of the Problem of Shortest Twilight. 281 



that they might not be supplemental instead of equal. Hence 

 his deduction is not authorized. It is, however, true, as may 



be seen by putting the corresponding values — - and — - + 2 a 



for p t and p tl in the above equations : vide Leybourn's Edition 

 of the Diaries, ut supra, or Delambre's Histoire de V Astro- 

 nomie Ancienne. 



It has been objected to the solution of Bernoulli, by De- 

 lambre, that it does not give the time of duration of the shortest 

 twilight, but the declination of the sun when the twilight is 

 shortest. By referring, however, to Bernoulli's own words, we 

 shall find that he proposed to determine " lejour de plus petit 

 crepuscule," and not to find the duration on that day. As an 

 objection, then, Delambre's amounts to this — that Bernoulli, 

 having proposed to himself one problem, did not substitute 

 the solution of another instead of the one he had proposed ! 

 This is travelling out of the way to find imperfections, most 

 assuredly. It would, indeed, at first sight seem difficult to 

 account for such an insignificant objection being urged at all ; 

 but doubtless it originated in the somewhat exaggerated ex- 

 pressions of admiration bestowed upon Bernoulli's solution by 

 Montucla, and the fact of the historian of mathematics having 

 greatly undervalued the original solution of Nunez. It is a 

 fact, however, that cannot have escaped the notice of every 

 one who has had occasion to consult Delambre's Histories, that 

 there is a feeling not altogether friendly displayed by him 

 towards both Montucla and Bailly ; and that points of com- 

 parative insignificance are often dwelt upon at very incom- 

 mensurable length, where he has discovered, or believes he 

 has discovered, those authors to be in error. The present is 

 one of them. Bernoulli did not, indeed, solve the other pro- 

 blems that were connected with the one he chose to attempt; 

 but having solved that one, all the others might have been 

 followed out without difficulty, — of which Bernoulli, of course, 

 was well aware. Nor is it more difficult in the general case. 



For cos (f,-?,,) = 1-2 sin 9 | (?,-£,)_ 



= 1—2 cot 2 A tan 2 \ p t —pu 

 But taking RN as the arc of a great circle joining the points 

 R, N, we have cos RN = cos p t cos p u + sin p t sin p n cos (£ y — £,,) 



• « t ,/* A> 1— cos RN 



sin i Wu— M = - a ■ 2 — 

 2 v u " 2 sin 2 p 



_ 1— cosftcospfl— sinp, sinp // + 2sinp / slnp,, cot g A tan 2 j p t — p n 



2 sm 9 p 

 Third Scries. Vol. 3. No. 16. Oct. 1833. 2 O 



