184 MR. IVORY ON THE THEORY OP ASTRONOMICAL REFRACTIONS. 



that Newton judged his own, it must be decided that they are all liable to objections. 

 They all involve some supposition that has no foundation in nature ; or they leave out 

 some necessary condition of the problem. It is allowed that the variation of heat at 

 different altitudes is unknown ; that we are equally unacquainted with the manner 

 in which is diffused the aqueous vapour that is always found, more or less, in the 

 atmosphere ; that the law of the densities has not been ascertained. But besides 

 these capital points, the accurate M. Poisson will suggest other properties that must 

 be attended to in an atmosphere in equilibrium : the conducting power of heat vary- 

 ing with the condition of the air ; its power of absorbing heat ; and the interchange 

 by radiation which takes place with the earth, with the etherial spaces above, and 

 with the stars visible above the horizon. So many conditions, placed beyond the 

 reach of our inquiry, may well puzzle the most expert algebraist to take them into 

 account. But it may be doubted whether this be the proper view of the problem. 

 The astronomical refractions at any observatory are mean effects of the atmosphere ; 

 and it maybe alleged that the proper way of accounting for them is to compare them 

 with other mean effects of the atmosphere at the same place. 



4. In 1715, twenty years after the researches of Newton, Brook Taylor published 

 in his Methodus Incrementorum, the first investigation of the astronomical refractions 

 on the supposition that the density of the air is variable. The differential equation 

 is accurately deduced from the principles laid down by Newton; which removed all 

 the difficulties of the problem in this respect. 



Kramp, in his Analyse des Refractions Astronomiques et Terrestres^ has elucidated 

 the elementary parts of the problem, and has greatly improved the mathematical so- 

 lution. His method is particularly commodious and useful in the case of the hori- 

 zontal refraction. For altitudes above the horizon the integrals are not susceptible 

 of being simply expressed, and seem to require the aid of subsidiary tables in apply- 

 ing them. 



The problem of the refractions being an important one in astronomy, many solu- 

 tions of it have been published by different geometers. Some of these are preferable 

 to others, because the method of calculation is easier in practice. For altitudes 

 greater than 16°, they may all be reckoned equivalent, differing from one another 

 only in the quantity assumed for the refractive power of air. They also mostly agree 

 in the horizontal refraction, which is taken from observation. But for altitudes less 

 than 16°, they are different: because, at these low altitudes, the refractions are 

 affected by the arbitrary suppositions used in constructing the tables. 



Thomas Simpson published a judicious dissertation on this problem. He distinctly 

 points out that, to a considerable distance from the zenith, the refractions are inde- 

 pendent of the manner in which the atmosphere is supposed to be constituted. In 

 comparing the two atmospheres that have densities decreasing in arithmetical and 

 geometrical progression, he remarks that the horizontal refraction comes much nearer 

 the observed quantity in the first atmosphere than it does in the second ; for which 



