A new departure of some little importance in connection with long distance navigation in the air has recently been taken by the publication on the part of the “Club Militar Naval" at Lisbon of an interesting memoir of the historic voyage from Lisbon to Rio by aeroplane, accomplished by two Portuguese naval officers in the spring of 1922.
The officers in question were Rear Admiral Gago Coutinho and Commander Sacadura Cabral, who in the Journal of the “Club Militar Naval” for the months October to December, 1922, gave a detailed account of the methods of navigation adopted, with practical examples as actually worked upon their adventurous voyage. This memoir has lately been reproduced in the form of a pamphlet, with translations in French, English, and German, so that practically the whole world of navigators is enabled to share in the benefits resulting from this epoch-making voyage. The opening chapter is naturally devoted to a discussion of the methods of keeping the reckoning independently of the heavenly bodies, in connection with which the effects of “drift,” naturally much more formidable in amount than is the case in ocean navigation, are considered at some length. To gauge the effects of “drift” it is essential to have access to some point of reference which may be considered at rest, and such a point is generally available, when the flight is over land. But at sea, except in the case of a becalmed sailing vessel, the point of reference has to be created. Recourse is in general taken to a smoke bomb containing a deposit of phosphide of calcium, which is opened before being dropped into the sea, when the water decomposes the phosphide, and the phosphorous gas liberated takes fire spontaneously on reaching the air. A white smoke is thus produced, visible at some distance, and marking a spot passed over by the aeroplane. The device, we are told, has not yet been fully developed by the expert, but another system is now being studied in which a substance is used of such a nature, that, being poured into the sea, it will leave a spot perfectly visible from the air.
In connection with the use of the bombs a small instrument called the Course Corrector was employed on the voyage. This instrument had been used by Admiral Coutinho and Commander Cabral on a previous voyage from Lisbon to Madeira in March, 1921, and had been exhibited by these officers at the International Congress of Aerial Navigation in November, 1921.
A more or less reliable method of finding the amount of “drift” is naturally a matter of the first importance in the navigation of the air, and attention may be directed here to an extract from a lecture by Squadron Leader H. E. Wimperis, O. B. E., before the Royal Aeronautical Society in April, 1919, and quoted in a work on aerial navigation by Lieutenant J. E. Dumbleton.* The extract is as follows:
Perhaps the best method is to take times and bearings of the object as it passes through the points E, F, and G, such that the time from E to F is equal to the time from F to G. Then if the angles ? and ß are small: i. e., not more than fifteen degrees, it is easy to find C D the course made good, by marking off A E proportional to the angle B, and A G proportional to L, and then joining F G.
* Principles and Practices of Aerial Navigation, London, Crosby Lockwood and Son. Price, twelve shillings and sixpence net.
The lecturer appears to have limited the case to small angles, such that their sines might be represented without sensible error by their circular measure, but this limitation is not really essential, and in the latest edition of Norie’s Nautical Tables a “Course Angle” Table is included, which, having given three bearings separated by equal intervals of time, will supply the course made good over the ground at sight, without recourse to any geometrical construction whatever. Such a table should be specially useful in the navigation of the air.
Astronomical Navigation
The general principle observed throughout the voyage was this, that as much as possible of the work connected with the necessary observations of the heavenly bodies should be effected in advance. For this purpose a number of points of reference, differing each from each by two degrees of latitude, and one degree of longitude, were selected beforehand for each section of the voyage.
For instance, as the Equator was approached, these points were fixed as follows:
Point | Lat: | Long: |
A | 14°N | 24°W |
B | 12°N | 25°W |
C | 10°N | 26°W |
D | 8°N | 27°W |
E | 6°N | 28°W |
F | 4°N | 29°W |
With an aeroplane traveling uniformly at upwards of seventy miles per hour, it was of course possible to hit off the time at which each point would be passed with sufficient accuracy to enable the declination to be corrected for an approximate Greenwich Date at the point. Having then latitude, declination, and hour angle deduced from the time shown by chronometer, the data would be complete for finding zenith distance, as required in the Marcq St. Hilaire system of fixing position lines followed throughout.
It only remained to throw the fundamental formula connecting the three sides of a spherical triangle with an angle into such a form as to permit as much as possible of the necessary computation to be performed in advance. The expression made use of, which was apparently specially deduced in view of the voyage, is the following:
the + sign for latitude and declination of same name, the — sign for different name, where
S = Log sine lat -f- Log sine dec
C = Log cot lat + Log cot dec
h = Hour Angle.
The values of S and C, it will be noticed, can be prepared in advance, and duly recorded for the point involved.
The following example, for March 30, 1922, will illustrate the process:
Lat 37°N
Dec 3° 31.3’ N
h 4h 59m 118 = 74° 48’
Coef C 1.33353
Coef S 8.5680
C 1.33353
L sec h .58139
(Diff) .75214 Nat No 5.651
1____
(Sum) 6.651
Log. .8229
C 8.5680
(Sum) L Sine alt 9.3909 Alt 14° 14’
The expression employed by the aviators, which appears to have been deduced from the fundamental formula for the special purposes of navigation in the air, has this advantage, that in the final stage only three openings of the tables are required. The same result might, however, be secured by methods in ordinary use to be found in the textbooks of navigation. The “All Haversine” process, for instance, an example of which is worked on page 159 of the American Practical Navigator (ed. 1919), might be brought into requisition; which possesses this recommendation, that only one table of logarithms is necessary, instead of two, the table of log secants and of natural numbers, as in the pamphlet.
The formula in this case is:
where is zenith distance, p polar distance, and c colatitude. The work by this formula would be as follows:
p 86° 29’
c 53
___ _______
p+c 139 29 Nat hav (p+c) .8801
p-c 33 29 Nat hav (p—c) .0830
_____
(Diff) Nat hav e .7971 Log hav ? 9.9015
Final Stage
The hour angle at observation having been found by means of the chronometer as 4h 59m 11s = 74° 48', the work to be completed in the air is as given below:
Log hav ? 9.9015 (from above)
Log hav h 9.5669
__________
(Sum) Log hav ? 9.4684
Nat hav ? .2940
Nat hav (p—c) .0830
____________
(Diff) Nat hav z .3770
z= 75° 46’
alt= 14° 14’
The use of three fresh logarithms for the final stage appears to represent the irreducible minimum in a method of this kind. In the words of the pamphlet, in connection with an alternative form proposed for the formula employed: “For the final stage, however, to enter the tables three times would be required, as is the case with all the known processes.”
This form of treatment by means of values for selected positions on the proposed voyage calculated beforehand probably offers all the simplification possible, so far as the work to be carried out in the air is concerned. The procedure adopted for azimuth, however, is not so easily intelligible. The expression employed includes both altitude and hour angle, and involves therefore as many openings of the tables as the calculation of the altitude.
The formula in this case is:
Cosec Az=Sec dec cosec h cos alt
And the work carried to three places of logarithms is:
Log Sec 3° 31’ .001
Log Cosec 74° 48’ .015
L Cos 14° 14’ 9.986
______
L Cosec Az. 10.002
Az = S 84° .30’ E
If the azimuth has to be calculated at all, the method selected by hour angle and altitude, generally known as the “Sine Method,” offers a reasonably simple method of finding its value, but it seems a little extraordinary that the Azimuth Tables which have now been in general use at sea for upwards of fifty years should not be turned to account in a case where the saving of labor in logarithmic computation is so important. In the present instance we have only to turn up in Burwood’s Azimuth Tables, Lat. 37° Dec. 3° 30' (same name) Hour Angle 5”, and we have at sight the value N 96° E or S 84° E approximately the value sought.
There is, moreover, a special objection to the use of this formula, in that it leads us into troubled waters in the neighborhood of the Prime Vertical, as indeed seems to be admitted in the following passage from the pamphlet:
This formula is not reliable in cases where the declination of the star is less than our latitude, but the usual tables of the favorable circumstances for the computation of the hour angle, showing the time at which the star cuts the East-West line, may be employed to avoid confusion. We only had to deal with this case during the journey from Cape Verde to the St. Paul Rocks.
Just as the Altitude Azimuth fails near the meridian so the “Sine Method” cannot be relied upon near the Prime Vertical. It is only the Time Azimuth, the problem on which the Azimuth Tables in common use are based, which, except for very high altitudes, when it is almost impossible to determine exact azimuth at all, can be depended upon in all cases.
An Alternative Method for Zenith Distance in the Tropics
Not satisfied with the simplification already effected in the general problem, the distinguished aviators, on entering tropical latitudes, hit upon a still shorter and simpler process for finding zenith distance, for use in those parts of the world where the sun at noon attains an altitude approaching ninety degrees. In this case nearly all the calculations necessary can be performed in advance, leaving nothing but a simple matter of subtraction of one angle from another to be carried out in the air. So far as the writer is aware the principle of the method is entirely new, nor is any reference to such a process to be found in the works of Bowditch, Norie, Raper, or in any other standard textbook. In the pamphlet the process is introduced as follows:
In the neighborhood of the Equator, and when the sun has a low declination, as was our case in April, as is easily deduced from the general formula for the resolution of the already mentioned nautical triangle, the altitude of the sun, which nears ninety degrees at noon, does not differ much from the complement of the hour angle. And, as the variation of the addition of the hour angle and altitude is slow, at least in the later hours, it becomes easy to compute a table for that addition. Therefore, starting from the hour angle, an altitude is immediately obtained by means of an interpolation at sight, and an elementary subtraction. Such was our case when on April 18, we approached St. Paul Rocks. The table used on that occasion and also an example, will give a quick idea of the practical execution of this process.
The data of the example were as follows:
Lat. 1° N Dec. 10° 45' N Hour Angle (from chronometer) 4h 54m 458
Observed Altitude (corrected) 16° 8'.
With the given values for latitude and declination the altitudes were computed for successive values of hour angle, giving results as shown below.
For 4h 40m, or 70° in arc, the altitude was found to be 19° 49', the sum of which quantities (70° + 19° 49’) = 89° 49'. Similarly for the following values, the hour angle being increased in each case by 10m, or 2° 30'.
The following is a section of the resulting table:
h | Sum | ||
4h | 40m | 89° | 49’ |
| 50 |
| 52 |
5 | 0 |
| 54 |
| 10 |
| 57 |
| 20 | 90 | 0 |
The altitude required having been duly observed at 4" 54m 458 the work is completed in the air as follows:
From table sum for 4h 55m | 89° | 53’ |
Hour Angle in arc | 73 | 41 |
(Diff) Altitude | 16 | 12 |
Observed Altitude | 16 | 8 |
Intercept |
| 4 |
The saving of labor effected by this extremely ingenious notion will be easily appreciated if we compare the work given above with that which would be required if we proceeded in the manner which at first suggests itself, by tabulating altitude and hour angle earlier and later than the time when it is anticipated that the altitude will be observed, and obtaining the calculated altitude by proportion. In the present instance we should have results as follows:
Hour Angle Altitude
4h 40m 19° 49'
5 0 14° 54'
5 20 10° 0
It is required to obtain the altitude at 4h 54m 458.
If x is the amount to be subtracted from 19° 49', the altitude at 4h 40m, we have
Thus the altitude required is 19° 49'—3° 37.5' = 16° 11.5'.
A comparison of the two processes brings out clearly the advantages of tabulating the sum of hour angle and altitude rather than the simple altitude itself. This happy idea of the Portuguese officers gives the calculated zenith distance immediately after the observed distance has been secured, and should be of great value not only to air navigators, but in ocean navigation also.
Some interesting details relating to the voyage are given in a statement drawn up by Admiral Coutinho. The number of nautical miles covered was 4,527, sixty-two hours twenty-six minutes being actually spent in the air, giving an average speed of seventy-two and one half miles per hour. The longest voyage was from Praia to St. Paul Rocks, 908 miles, accomplished in eleven hours, 21 minutes, and the shortest Las Palmas to Gando 15 miles, in twenty-one minutes. During thirty-six hours forty- four minutes spent out of sight of land the aviators computed ninety-six sets of sun observations, giving a mean of twenty-three minutes between the sets.
The work of the Portuguese naval officers is very happily summed up in the course of a foreword to the pamphlet contributed by the editor of the Journal of the “Club Militar Naval” in the following words:
But what one most admires is the progress our distinguished comrades have accomplished, as regards the methods of astronomical navigation.
The use of points of reference through the line of the intended crossing, and the able and skilful modifications of the formulas of Nautical Astronomy, by which the observer may, before starting, prepare the greatest part of his calculations, in this way leaving only another quite material part to be done in the air, are the two original conceptions by which the astronomical navigation, with a sextant, can be done in the air with accuracy and comfort, as happens on board ship, together with the quickness exacted in virtue of the great speed of the aeroplane.