XIV PEE FACE. 



his theory of refraction by giving a theorem from which it is 

 clear that Newton then understood how to form the differential 

 equation to the path of a ray of light through our atmosphere. 

 It is true that, for the sake of greater simplicity in this com- 

 munication to Flamsteed, Newton restricts the enunciation of 

 his theorem to the particular case where the density decreases 

 uniformly as the height increases, but it is obvious from the 

 form of'the enunciation of Newton's theorem that the method 

 is general, provided that the differential of the density which is 

 appropriate to any given law of diminution be employed in 

 finding the corresponding differential of the refraction. In an 

 interesting article in the Journal des Savants for 1836, M. Biot 

 directs particular attention to this subject, and tries to repro- 

 duce the method which Newton may be supposed to have 

 employed in order to calculate his table of refractions. M. Biot 

 closes his article in the following terms : 



"II est done prouve, par ce qui precede, que Newton a forme 

 1'equation differentielle exacte de la refraction pour les atmospheres 

 de composition uniforme; qu'il Fa appliquee exactement au cas ou 

 les densites des couches sont proportionelles aux pressions, ce qui 

 rend leur temperature constante; et qu'enfm, pour ce cas, il a obtenu 

 les vraies valeurs des refractions a toute distance du zenith, sans 

 avoir eu besoin d'employer les integrations analytiques qu'il a du 

 tres-vraisemblablement ignorer. II est done le createur de cette 

 theorie importante de 1'astronomie physique, qui serait probablement 

 aujourd'hui plus perfectionee, si Ton avait connu plus tot ses premiers 

 efforts." 



Judging from Newton's account of the time which he 

 employed in making these calculations, there must have been a 

 considerable mass of papers devoted to them which have not 

 been preserved. Fortunately, however, among the Portsmouth 

 papers we find a detailed calculation of the refraction corre- 

 sponding to the altitudes 0, 3, 12 and 30. In order to 

 make this calculation the path of a ray of light through the 

 atmosphere is divided into a number of parts subtending given 

 small angles at the centre of the earth. Hence are found by 

 the fluxional method quantities which are proportional to the 

 refractions suffered by the ray in passing over the successive 



