Solution to a Case in Sailing. 79 



Art. X.— Solution to a Case in Sailing ; by W. Chauvenet, 

 A. M., Professor of Mathematics in the U. S. Naval School, 

 Philadelphia. 



In a late treatise upon navigation, the following problem is 

 solved by middle-latitude sailing, and the solution is accompa- 

 nied with a note "declaring that this case "cannot be solved by 

 Mercator's sailing;"* and by referring to the various accessible 

 works upon navigation I cannot find that any rigid solution has 

 ever been printed, although it certainly cannot be regarded as 

 difficult. The true solution may have been overlooked because 

 it cannot be deduced by comparing the similar triangles em- 

 ployed in all popular works to illustrate Mercator's sailing ; or it 

 may have been neglected as unimportant because the case can 

 seldom occur in practice. The problem however is not without 

 interest, and a solution of it, theoretically accurate, though un- 

 necessary to the advanced mathematician, may be acceptable 

 to the student of navigation. 



Problem. — "A ship sails in the N. W. quarter 248 miles (d) 

 till her departure is 135 miles (p), and her difference of longitude 

 310 miles (D). Required her course (C), the latitude left and 

 the latitude come to ?'" 



Solution. — By putting 1= proper difference of latitude, P = 

 meridional difference of latitude, we have immediately from the 

 data 



l=</d 2 -p s , sin. G==S P=D cotang. C, (1). 



Now to find the two latitudes, represent them by L-f \l and 

 L— ^Z, and let the meridional parts for these latitudes be A and 

 A'. Then since an arc of Mercator's meridian from the equator 

 to any latitude (the radius of the earth being 1) is equal to the 

 Naperian logarithm of the cotangent of half the co-latitude, we 

 have 



A=Nap. log. cotang. &[90°-(L+£Z)] 



A'=Nap. log. cotang. £[90° -(L - %l)] 



which reduced to nautical miles by multiplying by the radius 



360x60 _ 

 R= — -kz — =3437.7, and expressed in common logarithms by 



* See Riddle's Navigation, (fourth edition,) pp. 177, 178. 



