Astronomical and Nautical Collections, 363 



Epoch o/'KraMi' and Laplace. 

 For the mathematical theory of refraction it may be said that 

 nothing of immediate importance was done from the time of New- 

 ton and Taylor, to that of Laplace and Kramp. It is true, that 

 the XVIth volume of the New Commentaries of Petersburg, for 

 the year 1771, contains an Essay of Euler, in which the particular 

 value of a fluent is first demonstrated, which is of singular import- 

 ance in abridging the computation of the horizontal refraction; 

 but it does not seem to have occurred to this great mathematician 

 in what manner his discovery might be rendered serviceable for 

 the solution of a physical problem. It was in the Memoirs of thd 

 Academy for 1782, that Laplace made public an essay on the in* 

 tegration of differential functions, which contain very high powers 

 of their factors ; and this essay Kramp considers as first develop* 

 ing the principle that led to the more accurate solution of the 

 problem. 



Professor Kramp had made himself known and respected in the 

 mathematical world, by his attempts to apply the principles of 

 mechanical hydraulics to the circulation of the blood in health and 

 in disease, and he was the author of some interesting essays on the 

 combinatorial analysis of Hindenburg, which excited at one period 

 so much attention in Germany, though none of its other results 

 appear to have been so satisfactory, as those which are contained in 

 the chapter on Numerical Faculties of the Analyse des Refractions. 

 The rapid and brilliant progress that is displayed in this chapter 

 through some of the most thorny paths of analysis will for ever dis- 

 tinguish its author among the original contributors to the advance- 

 ment of mathematical analysis ; but it is, perhaps, somewhat too rapid 

 to have avoided all traces of contact with the thorns that were to be en- 

 countered. The originality consists principally in the very great ge- 

 neralisation of the laws of the faculties of numbers, which have been 

 since more commonlycalledfactorials,and in their extension to facul- 

 ties with fractional indices, formed according to the analogy of frac- 

 tional powers, but which, in fact, though they may be shown to have 

 real values, are little less imaginary in their immediate structure, 

 than the square roots of negative numbers, and resemble still more 



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