398 



THE POPULAR EDUCATOR. 



known, c b can be computed in minutes ( c 1 6' in miles) without 

 difficulty. 



This reasoning only holds good where A c is so small that tho 

 cliff, of latitude between c and the point midway between A and c 

 can be neglected, and " mid. lat." treated as identical with " lat. 

 in." If A c represented diff. lat. resulting from an ordinary day's 

 sailing, we cannot fix c', and therefore c' B', in the same 

 simple manner; we must split AC into minute parts, say 1' 

 each, and calculate the meridional diff. lat. for each; the sum 

 -will be the meridional diff. lat. A c', by which we can compute 

 c' B'. The other sides of the triangle being measured in 

 nautical miles, B' c' will of course be in miles, equal in number, 

 ns already explained, to the minutes of longitude, in the depar- 

 ture. Or we may say that B' c' will work out in equatorial 

 degrees of longitude. 



Tables have been calculated of the meridional latitude in 

 vninutes called meridional parts corresponding to every degree 

 and minute of latitude, from 1' to 89 59', whence the meridional 

 diff. lat. can be found for any given diff. lat. Thus a ship sails 

 from 30 N., on a course and distance which give 10 = 600' as 

 her diff. lat., by which her "lat. in" is found to be 40. Her 

 inerid. diff. lat. is thus found : 



Meridional parts equivalent to 40 (lat. in) = 26227' 

 39 (lat. left) = 1888-4' 



Morid. diff. lat 734'3' 



The side C'B' of the right-angled triangle AB'C', of which 

 angle A and side A c' are known, has now to be computed. The 

 simplest formula is thus arrived at 



Tan.A= C l?-'; 



AC' 



.'. c' B' = A c' X tan. A ; 



or, diff. long, (in minutes) = merid. diff. lat. X tan. course (9) 

 If departure be given instead of course, wo may say 

 Diff. lat. : merid. diff. lat. : : departure : diff. long. 

 The increased latitude in minutes, or the " meridional parts " 

 corresponding to any given latitude, are thus arrived at : For 

 the first minute from tho equator we multiply 1 by tho natural 

 secant of 1' ; result is still 1, the nat. sec. not differing from 

 unity in seven places of decimals. The increased equivalent of 

 the second minute is 1 X nat. sec. of 2' = 1-0000002 ; increased 

 equivalent of 2' == sum of two increased lats., or 2'0000002. 



NAT. SECANT. MEK. PARTS. 



1-000000 = 1-000000; 



1-000000 + 1-000002 = 2-000002; 



2-000002 + 1-000004 = 3'000006 ; 



3-000006 -f 1-000007 = 4'000013; 



and so on, up to 5399', which is only 1' short of the pole. 



Tables of meridional parts are now, however, calculated on a 

 more accurate plan, which we cannot explain here. 



Eef erring to the specimen traverse table given under plane sail- 

 ing, it should be explained, now that the ways of finding longitude 

 by account have been shown, that although the nett departure 

 obtained by addition and subtraction is accurate enough for use 

 in finding the direct course and distance, it is safer to find the 

 diff. longitude separately for each course, and take the balance 

 at the end as the nett diff. long, on the day's work. Thus, if 

 the mid.-latitude rule be followed, do not total the departure 

 columns, but opposite each course note lat. left, lat. in, mid. lat., 

 and diff. long. E. or W., deduced by formula (7). If diff. long, 

 is to be found by Mercator sailing, leave out departure columns 

 altogether : opposite each course, besides distance and diff. lat., 

 put latitude in, meridional parts corresponding thereto, and 

 meridional diff. lat., found by subtracting merid. parts of lat. 

 in from merid. parts of lat. left (noted as lat. in against pre- 

 vious course). Then by (9) deduce diff. long, for each course, 

 which put in columns for E. or W. ; strike the balance at foot. 

 To find merid. diff. lat., the lat. left at commencement of first 

 course must be included in the table (at top) with its equivalent 

 merid. parts. 



The sea-charts used by navigators are always on what is 

 termed Mercator's projection, which is intimately connected 

 with the above theory of sailing. The earth, instead of being 

 a globe, is assumed to be a cylinder, which is unrolled as a flat 

 map. Hence every parallel of latitude is a circle, or rather 

 line, equal to the equator ; and hence degrees of longitude are 

 equal in all latitudes. As 60' of long, in a latitude where they 



really equal only 30 nautical miles are made to occupy the same 

 space as at the equator, where they equal 60 miles, the map is 

 evidently distorted, but the distortion is made symmetrical by 

 the device of increasing each minute of latitude in proportion 

 to the increase given to the minutes of longitude on its parallel. 

 Thus, though countries and distances lying towards the polea 

 are enormously exaggerated in size, they are comparatively 

 symmetrical in shape. The degrees of latitude are increased in 

 accordance with the table of meridional parts. Tho benefit 

 conferred upon mankind by Gerard Mercator, in tho invention 

 of his cylindrical projection of the earth's surface, can scarcely 

 be overrated. To get rid of the curved meridians, substituting 

 parallel straight lines for them, was a device which breathed a 

 new life into Navigation. 



Scientific and satisfactory as is the theory of Mercator sail- 

 ing, it is better to obtain the longitude by the mid.-latitudo 

 method, if the course exceeds, say, GO from the meridian, 

 because diff. long, is obtained by multiplying merid. diff. lat. by 

 tangent of course. If the angle of course exceed 45, then tan. 

 of course exceeds unity, so any error in estimating tho course 

 may lead to a large error in longitude. Thus by either mid. lat. 

 or Mercator sailing, we may consider the theory of plane sailing 

 completed and extended. 



VIII. To take a Departure. Although tho mariner estimates 

 and records his vessel's movements as well as he can, yet ho 

 loses no opportunity of fixing her position by observations of 

 land or sky. The bearing of the last headland, lighthouse, or 

 other prominent object expected to bo seen, is taken from two 

 positions, and noted, together with the course and distance run 

 from the first point of observation to the second. Unless one 

 of the observations is taken, by chance or design, when tho 

 object is exactly on the beam i.e., at right angles to tho course 

 or the course is of such length as to subtend 90 at the 

 object, the fixing the ship's distance from the object at tho last 

 observation will involve the computation of an oblique triangle, 

 the only one we have yet had to do with. 



Suppose a ship sailing down Channel observes Beachy Head 

 bearing N.W. After running 10 (nautical) miles W.S.W., the 

 Head bears N.N.E.JN. How far is the ship from tho Head at 

 the last time of observation ? 



Let c (Fig. 10) represent the Head, A ship's first, B ship's 

 second position. We have now to solve 

 an oblique triangle of which two angles 

 and included side are given, viz., AB 

 = 10 miles ; B A c = 6 points = 67 30' ; 

 and A B c = 4^ points = 50 37^'. This 

 can be done by Case 3, Section XXI., 

 Lesson V. on Trigonometry, in POPULAR 

 EDUCATOR. The distance being found, 

 the ship's position is marked on the 

 chart, and she is said to take her de- 

 parture from the object observed. The 

 position marked can be taken as lat. and long, left, and the day's 

 work reckoned from it, or, as is the usual practice, the bearing 

 of the ship from the object (just the opposite on the compass 

 card to the bearing of the object from the ship), and tho dis- 

 tance, are reckoned as giving the ship's first course, and are 

 entered in the traverse table as a course run. 



IX. Great Circle Sailing. The advantage of sailing on a 

 rhumb line is that the successive meridians are cut at one 

 angle ; in other words, one compass course may be steered 

 throughout the whole voyage. The rhumb line between any 

 two places, though a spiral on the globe, becomes a straight 

 line on the chart. This makes it very easy to lay down the 

 compass course to be steered to reach a given point on tho 

 chart ; but it has been known for centuries that the rhumb line 

 is not the shortest route between two places, though the fact 

 that it is steered on a uniform compass course gives the idea 

 that it must be a straight and therefore direct line. But the 

 fact that it cuts at the same angle successive meridians which 

 are not parallel is sufficient proof that its direction is constantly 

 changing, and that the ship's head never points in the same 

 direction, if properly steered, for two miles together. The real 

 direct course is an arc of the great circle passing through tha 

 two plr.cos (and the centre of the earth), ns will be readily 

 found ; by stretching a thread between them on the terrestrial 

 globe. People who look only at maps often wonder why our 

 ocean steamers should take Cape Eace, in Newfoundland, on 



Fig. 10. 



