H. O. 9
(See page 677, May, 1936, Proceedings)
Captain Radler de Aquino, Brazilian Navy.—The genesis of H. O. 9 (the New American Practical Navigator) or Bow- ditch, as it is known in the United States, was given by the great American shipmaster and mathematician himself in his original edition, issued in 1802.
Under the title of “Nathaniel Bow- ditch,” Lieutenant G. W. Logan, U. S. Navy, gave us in the U. S. Naval Institute Proceedings for December, 1903, a concise biography.
Ever since its publication H. O. 9 has always contained the most up-to-date information and formulas generally accepted as the best. As an official publication it cannot contain all the modifications and simplifications developed in these late years.
Of course, authors cannot “ignore first principles,” as Chief Boatswain Hopkins says they do, because these “first principles” are implicitly comprised within all modern formulas.
His suggestion to return to the fundamental formula
cos a = cos b cos c + sin b sin c cos A
or
sin h = sin L sin d + cos L cos d cos t,
as originally given by Albatani (880-928 A.D.), in the ninth century, for calculating the altitude h or hour angle t, and to the analogy of sines for the azimuth Z, gives the impression that nothing better has been presented.
Many distinguished authors have tried to revive this duo. The inconveniences of working the fundamental formula directly were shown in 1850 by William Chauvenet in his excellent Treatise on Plane and Spherical Trigonometry, dated U. S. Naval Academy, Annapolis, May 1, 1850. This requires, he says, “computing separately the two terms of the second member and adding (algebraically) their values to form the natural sine of h; but we should thus require to use, besides the table of log sines, also the table of logarithms of numbers, and the table of natural sines and cosines.” He further added: “In the use of these formulas, as indeed of all that follow, the signs of all the functions must be carefully observed.”
Captain Aved de Magnac, French Navy, in his “Tables simplifiant la Determination du Point d la Mer,” Brest, 1891, showed that A=sin L • sin d had the sign + when L and d were of the same name; the sign — when L and d were of contrary names; B = cos L cos d cos t had the sign + when 2<90° and — when 090°. A and B were added algebraically to find sin h and h was then taken from the table of natural sines.
In spite of his great authority the process was not accepted. Captain G. W. Logan, U. S. Navy, was also a great advocate of its use.
Versines and haversines, extensively used 100 years ago in the days of lunars, gave us the first possible simplifications of the fundamental formula. They required also three tables, viz.: log versines or log haversines, natural versines or natural haversines and log cosines. Later on the log versines and log haversines were placed alongside the natural versines or natural haversines in order to avoid finding the auxiliary angle 0 (see H. 0. 200), until the final simplifications I presented, in 1911, in my New Log and Versine Altitude Tables, London, 1912 and 1924, where the fundamental formulas were reduced to
versine (90° — h) = versine (L — d) -f versine 0
and
½ log sec L +log sec d +log cosec ½ t = log cosec ½ 0
for altitude
and
½ log cosec Z = (1/2 log cosec t+1/2 log sec d) — 1/2 log sec h
for azimuth.
The three necessary tables are comprised within 36 pages in 8vo giving the h and Z, without interpolation, following the century-old rules: contrary names, add; same names, subtract, and using the least number of figures. The use of log cosec §1 avoids the signs of cos t and permits the use of t from 0° to 360°. The “steps,” of course, are the same in all transformations.
As far as the writer knows, no further simplifications of the fundamental formula are possible. All transformations made after 1911 are more complicated.
Nothing compares, however, with the use of the formulas in tangents and secants only for finding h and Z for a Sumner line of position. Log tangents and log secants are given alongside one another in a table of only 18 pages in 8vo or those contained in any ordinary table of trigonometric logarithms can be used.
In 1877 Aved de Magnac, then a Lieutenant de Vaisseau, in the Pratique of the “Nouvelle Navigation Astronomique” showed that the formulas in tangents were the best for finding angles, as Chauvenet had shown before in his excellent Manual of Spherical and Practical Astronomy, vol. 1, pp. 31 and 32, dated St. Louis, January 1, 1863, where he gives “the following simple and accurate formulas” derived from the two fundamental ones, mentioned by Mr. Hopkins:
[EQUATION]
the same as those forming the basis of all my tan ± sec tables. They are absolutely general and can be used in all cases, as modern methods require.
These formulas can also be derived directly by dropping a perpendicular from the observed body upon the meridian of the observer, and using Lord John Napier’s well-known rules.
In any, method of navigation accuracy is the first requisite in all cases, simplicity the second, safety the third, and speed the fourth and last.
Of course all formulas, including those in Bowditch, originate from the same sources already over 1,000 years old.