IIG Prof. De Morgan on Fernel's Measure of a Degree. 



The reason will now be seen of my keeping cos ^ as a mul- 

 tiplier outside the brackets in the series for cos n 6. 



In the series for the sine, the coefficient of the mi\\ term is 



{n — m).{n — m — \) (?t — 2 ;» + 2) 



1.2.3 (?«— 1) 



A difficulty has been raised with regard to these series, 



that, if ^ + — be substituted for 6, one side of the equation 

 11 



will become cos {n9 + 2 7r}, which is the same as cos7i6, 

 and consequently that side of the equation will not be changed, 



while on the other side the powers of 2 cos l6-\ ) will be 



very different from the powers of 2 cos 9. The difficulty how- 

 ever is only apparent. The sum of the series is the same in 

 either case, whether it involve powers of 2 cos 6, or powers of 



2 cos (^ H ^). This maybe easily verified in simple arith- 

 metical examples. 



It is curious, that if the continued fraction 

 1 



2 cos ^ — 



2 cos ^ — Sec. 



be supposed to be continued infinitely, it is equal to cos 6 

 ± ^ — 1 . sin ^. 



University College, London, 

 December 22, 1841. 



XIX. Additional Note on the History of Fernel's Measure 

 of a Degree. By Professor De Morgan. 

 To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 



TO my last communication on this subject I add the follow- 

 ing, which I cannot help thinking is completely decisive 

 on the subject. You will remember that the assertion made 

 by me is, that Fernel's G8"096 Italian miles are not (as De- 

 lambre and others make them) more than 69 English miles, 

 but really less than 64^ English miles. In my last I left upon 

 any one who would dispute this assertion the onus of proving 

 that Fernel's pace was longer than five English feet: I will 

 now disprove this alternative, 



Fernel himself has a table of his own measures, in which, as 

 was nsual, he makes the geometrical pace to be five feet, and 

 the Italian mile 1000 paces; and he always styles his degree 



