625 



GREAT CIRCLE OR TANGENT SAILING. 



GREAT CIRCLE OR TANGENT SAILING. 



626 



clearness) the line D X would represent the difference of longitude, 30 



and P x the distance, while the spheric angle x p D would be the 



Fig. 3. 



required course ; and being measured on c B, would give 50", or (S. 50 

 W.), and the distance p x would be found as in fg. 2. 



Suppose, further, the next ascertained position of the ship to be in 

 lat. 30" N., and the diff. of long. 30 (as before). The under pole p being 



Fig 4. 



moved to 30 N., would show the problem as in Jig. 4, which would be 

 solved as in the above, giving the course about 60* or S. 60 W. 



The mathematician will have noticed that the useful problem of 

 great circle sailing is solved in the spherograph by means of a triangle, 

 differing from that generally used in its solution by calculation, an 

 example of which it is essential to give. For while by the latter we 

 use the 'two zenith distances with the difference of longitude, in the 

 instrument we simply take the latitude in, and the latitude bound to, 

 and the difference of longitude. Instead of using triangle X z p as in 

 calculation, we use the rational triangle X p D. 



Fig. 5 represents the problem given in fy. 2, as adapted to the 

 following computation : 



Let z P be the co. lat. 40, and z X be the co. lat. 80, and the angle 

 p z x be the diff. of longitude 60. To fiud the distance P X, and x p z, 

 the complement of the first great circle course. 



= 40' 

 = 80 



120 sum 



40' 

 80 



40 diff. 



Angle j? z x = 60' 



30 i contained angle. 



60 \ sum . 20 \ diff. 



As sine J sum 2 sides 60 ' ar. co. 0-062469 



To sine {diff. . . 20 9-534052 



Cot. 1 contained angle 30 10-238561 



A cos J sum 2 sides . 

 To cos J diff. . 

 Cot. I contained angle 



J'sum 72" 55' 

 4 diff. 34 22 



80 



2ti 



9-835082 = tang. J diff. of the other two 



angles, or 34 22'. 

 0-301030 

 9-972986 

 10-238561 



10-512S77=tang. J sum of the other two 

 angles, or 72 55' 



Bum 107 17 =co. course xrz, and 180' 107 17'=tUc course a r x, or 



7 2' 43'. 



Diff. 38 33 



.'. To find the distance P x 



Assinexrz . 107 '17' ar. co. 0-020065 

 To sine z x . 80 9-993551 



So sine pzx . 60 9-937531 



9-951 147 =sine 63 20' 

 60 



3800 



the distance in geo- 

 graphical miles. 



The instrument, however, has this advantage over all other methods, 

 namely, it taken the simple data alone as they occur in practice, with- 

 out there being necessity for the terms hitherto found convenient, 

 such as lat. of vertex, long, from vertex, &., or indeed even the co. 

 latitude*, co. courses, &c., and the tyro is thus only using terms in the 

 spherograph which he llioroughly comprehend*. 



We have now shown that it is really easier to navigate on the great 

 circle than by any other method. The applying of parallels to a com- 

 mon chart in order to obtain what is actually an erroncvtis course, 

 occupies more time than using the spherograph to find the course 

 which leads directly to the place bound to. 



If anything could be more simple, it is to be found in the tables 

 which accompany the instrument, and which are derived from spheric 

 calculation, and have no connection with the diagrams, except that 

 they verify the accuracy of instrumental projection. We give an 

 example as worked in these tables. Taking the question in the above 

 fig. 4, namely, lat. of the ship, or " Lat. in" 30, diff. of long. 30, diff. of 

 lat. 20, we will suppose a block of figures (omitting such as are unne- 

 cessary for our illustration) to be as under : 



Diff. of longitude. 

 5 10 15 20 85 80 35 40 45 50 55 60, &c. 



Ship's Diff. of 



lat. in lat. 



30 5 



10 



15 



20 



IS 



SO 



The intersection of the vertical and horizontal lines at " 60," having 

 reference to the data, shews the great circle course to be S. 60 W., as 

 measured at B (Jig. 4). 



The not easy consideration'of composite sailing [SAILINGS], is hence- 

 forth entirely superseded by the following general rule in determining 

 the course of a voyage, and its importance to maritime commerce 

 cannot be over-rated. A navigator with .the chart before him, and 

 possessing a spherograph, will in future reason thus with himself : 

 " Being anxious to make the shortest possible voyage, I must not let 

 great circle sailing take me into difficulties as to dangers, baffling winds, 

 ice, &c. ; no, I shall for very many reasons like to place my ship there 

 and there, on my route " (at the same time marking dots on his chart 



