MEKCATOR'S SAILING.] 



NAVIGATION 



1061 



latitude ; and consequently the resultant single course 

 will not, in strictness, be the true one. 



In a day's run. the error will no doubt be in general of 

 but little moment, and not worth taking note of practi- 

 cally, when we consider that the courses steered are not 

 rigidly determinate ; but it is well to apprise the learner 

 of the mathematical shortcomings of our proceedings ; 

 and more especially to forewarn him, that if longitude 

 be deduced from the balance of departures, even for a 

 single day's run. in high latitudes latitudes, for instance, 

 above 52 or 53 the result may be sensibly erroneous ; 

 and in latitudes of from 60 to 70, the error may be such 

 as to endanger the safety of the ship. Unfortunately, 

 sailors are, iu general, so much guided by prejudice, and 

 BO unwilling to adopt innovations, as they think certain 

 improvements to be, that the common practice still is to 

 confide as implicitly in the resultant departure of a set of 

 departures, as in the departure made iu a single course. 

 And it is probably in deference to this erroneous impres- 

 sion, that even the most popular books on navigation are 

 silent on the subject. In such books, the middle latitude 

 distance of the meridians is still taken for the correct 

 departure ; though, as already noticed, the resulting 

 error in longitude may amount to so much as 30 miles. 



The place of a ship in reference to any given meridian, 

 can be determined correctly only by correctly finding her 

 longitude ; and this is done by Mercator's sailing, in a 

 very ingenious manner, as we shall now show. 



INVESTIGATION OF THE THEORY OF MERCATOR'S 

 SAILING. When a ship sails upon an oblique rhumb, it 

 Already been shown that the difference of latitude, 

 the departure, and the distance run, are all truly repre- 

 sented by the sides A C, C B, A B of a plane triangle, the 

 angle A being that of the course (Fig. 20). The departure, 

 C B, is not the representative of Fig. 20. 



any line on the sphere : it is the [ _. 



equivalent of all the minuto 

 departures in the diagram at page 

 1052, united in one continuous 

 line. Let A6c, in the annexed 

 diagram, be one of the elementary 

 trundles figured in the representa- 

 tion just referred to, cb being one 

 of the elementary departures, and 

 Ac the corresponding difference of 

 latitude. Now c b being a small 

 portion of a parallel of latitude, it 

 will be to a similar portion of the 

 equator, or of the meridian, as the 

 cosine of its latitude to radius, as 

 was proved at page 1052; and this similar portion of the 

 equator, or of the meridian, measures the difference of 

 longitude between c and 6. Suppose the distance Aft 

 prolonged to 6', t.ll the departure c' V ia equal to this 

 difference of longitude : then we shall have 



c 6 : e' b' : : cos. lat. of c 6 : 1 (the trig, radius). 



But cb : cfb' : : Ac : Ac 1 (Euclid, Prop. 4, VI.) 



.". Ac : A tf : : Cos. lat. of C 6 : 1 .'. 



.'.Ac'- p^-j . =AcXec. lat. of cb. . .(I) 



COB. lat. of c6 



That is, the proper difference of latitude, A c, must be in- 

 creased to A </ <= A c X sec. lat., in order that the proper 

 departure, e b, may be increased to an amount c' b', equal 

 to the difference of longitude of c and 6 ; in other words, 

 the ship having made the small difference of latitude A c, 

 and the corresponding departure c b, must continue her 

 course till the difference of latitude has increased to 

 Ac X sec. lat. of c, in order that her increased departure, 

 c? V, may give her difference of longitude made in sailing 

 from A to 6. 



Suppose, now, that all the elementary distances are 

 prolonged in this manner ; it is then obvious that the 

 sum of all the corresponding increased departures will 

 necessarily be the whole difference of longitude made by 

 the ship during its course from A to B. Hence, to re- 

 present the difference of longitude between A and B, we 

 n i nit prolong the difference of latitude AC, till the length 

 A C' becomes equal to the sum of all the increased 



elementary differences of latitude ; this done, it follows 

 that the departure C' B', due to this increased difference 

 of latitude, will represent the difference of longitude 

 made in sailing from A to B. 



It is self-evident that, as the departure C B, actually 

 made by the ship, is always less than the difference of 

 longitude made, there must be some more advanced 

 departure C' B', that would be exactly equal to this 

 difference of longitude. The object of the foregoing 

 reasoning, is to discover how the increased difference 

 of latitude, due to this departure, is to be ascertained. 



Now the finding the length of A C' implies the finding 

 of all its elementary parts : suppose we take each of the 

 elementary parts of A C equal to 1', that is, that we 

 regard Ac to be 1 nautical mile ; then Ac' will be equal 

 to 1' X sec. lat of c. And, generally, if i be the latitude 

 of any point in A C, the length of a minute of latitude, 

 terminating in that point, will be increased to 1' X 

 sec. I. 



These enlarged portions of latitude are called MERI- 

 DIONAL PARTS ; so that we have 



Meridional parts of 

 1' = sec. 1' 

 2' = sec. 1' -f sec. 2' 

 3' = sec. 1' + sec. 2' -f sec. 3' 



4' = sec. 1' -f- sec. 2' -f- sec 3' -f- sec. 4' 

 5' = sec. 1' -(- sec. 2' + sec. 3' -j- sec. 4' + 

 &c. <kc. 



sec. 5' 



Consequently the meridional parts, on the proper enlarge- 

 ment of every portion of the meridian, measured from 

 the equator up to any latitude, may be calculated by help 

 of a table of natural secants : thus- 

 Meridional parts of 



Nat. See. Mrr. Part>. 



1' = 1 0000000 = 1 0000000 



2' = 10000000 + 1-0000002 = 2-0000002 

 3' = 20000002 -(- 1-0000004 = 3-OOJOOOG 

 4' = 3-0000006 + 1-0000007 = 4-0010013 

 5 = 40000013 + 1-0000011 = 5-0000024 

 i-c. <tc. &c. 



If, therefore, a ship leave the equator, and sail upon 

 any course till her latitude becomes 1', and we desire to 

 know how much she must increase her latitude in order 

 that her corresponding increased departure may be equal 

 to her advance in longitude, we find, from the above, 

 tliat no further advance in latitude is to be made ; for 

 the difference between the departure corresponding to the 

 1' of latitude already made, and the advance in longitude 

 is, as might be expected, insensible ; or, to speak more 

 rigorously, the difference is too minute to have any 

 numerical value within the limits of seven places of 

 decimals. If her latitude become 2', her corresponding 

 departure falls short of her advance in longitude by 

 a quantity so small that she has only to increase her 

 latitude by -0000002 miles to render her departure exactly 

 equal to her difference of longitude. 



In like manner, when 3' of latitude are made, a further 

 advance in latitude to the extent of -0000006 miles 

 is all that must be made to render her departure the same 

 as her difference of longitude due to the 3' of latitude. 

 And in this way may a table of meridional parts be cal- 

 culated, minute by minute. If we enter such a table 

 with the latitude sailed from, and the latitude arrived at, 

 and subtract the meridional parts for the lower latitude 

 from the meridional parts for the higher, the remainder 

 will be the meridional difference of latitude, or the line 

 AC', in the preceding diagram. If the latitude in be on 

 the contrary side of the equator to the latitude left, then, 

 of course, the sum of the meridional parts for the two 

 latitudes will be the meridional difference of latitude. 



Having thus obtained from the table the meridional 

 difference of latitude (that is, the line A C'), the difference 

 of longitude (that is, the line C'B'J is then deduced by 

 this proportion; namely 



As radius (1) is to the tangent of the course, 

 so is the meridional difference of latitude to tho 

 difference of longitude ; or, if instead of the course, 



