186 MR. IVORY ON THE THEORY OP ASTRONOMICAL REFRACTIONS. 



The two atmospheres with densities decreasing in arithmetical and geometrical pro- 

 gression, which it now appears were imagined by Newton, and which have been dis- 

 cussed by Thomas Simpson and other geometers, are found, when the same elements 

 are employed, to bring out horizontal refractions on opposite sides of the observed 

 quantity. Laplace conjectured that an intermediate atmosphere which should par- 

 take of the nature of both, and should agree with observation in the horizontal re- 

 fraction, would approach nearly to the true atmosphere. It must be allowed that 

 these conditions, which may be verified by innumerable instances between the two 

 limits, are vaguely defined ; and in order to ascertain the real meaning of the author, 

 recourse must be had to the algebraic expressions. When this is done it will be found 

 that the atmosphere intended is one of which the density is the product of two terms, 

 one taken from an arithmetical, and the other from a geometrical progression ; the 

 effect of which combination is to introduce a supernumerary constant, by means of 

 which the horizontal refraction is made to agree with the true quantity. No one will 

 deny the merit and the ingenuity of Laplace's procedure ; but though very skilful, 

 and guided in some degree by fact, it is liable to all the uncertainty of other arbitrary 

 suppositions, as indeed the author allows. Dr. Brinkley has given the character of 

 the French table fairly when he says, that it is only a little less empirical than the 

 other tables. On divesting Laplace's hypothesis of vagueness in the language, and 

 expressing it in the unequivocal symbols of algebra, it does not appear to possess any 

 superiority over other supposed constitutions of the atmosphere in leading to a better 

 and less exceptionable theory ; at least the Mec. Celeste has been many years before 

 the public, during which time not a few geometers have laboured on the subject of 

 the refractions ; but no improvement originating in the speculations peculiar to La- 

 place has occurred to any of them. 



Having said so much on the theory of the French table, it may be proper to add a 

 word on its accuracy. If it be compared from 80° to 88° of zenith distance with Bes- 

 sel's observed refractions, there will be found a small error in excess, continually in- 

 creasing, and amounting at last to 4". This shows that, in combining the two atmo- 

 spheres, too much weight has been given to that with a density varying in geometrical 

 progression, in consequence of which the air is not rarefied enough in the interpolated 

 atmosphere. With respect to the two degrees of altitude next the horizon, no accurate 

 judgement can be formed, for want of observed refractions that can be depended on. 



The astronomical refractions have also occupied the attention of the astronomer of 

 Koenigsberg, who has contributed so largely to the improvement of every part of 

 astronomical science. For the purpose of representing the observations of Bradley 

 with all the accuracy possible, Bessel investigated a table of refractions which ap- 

 peared in the Fundamenta Astronomice in 1818. He assumes a theoretical formula; 

 but as every arbitrary quantity is determined by a careful comparison with real ob- 

 servations, what is supposititious may be considered merely as an instrument of inves- 

 tigation, which is finally laid aside, leaving the result to rest on the foundation of 



