380 



THE POPULAR EDUCATOR. 



Fig. 4. 



LESSONS IN NAVIGATION. II. 



TLANE SAILING TEAVEESE SAILING OR COMPOUND 

 COURSES PARALLEL SAILING. 



III. Plans Sailing. The main problem of Navigation, as 

 has been already stated, is to find the right position of the 

 ship upon the chart, after a run of known or rather estimated 

 length and direction ; in other words, to track her course 

 upon the chart from her last known position. The chief diffi- 

 culty arises from the fact that, while charts are flat, the globe 

 is round, and therefore is never represented on paper without 

 distortion, as well as from the fact that meridiars of longitude 

 converge towards each other at an angle which varies with the 

 latitude, making it extremely difficult 

 to fix the longitude of the ship from 

 mere knowledge of her course. 



A ship may sail (first) on a meridian 

 i.e., due north or south; (secondly) 

 on a parallel of latitude i.e.> due 

 east or west ; and (thirdly) in a course 

 compounded of the other two. The 

 last is naturally the commonest case. 

 If a ship sail due north or south from 

 a given place, for a known number 

 of geographical miles, it is clear she 

 can fix her position by simply taking 

 a spot, on the same meridian, as 

 many minutes north or south of 

 her starting-point as she has sailed 

 miles. Again, should she sail due 



east or west (this is called parallel sailing), it is clear that, by 

 a not very difficult calculation, to be presently explained, she 

 can fix her position east or west along the parallel on which her 

 whole course has lain from the starting-point, the relation 

 between miles and minutes of longitude on any given parallel 

 of latitude being easily arrived at. But now suppose the ship 

 to sail in a line cutting both meridians and parallels at an 

 angle say 40 east of north, therefore crossing the meridians 

 at that angle. Though she cuts them all at the same angle, 

 her course will not be a straight line (for the meridians them- 

 selves are not parallel), but a curve of some complexity, as A B 

 in Fig. 4. This line that is, a ship's track cutting successive 

 meridians at the same angle (other than a right angle) is 

 called a rhumb line, and is, in fact, a spiral, which will wind an 

 infinite number of times round the pole before reaching it. 

 The line A c is said to represent the difference of latitude made 

 while running the distance A B. The angle A is the course 

 i.e., the angle which the ship's track makes with the meridian. 



Here, then, we have some of our most important expressions, 

 as course and distance sailed, which are usually known quanti- 

 ties, and difference of latitude between the place left and the 

 place arrived at usually one of the quantities sought em- 

 bodied in a right-angled triangle, which has, however, the 

 defect of not being a plane triangle. A little consideration 

 will show how they can be embodied instead in that useful and 

 easily-computed figure, a right-angled plane triangle. 



Let us suppose A B divided into an infinite number of parts, 

 A b, b V, V B, by equidistant meridians ; these parts will not sen- 

 sibly differ from straight lines. Then, since the angles at A, b, b' 

 are equal, and c, cf, c" are all right angles, the triangles A c b, 

 b c' b', b' c" B are similar plane triangles (their infinite smallness 

 being assumed). Whence A 6 : A c : -. b b' : b d -. -. b' B : ?/ c". 



.'. (Euc. V. 5) A 6 : A C : : A B : AC + b c' + &' c". 

 But A c + b c + b' c" = A c = diff. of latitude (between A and B). 

 .'. A b : A C : : A B : diff. lat. 

 Let us now draw the ^"ane triangle ABC 

 (Fig. 5), making A = A in Fig. 4, c a right 

 angle, and A B = A B in Fig. 4 ; then A b : 

 A c : : A B (Fig. 5) : A c (Fig. '5). Com- 

 paring these two proportions, we see that 

 if A (Fig. 5) represent course (angle of 

 track with meridian), and A B represent 

 distance sailed on that course, then 

 A c = difference of latitude. 



The infinitely . small but infinitely nu- 

 merous portions of parallels of latitude c b, d b', c" B, added 

 together, make up what is called the ship's departure her 



Departure. 



rig. s. 



"departure," that is, east or west from, the meridian on which 

 she started. The departure, it will be noticed, is neither C B nor 

 A d, but something between them, and is, in fact, an imaginary 

 quantity, and not a direct measure of the ship's change of longi- 

 tude, though a potent means by which it is found indirectly. 

 It is best described as the sum of all the minute departures made 

 by the ship in passing from one meridian to another, the 

 meridians being supposed infinitely close together. And as we 

 proved A c = diff. lat., we can prove that c b + c' b' + c" B = c B 

 in Fig. 5. Thus our plane right-angled triangle now includes, 

 represented by the sides of an angle, distance, diff. lat., departure, 

 and course, and any two of these being given we can find the rest. 

 The above is called the " Theory of Plane Sailing," and the 

 mariner is enabled by it, his course and distance from a given 

 point being known, to calculate his change of latitude and his 

 "departure" upon the latter of which he bases a further 

 inquiry as to his longitude. The following are the practical 

 rules (see Trigonometry) by which the unknown quantities are 

 found from the known : 



Dis.=diff. lat. X sec. course (4) 

 Tan. course = 5T5 .^ (5) 



Dep. = dist. X sin. course... (1) 

 Diff. lat. =dist. X cos. course (2) 

 Dist. = dep. X cosec. course (3) " <jiff. j a t_ 



The correctness of the above formulas appears on simple inspec- 

 tion of Fig. 5, bearing in mind that course = angle A. 



In books of mathematical tables will be found a " table of 

 difference of latitude and departure " for all distances sailed 

 within reasonable limits, for all courses from 1 to 89 ; also 

 for all the points and quarter-points from to 7f , in case the 

 mariner should prefer to estimate his course as so many points,, 

 instead of degrees, from the meridian. Look down the column 

 headed by the given course, and opposite to the given distance 

 will be found both diff. lat. and departure. Thus, if a ship 

 sails 60 (nautical) miles from lat. 30 N., on a course inclined 

 40 from the meridian, she will have advanced 46 minutes of 

 latitude, and will have made a departure of 38'6 miles. She 

 will thus have reached either to 30 46' N. lat., or 29 14' N., 

 according as she has sailed towards the north or the south ; 

 and her departure is reckoned as east or west, according to the 

 direction taken. The use of the table of course saves the labour 

 of calculating ; but it is well to remember the f ormulaa by which 

 it was constructed. 



IV. Traverse Sailing, or Compound Courses. It has already 

 been stated that a ship will often make several courses of vary- 

 ing length and direction during the day, whan the problem for 

 the mariner becomes, How far has the ship come, in a direct 

 line, from yesterday's position, and what angle does that line 

 make with the meridian ? Suppose the day's work (from noon 

 to noon) represented on the log-board as below : 



