VOL. XXVI.] PHILOSOPHICAL TRANSACTIONS. 47^ 



-t -i 



In our problem, where qn = a? »•, because j/'" = a:, if fq, the exponent of 

 the density of air, be called z, it will be z = , . =, or z = - — by 



taking r and cp for unity; and supposing the hyperbolic wave to be equilateral, 

 and consequently the ray a similar equilateral hyperbola, then 



y = -, and because t = 1, it will be z = 'j--=='. 



But because both Mr. Herman, in Act. Lips. 17 o6, and Dr. Gregory, in his 

 Astron. lib. 5, demonstrate, that the curve which determines the degrees of the 

 density of the air, is the logarithmic ; so that the heights oq, oq, or ^, are the 

 logarithms of the exponents of the density of the air at the points q, q ; it is 

 plain that the curvature ng, of the continually refracted ray Nn, is described by 

 such a law, that the cosines of incidence and refraction, raised to the power 



— — —, have a ratio compounded of the ratio of the right sines, raised to the like 



power, and of the ratio of the logarithms of the densities. 



Moreover, though we should allow that the common law of the refraction 

 of light gives the sines of incidence and refraction, proportional to the den- 

 sities of the mediums, yet that proportion may not possibly be so exact, seeing 

 the ratio of the sines in the refraction out of air into glass, is nearly sesquial- 

 teral, and air upwards of 1000 times more rare than glass; but when geometri- 

 cians found that the sine of refraction became greater, in passing into another 

 medium, according to the greater facility with which light penetrates that me- 

 dium in the common hypothesis, or according to the greater difficulty in the 

 hypothesis of Descartes ; who, on the contrary, supposes that light sutlers a 

 greater refraction, on account of the greater difficulty, in a rarer than in a 

 denser medium, (as heavy bodies, because of the greater difficulty in penetrating 

 denser bodies, are more refracted in these, by receding from the perpendicular) 

 and that both laws agree in this, viz. that according to the greater rarity oi 

 the medium, the greater would be the refraction ; hence it came to prevail, 

 that the sines were said to be proportional, not to the facility or difficulty of 

 their passage, one or other of which is called into question by some, but to the 

 rarity of the medium, in which all agree, though the true proportion does not 

 entirely answer in geometrical rigour; therefore, whenever mention is made of 

 density or rarity, perhaps the facility of passage in the common hypothesis, and 

 difficulty in the Cartesian, might be substituted for it : excepting where it is said, 

 that the rarity varied by the weight of the incumbent air answers to the heights, 

 as numbers to their logarithms, which is strictly true. 



