666 



SCIENTIFIC THOUGHT. 



29. 

 Reciprocity 



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 projective 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- 

 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, loc. cit., p. 33. 



2 The principle of Duality seems 

 to have been first put for ward in its 

 full generality by Gergonne, in- 

 spired probably by the theory of 

 Reciprocal Polars (see note, p. 663) 

 enunciated by Poncelet, who many 

 years afterwards carried on a vol- 

 uminous polemic as to the priority 

 of the discovery. ' ' Gergonne saw 

 that the parallelism (referred to 

 above) is not an accidental conse- 

 quence of the property of conic 

 sections, but that it constitutes a 

 fundamental principle which he 

 termed the 'principle of duality.' 

 The geometry which is usually 

 taught, and in which a line is con- 

 sidered to be generated by the 

 motion of a point, is opposed by 



another geometry equally legiti- 

 mate in which a point is gener- 

 ated by the rotation of a line. 

 Whereas in the first case the line is 

 the locus of the moving point, in 

 the latter case the point is the 

 geometrical intersection of the 

 rotating line. In this generality 

 the principle of duality has been in- 

 corporated into modern geometry " 

 (Hankel, loc. cit., p. 21). Gergonne 

 says of the new principle (1827, see 

 Supplement toivol. ii. 2nd ed. of 

 Poncelet's ' Traite,' p. 390) : " II ne 

 s'agit pas moins que de commencer 

 pour la ge"ometrie, mal connue 

 depuis pres de deux mille ans qu'on 

 s'en occupe, une ere tout- a -fait 

 nouvelle ; il s'agit d'en mettre tous 

 les anciens traites a peu pres au 



