The problem of readily solving the astronomical triangle by means of tables without using logarithms has naturally always had a certain amount of fascination for the navigator with a view to lessening his work of computation, besides stimulating the ingenuity of experts to produce the required tables for this purpose. How much, however, any of the many volumes of tables so far produced are a really satisfactory substitute for the already existing tables of logarithms is rather open to question.
The problem is, viz.: with latitude, declination, and altitude, hour angle and azimuth, given any three of them to find the other two to the nearest minute of arc.
In comparing tables with methods using logarithms it is to be remembered first of all that whichever we use there are certain preliminaries that are common to both, e.g., elements have to be taken out of the Nautical Almanac, altitude corrected, hour angle worked up, etc. These take exactly the same time whether using tables or using methods by logs; it is only in the subsequent calculations that there may be any difference, and although undoubtedly a position of sorts is obtained more quickly by tables than by logs, it may prove to be anything but a convenient method for plotting the position line and result in no saving of time or trouble in the end. With recognized methods using logs, no interpolation or correction whatever for the odd minutes of declination is necessary, while the position found will be the most convenient one for drawing the position line. The advantages usually claimed of being relieved of the necessity of using logs at all is not one that appeals to the modern qualified navigator, to whom, having been taught as a part of the ABC of his profession the use of them and their application to the ordinary methods of navigation, the labor of taking from the nautical tables the few logs required and adding them together is both simple and almost mechanical, hardly takes more time than the necessary operations by tables, and is less liable to error. Moreover, he has no desire for the addition to his library of extra books of tables, generally bulky and expensive; his tables of logarithms that he already possesses are the best tables not only for navigation problems but also for many Others connected with his profession.*
Is the distinguished British Admiral correct?
General Principle
The general principle upon which this method is based is as old as spherical trigonometry, and naturally it was the only way of solving spherical triangles until, as Dott. Giuseppi Pesci informed us in 1906, Albatani (880928 A.D.) discovered the wellknown relation (erroneously attributed to Euler) between the three sides and an angle of a spherical triangle
cos a = cos b cos c + sin b sin c cos A
or, if we consider the triangle of position
sin h =sin Lsin d + cos L cos d cos t.
Figure 1 represents the projection of the triangle of position PMZ upon the horizon and shows clearly the relations between the coordinates d and t, a, b, and C, and h and Z for a given latitude L.
General Formulas
From Lord Napier’s wellknown theorems in Spherical Trigonometry we obtain the following groups of formulas^{1} which bind together several of the elements of the two rightangled triangles:
(1)sin d = cos a sin b
sin h = cos a sin B
(2)sin a = cos d sin t
sin a = h sin Z
(3)cot t = a cos b
cot Z = cot a cos B
(4)cot b = cot d cos t
cot B = cot h cos Z
Greenwich Hour Angle West
The following expressions give us the value of t_{G}—the Greenwich hour angle West:
t_{G} = G.C.T. 12^{h}Eq. of T. for the
and
t_{G} = G.S.T. — R.A. for the *s, the
and the planets where
G.S.T. = G.C.T. + (R.A.M.S. +12^{h})
+ Corr. for G.C.T.
The American Nautical Almanac gives the Greenwich hour angle West directly in arc from the G.C.T. for the sun, moon, Venus, Mars, Jupiter, Saturn, and 55 navigational stars—a great convenience when only G.C.T. chronometers are available.
His Majesty’s Nautical Almanac, the Berliner Jahrbuch, Brown’s, and the Argentine Navy Nautical Almanacs give R and E in order to facilitate the finding of t_{G}. R is equal to R.A.M.S.+12^{h} and E is equal to 12^{h}±Eq. of T.
If only sidereal time chronometers are used, the finding of t_{G} is greatly simplified, as t_{G} is always G.S.T.—R.A., whatever the observed body is. Even for the sun there is an advantage.
Local Hour Angle West
LHA or t (the local hour angle West) is found by combining the Greenwich hour angle West GHA or t_{G }with the longitude G:
t = tt_{G} ± G
(— when West and + when East) i.e., same names: subtract; contrary names: add.
When t_{G} is smaller than G add 360° to t_{G}. If t_{G} + G is larger than 360° drop 360° from it.
To Find the Value of b—Towson’s Augmented Declination
When the LHA or t is comprised between 0° and 90° or 270° and 360°, b is less than 90°: we enter the tables from the top and take b from the top.
When LHA or t is comprised between 90° and 270°, b is greater than 90° and we enter the tables from the bottom and take b from the bottom, instead of b at the top, or, in other words, when entering tables with LHA or t from the bottom take b from the bottom, in righthand corner (blackfaced type).
b always takes the name of the declination, if you care to name it, which is convenient but not dispensable.
To Find the Value of C from b and L
Case I. d and L contrary names: add; C=b + L.
Case II. d and L same names: subtract; C=bL.
C=b + L or bL is always less than 90° and the tables are entered from the bottom, as indicated.
By these two rules C, always less than 90°, is obtained from b and L with the utmost simplicity and rapidity.
To Name Azimuth N. or S. when Reckoned from 0° to 90° E. or W.
If latitude and declination are of contrary names, azimuth will be of opposite name to latitude.
If latitude and declination are of same names, azimuth will be of opposite name to latitude if b is less than latitude, but of same name as latitude if b is greater than latitude.
E. or W. according as heavenly body is rising or setting. (Admiral PureyCust, loc. cit.)
This is the way azimuth is reckoned in all the editions of my inspection tables.
To Name Azimuth when Reckoned from the Elevated or Depressed Pole from 0° to 180° E. or W.
Always take Z from bottom of tables (blackfaced number in the righthand corner), except when b is same name and greater than latitude, in which case take Z, less than 90°, from the top of the tables. (Lieut. A. A. Ageton, U. S. Navy, in H. O. 211). Commander M. R. Pierce, U. S. Navy, in H. O. 209 would always take it from the bottom, but reckoning from the elevated or depressed pole. Commander J. Y. Dreisonstok, U. S. Navy, in H. O. 208 always reckons Z from the elevated pole, combining Z’ and Z”. In our methods this combination is avoided.
The All Log Tangent ± Log Secant Method
In spite of the several short methods devised in recent years, there are still many navigators who prefer to work from their D.R. position. As the navigator’s work at sea consists largely of computing lines of position, this table has been arranged primarily for computation of altitudes and azimuths from the D.R. position.
The time and labor required to work a sight from D.R. position have been materially reduced. In point of time, this method compares favorably with the various short methods.
The distinctive features of this method of navigation are:
(1) The D. R. position of the ship is used for working sights and plotting lines of position. (2) There is little interpolation. (3) The azimuth is most positively determined, i.e., the quadrant in which the body is, is always known a priori. (4) The solution is short, simple and uniform under all conditions. (From Preface of H. O. 211 by Rear Admiral W. R. Gherardi, U. S. Navy, Hydrographer.)
We might add for the all log tangent ± log secant table:
(5) This new table has only 18 pages and the log tangents and log secants are given alongside of each other for every minute of arc from 0° to 360°. Log cotangents and log cosecants are given from below. All logarithms increase from 0° to 90°. Tan from — ∞ to + ∞ and Sec from 0 to + ∞. (6) Strictly speaking no special tables are necessary. Any ordinary set of tables of trigonometric logarithms will do. With these and the basic formulas anyone can devise his own method of solution by combining arcs with logarithms as Professor Souillagouet did, for the first time, in 1891. (7) The disposition of the tables permits us to use the LHA or t from 0° to 360° or t from 0° to 180° E. or W. and Z from 0° to 360°, clockwise, from N. or S., Z from the elevated or depressed poles from 0° to 180° E. or W. or Z from the elevated or depressed poles from 0° to 90° E. or W. (8) The use of tangents divides the table into two symmetrical parts, each indicated by 9 (arcs less than 45°) and 0 (arcs greater than 45°), which, of course, facilitates entries into the tables and avoids confusion between Tan and Sec, which might happen when coSec and Sec are used without the 0. (9) The use of tan and sec leaves the formulas in cosec and sec available for proof or check, very convenient after 6 entries in tables and so many operations. (10) The rules for combining the auxiliary arc b with the D. R. latitude to find C are the same as the centuryold rules, used with the versinecosine or haversinecosine formulas, i.e., same names: subtract; contrary names: add.
General Formulas in Logarithmic Form
By changing the sines into cosecants, the cosines into secants and the cotangents into tangents, and applying logarithms to all formulas, we will have:
Formulas in log cosec and log sec
(1) log cosec a = log cosec t + log sec d
(2) log cosec b =log cosec d  log sec a
(3) log cosec A=log sec a + log sec C
(4) log cosec Z =log cosec a  log sec h
(1) and (2) give log cosec a and b, while (3) and (4) give h and Z.
Formulas in log tan and log sec
(5) log tan b = log tan d + log sec t
(6) log tan a = log tan t  log see b
(7) log tan Z =log tan a + log sec B
(8) log tan A  log tan B  log sec Z
(5) and (6) give b and log tan a, while (7) and (8) give Z and h.
The formulas in their logarithmic form show clearly the operations which have to be performed to pass from d and t to the auxiliary arcs a, b, and C and from these arcs to h and Z required.
The formulas in their logarithmic form also show the following advantages of those in tan and sec over those in cosec and sec:
(a) the perpendicular or auxiliary arc a is common to both triangles and eliminates itself automatically and exactly in the calculations and b and C are the only auxiliary arcs used;
(b) All arcs are better determined by their tangents than by their cosecants and, as their differences are greater, a reduction of volume of the tables to onehalf (18 pages for tan and sec instead of 36 pages for cosec and sec), is permissible;
(c) As we enter the table only once with d the effect on h_{e} of taking d to the nearest minute of arc is decreased (not greater than dΔ cos M);
(d) As t and C are adjusted to the nearest minute of arc, the corresponding sec and tan are exact.
The All Log Tangent ± Log Secant Table
A table of log tangents and log sines, as originally given by Edmund Gunter in his “Canon Triangulorum,” published in London in 1620, is also contained in Adrian Vlacq’s “Trigonometria Artificial,” Gouda, 1633, just over 300 years old, in the first complete set of logarithmic tables, and the present tables were taken from Ligowski’s “Sammlung fünfstelliger Logarithmischer, Trigonometrischer und Naulischer Tafeln,” 4th ed. Kiel, 1900, and checked by De La Lande’s, Houel’s, Fontoura da Costa’s, and Alessio’s tables.
The eminent French astronomer, Jerome de La Lande, in his “Tables de Logarithmes,” Paris, 1805, anticipating the modern conclusions upon the number of necessary decimal places, declared:
Je n’ai mis que six chiffres (with the characteristic) aux logarithmes, parcequ’on n’a pas besoin du 7 quand on ne calcule les angles qu’en minutes et les nombres a 4 chiffres; j’ai à cet égard 50 ans d’experience; et Vlacq, dans sa grande édition de l’Arithmétique logarithmique, en 1628, et dans sa Trigonometric artificielle, en 1633, l’avait compris avant moi, car il avait séparé les six premiers chiffres par une virgule, afin qu’on pût s’en tenir lá dans le plus grand nombre des calculs.
In my book for the first time log tangents and log secants appear together alone in only 18 pages. Log cotangents and log cosecants are found from below and no log sines or log cosines are necessary.
It is always better to transform the formulas in sines and cosines into formulas in cosecants and secants.
The Uniform Method of Solution
The work form shown may be used for the navigational stars, planets, the sun, and the moon.
Typical solutions for the stars, planets, sun, and the moon are shown.
The following steps are taken to obtain the altitude and azimuth in one combined solution:
(1) Find the GHA from 0° to 360° in degrees and minutes.
(2) Apply to the GHA the D. R. longitude, slightly modified in order to obtain the LHA from 0° to 360° to the nearest minute of arc.
(3) Enter on form declination to nearest tenth of a minute and name it N. or S.
(4) Correct h_{s} to obtain h_{o} to nearest tenth of a minute of arc.
(5) Tan or T and Sec or S on the form merely indicate from which column to take the corresponding functions.
(6) Enter table with dec, to the nearest minute of arc and take from Tan column the tabulated number. Enter table from above or below with LHA or t and take tabulated number from Sec column, writing down the Tan found alongside this Sec. Add the first two numbers. (Do not interpolate.) Find b corresponding to Tan and write down the Sec found alongside.
(7) Give b same name as declination. Combine b with the D. R. latitude, slightly modified to give C to the nearest minute of arc, adding b and latitude if contrary names and subtracting the smaller from the larger if the same name.
(8) In following the form the necessary operations to find altitude and azimuth become evident and natural.
The difference between the computed h_{c} and the observed h_{o} altitudes is the altitudedifference “a”; “a” is + and measured from the D. R. position toward the body when h_{o} is greater than h_{c}, and — or away when h_{o} is less than h_{c}.
The Sumner line of position is plotted from the D. R. position, as slightly modified above, as shown on Fig. 2.
In Problems I, II, and III, LHA or t is less than 90° and b is taken from the top of the table.
When LHA or t is comprised between 90° and 270°, as it is in Problem IV, i.e., t is greater than 90°, b is greater than 90°; b therefore is taken from the bottom of the table, in the righthand corner (blackfaced type).
The delicacy with which the azimuth Z may be determined, even when the body is very near the prime vertical, is evident. When b is a little less than the latitude and the same name, Z is taken from the bottom of the table. When b is a little greater than the latitude and the same name, Z is taken from the top of the table. Z is reckoned from the elevated pole.
When b and latitude are of contrary names, Z is taken always from the bottom of the tables and reckoned from the elevated pole.
Problem IV illustrates the case when b is the same name as and greater than the latitude, and Z is accordingly taken from the top of the tables and reckoned from the elevated pole.
A Determination of Position from Simultaneous Altitudes of Sun And Moon
Use of Tan ± Sec Tables
“March 2, 1933 about 2^{h}30^{m} P.M. in latitude 40°37'.5 N., longitude 50°40'.6 W. by D. R., observed simultaneously altitudes of sun’s lower limb bearing southwestward and moon’s lower limb bearing southeastward. Sextant altitude of sun 33°03'.9; of moon 57°31'.5; I. C. +0'.9; height of eye 550 ft.; watch time of observation 14^{h}29^{m}08^{s}; CW 3^{h}22^{m}18^{s}; chro. fast 0^{m}52^{s}.0. Required the fix or true position.” (From the American Nautical Almanac for 1933, taking aircraft’s position found, i.e., the intersection of the two Sumner lines of position, as the D. R. position, in order to show the accuracy of the new method and table.)
Supposing the calculations shown in the American Nautical Almanac to be exact we should have found 0'.0 for both values of “a”. The differences 0'.4 and 0'.1 are within the limits of accuracy accepted and are in part due to the fact we took L 40°37' instead of 40°37.5. There is no need to find a and B within the parenthesis.
This shows how illusory would be any attempt to obtain more accurate results, short of interpolating in all steps of the calculations.
As a partial check, a new calculation of h_{c} could be made by means of the formula
log cosec h = log sec a + log sec C
This figure is a reproduction of one which accompanies the example contained in the American Nautical Almanac for 1933, with the necessary additions for our article. The altitudedifferences 20'.2 and 29'.1 were calculated by means of H. O. 208 (1928 edition, Dreisonstok’s method) by the U. S. Hydrographic Office.
The third edition of H. O. 208, published in 1931, was extended very properly to all latitudes. In the fifth edition of 1935 the data for the values of Lat. 66° to 90° were taken out of the tables.
For further exercise take any one of the positions marked A, C, H, D, and E or any other as the D. R. position and naturally we will find the same Sumner lines of position; ED for the sun sight and CH for the moon sight.
Accuracy of Solution and a Complete Check
The calculated altitude h and azimuth Z are always well determined by means of tangents. With five decimal place logarithms no interpolation is necessary to obtain h and Z to the nearest minute of arc in nearly all cases.
In order to obtain very accurate results without interpolation and within a predetermined degree of accuracy—say one half of one minute of arc, it will be necessary to transfer to the nearest tabular Sec b or Sec Z, the difference between the value of Tan b or Tan Z found and their nearest tabular value, when b and Z are greater than 60° (theoretically when greater than 45°: Alessio’s criterium checked and improved upon by Dr. G. Pesci, in 1910). If the difference is smaller than 7 units of the fifth decimal there is no need to transfer it.
When b and Z are less than 45° the difference in Sec b and Sec Z will always be less than 7 units of the fifth decimal and the usual procedure to obtain Tan a and Tan h is always correct.
Always remember that exactly:
Sec b = Tan b + coSec b and
Sec Z = Tan Z + coSec Z.
Very accurate results can be obtained also when b and Z are greater than 45° by using the following formulas for Tan a and h:
tan a = cot d ÷ (cosec b cosec t)
or
Tan a =coTan d  (coSec b + coSec t)
cot h = tan a sec C cosec Z
or
coTan h = Tan a + Sec C + coSec Z.
They avoid careful interpolation, when more accurate results are desired.
b and Z are always well determined by Tan b =Tan d + Sec t and Tan Z= Tan a + Sec B to the nearest minute of arc, without interpolation.
Table Showing the Accuracy Obtainable with a Log Cosecant and a Log Tangent
Arc—> 
1° 
5° 
10° 
20° 
30° 
40° 
45° 
50° 
60° 
70° 
80° 
85° 
89° 

cosec 
0".08 
0".42 
0".8 
1".7 
2".7 
4".0 
4".6 
5" 
9" 
12" 
30" 
55" 
273" 
sec 
tan 
0".08 
0".41 
0".8 
1".5 
2".l 
2 ".3 
2".4 
2".3 
2".1 
1".5 
0".8 
0".41 
0".08 
cotan 

89° 
85° 
80° 
70° 
60° 
50° 
45° 
40° 
30° 
20° 
10° 
5° 
1°< 
Arc 
Another “Criterium” for Greater Accuracy
In order to obtain more accuracy than the AlessioPesci criterium gives, when passing from Tan to Sec, without finding the arc, the writer proposes putting at the beginning and end of each one of the 18 pages of the Tan ± Sec tables the factor ΔS÷ΔT. Then multiplying it mentally by the ΔT found, we would have a more accurate value of ΔS. The procedure for h and Z would be uniform and universal, except for the final correction to Tan h, in order to obtain a more accurate value of h. As the arcs go from 0° to 90°, ΔS÷ΔT runs from 0.0 to 1.0. One decimal is sufficient. Practically it gives the same accuracy as a rigorous interpolation and avoids the necessity of proportional parts in the tables.
Accurate results within 2".5 (two and onehalf seconds of arc) are always obtainable, if careful interpolation is used, for there is nothing better than a tangent for finding an arc, as the above table shows.
For a study of the extreme accuracy with which h and Z can be obtained from these Tan and Sec tables, see an article by Dr. Guiseppe Pesci in the Rivista Marittima (Italian) for February 1910: “Sul Calcolo delle rette SaintHilaire.”
A check, if desired, can be obtained by means of the following formula:
log sec d + log cosec t  (log sec h + log cosec Z) =0
If instead of 0 (zero) we find the difference to be greater than 24, the calculations of h and Z, according to Dr. Pesci’s reasoning, are in error. For the sun sight we find only 8 and for the moon sight only 1, which shows the extreme accuracy of the calculations by means of the Tan ± Sec tables.
Case II. Problem III. Lat. and Dec.: Same names; C=b  L. Body near the prime vertical.
At 7.31 A.M. on December 31, 1931, the airship U.S.S. Akron was in D. R. position, Lat. 42°10/30" S., Long. 68° 37'48" W. With bubble sextant, observed the sun on the prime vertical: W. 7^{h}31^{m} 00^{s}. CW 5^{h}02^{m}46^{s}. chro. fast 02^{m}46^{s}. h_{s }35°58'.0.
Case II. Problem IV. LHA or t comprised between 90° and 270° (or 90° and 180° E. or W.):
C=b—L.
On May 15, 1931, about 7:36 p.m., the U.S.S. Pennsylvania while in D. R. position Lat. 40°43' N., Long. 68°30' W., observed the star Vega (∞ Lyrae) as follows: W. 7^{h}36^{m}128, CW 4^{h}59^{m}128, chro. slow 1^{m}01^{s}, h, 14°45'.7 Ht. of eye 35 feet. I.C. 0'.0.
Simplifications
Certain simplifications will become apparent with the continued use of the table.
When entering the table with LHA (or t) and C and Z always take functions from both Tan and Sec columns in one entry.
When the LHA or t is exactly 90° or 270°, b is also 90° and then a cannot be computed by formula log tan a=log tan t  log sec b; but by adding or subtracting 1' from t, this makes it possible to follow the same procedure. However, a is then practically equal to 90° d and does not have to be computed.
When the observed body is near the prime vertical (say Z over 75°) there will be a saving of figures, if we find h from
coTan h = Tan a +Sec b +coSec Z.
This last formula is simply a combination of the two logarithmic formulas:
Tan Z = Tan a +Sec B
and
Tan h = Tan B  Sec Z.
When Z is exactly 90° or 270° or nearly so, and C less than 30', as it is in Problem III, the calculations need not go any further than Tan 0 13808 (the log tan a) because h_{c} is then the complement of a, and Z is exactly or practically equal to 90°. When d is also small, less than 1°, a is then equal to t and no calculation is necessary to find h and Z.
Advantages
The calculations might look a little long to one not accustomed to the tables, but since the process is absolutely uniform and universal, and since a good D. R. position is hardly ever over 10' distant from the true position, we can always find an accurate altitude and azimuth to the nearest minute of arc.
The following advantages are claimed for the Tan± Sec method and tables:
(a)The tables are very simple and accurate, only 18 pages, giving h and Z to the nearest minute of arc. A very accurate check of the gyro or magnetic compass is also obtained, as Z is always found to the nearest minute of arc.
(b) There are no interpolations for arcs to the nearest minute of arc.
(c) The process is general or universal, straightforward, and uniform throughout. There are no limitations whatsoever, as to Dec., Lat., H.A., Alt. or Az. (All others have their weak spots, their failures, their complications, their limitations and their inaccuracies.)
(d) The use of an accurate D.R. position always gives a small altitudedifference.
(e) An observation of altitude will always give an accurate “point Marcq”—the most probable position of the observer when only one observation of altitude is available.
(f) A simultaneous observation of azimuth will give a fix (an accurate determination of the true Lat. and the true Long, of the observer), by direct calculation. See page 1727 of the December, 1936, Proceedings.
(g) The disposition of the log tan and log sec (and also the log cotan and log cosec) facilitates the use of the LHA or t from 0° to 360° or from 0° to 180° E. and W. and also the finding of b when it is comprised between 90° and 180°. When the LHA or t is comprised between 0° and 90° or 270° and 360° the tables are entered from above and b, less than 90°, is taken from above. When LHA or t is comprised between 90° and 270° or between 90° and 180° E. or W., we enter the tables from below and b, greater than 90°, is also taken from below—the blackfaced figure in the righthand corner.
As C is always less than 90°, the tables are entered with C from below or C is taken from below. By taking B =90° — Cone can always enter the tables from above.
(h) The cost of the tables when printed in English will be low. The Brazilian edition, in Portuguese, is available and costs, in Rio de Janeiro, only 10$0 (ten milreis), about 59 U. S. cents, in a strong paper cover.
(i) The method and the tables are as old as log tangents and log secants themselves, i.e., as old as Edmund Gunter’s “Canon Triangulorum.” 0') Rule—the same as the centuryold one: contrary names: add; same names: subtract.
(k) Working from an “assumed D. R.” position simplifies and renders more exact the solution.
(l) Logarithms are added and subtracted alternately for the azimuth Z and the altitude h.
(m) No other process will always guarantee these accurate results.
As I have given public notice to all my Captains that I expected implicit obedience to every signal made under the certain penalty of being instantly superseded, it had an admirable effect as they were all convinced after late gross behaviour that they had nothing to expect at my hands but instant punishment to those who neglected their duly. My eye on them had more dread than the enemy’s fire and they knew it would be fatal. No regard was paid to rank. Admirals as well as Captains, if out of their station were instantly reprimanded by signals or messages sent by frigates: and in spite of themselves I taught them to be what they had never been before—Officers: and showed them that an inferior fleet, properly conducted, was more than a match for one far superior.—Rodney to Lady Rodney
* From Sumner’s Method, by Admiral Sir H. E. PureyCust, R.N., K.B.E., C.B., Hydrographer of the British Navy, 190914. 2d edition, London, 1928.
1. These formulas were given by the writer in their present form for the first time in the Naval Institute Proceedings, June, 1908; tan, sec, and cosec are the natural functions; Tan, Sec, and coSec their logarithms, as indicated by Gunter in 1624.
2. A Navegaçāo Hodierna com Logaritmos de 1633! (Aérea, Maritima e Radiogoniometrica) com as neces sarias TabuasLinotipadas. Supplementó da Revista Maritima Brasileira de SetembroOutubro de 1933. Rio de Janeiro. 2d edition in January, 1934, 3d edition in January, 1935.