666 SCIENTIFIC THOUGHT. 



from a point drawing lines or rays on the plane and in 

 space, and we can cut these by lines in a plane or by 

 planes in space. And it can be shown that "if one 

 geometric form has been derived from another by means 

 of one of these operations, we can conversely, by means 

 of the complementary operation, derive the second from 

 the first." l 



The protective geometry of Poncelet contains the two- 

 fold origin of the principle of duality in his method of 

 projection and section, and in his theory of the reciprocity 

 of certain points and lines in the doctrine of conic sections, 

 . called the theory of reciprocal polars. But the mathe- 



Reciprocity. 



matician who first expressed the principle of duality in a 

 general though not in the most general form was 

 Gergonne, who also recognised that it was not a mere 

 geometrical device but a general philosophical principle, 

 destined to impart to geometrical reasoning a great 

 simplification. He sees in its enunication the dawn of 

 a new era in geometry. 2 



1 Cremona, foe. cit., p. 33. | another geometry equally legiti- 



1 The principle of Duality seems mate in which a point is gener- 



to have been first put forward in its ated by the rotation of a line, 



full generality by Gergonne, in- Whereas in the first case the line is 



spired probably by the theory of the locus of the moving point, in 



Reciprocal Polars (see note, p. 663) the latter case the point is the 



enunciated by Poncelet, who many geometrical intersection of the 



years afterwards carried on a vol- rotating line. In this generality 



uminous polemic as to the priority the principle of duality has been in- 



of the discovery. " Gergonne saw corporated into modern geometry " 



that the parallelism (referred to (Hankel, foe. cit., p. 21). Gergonne 



above) is not an accidental conse- says of the new principle (1827. see 



quence of the property of conic Supplement to vol. ii. 2nd ed. of 

 sections, but that it constitutes a , Poncelet's ' Traite,' p. 390) : " II ne 

 fundamental principle which he j s'agit pas moins que de commencer 



termed the 'principle of duality.' pour la geometric, mal connue 



The geometry which is usually depuis pres de deux mille an.* qu'on 



taught, and in which a line is con- s'en occupe, une ere tout -h- fait 



sidered to be generated by the nouvelle ; il s'agit d'en mettre tous 



motion of a point, is opposed by les anciens traites a peu pres au 



