LESSONS IN NAVIGATION. 



I 7 



LESSONS IN NAVIGATION. 111. 



1: LATITUDE SAILING MEUCATOR BAIUXO TO TAKE 

 A DEPARTURE GREAT CIRCLE BAM 



./. Tlio occasions on which a ship 

 wc'l, ami can toko advanta^.- 



moans of finding h> r I-. -iiu.l.- ju.it exi>lun> . cuurmj 



rare, i , mid-latitude Hailing, tlio cliff, long. 



can !> act found l>y piano Mailing. 



nlii-iv.l, in connection \\itli I'i-j. '. that tint 



UM on tin? coin-.-;.! A n was explained to bo an imaginary 



lino or value, greater than tho arc of parallel c n, and less than 



equal to aotnr similar parallel lying between 



till-in. If wo can find a parallel on which tho departure <-;ui lie 



taken as tho true chu.i^e of longitude, wo can ascertain tin; 



latter by the. last ruli , the departure being taken 



.ust or west. Tho difficulty is to say exactly whore 

 such i>:in;i!el ia to coino, but tho general practice ia to put 

 between " lat. left" and "hit. in," a practice 

 never strictly correct, and sometimes glaringly wrong. In low 

 latitudes, where meridians incline to each other but slightly, 

 and in all cases whore tho difference of latitude is slight in 

 comparison with the departure made, the rule may be trusted 

 for practical purposes ; but in high latitude, whore tho meri- 

 dians converge rapidly, or wherever the diff. lat. is considerable, 

 it cannot bo relied on. 



Suppose a ship, steering southerly, makes a difference of lati- 

 tude of 4, from 22 to 18 P N., and a departure from long. 

 30 W. of 160 miles west ; what is her difference of longitude 

 according to tho mid.-latitude rale ? 



Half-way between 22 and 18" N. is 20 N. ; 160 miles on an 

 arc of a parallel in 20 N. hit. is therefore tho measure of the 

 difference of longitude. In other words, tho ship's oblique 

 course a curved lino, the exact position of whose finial point 

 is unknown to us has been resolved into two straight courses, 

 one pointing duo south, the other due west. Tho length of both 

 is known in miles, but this gives no direct clue to the measure 

 of longitude contained in that pointing west, unless we can tell 

 in what latitude it is to be measured. Knowing this under 

 the rule, we apply formula (6) 



. 16 ; o= 16 = 17-3' =2 50-3' W. 

 cos. 20 '9397 



Diff. long. = _** 

 cos. lat. 



Present position of ship is therefore 18 N. lat., 32 50'3' W. 

 long. 



Formula (6) may bo recast for middle-latitude sailing, as 

 follows, as tho last example shows : 



Diff. long. = de P arture .. (?) 



cos. mid. lat. 



These relations, like those in plane sailing, are involved in 

 the construction of a right-angled tri- 

 angle, and the two can be united in a 

 single figure, which shows at a glance 

 tho relations subsisting between course, 

 distance, diff. lat., and diff. long. tho 

 four things which we alone care to know 

 and the intermediate expressions de- 

 parture and mid. lat., which serve only 

 to link the others together. Seo Fig. 8, 

 where A B is tho distance representing 

 tho rhumb lino in Fig. 4, and where in 

 fact the triangle A B c is a reproduction 

 of Fig. 5. Hero all the lines (except c D, 

 which represents nothing) are measured 

 in a like denomination, viz., minutes; for 

 the nautical miles, in which c B and A B 

 are measured, are identical with minutes 

 of latitude, and D B of course is measured 

 in minutes of longitude. 



It has been said that the particular latitude in which the 

 departure just equals tho difference of longitude duo to tho 

 ship's oblique course, is not quite truly found by tho mid-latitude 

 rulo. There is, however, a table, called Workman's Table of 

 Corrections to be added to Mid. Latitude (not commonly included 

 in books of tables), by which tho true latitude can be found. 

 In the column headed by the number of degrees nearest to the 

 diff. lat. found, and opposite to the degree of latitude given as 

 "mid. lat." by tho rule, will bo found tho number of minutes 



Fig. 8. 



to be added to the latter to give the true hflladn in wbieh 

 departnre will represent diff. long. In the ease just worked ftrt t 

 where diff. lat. = 4", snd mid. lat. (by rale) - 80* .N 

 by Workman's Table that 9 JMmH be fH+d, and t*fir Uif Tim- 

 lation should have stood 



diff. long. = Iff 



eosTS 



The difference in the result would have been *" t f"*f t niit. 

 But suppose tho ship had made as much as 10" diff lat. from 

 tho high latitude 65* N., then mid. lat., by rale, would have 

 boon taken as 60 N., whereas by the table it should b* 

 60 10' N. With a departure of, say, 600 miles, this inaocu- 

 racy would cause dangerous error in the longitude, as may b* 

 seen by working (7) with tho corrected and nneorrected value* 

 of mid. hit. Nevertheless, by using Workman's Table we may 

 safely find tho longitude by " mid-latitude tailing," which wo 

 may regard as an appendix by which the theory o/j/Iaiic tailiny 

 is rendered complete. 



VII. Mercator Sailing. Besides mid.-latitade safling, there U 

 another plan by which tho difference of longitade may be de- 

 duced from tho departure, and as it is scientifically acourata 

 if correct data bo given, it is usually to be preferred. This is 

 Mercator sailing, so nai.icd after a Flemish chart-maker, Gerard 

 Mercator* (1512-1594), whose charts were based upon the prin- 

 ciple now to be explained. 



As a nautical milo everywhere, except at the equator, mooed* 

 a minute of longitude, it follows that the number of rafl+* of 

 departure is always less than tho number of minutes of longitade 

 traversed. Henco it is clear that some greater departure (ex- 

 pressed in miles) than that which properly belongs to the course' 

 and distance traversed will exactly express the real departnre 

 in minutes of longitude in other words, will give us the differ- 

 ence of longitade. All wo have to do, then, is to exaggerate 

 the diff. of latitude in. some known proportion, and take toe 

 corresponding exaggerated departnre, in miles, as the diff. of 

 longitude in minutes. Of course the exaggerated diff. lat. is 

 merely a means to an end, and the ship's position, as to latitude, 

 is not to be fixed by it, but by the true diff. lat. found by plane 

 sailing 1 . Wo will now prove tho rule by which the requisite 

 addition to diff. lat. is found, premising that, owing to the in- 

 creasing convergence of meridians towards the poles, the pro- 

 portionate addition to be made increases with increased distance* 

 from the equator. 



In Fig. 9, let A b c be one of tho infinitely small triangles in 

 Fig. 4. The diff. lat. AC being in- 

 0' finitely short, c, the " latitude in " 

 does not differ sensibly from mid. 

 latitude. Hence we may consider c 6. 

 the departure, as the parallel from 

 which diff. long, may bo deduced. (For 

 convenience we shall speak of these 

 " infinitely small " lines as containing 

 miles and minutes ; millionth* of milea 

 and minutes would do as well, of 

 course.) Now suppose AC" exag- 

 gerated" to tf, so that the increased 

 departure c' b' shall be the arc of the 

 equator corresponding to c b, or in 

 other words, shall contain as many 

 miles as c b contains minutes, c V ia 

 thus tho diff. long. made. Therefore, by formula (6) or (7), c 6' 



cb 



= . 



cos. lat. of c 6 



Expressing this as a proportion, c* b' -. 1 : : r b : cos. lat. of co 

 and transposing, cb . c 1 V -.-. cos. lat. of c b : 1. 



But C b : <f b' : : A C : A C* ; 

 /. A C : A C* : : COS. hit. of C 6 : 1 ; 



.-. A d = - A - - AC x see, lat. of c 6 (8 



cos. utt. of c o 



A c' is called the " meridional difference of latitude," as dis- 

 tinguished from A e, the diff. lat., and it is obtained, as wo 

 have just seen, by multiplying the diff. lat. by the secant of the 

 latitude of cb (the "latitude in"). The position of c' being 



Properly Qttrard KnuTmtn, of which name Mercator is the Latin 



tramialitm. 



Pig. 9. 



