A FIX AT sea or in the air can be obtained by the methods and tables indicated below by means of the simultaneous observations of altitude and azimuth of a single celestial body in a very simple and practical way: ^{[1]}

The altitude is taken by any bubble sextant or octant.

The azimuth is taken simultaneously by any well-adjusted gyrocompass or any well-adjusted magnetic compass.

The time is taken by means of double split-second stop watches from the sidereal, civil, or solar time chronometers. The stop watches give the true or “no correction” sidereal, civil, or solar time at sight.

The calculations are made by any ordinary table of trigonometric logarithms containing log tan, log sec, log cosec for every 1' or less.

The plotting is ordinarily done on a Mercator chart.

The writer has busied himself with this problem for over 30 years, while the question of determining the azimuth with an accuracy comparable to the accuracy of the altitude was slowly being solved by various inventors: Sir William Thomson (Lord Kelvin) for the magnetic compass and Dr. Anschutz and Mr. Elmer Sperry, for the gyrocompass—just to mention three modern leaders.

Ever since 1903, when the writer discovered that the value of a—the perpendicular dropped from the observed body upon the meridian—could be found not only from the declination d and the local hour angle t, but also from the altitude h and the azimuth Z, the problem of a “Fix from Altitude and Azimuth” was theoretically solved, as later he found it had already been by many prominent English authors in the seventeenth century, when the azimuth was found at sea more accurately than the hour angle or time. Hence Andrew Wakely’s The Mariner’s Compass Rectified: Containing Tables Shewing the True Hour of the Day, the Sun being upon any point of the Compass, London, 1665, (1st ed.). This work contains the first set of Inspection Tables ever published. The arguments were latitude, declination and azimuth to find the hour angle to the nearest minute of time. The author called each page a “Sun-Dial for the Latitude of—.” Latitudes ran from 0° to 60°.

The late Captain J. P. Ault, late commander of the non-magnetic yacht Carnegie and one of the foremost sea and air navigators of the world, in his pioneer experimental work in the air declared:

The problem confronting the navigator, either at sea or in the air, is to measure the altitude of one or more celestial bodies as accurately as possible and then to compute and plot his most probable position. From this measured altitude a circle of position is determined, the observer being located somewhere on this circle, at any point of which the sun is at the altitude observed. An approximate knowledge of the azimuth of the object observed will designate the portion of the circle of position on which the observer is located, and usually the circle is so large that a portion 60-100 miles long may be considered as a straight line without appreciable error. This line is known as the Sumner line, after Captain Thomas H. Sumner, an American shipmaster who discovered the method which involves the use of this line in navigation. If two celestial bodies are available, two position lines can be determined, and their intersection completely fixes the geographical position of the observer. If only one object is available, then the observer knows only that he is located somewhere on this position line, and to completely fix his position it is necessary to observe also the azimuth of the celestial body, or the bearing of some known object on land.

The altitude, usually measured with some form of sextant, is the angular distance of the body above the horizon. For ocean navigation, either on the surface or in the air, the sea horizon is usually available but for aircraft flying over land and, at times, over the ocean, some form of artificial horizon must be provided.

After the altitude is measured, the next proceeding is to make the calculations and to draw the position line on the chart, or by some method, determine the position of the observer. The usual methods for this work require too much time for aircraft, so that new and rapid methods are necessary.

The trip from Langley Field to Washington on Monday, September 23, was made to try out the method based on the use of Aquino’s tables, as outlined in my report of September 10,1918.

Owing to the uncertainties of dead reckoning during airplane flights over the ocean or above the clouds, where there is no means at present to determine the amount of drift, it may be of value to obtain the azimuth of the body observed as well as the altitude. For this purpose a simple pelorus or azimuth sighting-device was mounted on the artificial horizon. It is seen from the trigonometrical conditions and from Aquino’s tables that if the altitude and azimuth can both be measured with a fair degree of accuracy, then the navigator need not know his dead-reckoned position at all, except to determine his magnetic declination.

The first known instance of an airplane pilot being informed of his position by astronomical methods should be recorded here. During my flight to Washington from Langley Field September 23,1918, the visibility was very poor. The pilot, Lieutenant Charles Cleary, wished to verify his position, so he slowed down and asked if the river below us was the Potomac. I had just completed drawing in position line No. 5 (see my report of October 1, 1918), which intersected our track at the Potomac River, so I was able to inform him that my observations placed us at the Potomac River. These results show that most excellent azimuths can be obtained with an airplane compass. Thus aerial navigation by daylight will be a much more certain proposition by measuring both the altitude and azimuth of the sun rather than by measuring the altitude only.

The writer is anxious to know if anybody did “inform an airplane pilot of his position by astronomical methods” before Captain Ault did on September 23, 1918.[2]

Methods of Solution

Captain A. Fontoura da Costa of the Portuguese Navy, Professor at the Portuguese Naval Academy, in his interesting study: The Present and the Future of Position Finding at Sea (0 actual e o futuro ponto no mar), Lisboa, 1930, p. 39, said: “Radler de Aquino, according to Commander Newton, has already worked out a fix by combining the altitude of a celestial body with its simultaneous azimuth by a gyrocompass.” He thinks it could be called an “Altazimuthal Fix.” He further says, after giving up the idea of presenting a simple solution of the problem:

It is true that the determination of the true latitude and true longitude could be obtained by direct logarithmic calculation, once we have the three known elements of the triangle of position. But in this case, the formulas are complicated, the simplicity and the rapidity, always required, are excluded and at the same time the advantageous geometrical interpretation of the lines of position disappears.

Captain F. Marguet, of the French Navy, Professor at the French Naval Academy, having perused Fontoura’s excellent and valuable Navegaqao Radiogoniometrica, Lisboa, 1927, and many other publications, declared in his excellent article “Le ‘Point' Azimutal" in the January, 1935, issue of the Revue Maritime, page 60: “Thus we will find the fix (coordonnés exactes), by a general method, rigorous and readier (plus rapide), than by drawing a Sumner line of position by the ordinary process.”

Mr. Edward J. Willis in his valuable and most interesting “The Line of Azimuth” (an appendix to The Methods of Modern Navigation, Richmond ,Va., U.S.A., February 5, 1927, revised in January, 1928), showed how to calculate the latitudes and longitudes of several positions on the Sumner line of position by means of two formulas based on Z_{0}, deduced from the “rate of change of altitude” of the observed body. His method is implicitly comprised within the altitude and azimuth fix, as the “rate of change of altitude together with the latitude” gives us the azimuth, as if it had been observed. His formulas are exclusively based upon sines and cosines and the “rate of change of altitude” is not always obtainable.

The rate of change of altitude is closely associated with the azimuth. If at sea, we could observe azimuth with the same degree of accuracy with which we can altitude, we could obtain two simultaneous lines of position from a single object, and thus the ship’s position. The same is true of the rate of change of altitude. If we knew it and also the altitude at the same instant, a position would be obtainable. [Page 1 of the 1927 edition.]

The main point to appreciate is that no matter what value is given Z_{0} (and hence the rate of change of altitude), the resulting position will be on the Sumner circle.

An accurate determination of Z_{0} defines a point on this circle, and hence the position of the observer. An approximate determination of Z_{0} defines a short arc of the Sumner circle. This arc being short can be ruled on the chart as a short line of position. If the value of Z_{0} is changed, the effect is only to move the obtained position along the Sumner circle. A slight change in Z_{0}, such as a small error in its determination, moves the obtained position along the line of position. [Page 8 of the 1928 revised edition.]

As sin Z=sin Z_{0} sec L, the finding of Willis’ Z_{0} from the “rate of change of altitude” will give us the necessary azimuth, when we cannot observe it directly. The method used with care will give results comparable to those obtained by a direct observation of azimuth.

These valuable quotations from three leading authorities and nautical exponents will show the value of the simple solution proposed below, whereby the true position is found and the Summer line of position drawn, when necessary.

The accuracy of the dead-reckoning position is very great nowadays, and due to the improvements in all navigational apparatus and equipment and greater care on the part of the navigator, it seldom will be more than 10 miles away from the ship’s true position.

Whenever the “altitude-difference” is greater than 20' very accurate results will be obtained by using the “point Marcq,” as a new dead-reckoning position and working out the sight again.

Most methods and tables based upon auxiliary or assumed positions are doomed and will soon be obsolete, in view of the simplifications presented by the new solutions based upon the dead-reckoning position and the general trend in methods used for sea, air, submarine and radio navigation.

[FIGURE]

Those based upon general formulas will still be used where approximate solutions are desired, as in altitude and azimuth for sighting and star identification, latitude and longitude factors and all other problems where great accuracy is not required.

Although well known, the writer gives a figure of the triangle of position PMZ projected upon the horizon and the necessary formulas in their latest logarithmic form for finding the hour angle t with h, Z and d, C from h and Z and finally b from d and t.

The general formulas from Napier’s or Mauduit’s Rules,[3] in log tan (tan), log sec (sec) and log cosec (cosec), are:

log cosec t = (log sec h +log cosec Z) —log sec d for the hour angle t

log tan B = log tan h -flog sec Z for the value of C = 90° —B

log tan b = log tan d -flog sec t for the value of b.

The combination of t with the GHA or over in case an error has been made or t_{Gr }in arc will give the true longitude.

The combination of C and b will give the true latitude. When *t*>90°, b is also >90° i.e., when t is comprised between 90° and 180° E. or W. (or between 90° and 270°), b is also comprised between 90° and 180°.

No cases, nor rules, nor precepts are necessary with these formulas—a decided advantage in practice.

These formulas lend themselves to mechanization and slide-rule practice, but the writer at present finds the tables more convenient and, in writing down every operation, we have a record to go

Example for All Sights

(165 navigational stars, 4 planets, the sun, and the moon)

Example.[4] [5] On July 2, 1932, during evening twilight, in position by D. R. Lat. 36°-49'N., Long. 75°-12' W., the navigator of the U.S.S. Detroit and his assistant took simultaneous observations of altitude and azimuth of a Scorpii (Antares), with a “no correction” double split-second sidereal stop watch, a bubble sextant and a gyrocompass for a fix, as follows:

[EQUATIONS]

If the azimuth Z is a few minutes of arc in error the position found will be on the Sumner line of position on one side or the other of the true position of the observer. A knowledge of Isoaz Navigation (navigation by means of one or more lines of equal azimuth) will be necessary to take full advantage of isolated observations of azimuth.

The fix in Dutton’s excellent treatise is in Lat. 36°-49'.5 N. and Long. 75°-17'.0 W. This fix was found from two lines of position by the ordinary cosine-haversine formula for the two altitudes and a different method for the two azimuths found separately from different tables.

Computation of Latitude for Determination of “Towson’s Point and of Azimuth when Observation Is Impossible

[EQUATIONS]

The direct calculation of the true latitude and the true longitude of the intersection of two circles of position is given, in detail, in Facio’s Navigation Improved, London, 1728, and referred to by John Robertson in his Elements of Navigation, Vol. II, page 345, London, 7 editions in all from 1754-1805. The writer has only the 6th edition of 1796 and presented the 3d edition of 1772 to the U. S. Naval Academy at Annapolis, in 1926.[6]

The Determination of “Towson's

Point"—T when the Azimuth

Cannot Be Observed

When it is impossible to observe the azimuth, even approximately, it will be necessary and advantageous to calculate it from the declination, the altitude and the local hour angle t, deduced, as usual, from the Greenwich hour angle t_{Gr}, and the dead-reckoning longitude. (See computation in example above.)

The latitude thus found will be the latitude of T—“Towson’s point”[7]—the intersection of the dead-reckoning meridian with the Sumner line of position drawn through it. This process should not be used when the azimuth is greater than 70°.

With these two processes the “altitude- difference” is always zero and the Sumner line of position can always be drawn directly and accurately on the chart through the position found.

The distance between dead-reckoning position and the true position will rarely be greater than 10', and there will be many opportunities during the day, during twilight, and at night to rectify the dead-reckoning position by means of Venus, Jupiter, Mars, and Saturn and the 165 navigational stars (20 of the 1st magnitude and 59 of the 2d magnitude) given in the American Nautical Almanac, besides the sun and the moon—the least desirable of all celestial bodies, except for illuminating purposes.

Our figure, page 1732, shows the advantages of a carefully computed dead-reckoning position.

Draw a circle from the dead-reckoning

[FIGURE]

(Cape May to Cape Hatteras) U.S.C. & G.S.

The Determination of the “Point Marcq” M

“ . . . son grand avantage, à nos yeux, consiste dans sa généralité, elle s’applique à tous les cas.” Marcq Saint-Hilaire. “Calcul du Point Observé,” Revue Maritime et Coloniale. September 1875, p. 740.

[EQUATION]

position with a radius of 10' or with the probable error of the dead-reckoning position. The segment of the Sumner line of position within the circle of doubt will be the useful part of the line and it contains the true position of the observer.

M the foot of the perpendicular dropped from dead-reckoning position upon the Sumner line of position is the “point Marcq”—the observer’s most probable position when only one observation of altitude is available.[8]

Accuracy, Simplicity, Safety, Value, and Speed of this Method

The accuracy, simplicity, safety, value, and speed of the method can be shown better by the same example (see below) worked out to the nearest tenth of a second of arc by means of Callet’s Tables of Seven Decimal Place Logarithms, Paris, 1883.

*Comparison of Results*

The differences between the results calculated by means of the ordinary tables to the nearest minute of arc, without interpolation, and Callet’s to the nearest tenth of a second of arc are only +8".5 for the azimuth and +14" for the altitude. The “altitude-difference” will generally be very small and Z_{N} or Z_{D}._{R}. very near the value of Z for the “point Marcq” from the dead-reckoning position. Callet’s tables have no log secants and we find the secant from the log cosines.

As Callet gives the logarithms for every 10", accurate interpolations were used to find the logarithms for every tenth of a second of arc.

By interpolating in the ordinary tables of five decimal place logarithms we would find *h*_{D.R.}. 19°-26'-46" and Z_{N} 148°-18'- 51 ".5—the same as with seven decimals.

Although these calculations can be done by any table of trigonometric logarithms the computation is greatly facilitated by the use of a specially arranged table of log tangents and log secants placed alongside one another and contained in the writer’s A Navegação Hodierna com Logarithms de 1633! (Aérea, Maritima e Radiogoniometrica) com as necessarias Tabuas Linotipadas. Suplemento da Revista Maritima Brasileira de Setembro-Outubro de 1933, 2d edition in January, 1934, 3d edition in January, 1935. Only 18 pages—the shortest in existence.

For further tests of the extreme accuracy of the above method and tables, use any position on the Sumner line LL' and the calculated altitude will always be within 0.’5 of the true altitude 19°-24.'2 and the azimuth will be found to the nearest minute of arc, without interpolation. No other process will always guarantee these accurate results.

The United States of America must have a Navy or Cease to be a Commercial Nation, it is therefore highly necessary that a proper system of Subordination, attention to duty, and respect be introduced in its first out set, otherwise the Consequences are easy to be foreseen (by Men of discernment) especially when at this day, there is Scarcely Officers of tollerable abilities to be found {who have had any experience) for the few heavy Ships we have built, and among those few Scattered from one end of the United States to the other, we find them frequently given up to that detestable vice drunkeness, which must always render every officer unworthy and unfit for Service particularly for any important Service; in fact every drunkard is a Nuisance and no drunkard ought to be employ’d and if employ’d Shall ever remain an Officer with me; I do not mean to insinuate that a Convivial fellow is a drunkard, who may become Cheerful in Company, the distinction is too great to make it necessary for me to draw any line on that Subject.—The sentiments Contain’d in this letter, you will be pleased to make Known, to every one who may be appointed to the Ship, in order that we set out fair, and a Steady adherence to the same will be expected by me.—Thomas Truxtun to Lieutenant Gross, August 30, 1797.

^{[1]} This problem was proposed for the first time and solved theoretically and instrumentally upon a globe, in 1537, by the eminent Portuguese cosmographer and mathematician Pedro Nunes when he found the latitude by means of two altitudes of the sun and the angle made by the azimuth circles passing through the sun, when the altitudes were taken. See Tratado da sphera com a Theorica do Sol e da Lua, Lisbon, December 1, 1537.

**[2]** By J. P. Ault, Navigation of Aircraft by Astronomical Methods. With one plate and three figures. (Extracted from Publication No. 175, Vol. V, of the Carnegie Institution of Washington, pp. 315-337. Contains Captain Ault’s Reports from September 3 to December 11, 1918, and also Dr. J. A. Fleming’s. J. P. Ault, On Determination of Position of Airplanes by Astronomical Methods. (Presented before the American Physical Society at Washington, April 25, 1919.) Abstract published on pp. 303 and 304 of the Annual Report of the Director of the Department of Terrestrial Magnetism, Carnegie Institution of Washington, for 1919.

**[3]** William Chauvenet, A Treatise on Plane and Spherical Trigonometry. 9th edition. No date. Philadelphia, pp. 169 and 170. 1st ed. 1850.

[4] Taken with the necessary alterations, from Captain Benjamin Dutton’s excellent treatise on Navigation and Nautical Astronomy, 5th edition.

[5] Directly from a double split-second stop watch regulated upon a sidereal chronometer.

The use of sidereal chronometers only greatly simplifies the problem of finding GHA or to, and renders the process uniform for all celestial bodies.

[6] For the history and development of these methods see an article by the writer: “Modern Methods in Sea and Air Navigation” in the United States Naval Institute Proceedings for January, 1927, pp. 17-34, and accounts of Pedro Nunes’ problem (1537) and Robert Hues’ problem (1592), solved instrumentally upon globes long before Facio’s time, and also an article by the writer: “Modern Methods in Sea, Air and Radio Navigation” in the Nautical Magazine of Glasgow for January, 1934.

[7] Tables for the Reduction of Ex-Meridian Altitudes, by John Thomas Towson, F.R.G.S. 20th edition, 1906. Published by J. D. Potter. On page 4 he declares: “This problem was solved by Robertson in his Elements of Navigation, Book V, problem 33. London, 1754-1805.” (Seven editions in all.)

[8] Calcul du Point Observe. Méthode des hauteurs estimées. Marcq Saint-Hilaire used the estimated or dead-reckoning position and in his calculations he dropped a perpendicular from the observed body upon the meridian and worked the right-angled spherical triangle formulas by means of logarithms. He was the first to give us the “point rectify,” “point Marcq” or “point rapproché”—the most probable position of the observer, when only one observation of altitude is available.

[EQUATION]

(3) cos Z=tan (D'—L) -tan H_{e} where *D*' is our b, E the altitude, D the declination, P the hour angle and Z the azimuth.

The critical points and the limitations of sines and cosines are too well known. The same may be said of cosecants and secants.

In view of these limitations Marcq Saint-Hilaire gave another method which he considered simpler and more accurate. The altitudes employed in his example were 83°-37' and 84°-14'.5. The use of tangents avoids the necessity of a change in method.