660 



SCIENTIFIC THOUGHT. 



in the prisons of Russia meditated on the real cause of 

 the power which algebraical analysis possessed, on the 

 reason why geometry proper was deprived of it, and 

 what might be done to give it a similar generality. In 

 pursuing this line of thought he was led to discover the 

 cause of the existing limitation of purely geometrical 

 reasoning in its rigidity, inasmuch as it was arrested as 

 soon as its objects ceased to have a positive or absolute, 

 that is a physical, existence. 1 Opposed to this limitation 

 was the freedom of the analytical method, which, operating 

 with indeterminate symbols, could, by letting them change 

 gradually, include not only what was explicitly given, 

 but also that which was merely implied ; not only the 

 finite, but likewise the infinite ; not only the real, but 

 likewise the fictitious or imaginary. In order to gain a 

 similar generality in purely geometrical or descriptive 

 science, a similar flexibility would have to be introduced. 

 Poncelet was thus led to the enunciation of his celebrated 

 and much -criticised "principle or law of continuity." 2 



1 See the "Introduction" to the 

 1st volume of the ' Traite des Pro- 

 priete"s projectives des figures,' pp. 

 xi, xii. I quote from the 2nd edi- 

 tion of 1865. The 1st was published 

 in 1822. The researches date from 

 1813, the year of Poncelet's im- 

 prisonment. See " Preface de la 

 premiere Edition." 



2 Ibid., Introduction, p. xiv. 

 On the principle of continuity 

 in geometry, see an article in 

 vol. xxviii. " Ency. Brit.' by the 

 Rev. Charles Taylor, and the re- 

 ferences given therein ; also Prof. 

 E. Kotter's Report on the 

 "Development of Synthetic Ge- 

 ometry " in vol. v. of the 

 ' Jahresbericht der Deutschen 

 Mathematiker Vereinigung, ' p. 



122, &c. : " Originally the ex- 

 positions referring to the prin- 

 ciple of continuity were intended 

 to occupy much greater space. . . . 

 In consequence of correspondence 

 with Terquem, Servois, and Brian- 

 chon, Poncelet desisted from the 

 publication of it. ... However 

 cautiously Poncelet advanced his 

 principle " in the ' Essai sur les 

 proprietes projectives des sections 

 coniques ' (presented to the 

 Academy in 1820) " it never- 

 theless aroused the doubts of 

 Cauchy, who in his report on 

 Poncelet's paper warns against the 

 too hasty application of the 

 principle. Gergonne accompanied 

 the reprint of this report with 

 notes, in which he characterised 



