THE LAW OF REFRACTION 



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where i is the angle of incidence and r the angle of refraction. 

 Kepler, it is true, does not give the formula explicitly, partly 

 because he is interested more in i — r than in r, regarding the 

 deviation as more important than the refraction ; but it is easy 

 to write the formula down from his procedure. 

 The formula, like the modern formula, 



sin i = /M sin r 



is a one-constant formula. Like the latter, it reduces to t = /u. r, 

 when i and r are small, for then sec r = i. The following table 

 gives Kepler's verification of his formula : 



The column headed i gives the angles of incidence and the 

 column headed r the corresponding angles of refraction as 

 given by Vitellio ; it will be observed they differ from Ptolemy's 

 values in three cases, but agree in the very bad observation at 

 the end. The third column gives the excess of r as calculated 

 by Kepler on the basis of fx, = 1*317 over the observations. 

 In the fourth column I have calculated the excess of r as given 

 by sin i = fi sin r over the observations on the basis of /u, == i*333, 

 the correct value for water. 



It will be observed that Kepler's formula agrees much 

 better with the observations than the modern one does, because 

 the last experimental value, which is common to both Ptolemy 

 and Vitellio, is very far out. 



The fact that Kepler obtained a one-constant formula for 

 the law of refraction seems to have escaped the notice of all 

 modern writers. I have seen various statements to the effect 

 that he used a two-constant formula and several rather 

 patronising references to his failure to obtain the correct law. 

 But, under the circumstances, his formula is a triumph ; it is 

 unfortunate that he did not check the observations before 

 devoting so much labour to working them out. 



As regards the differences between his calculated results 

 and the observations, he states quite emphatically that the 



