400 SCIENCE PROGRESS 



Ptolemy's data, we obtain the results in the third, sixth, and 

 ninth columns of the table, which give means of 1-311 for air 

 to water, 1-485 for air to glass, and 1-109 for water to glass. 

 The correct value for air to water is 1*333. The value for 

 glass varies according to its composition ; we do not know what 

 Ptolemy's result should have been, but his value is probably 

 low. On the basis of his first two results, his third result 



should have been ~ — = 1-133, so it is 2 per cent. out. But 



1-311 '^^' ^ 



the agreement is good when we consider the nature of the 

 apparatus by which the observations were made. Ptolemy, of 

 course, did not discover the law of refraction, and knew nothing 

 of the index of refraction. He merely left his results in the 

 form of tables. But he applied them correctly to the explana- 

 tion of astronomical refraction, i.e. the apparent displacement 

 of a star towards the zenith by the refraction of its rays in its 

 passage through the earth's atmosphere. He states that the 

 height of the atmosphere is unknown, but that it must begin 

 below the sphere of the moon. 



Alhazen, who flourished in Arabia in the eleventh century, 

 also made measurements on the angles of refraction and by a 

 more elaborate arrangement than Ptolemy. He established 

 the view that vision is performed by rays which proceed from 

 the object to the eye. The next investigator of the law of 

 refraction was the Polish philosopher Vitellio, whose book was 

 published at Nuremberg in 1535. But neither Alhazen nor 

 Vitellio discovered the law of refraction. There is some doubt 

 also as to whether the values for the angles of refraction given 

 by Vitellio were the result of independent measurements, or 

 whether they were merely copied from a variant of Ptolemy's 

 values. 



Kepler attacks the question of the law of refraction in chap, iv 

 of the Paralipomena ad Vifellionem, which was printed at 

 Frankfort in 1604. The problem is to get a mathematical 

 expression which will fit Vitellio 's table of the refraction from 

 air to water. He starts off with the idea that the solution is 

 to be found in some of the geometrical properties of the conic 

 sections ; hyperbola, ellipse, and parabola are investigated 

 tediously and laboriously with httle result. Then Kepler 

 breaks out into a characteristic exclamation : " O Deum 

 immortalem quantum mihi temporis et operae perdidit Gebri 

 fiducia ! " A few pages further on, however, we come upon a 

 practical method for calculating the angle of refraction corre- 

 sponding to any angle of incidence, which can be expressed by 

 the formula : 



. fM r 



fju — [fjt, — i) sec r 



