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♦ KNOW^LEDGE ♦ 



[July 1, 1887. 



Mercator's chart ; and varioiis approximative constructions 

 have been suggested from time to time. 



To meet the second difficulty, Mr. Towson, of Liverpool, 

 suggested composite saihng, by which a great circle course is 

 taken from the port of departure to touch the highest lati- 

 tude deemed safe, and another great circle course is taken 

 which, touching that highest latitude, passes through the 

 port of arrival ; the journey pursued is along the former 

 great circle course to the limiting latitude-parallel, then 

 along this parallel till the second great circle course is 

 reached, and thence along the latter great circle course to 

 the desired haven. Mr. Towson published also a valuable 

 series of tables for t'lcilitating the calculation of the compo- 

 .site course from port to port. The Eev. George Fisher, 

 chaplain of the Greenwich naval schools, devised a graphic 

 method for approximating to the composite cour.se on a 

 Mercator's chart. 



These methods of calculation and of construction have 

 not come into general use. It has been found impossible to 

 introduce the general use of great circle sailing as hampered 

 by these requirements, especially in the case of sailing vessels, 

 where fresh calculations or constructions, by no means 

 simple, would have to be made whenever a ship had been 

 driven out of her course by stress of weather. 



A chart on which the great circle course between any two 

 points can be at once laid down would obviate these olyec- 

 tions. And at first sight it seems as though Mr. Godfray's 

 proposition met this want ; for, as I have said, the gi'eat 

 circle course on a gnomonic cliart is a .straight line. But 

 ,a gnomonic chart cannot show even a full hemisphere. 

 The point of projection is .at the centre of the sphere, as 

 shown at O, Fig. 1, the projection being made on a tangent 



plane as oPi. Supposing one of the poles to be at P, the 

 centre of the pi'ojection, the points A and B would be pro- 

 jected at a and h rcspectivelj'. The points D and E could 

 not be projected on the plane «.Pi at all. The scale of tlie 

 cliart increases rapidly from P outward, becoming infinite 

 for points 90 degrees from P. The projection then, though 

 it might serve for ports in one hemisphere, is evidently not 

 available generally. Moreover, the gnomonic projection 

 would not indicate the bearing (anywhere on the course), the 

 vertex, or the distance. 



Instead of this practically useless method of constructing 

 charts for great circle sailing, I propose the use of the stereo- 

 graphic projection, whereby (I) the construction for mark- 

 ing in the great circle course between any two points is 

 made exceedingly simple ; (2) the whole course is obtained 

 at once; (3) the " bearing" at each point of the course is 

 shown as plainly as the bearing of the I'humb course on a 

 ISIercator's chart; (4) the " composite course" where wanted 

 Ciin be obtained by a simple construction ; and (5) the dis- 

 tance fi'om point to point can be easily determined. 



I need not here discuss the principles of the stereo- 

 graphic projection at any length. I will simply note those 

 points which make the projection convenient for the pro- 

 posed purpose. 



In the stereographic projection of the s|)hore, the point of 

 projection 0, Fig. 2, is on the surface of the sphere, at the 



extremity of a diameter PCO, through P, the centre of 

 projection. Thus if d P _/" represent the tangent plane 

 through P, the points A and B on the sphere would be pro- 

 jected on d P /"at a and //, where OA and OB produced 

 meet d P /" If D and E are OO degrees from P (as in 



Fig. 1), their projections fall on d P./ at d and c. A point 

 as P, still nearer to 0, will be projected on the tangent 

 plane as aty. 



If P be either pole, the projection of the sphere on the 

 plane d Vf is a very simple matter ; for all the meridians are 

 projected into straight lines through P, and all the latitude- 

 pai'allels into circles around P as centre. The radii of these 

 circles can be obtained by construction, as shown in Fig. 2. 

 But in practice it is far better to use their known lengths 

 as indicated in trigonometrical tables. Thus if PI! is an arc 

 of 60°, we know that the angle POB contains 30° ; so that 

 P /* is equal to PO tan. 30°. Thus, for the parallels correspond- 

 ing to latitudes 8.')°, 80°, 7-5°, and so on, we take from the 

 trigonometrical tables the natui'al tangents of 2J,°, 5°, 7i°, 

 and so on ; and the.se numbei's, with any convenient unit of 

 length, give \is the radii of the circles we are to describe 

 round P. For inst.anco, if we wish the equator to have a 

 radius P c (equal to PO) five inches in length, we di'aw a 

 line five inches long, divide it into ten equal parts, and one 

 of these again into ten parts (or preferalily make a plotting 

 scale for the smaller divisions) ; then regarding one of the 

 tenths of the line, i.fi., one-half an inch, as our imit, we 

 take, for our successive radii, lines having the following 

 lengths : 



For Latitude S."")" 0437 which is the tang, of 1],° 

 „ 80° 0-87.') „ ' .5° 



'■'° i-sn „ 7J,° 



and so on. Then a series of radial lines drawn to divisions 

 5° apart round any one of these circles give the meridians, 

 and complete our projection. The chart should have out- 

 lines of continents, islands, &c., marked in, for convenience, 

 though in reality this is not essential, because the longitudes 

 and latitudes of ports, Ac, are alone really needed lor deter- 

 mining the great circle course, and the course obtained by 

 the simple constructions I shall indicate could always be 

 plotted in on the Mercator's chart, to whose use seamen are 

 more accustomed than to that of any other kind of chart. 



The properties of the stereographic projection, which 

 enable us at once to project a great circle course, and to 

 determine bearings, distances, kc, on a stereographic chart, 

 are the following : 



(«) Every circle on the sphere, great or small, is projected 

 into a circle. 



(6) All angles, bearings, kc, on the sphere are correctly 

 presented in the projection (a property found also in Mer- 

 cator's projection). 



With these properties I combine the following properties 

 of great circles on the sphere : 



(1) Since every diameter of a great circle is a diameter 

 of the sphere, each point on a great circle is antipodal to 

 another point on the same circle ; or, in other words, if a 



