One object of the navigation course at the Postgraduate School is to investigate devices for aids to, and methods of, navigation. After trying various methods of navigation, officers of my class were of the opinion that:
(1) Ogura’s altitude tables as used by Weems were best for computing the altitude.
(2) The Dreisonstok azimuth tables in “H.O. 208” were best for computing azimuth.
(3) The Rude Star Finder was the best device or method for star identification.
In order to compare the numerous methods, tables, and solutions, the instructor gave us a night’s run with three fixes of three stars each. All available methods of navigation were used by the several members of the class employing the same data. It was found that all fixes obtained were practically identical. The conclusion reached was that all the solutions were equal as far as accuracy was concerned. The next consideration was, whether it was better to use an assumed position of an even degree of latitude and longitude such that the local hour angle would be an even degree, or to work from the deadreckoning or navigator’s estimated position. By using the navigator’s estimated position, the computation is longer and the intercepts short. By using the assumed position, the computation is shorter but the intercepts are long. Considerable discussion arose over the relative advantage of the two different methods. Many preferred a system resulting in short intercepts, but no satisfactory “short” method existed for this computation. So Lieutenant A. A. Ageton developed the method now known as “H.0.211.” An analysis by the class of the several publications and methods then in use indicated that the existing systems could be improved. The composite criticism of the several publications and methods may be epitomized as follows:
Bowditch: Satisfactory as a text and reference book; obsolete as a practical navigation book.
“H.0.203-204”: Excellent reference tables; too large and cumbersome for facile use.
Aquino: Considered by many the best system yet devised. However, many American navigators are confused by the multiplicity of headings for the single set of tables and hesitate to use this method for fear of making mistakes.
“H.O.208”: Best for azimuth; altitude tables involve two cases; azimuth tables involve four cases.
Weems’s Line of Position Book: Best for computation of altitude. Rust’s azimuth diagram, while approved by some aviators, is not preferred by mariners.
The author undertook the compilation of a set of tables that would combine the advantages of the Ogura altitude tables and the Dreisonstok tables for azimuth. In general, this set of tables permits the use of Dreisonstok’s azimuth tables with Ogura’s altitude tables, without interpolation, and with but a single simple rule for use with the azimuth table. The use of this set of tables results in a shorter process than does that of “H.O.211” and avoids the troublesome cases found in “H.O.208.” To illustrate, when using “H.O.208,” the following precepts determine the application of the several rules:
Dec. and lat. same name
t <90°-b+d <90°: B=b+d Z=Z'+Z"
t<90o b+d>90°: B = 180° —(b+d) Z=Z' -Z"
t>9 0°, B =d—b Z=Z"-Z'
Dec. and lat. contrary name
B=b—d Z=Z' +Z"
These precepts give two cases for altitude and four cases for azimuth.
With the simplified tables, the solution of all problems is the same, with the exception of the unusual circumstance in which the declination is of the same name as the latitude and greater than K, when the factor Z is marked minus. By 10 lines of simple, easily followed instructions and one sample problem, the three pages of explanation and eight problems of “H.O. 208” are eliminated. After these tables were used by several members of the class, it was found that the single exception to the general rule was far more easy to “spot” than to determine the elements of the cases in “H.0.208.” This is of considerable weight to a navigator.
The arrangement of tables is shown in the accompanying illustrations. This is not the best for a smooth working system, but was necessitated by the available tabulated functions of the astronomical triangle. In Table I, the A and K should be interchanged. Table II should be reversed and inverted in order to permit entering Table II with at the top of the page. This would put 0° of Z" at the upper left-hand corner of page 1, and 89°-60' of H„ at the lower left-hand corner of page 1. This change would result in Z" being found at the top of the page, and Hc at the bottom. The illustrated arrangement necessitates entering Table II at the bottom of the page with K~d and, in addition, the values of K~d run from 89° to 0°. This is awkward.
The advantage of the simplified tables can be fully appreciated by comparing the following instructions with those given on pages 68-75 of “H.O.208”:
(1) Obtain G.H.A. and convert into degrees of arc.
(2) Assume an even degree of latitude, nearest D.R. latitude. Call it lat.
(3) Assume a longitude in degrees of arc near D.R. longitude such that, when applied to the G.H.A., will give a L.H. A. in even degrees.
(4) Measure L.H.A. east or west to 180° from local meridian. Call it t.
(5) With I and lat., enter Table I and pick out K, A, C, and Z'.
(6) Mark K same name as lat. Take algebraic difference of K and dec. (If signs alike, subtract; if signs opposite, add.)
(7) With K~d, enter Table II from bottom and pick out B and D.
(8) Altitude.—With A +2?, enter Table II, column B. Hc is at top.
(9) With C+D, enter Table II, column D.Z" is at the bottom; correction is at the right.
If declination is greater than K, and of the same name, mark Z" minus.
(10) Azimuth.—Z' +Z" (algebraically) = azimuth. Measure from nearest pole.
Anyone familiar with the use of “H.O. 208” will discern the smoother operation of the “simplified tables” by following the two problems given below. Take for example, problem 7, page 74, “H.0.208”. In this problem, the L.H.A. falls between 90° and 270°; so rules 5, 12, 13, and 14 are used in the solution by “H.O.208.” By the “simplified tables,” the solution is as shown at the top of the next page.
This problem illustrates the advantage of the “simplified tables” over “H.O.208” for altitude, and the elimination of all rules for the application of the plus or minus sign to Z'. Referring to the illustration of Table I of the “simplified tables,” it is seen that when t is less than 90°, it is at the top of the table and K, A, C, and Z'(+) are also picked from the top of the table. When t is greater than 90°, it is at the bottom of the table and K (180°—K), A, C, and Z'( —) are picked from the bottom of the table. It can be seen from this, that Table I automatically eliminates the necessity for any thought on the part of the computer as to when to apply the minus sign to Z'.
Under what circumstances is the minus sign applied to Z"? Rule 14 of “H.O.208” says,
In finding the quantity d~b with which Table II is entered, should this amount exceed 90°, use supplement (i.e., subtract it from 180°) and use this quantity with which to enter Table II. Give the resultant Z" a negative sign in this event.
This is illustrated by problem 2, page 71, “H.O.208.” The elimination of this rather awkward step is shown by the solution of the same problem by the use of the “simplified tables.”
This problem also illustrates the only rule, exception, or “case” involved in the “simplified tables.” The instructions for the use of the simplified tables prescribe that, if the declination is greater than K and of the same name, Z" is given a negative sign. Fortunately, the computer does not have to keep this rule in mind. The signal to mark Z" minus is quite pronounced and unmistakable. In using the form shown in the two examples, K is on the line above d. K and d are then in the normal position to add (when signs are contrary) and to subtract (when signs are alike), except in the circumstance in which Z" is given the negative sign (d the same name and greater than K). The extra mental effort involved in subtracting the upper line from the lower line is the “signal” for marking Z" minus. Thus, the single deviation from the general case that will solve all problems is forcefully brought to the attention of the computer.
Another distinct advantage of the “simplified tables” is found in the determination of t. Rule 13 of “H.0.208” states:
(a) If the L.H.A. exceeds 90° west, use supplement as t.
(b) If it exceeds 180° west, reject 180°, use remainder as I.
(c) If it exceeds 270° west, use explement as t.
(d) If it exceeds 360°, then treat as in (a).
The t used by the “simplified tables” is obtained as follows:
Measure L.H.A. east or west 180° from local meridian. Call it t.
Figure 1 illustrates the basic difference between the solutions of the astronomical triangle by Ogura and by “H.0.208.” The cases used in the method explained in “H.O.208” for computing the altitude seem to arise from the fact that the supplement of B or (d~6) is used in the solution of the triangle MZO instead of B itself. In Ogura’s method, K~d is the side MO of the triangle MZO, and the value of this side can never exceed 90°. Therefore, by substituting Ogura’s K (“H.0.208” already uses Ogura’s A), we avoid the troublesome case in which d~b exceeds 90°. The figure illustrates the parts used by Ogura and by “H.0.208.” In the case shown, the complement of B is 90° ~B=d+b.
Captain Aquino, in the May, 1930, Naval Institute Proceedings gives us a hint about the point made in the last paragraph. He states, “ . . . Ogura, Weems, and others take my b and my C which simplify the rules and precepts and render them mnemonical.” Captain Aquino also points out a discrepancy between the tabular values of Ogura’s tables and “H.0.208.” This probably results from the fact that Ogura’s tables are computed with 7-place logarithms which gives them additional accuracy. For this reason, the tabular values of A were taken from Ogura rather than from “H.O.208.”
No major improvement in the problem of computing the altitude and azimuth is claimed by this new arrangement of tables. All that can be claimed for any “new” short method is that the arrangement of tables and the use of functions of the astronomical triangle will result in a solution that is simpler and less liable to cause errors on the part of the computer. The “simplified tables” are identical to Ogura’s tables in Weems’s Line of Position Book for the altitude factor. They eliminate the “cases” and simplify and clarify the “H.O.208” solution of the azimuth factor.