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U. S. NAVAL INSTITUTE, ANNAPOLIS, MD.

GREAT CIRCLE SAILING-A FEW “WRINKLES”

TO SAVE TIME
By COMMANDER H. G. S. WALLACE, U. S. Navy

Now that so many ships are making trips across the Atlantic, a few notes on great circle sailing may be of timely interest.

There are in common use two methods of getting the great circle distance between two points. These are: (1) By measurement from the great circle chart; (2) by computation from formulæ derived from Napier's rules.

The first of these methods, measurement from the great circle chart, requires (usually) a rather long preliminary study of the method (or methods, as two are given in the printed instructions on the chart) before one is sufficiently familiar to start on the case in hand. In addition to the inconvenience and delay, the result attained is only approximate, and has a probable error in the neighborhood of 10 to 20 miles.

The second method, computation from Napier's rules, requires the navigator to burden his memory with a rarely used formula, or else to look it up in Bowditch or elsewhere before using; and after getting it, he must be familiar with the signs of the various functions in the various quadrants, and carefully observe them, or the result will be liable to error.

It requires no argument to prove that any method which involves little-used formulæ, or which requires a preliminary study, or which fails to give a reasonably correct answer, is faulty in actual service. The chances of mistakes are already too many, and anything which adds unnecessarily to these should be avoided.

Happily there is an easier method, one which is available to the average navigator without any preliminary study, and which gives results that are not only accurate, but quickly attained. It is nothing else than our old friend, the Marcq Saint-Hilaire.

The analogy between the astronomical triangle and the terrestrial triangle is very close.

By comparing the two triangles in the figure, it appears at once how the Marcq Saint-Hilaire method is applicable, as it is simply necessary to substitute (1) the latitude of the destination in place of the declination of the heavenly body, and (2) the difference of longitude in place of the hour-angle. It is not even necessary to reduce the difference of longitude to time, as the haversine tables in Bowditch give the functions for one as readily as for the other. The zenith distance, reduced to minutes of arc, is the desired great circle distance between the two points.

CO-LATITUDE

DECLINATION

DIFF. OF
LONGITUDE

M'

CO-LAT.

DD

CO-LAT

DISTANCE

GREAT CIRCLE DISTANCE

ZENITH

AZIMUTH

INITIAL
COURSE

ASTRONOMICAL TRIANGLE.

P= Pole.
2= Zenith.
M=Heavenly body.

TERRESTRIAL TRIANGLE.
P = Pole.
2= Zenith.
M'= Destination

00 N.

00 W.

Ai 47

IO

Example.---Find the great circle distance between Lat. 45" 40' N. Long. 2° 50' W., and Lat. 41° 00' N. Long: 50° oo' W. L 45° 401 N.

de '2° 501 W. L 41

da 50 L,~L,

4° 407
An 47° 10' log hav 9.20430
L1 45 40 log cos 9.84437
L 41

00 log cos 9.87778
log hav 8.92645.

N. hay .08443
L, L,
4° 40'...

N. hay .00166
Zenith distance 34° 07.5'..... N. hav .08609
Great circle distance 2047.5 miles.

The initial course is readily obtained by solving the same triangle to get the azimuth, using any of the well-known methods. In this case the difference of longitude must be reduced to time, as azimuth tables and diagrams do not have the hour-angle expressed in arc.

The easiest method of actually following a great circle is usually by drawing a straight line between the two points on a great circle chart, picking off points on the line every 5° or 10° apart, and plotting these points on a Mercator's chart, then connecting by a fair curve. If this method is followed, the course at any point may be taken directly from the chart by means of the parallel rulers. This is easier than computing the successive courses, and is equally satisfactory in practice.

The writer accidentally hit on the above method for computing the great circle distance, and wondered whether so easy and obvious a method could have escaped observation. It is not given in Bowditch, but it has since developed that the late edition of Muir (1918) contains the formula on p. 267, and an example is given on p. 270. It is, however, not pointed out that the formula is the same as that of the ordinary Marcq Saint-Hilaire sight, and the close analogy between the two might readily escape observation, even from a person making a fairly careful study of Muir.

It is believed that in any case the method will prove of interest to those who, not having a copy of Muir, have not seen it, or, having seen it, have not happened to notice the analogy.

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