180 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



Dismissing his first hypothesis, Newton next turned his attention to the 22nd pro- 

 position of the third book of his Principia. If the atmosphere consist of an elastic 

 fluid gravitating to the earth's centre in the inverse proportion of the square of the 

 distance, and if it be admitted that the densities are proportional to the pressures, 

 Newton, in the proposition cited, proves in effect that the densities will form a de- 

 creasing geometrical series, when the altitudes are taken in arithmetical progression*. 

 He writes to Flamsteed that an atmosphere so constituted is certainly the truth. 

 Newton evidently intended by this assertion to mark a distinction between pressure, 

 which is a cause of the variation of density that actually exists in nature, and his first 

 assumed law of the densities, which is entirely arbitrary. Setting aside hypothesis, 

 he now advanced so far in the true path of investigation ; and if the manner in which 

 heat is diffused in the atmosphere and the consequent decrease of density were not 

 known when he wrote, he advanced as far as the existing state of knowledge enabled 

 him to do. It is certain from his letters, that, after much time and labour, he at last 

 succeeded in calculating a table of refractions on the principle that the density is pro- 

 portional to the pressure. Such a table he communicated to Flamsteed, although it 

 is not found in the letters lately published ; and there is every reason to think it the 

 same which he gave to Halle y, and which that astronomer inserted in the Philo- 

 sophical Transactions for 1721. Two elements only are sufficient for computing all the 

 numbers in a table of refractions constructed by assuming that the density is propor- 

 tional to the pressure, namely, the refraction at 45° of altitude, and the height of the 

 homogeneous atmosphere, which is deducible from the horizontal refraction. The 

 table of Halley, therefore, contains in itself all that is required for ascertaining 

 whether it was calculated or not by the principle alluded to in the letters of Newton 

 to Flamsteed. Kramp seems to be the first who sought in the table for the manner 

 of its construction ; and his discoveries in this branch of science enabled him to as- 

 sign the height of the homogeneous atmosphere, which is one essential element. The 

 refraction at 45° of altitude, which is the other element, is found in the table equal 

 to 54", or, in parts of the radius, to '0002618; and Kramp found 4377^ toises for /, 

 the height of the homogeneous atmosphere ; so that, if a be the radius of the earth in 



toises, we have 



a = -0002618, 



i = — = -0013356; 



and the two elements, a and i, are sufficient for computing the whole table, if it be 



* Newton demonstrates strictly that the densities will be in geometrical proportion when the distances from 

 the earth's centre are in musical or harmonica! proportion, that is, when they are the reciprocals of an arith- 

 metical progression ; but in a series of this kind, if the first term bear an almost infinitely great proportion to 

 the differences of the following terms, as is the case of the radius of the earth when compared to elevations 

 within the limits of the atmosphere, the differences of the terms or the elevations may, without sensible error, 

 be reckoned in arithmetical progression. 



t Anal, des Refractions Astronomiques, p. 19. 



