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Tables of Computed Altitude and Azimuth, No. 214. Vol. IV. By Commander Richard H. Knight, U. S. Navy, and Lieutenant Robert E. Jasperson, U. S. Navy. Washington: Hydrographic Office, U. S. Navy Department.
Reviewed by Lieutenant Arthur A. Ageton, U. S. Navy
These navigation tables are a splendid accomplishment. It is difficult to conceive how the solution of the astronomical triangle from an assumed position using L.H.A. and latitude to the nearest degree of arc can be further abbreviated.
Volume IV is the first of six volumes of tables eventually to be published by the Hydrographic Office. The present volume contains latitudes from 30° to 39°, same and contrary names of declination. The scope of this monumental work can be readily appreciated when it is realized that one volume comprises 260 pages of tables in a large format. Volume IV is handsomely bound in a flexible semi-waterproof binding. On the front cover pages are included altitude correction tables for all bodies and an “Arc to Time” conversion table.
The main table consists of tabulated altitude, Δd, Δt, and azimuth for every degree of hour angle and latitude and every half degree of declination. Δd and Δt are the rate of change of altitude to two decimal places for an increase of one minute of arc in declination or hour angle. At the end of each degree of latitude is included an excellent 2-page abstract from H. O. 127 for identification of stars from the latitude in question.
Calculation of the co-ordinates L.H.A., declination and latitude are, at present, an inescapable minimum common to all methods of navigation. Having computed these functions, H. O. 214 is entered once and the tabulated altitude, Δd, and azimuth are picked out. Δd is used in a clever multiplication table on the back cover pages to obtain a correction for altitude for the minutes of declination. Thus with four figures and one addition, the computed altitude from an assumed position is calculated. It is safe to say that this is the shortest and easiest method extant from the assumed position. If the navigator desires, Δt may also be used to enter the “Multiplication Table” and further to correct the altitude so as to plot all of several sights from the same even degree of latitude and the D. R. longitude.
After working several problems, it occurred to me that these tables might possibly be employed to calculate the computed altitude from the D. R. position. This would involve a “3-dimensional” interpolation similar to that required in using the Red Azimuth Tables, only here a much higher degree of accuracy would be necessary.
Near the meridian, when latitude and declination are the same name and nearly equal (or in any case where (L~d) is small), the altitude increases very rapidly and consequently Δd and Δt vary considerably from one tabulated latitude to the next degree of latitude. It is under such conditions that use of the table from the D. R. position would be expected to develop any large inaccuracy.
However, I have worked quite a number of problems in this section of the tables and by interpolation between the two values of Δd and Δt, say for Lat. 30° and 31°, I have obtained no resulting H0 more than 0'.9 different than the altitude computed by the cosine-haversine formula. When it is recalled that cosine-haversine is not particularly accurate at high altitudes (small zenith distances), this difference in altitudes by the two methods is not so large as to be important; in fact, may even indicate the greater accuracy of H. 0. 214.
The tables as now constructed can be used to advantage from the D. R. position. The data for the following example are taken from the problem given on page 244 of Dutton’s Navigation and Nautical Astronomy, 1934.
t 36°-23.1 W ΔL = diff. alt / 60 = 25.3/60
d 14°-57.1 N
L 36°-49.0 N ΔL = 42
Lat. | Alt. | Δd | Δt | ΔL | Z |
37 | 50-52.9 | +67 | -72 |
| 115.5 |
36.8 |
| +67 | -72 | -42 |
|
36 | 51-20.2 | +65 | -74 |
| 114.4 |
-19.1
H0 51-01.1
C (27.1 Δd) +18.2
(23.1 Δt) -16.7
(49.0 ΔL) -20.6
Cor. -19.1
The answer in Dutton computed by the cosine-haversine formula is H0 51°-00'.6, an error of 0'.5.
If ΔL were tabulated in H. O. 214, the figures shown above in bold face type would not be necessary for solution of this problem.
To work this same example from an assumed position, the following figures are required:
t 36° W | Alt. | Δd | Z |
d 14°-57’.1 N | 50-54.9 | +67 | 115.5 |
L 37° N | C +18.2 |
| |
| H0 51-13.1 | Zn | 244.5 |
From a purely mathematical standpoint, is this approximate “three-dimensional” interpolation here advocated entirely accurate? And if not, what maximum error may be expected? Professor John Tyler of the Department of Mathematics, U. S. Naval Academy, has made a study upon which the following observations are based.
The general equation for variation of altitude in the astronomical triangle is
dh = cos Z dL + cos M d(d) — sin Z cos L dt
Assuming a maximum dL, d(d), and dt of 30 minutes and assuming that interpolation between latitudes in H. O. 214 for Z under the most unfavorable condition will give an error in Z of no more than 1°, then
Evaluating the term cos Z dL, the error in dh/dL varies from 0' to 0'.53 in a sine curve.
Evaluating the term cos M d(d), the error in dh/dd varies from 0' to 0'.53 in a sine curve displaced by 90° from the first term and not symmetrical about the zero axis.
Evaluating the term sin Z cos L dt, the error in dh/dt varies from 0' to (—) 0'.53 in a cosine curve.
A summation of these curves algebraically would indicate that the maximum error of dh in this formula based on the above assumptions Would be less than 1' of arc, a conclusion which trial and error in the most vulnerable section of the table has borne out.
However, in 85 per cent of the main table, interpolation between latitudes will give an azimuth correct to the nearest tenth of a degree, which will reduce the error in dh to one-tenth that based on the assumptions previously stated. Therefore, from the mathematical point of view, it may be concluded that it is fundamentally sound to use H.O. 214 for computing Hc from the D. R. position. The maximum error in the most vulnerable sections of the table will probably be less than 1' of arc.
Despite the several short methods available, there are still many navigators who for various reasons prefer to work from their D. R. positions. As can be seen by examination of the examples given above, the method of H. O. 214 from the assumed position is the shortest in existence and probably will prove to be the shortest that can be devised. Even with ΔL not tabulated in the tables, I do not believe there exists a shorter or easier method for solution of the astronomical triangle from the D.R. position. If ΔL be tabulated in the tables, I doubt that any method can be devised by which one can compute H0 from the D.R. position more briefly or more easily than by use of H.O. 214. When it is realized that precomputation of altitude curves for use in navigation in the air is much more readily accomplished from the D.R. position than from an assumed position, it will be appreciated that use of H. O. 214 for solution of problems from the D.R. position may well be of considerable importance.
Many navigators will shy away from a 6-volume navigation table of some 1,600 Pages but they should not do so, for the volumes are logically arranged in numerical sequence from 0° to 60° of latitude with each degree thumb indexed. Since only one volume would be used at a time, the large extent of the table presents no practical difficulty. Aboard ship there is ample storage space and in the air usually no more than two volumes would be required on a flight.
The cost, $2.25 per volume, $13.50 for the set, may prove prohibitive to some navigators and students but the cost is reflected in the careful computation of the elements and their excellent presentation.
Similarity to the methods of Altitude or Position Line Tables by Frederick Ball, M.A. (London, 1907), Altitude-Azimuth Tables by Davis (London, 1921), and H. O. 201 (Washington, 1919) is noted. However, there are important and distinctive innovations. In all these earlier methods, functions are tabulated only for every degree of declination from 0° to 24°, which necessitates an inversion of the normal use of the tables when solving a problem with a declination greater than 24°. In the Davis Tables and in H. O. 201, azimuths are tabulated together with the altitude in the main table, in much the same fashion as in H. O. 214. Inclusion of Δd and Δt in this latest Hydrographic publication has made interpolation by use of the “Multiplication Table” infinitely easier than it was with any of the previous methods.
The practical navigator’s continuous demand for briefer and ever briefer methods of navigation may well be met in the present tables. And if a column of AL be tabulated, I venture to say that here in H. O. 214 will be found the final answer to the problem of celestial navigation, both from the assumed and the dead-reckoning positions.
Elizabethan Seamen. By Douglas Bell. Philadelphia: J. B. Lippincott & Co. 1936. $3.00.
Reviewed by Major John W. Thomason, Jr., U. S. Marine Corps
To persons who use the sea, as well as to those who do their cruising under a reading lamp, there is unfailing fascination in the story of the remarkable captains who served the English Queen afloat. Even before her time an English master mariner, Robert Thorne, set down the words: “There is no land uninhabitable nor sea unnavigable.” She lived to watch those valiant words made good by her servants. There was Captain Chancellor of the Company of Merchant Adventurers, who tried for the Northeast Passage, cast anchor first of all Englishmen in the bleak White Sea, and visited the court of Ivan the Terrible at Moscow, from whence came a very respectable amount of Russian trade. There was long-legged Martin Frobisher, who took the Gabriel, 25 tons, and the Michael, 20 tons, across the Western Ocean and on the northwest track around Labrador and Baffin Land. There were the ships who went south, past Africa and around the Cape of Good Hope, into the oceans the Portuguese claimed. And there were Drake and Hawkins, the greatest of them all, who raided the New World treasure-houses of the Most Catholic King.
Mr. Douglas Bell has collected the high passages of the sea-faring Elizabethans into a single volume, gracefully written. The stories are well told. There is no new material, except for the statement (in the discussion of the Great Armada episode) that Elizabeth’s ships were adequately maintained and well supplied; this is at variance with other accounts, which insist that the Gloriana’s incurable parsimony made it very hard for her captains to keep their magazines in powder and shot and their men provided with beef and beer.
Thumb-Nail Reviews of Interesting Foreign Books
Years of Endurance. By Surgeon Rear Admiral John R. Muir, R.N. London: Philip Allan & Co., Ltd.
Medical and living conditions in peace and war, including the actions of Dogger Bank and Jutland.
War and Trade in the West Indies 1739-1763. By Richard Pares. London: Milford.
An exhaustive and scholarly study of England’s two greatest colonial wars in the West Indies.
Marlborough. His Life and Times. Vol. III. By the Rt. Hon. Winston Churchill. London: Harrap.
The third and next to last volume of Winston Churchill’s epic history of his great ancestor.
The Modern Book of Lighthouses, Lightships, and Lifeboats. By W. H. McCormick. London: A. & C. Black.
Lighthouses from the Pharaohs till today. Accounts of thrilling rescues from shipwreck. Profusely illustrated.
From Sailing Ship to Submarine. By Admiral Scheer. Verlag Quelle & Meyer, Leipzig.
Some new reminiscences of the man who commanded the German High Seas Fleet at its zenith.
Austria-Hungary’s Last War, 1914-1918. 6th Vol. The year 1917.
Official work of the War Archives. The presentation is taken up on the Italian front. The German command is holding itself in readiness for the oncoming heavy attack in the West; Field Marshal von Conrad pushes out on a double offensive from the Tyrol and on the Isonzo; General Cadorna must appreciate their danger and wants to bide his time, remaining deaf to the entreaties of the Allies who urge the offensive without wishing to contribute anything toward it. Not until the middle of May, 1917, does the time seem more opportune and the 10th Isonzo battle results in the serious breach at Gorz and Tolmein. The Central Powers reply with the murderous counterattack up to the Tagliamento, which almost cost the Italian army its life. Further hopes must remain quiet for reasons of strategic economy but the success is big enough and finds real appreciation in this work. Marine-Rundschau, November, 1936.
Mineral Deposits as Factors of World Politics and Military Power. By Ferd. Friedensburg. Stuttgart: Ferd. Enke. 1936. 260 pp.
Raw materials have become the watchword. This study is devoted particularly to mining and minerals of the inhabited world, pointing out what it embraces and how it is divided, giving its military significance and resultant political conflicts in a thoroughgoing manner.—Marine-Rundschau, November, 1936.