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XXXIII. On Bernoulli's Solution of the Problem of Shortest 

 Twilight. ByT. S. Davies, Esq. F.R.SS. L. $ E.,F.R.A.S., 



npHE problem of the shortest twilight was first solved, and 

 A very elegantly too, by Nunez, in a little quarto tract of 

 142 pages, printed at Lisbon in 1542; and it has engaged the 

 attention of many distinguished authors since that time. John 

 and James Bernoulli as well as D'Alembert and L'Hopital 

 discussed the problem : but the researches of the two illus- 

 trious brothers are nowhere, that I know of, to be found in print. 

 The result is, indeed, stated in one of John Bernoulli's letters 

 to the Journ. des Savans: but he says nothing that can lead us 

 to discover the particular details of his solution. He how- 

 ever certainly bestowed a good deal of trouble upon it; as he 

 says, "J'ai resolu le probleme de trouver geometriquement le 

 jour du plus petit crepuscule, ce qui a occupe monfrere, Profes- 

 scur de Mathematique a Bale, et moi, depuis plus de cinq 

 ans, sans en pouvoir venir a bo?it." He tells us, too, that he 

 effected his solution by means of the differential calculus. 

 D'Alembert, in the Encyclopedie Methodique (Art. Crepus- 

 cule), undertook to give a complete solution on the supposed 

 plan of Bernoulli : but he obtained also an equation of the 

 fourth degree, the roots of which embarrassed him considerably. 

 Two of the roots, it is true,, he dismissed from his formula 

 pretty readily; but not, as it appears to me, upon grounds alto- 

 gether satisfactory. He does not attempt to show either how 

 they came there or what was their actual signification ; nor 

 does he even attempt to do so with the remaining root of the 

 remaining pair, though he employs more than three folio co- 

 lumns in showing (by special instead of general reasonings) 

 that it does not belong to the minimum, and he even says, 

 and attempts to prove, that it does not refer to the maximum 

 twilight problem. 



The intricacy of D' Alembert's solution seems to have de- 

 terred successive geometers from attempting to develop in a 

 more advantageous form the probable method of Bernoulli, 

 these being the only solutions (L'HopitaPs perhaps excepted, of 

 which I cannot speak, not having his book within reach,) 

 which I have found, in which the modern method of maxima 

 and minima is employed. Every other solution with which 

 I am acquainted proceeds upon certain geometrical considera- 

 tions derived from the nature of the figure itself rather than 

 from the, equations of the problem: and it must be admitted 

 that in point of facility these have very considerably the ad- 



* Communicated by the Author. 

 2 A 2 



