

 tion standing in both cases in the 

 relation of pole and polar to each 

 other. Poncelet uses "this trans- 

 formation of one figure into its 

 reciprocal polar systematicallj' as a 

 method for finding new theorems : 

 to every theorem of geometry 

 there corresponds in this way 

 another one which is its ' polar,' 

 and the whole of geometry was 

 thus split up into a series of 

 truths which run parallel and 

 frequently overlap each other " 

 (Hankel, loc. cit., p. 20). 



