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 ' 122, &c. : "Originally the ex- 

 ist volume of the ' Traite des Pro- . positions referring to the prin- 

 prie'te's projectives dea figures,' pp. ciple of continuity were intended 

 xi, xii I quote from the 2nd edi- i to occupy much greater space. . . . 

 tionof!865. The 1st was published j In consequence of correspondence 

 in 1822. The researches date from ! with Terquem, Servois, and Brian- 

 1813, the year of Poncelet's im- i chon, Poncelet desisted from the 

 prisonment. See " Preface de la | publication of it. ... However 

 premiere edition. " cautiously Poncelet advanced his 



2 Ibid., Introduction, p. xiv. principle " in the ' Essai sur les 

 On the principle of continuity proprietes projectives des sections 

 in geometry, see an article in I coniques ' (presented to the 

 vol. xxviii. ' Ency. Brit.' by the Academy in 1820) "it never- 

 Rev. Charles Taylor, and the re- theless aroused the doubts of 

 ferences given therein ; also Prof. Cauchy, who in his report on 

 E. Kotter's Report on the Poncelet's paper warns against the 

 " Development of Synthetic Ge- too hasty application of the 

 ometry " in vol. v. of the principle. Gergonne accompanied 

 ' Jahresbericht der Deutschen the reprint of this report with 

 Mathematiker Vereinigung,' p. notes, in which he characterised 



