ly informed Flamsteed that he did not intend to publish it, in consequence 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 communicated 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 determining 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 atmosphere 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 conception 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 expressions 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 constant, by means of which the horizontal refraction is made to agree with the true quantity. The author considers, with Dr. Brinkley, that the French table, founded on La Place's investigation, is only a little less empirical than the other tables, and that the hypothesis of La Place does not appear to possess any superiority over other supposed constitutions of the atmosphere in leading to a better and less exceptionable theory.
After eulogizing Bessel's tables of mean refractions, published in his Tabulæ Regiomontanæ, the author refers to his own paper in the Philosophical Transactions for 1823. In this paper the refractions are deduced entirely from the very simple formula,—
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