“THE ‘NEWEST NAVIGATION’ AND AVIATION—ALTITUDE AND AZIMUTH TABLES—THE SIMPLEST AND READIEST IN SOLUTION.” By Commander Radler de Aquino, Brazilian Navy.
Review by Lieutenant Commander R. E. Rogers, U. S. Navy
The Aquino methods, popularly so called, and as outlined in this latest edition, offer a perfect opportunity for definitely reducing the work incident to the solution of observations, with a corresponding saving of time. The merits of the various methods of solving the astronomical triangle on the St. Hilaire principle have been widely discussed, and while it is possible that there is a single best method for average conditions, the fact that several distinct methods are in current use indicates that the best method for a particular individual depends largely upon the personal predilections of the individual concerned, on the general basis that he does best the thing he likes to do. Those who possess the temperament of a precisian or a talent for accuracy in arithmetical work, and the gift of patience, are likely to elect to work their sight by a rigorous formula, involving considerable logarithmic work or its equivalent in natural functions, but possessing the merits of rigid accuracy and of general application. Others, finding logarithms tedious and arithmetic laborious, and others still who have a penchant for speed, naturally resort to less involved and shorter processes—at a small, and some say negligible, sacrifice as regards accuracy. On this point it is frequently argued, although evidently somewhat fallaciously, that since the observed altitude is subject to an error of a minute (a point also in dispute) nothing is lost by working to the nearest whole minute of altitude, tor those who subscribe to this doctrine, however, and in general for those who are willing to accept a possible error in computed altitude of half a mile, there appears no argument in favor of the relatively long rigorous solutions except the incontrovertible one of a personal preference.
It appears likely that the author, in preparing this book, has taken into consideration these various personal proclivities. As one qualified to judge, he might reasonably have designated a best method, and have undertaken as his mission its establishment accordingly. Actually, however, there is simply offered a wide variety of methods, for the most part for use with the accompanying altitude and azimuth tables, but in addition there is furnished a method, with its set of tables, providing for a rigorous, and of course relatively long, solution.
The following incomplete tabulation represents the practical scope of the new edition:
Complete descriptions of methods and use of tables; a section on chart methods, including a convenient device for accurate chart work by means of cross-section paper; there is also a demonstration of a most ingenious and practical method of plotting lines of positions without use of predetermined azimuth, recommended for high latitudes; traverse tables (of somewhat handier use than those of Bowditch); corrections to observe altitude for all bodies; and a new table of dip and visibility, for altitudes to 20,000 feet, for aerial navigation. Another set of tables, referred to above, affords a rigorous solution for all observations, by means of logs and natural functions, the methods and tables being strictly analogous to the cosine-haversine Solution, with which method it appears to have equal merit. Altogether, the book is fairly comparable to the familiar Bowditch in point of general utility. With the Nautical Almanac, it provides all equipment necessary for the solution of any problem relating to nautical astronomy—separate azimuth tables being dispensed with. The most important element is the justly famous altitude and azimuth tables, and the accompanying descriptions of the wide range of their use, which is practically as wide as spherical trigonometry itself. Specifically, in addition to the direct solution for altitude and azimuth, the tables provide equally readily for the converse solution for hour angle and declination (for star- identification) and incidentally for great circle problems.
The principle of the processes peculiar to the Aquino system is based on the solution of the right spherical triangle. A perpendicular being drawn either from the observed body or from the zenith to the opposite side of the astronomical triangle, the two known or assumed elements in each resulting right triangle are used in the tables as arguments for picking out the remaining required parts. As indicated in the title of the tables, the single solution leads directly to the finding of both altitude and azimuth—the two are located simultaneously in adjoining columns. Several methods are available, each with its peculiar advantage. The newest development, and one which is judged to be a strictly Aquino process, appears in the book in rite form of an insert, but is again demonstrated in the new Appendix IV. Interpolations are entirely dispensed with, an easy arithmetical process taking care of the usual odd minutes of declination. While this particular process is presented simply on its merits, without special recommendation, it is, to quote from the author’s discussion of an earlier process, obviously the “simplest and readiest in solution.” The method involves an assumption of longitude, other than the D.R., to conform to tabulated values of hour angles, and the assumption of latitude to the nearest degree, if interpolations are to be entirely eliminated. From rite point of view of many this requirement constitutes a defect, in view of the occasional large altitude differences obtained. The author fully covers this point, and makes provision for rectification of lines in cases where a practical error would otherwise result. In a doubtful case, however, a more satisfactory procedure would evidently be to apply a rigorous
Solution, using the actual D.R. data. In this connection it is worthy of note that the adoption of Aquino methods is not a bridge-burning process. These methods, based as they are on the soundest principles, may be used in connection with any other system, new or old.
While the Aquino methods come altogether under the Marc St. Hilaire classification, in that a measured altitude is compared to the altitude by computation, the methods involve a construction element that does not occur in the so-called rigorous solutions. Substantially, a new triangle is built up, the component parts being so selected as to facilitate computation. Each step has a clear relation to what precedes and to what is to follow—a favorable contrast, from most points of view, to logarithmic solutions, where all is fairly taken for granted from the first log haversine to the final picking out of “3” from its natural function. While this element of difference is not vital, it has a bearing on the matter of preference in solutions, and is at least analogous to the relative merits of solving a traverse, or dead reckoning in general, by plotting as against computation. Plotting methods, both for traverse and for handling lines of position, have grown steadily in favor, and there is no reason to doubt that these methods produce the best results, and yet computation methods are in themselves more accurate, in a mathematical sense. But in plotting it is possible to see more clearly what is taking place, and problems are simplified and are thus made more interesting. Otherwise stated, there is a saving of conscious effort.
It is considered that demonstrations of methods are out of place here. Estimates of time saved, further, are somewhat inconclusive. There is a wide range in the rapidity of computing even among navigators whose experience and methods are identical. An authoritative review of a previous Aquino edition, comparing with logarithmic solutions, estimated the saving of time in the complete solution of a sight to be three and one-half minutes. It is reasonably safe to estimate that later Aquino methods save another half minute. But the actual saving of a few minutes of time in solving a sight does not as previously implied, appear to be the criterion of an individual’s choice as to method. While Aquino methods appear to lead in this respect, the time-saving feature is regarded as secondary in importance to the advantages inherent to an analytic solution, such as is afforded by these methods.
Altogether apart from the special features here discussed, the new book, considered as a treatise, is a highly valuable contribution to the science of navigation and will afford deep interest and certain profit to anyone concerned in its theory or practice.