KIDDLE LATITUDE SAIUXQ.] 



NAVIGATION. 



1059 



2. To find tlie middle latitude. 

 6,0) 85,7 



14 17' S. 

 61 18' N. 



diff. lat 

 lat. left 



37 1' N. = latitude in 



;5i c 



37 



= 88 19 7 = Sum of the latitudes 



Qoo -I h/ 



- = 44 9' = middle latitude. 



3. To find the difference of longitude (Proportion 3). 

 As cos. mid. lat, 44 9i' Arith. Comp. -1442 



tan. course 

 : : diff. lat. 



33 8' 

 857-5 



diff. long. = 780-1 

 6,0)78,0 



9-8147 

 . 2-9332 



, 2-8921 



13 

 9 



50' 



W. diff. long 

 W. long, left 



22 50 1 W. long, in 



Hence the place of the ship is lat 37 1' N., long. 22 

 50' W. We shall now work the last proportion by using 

 the middle latitude as corrected by Workman's table on 

 a following page. 



Under the diff. lat. 14, at the top of the table, and in 

 a line with 44", the middle latitude, we find the correc- 

 tion 27' ; which, added to 47 9J, gives 44 36J' for the 

 corrected middle latitude. 

 As cos. corrected mid. lat, 44 36 J' Arith. Comp. -1476 



: tan. course 33 8' . . 98147 

 :: diff. lat = 857 '5 . . 2-9332 



: diff. long. = 786-2 . . 2'8955 



.'. diff. long. = 13 6'.'. 9*50'+ 136'=2256-W.long. in. 



Hence the error in longitude, by taking the unconnected 

 middle latitude, is 6 miles. 



The construction of the table here made use of, cannot 

 be explained till we come to Mercator's Sailing. But 

 the necessity for some such table may be readily made 

 apparent ; for since, as shown before, 



diffi long. = departure X sec. mid. lat, 



it follows, that if there be any error in our estimation of 

 the departure that is, in our considering it to be exactly 

 equivalent to the middle latitude distance between the 

 meridians there will be a still greater error in the result- 

 ing difference of longitude, because a secant is always 

 greater than unity, when the angle to which it relates is 

 of any value at all ; and the greater the angle the greater 

 the error. In high latitudes, therefore, where the middle 

 latitude is considerable, the error in longitude, if left 

 uncorrected, may be seriously wide of the truth. For 

 example, when the difference of latitude is 20, and the 

 middle latitude 72, the error in longitude would amount 



i to nearly half a degree ; that is, to nearly 30 miles. Mr. 

 Workman's table shifts the middle latitude parallel a 

 little higher up, as the mid. latitude parallel a little ex- 

 ci'cds the departure in length ; as will be shown pre- 

 sently. 



1. 2. A ship from latitude 49' 57' N., and longitude 

 6 11' W., sails, between the south and west till she arrives 

 in latitude 38 27' N., and finds that she has made 440 

 miles of departure : required the course steered, the dis- 

 tance run, and the longitude in. 



1. To find the diff lat. and mid. lat. 



Latitude left, 40" 57' X. 

 Latitude in, 38 27' N. 



Sum 88 24' - 

 Diff. 11 SO 7 



2 = 44 12' = mid. lat. 

 = G'JO miles = diff. lat 



As diff. lat = 

 : departure 

 : : radius . 



2. To find the course. 

 By logarithms. 



690 



: 440 



2-8388 

 2-6435 

 10 



: tan. course 32 32' 

 Hence the course is 32 32f. 



Without logarithms. 



9-8047 



tan. course dep. -7- diff. lat. = 



69)44 (-6377 = tan. 32 32' 

 414 



26 



207 



53 

 483 



' 47 



44 

 69 



3. To find the distance. 

 By logarithms. 



As sin. course, 32 32' 

 : radius . 

 : : departure = 440 



9-730G 

 ]0 

 2-6435 



: distance = 818-3 . . . 2-9129 

 Hence the distance is 818 miles. 



Without logarithms, 

 dist. = dep. -=-sin. course 

 sin. 32 32' = 5,3,7,7,9) 440 (818 -2 

 43023 



977 

 533 



439 

 430 



9 



4. To find the difference of Longitude (Proportion 1). 

 As cos. correct mid. lat. 44 28' 9 -8535 

 : radius ..... 10 

 :: departure = 440 . . . 2-0435 



-. diff. long. = 616-6 



Longitude left 

 Diff. long. 617 miles, 



2-7900 



6 11' W. 

 10 17' W. 



Longitude in . . . .15 28' W. 

 If the mid. lat. had not been corrected, the long, would 

 have been about 3' in error. 



3. A ship from latitude 51 18' N., longitude 9 50' W., 

 sails S. 33 19' W. until her departure is 564 miles : 

 required the latitude and longitude in, and the distance 

 sailed. 



1. To find the difference of latitude. 



By logarithms. 



As tan. course 33 19' . . 9-8178 

 : radius ..... 10 

 :: departure = 564 . . 2-7513 



: diff. lat. =858 . 



Hence the diff. lat. is 858 miles. 



2-D335 



Without logarithms, 

 diff. lat. = dep. -i- tan. course, 

 tan. 33 19' = -065,7)564 (853 

 5256 



384 

 329 



55 

 62 



