Mr. Ivory on the Theory of the Astronomical "Refractions. 149 



a surface of increased density, it receives an impulse which may be 

 considered as instantaneous; and this impulse being distributed over 

 the breadth of a stratum of uniform density, ascertains the centripetal 

 force tending to the earth's centre, by the action of which the tra- 

 jectory is described. 



It appears, that Newton himself was the first to apply this new 

 method to the problem of the astronomical refractions. In his first 

 attempt he assumes that the densities decrease in ascending, in the 

 same proportion as the distances from the earth's centre increase. 

 On this supposition the author investigates a formula, which M. Biot 

 has also obtained, and which is equivalent to the construction com- 

 municated by Newton to Flamsteed. On this basis a table was 

 computed and communicated to Flamsteed; but Newton subsequent- 

 ly informed Flamsteed that he did not intend to publish it, in con- 

 sequence of a serious objection to the supposed scale of densities. 

 Adopting the principles in the twenty-second proposition of the 

 second book of his Principia, Newton, it appears, succeeded at length 

 in computing a second table of refractions, which he likewise com- 

 municated to Flamsteed, and which, there is every reason to think, 

 is the same which he gave to Halley, and which was inserted by that 

 astronomer in the Philosophical Transactions for 1721. As the de- 

 termining whether the two tables are identical is a question of much 

 interest, the author enters very fully into it, and, from the results 

 of elaborate calculations, concludes that Halley's table is no other 

 than the one which Newton calculated on the supposition that the 

 densities in the atmosphere are proportional to the pressures. He 

 remarks that, as far as the mathematics are concerned, the problem 

 of the astronomical refractions was fully mastered by Newton. 



After referring to the labours of Brook Taylor, Kramp, and Thomas 

 Simpson, the author again adverts to Newton's views, remarking 

 that, in assigning the rarefaction of the lower region of the atmo- 

 sphere by heat as the cause why the calculated refractions near the 

 horizon so much exceeded the observed, as was found to be the case, 

 Newton had assigned the true cause ; but that he had no clear con- 

 ception of the manner in which the density in the lower region is 

 altered by the agency of heat; and he considers that nearly the same 

 ignorance in that respect still prevails. 



The two atmospheres, with densities decreasing in arithmetical 

 and geometrical progression, which, it now appears, were imagined 

 by Newton, and which have been discussed 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 

 quantities. La Place conjectured that an intermediate atmosphere 

 which should partake of the nature of both, and should agree with 

 observation in the horizontal refraction, would approach nearly to 

 the true atmosphere. If recourse be had to the algebraical expres- 

 sions of La Place, it will be found that the atmosphere he proposes 

 is one of which the density is the product of two terms, the one 

 taken from an arithmetical, the other from a geometrical series; the 

 effect of which combination is to introduce a supernumerary con- 



