June 2, 1898] 



NATURE 



105 



/i/ ^« ° 7/^ 



which the rhumb line (or loxodromic curve which on the 

 sphere is a spiral approaching nearer to one of the poles 

 at every convolution) cuts every meridian that it crosses 

 at the same angle. Mercator does not seem to have 

 understood the principles on which his charts should be 

 constructed, for he left no description of them, nor were 

 they even accurate, and it was left to an Englishman, 

 Wright, to demonstrate that, as in making the meridians 

 parallel the meridian distances were being increased in 

 proportion to the secant of the latitude the lengths of the 

 degrees of latitude must be increased in the same ratio. 

 This is obvious from the fundamental formula of parallel 

 sailing. On this principle Wright proceeded to construct 

 a table of meridional parts, by means of which we get a 

 meridional difference of latitude which bears the same 

 proportion to the difference of longitude as the true 

 difference of latitude bears to the departure. We have 

 then two similar triangles with the course as a common 

 angle, either of which can be resolved by the rules of plain 

 trigonometry. Now, whilst this method is in all cases 

 theoretically accurate, in finding the difference of longitude 

 in a low latitude corresponding to the distance run and 

 the difference of latitude, if the course 

 be near east or west, its tangent being 

 large will rapidly multiply any error in 

 the meridional difference of latitude 

 (due to neglecting decimals, for the 

 parts are generally given to the 

 nearest whole number), and thus pro- 

 duce a large error in the difference 

 of longitude, whereas the departure 

 multiplied by the secant of the middle 

 latitude would not be open to the same 

 objection ; besides, the course would 

 approximate to a parallel, and so 

 small would be the error from 

 treating the middle latitude as such, 

 that the result would be practically 

 if not scientifically accurate For 

 reasons of a similar nature the 

 course and distance run from day 

 to day, if sailing near a parallel, are 

 better found by middle latitude sail- 

 ing, especially in low latitudes, unless 

 the ship crosses the equator, when 

 the portions on each side of it ought 

 to be obtained separately if this method 

 be used. In all cases where the fore- 

 going conditions do not obtain, 

 recourse should be had to Mer- 

 cator's sailing. In a doubtful case the course and 

 distance might be calculated by both methods, and the 

 results compared. For the purposes of steering, the 

 course is only required to the nearest degree and, as a 

 general rule, for computing the distance to the nearest 

 minute. If, however, the course be near east or west, its 

 secant, being large and changing rapidly, is required to 

 the nearest second to obtain the distance accurately. As 

 the seconds are of no use, except to get the secant 

 exactly, they may be done without by observing that the 

 required secant will exceed its tangent, which is in the 

 computation already by the same amount as the nearest 

 tangent in the tables is exceeded by its secant. 



Except the ship is being navigated along the equator 

 or a meridian, none of the foregoing methods give the 

 shortest distance between two points on the globe, nor 

 the courses to steer to attain it. This can only be 

 accomplished by great circle sailing. A knowledge of 

 great circle sailing is much older than is generally sup- 

 posed, though it is only of late years that it, or a 

 modification of it, has been at all generally practised, 

 and even now it is not as much used as it ought to be. 

 The earliest record that I have been able to find of the 

 application to navigational purposes of a principle that 



NO. 1492, VOL. 58] 



must have been long known to mathematicians and 

 astronomers, is in a work on navigation by Captain 

 Samuell Sturmey, published in the middle of the seven- 

 teenth century, in which the gnomic chart is described. 

 The gnomic chart is to great circle sailing what 

 Mercator's chart is to the sailing of that name, and this 

 old navigator gives rules how to convert a log slate into 

 a chart on this projection so that the great circle courses 

 can be read off with a protractor. Whilst great circle 

 sailing can never have been forgotten, even if little 

 practised, the gnomic chart seems to have dropped out 

 of men's memories, for two centuries later it was redis- 

 covered simultaneously by Mr. Godfray, of Cambridge, 

 and Captain Bergen. Within the next few years Knorr, 

 Hillarett, Jensen and Herrle all brought out gnomic 

 charts more or less like Godfray's, of which Herrle's 

 seems the best and most convenient for finding the 

 distance as well as the course. Before, however, the 

 gnomic charts were reinvented, Towson introduced a 

 diagram and set of tables for facilitating great circle 

 sailing. By means of the diagram the vertex of the 

 required great circle is found, and then taking the 



. — Showing the composite track from the Cape of Good Hope to Cape Otw.-iy, with 45° as 

 maximum latitude. The composite track is from .\ to V, V to V, and thence to B. 

 Cos AV = >in lat. A, cosec lat. V. I Cos APV = tan lat. A cot lat. V. 



Cos BV = sin lat. B, cosec lat. V. I Cos BPV = tan lat. B cot lat. V 



APB - (APV + BPV) = VPV which x cos 45° = VV. 

 Sin A = sec lat. .\, cos lat. V and sin B = sec lat. B, cos lat. V. 



successive courses and distances out of the tables is a 

 mere matter of inspection. A few years later Deichman 

 endeavoured to improve on Towson's diagram, and 

 Brevoort brought out a somewhat similar diagram to 

 accomplish the same object. Lecky has pointed out 

 that great circle courses, within certain limits, may be 

 taken out by inspection from Burdwood's (and other) 

 azimuth tables, and almost without limit from his own 

 A, B, and C tables. Lecky, too, gives short rules for 

 computing the first course and distance. With all these 

 methods open to the navigator, great circle sailing ought 

 to come to the front. One of the drawbacks to it is that 

 in the parts of the world where it would save most dis- 

 tance, it leads through inclement regions and amongst 

 ice, and not the least of Towson's merits was showing 

 how to combine it with parallel sailing so that, with- 

 out going to a higher latitude than was desired, the 

 shortest track could be followed. He finds either by 

 calculation or his tables the two great circles passmg 

 through the points of departure and destination whose 

 vertices just touch the limiting parallel. The vessel 

 is navigated along the first arc till the parallel is 

 reached, along which she is kept till the vertex of the 

 second circle is attained when she takes the great 



