It has been said “The easiest method is the one that was first learned.” That is true for the person who is contented to keep in the rut, but for those who are inquisitive, progressive, or not satisfied with what they know, but want to know more, it is not true.
It is also said there is nothing new in navigation. This is true if it refers solely to the solution of the astronomical triangle, but when it comes to the methods of solution and the application of the parts found to the practice of navigation then it is not true.
Captain Sumner, as we all know, discovered the fact that if in working a time sight for longitude two or more latitudes were assumed, the resulting positions, if plotted on a Mercator chart, would all lie on the same straight line. This was really a discovery for it came when he was trying to fix his position, entering the English Channel during foggy weather. This method is now known as the chord method of finding a line.
The French Admiral Marcq St. Hilaire brought into general use in the French Navy his method of finding a line, which we now know as the tangent method.
It is well understood that it is impossible with a single sight of a heavenly body to fix exactly a ship’s position. This can be easily explained by the fact that, after leaving a known spot, the current, course, and speed can only be approximately known, therefore the position by dead reckoning becomes more and more uncertain as the time and distance increase. In the astronomical triangle, when a heavenly body is observed, we have with reasonable accuracy the altitude and the declination, but no other data, for if we use the time sight formula, latitude has to be assumed; if we use <f>" <j>', the longitude; and, if the altitude formula, the latitude and longitude.
The use of the compass azimuth cannot be considered at all owing to the imperfection of the instruments of observation and lack of knowledge of the exact variation at any particular point. It might be said that in case the gyroscopic compass becomes as perfect as it bids fair to be we may be able to use the compass azimuth, Z, in finding our position.
The older officers in the service were taught at the Naval Academy the method of finding a Sumner line by means of two time sights, using two assumed latitudes. As the years rolled by, the later graduates were taught that the Sumner line could also be obtained by working one time sight and taking the azimuth of the body from the tables; and all were taught that these methods were only applicable when the hour angle was greater than 22i°. The present graduates are not only taught the above, but also that the line can be obtained by assuming a latitude and longitude, finding the altitude, li, by formulas and the azimuth from the tables or from a diagram or nomogram. All were taught that by assuming two longitudes and by using the <£" </>' formulas a line and position could be obtained when the hour angle was less than 22\°.
Now, to sum up the methods of finding the coordinates of a line, we can use:
 Time sight formula assuming two latitudes, result two longitudes. Chord method.
 Time sight formula assuming one latitude, result one longitude and azimuth. Tangent method.
 <f>" 4>' formulas assuming two longitudes, result two latitudes. Chord method.
 <f>" <f>' formulas assuming one longitude, result one latitude and azimuth. Tangent method.
 Altitude formulas assuming latitude and longitude, result one altitude and azimuth. Tangent method.
These five methods are rigorously exact as far as the solution of the triangle is concerned. The formulas as obtained directly from the triangle are as follows:
 sin = Vcos j sin (s — h) sec L cosec p
 sin Vcos .y sin (s — h) sec L cosec p cos \Z — Vcos .y cos (s—p) sec L sec h
 tan cj> —cot d cos t
cos <£' = cos <f> sin li cosec d L — 90^{0} — (<£±<f>')
 tan </> = cot d cos t
cos (f>^{r} = cos (p sin h cosec d
L — go° — 0±<£')
cot Z = cot t cos (L + <£) cosec <j> or sin Z = cos d sec h sin t (5) sin h — cos (L~d) — 2 cos L cos d sin^{2} U tan <£ = cot d cos t cot Z = cot t cos (L + <f>) cosec <f>
The above formulas are the original ones from the triangle and can be solved with any log table. Now let us see what changes in these have been made to permit an easier solution of the problem.
 This formula has been changed to
_{haver }_{t} _ V1ia~ver~[^~~ UTd) \ liaFer fi+TL  d) ]
cos L cos d ? '
and a special table of haversines has been devised.
 haver t= Vhaver ~z— (L — d)  haver(L — d) Jsec d sec L haver Z = Vhaver />— (h — L)]haver[/>+ (h — L) Jsec h sec L
 (j> = go° —
tan <t>" = tan d sec t cos cj>' = sin d>" sin h cosec d L = cf>" ± <j>'
(4)
tan <^" = tan d sec t cos — sin 4>" sin h cosec d L = <f>" ± 4>'
cot Z = cot t sin (</>" — £) sec
 haver 6=cos L cos d haver t haver ir = haver (L~cO+haver 0
haver Z= Vhaver[/>— (h — L)“]haver[/)+ (h — L) Jsec h sec L
The above formulas simplify the actual computation considerably and therefore should be used in place of the first set. The haver sine tables then are the first help in the solution. The parts of the triangle which have to be found with the greatest accuracy are t, L and h, because it is upon them that the position of the ship depends. The other part Z is at present only used for determining compass errors.
As the instruments provided on board ship, the compass and azimuth circle, are scientifically crude, not reading closer than even degrees, although interpolation can be made to 15' of arc, it can readily be seen that tables and diagrams of reasonable size can be constructed so that the value of Z can be found well within the error of 15'. Therefore the next step in the simplification of working out a line of position would be the construction of the above tables and diagrams, and these have been constructed where the given parts of the triangle are L, d and t. These eliminate all formulas for finding Z.
In (i) we have no need of Z.
 The data gives us L and d and the solution t, therefore
the tables can be used for Z.
 There is no need of Z.
 The data gives t and d, the solution L, and the tables can
be used.
 The data gives t, d and L, and the tables can be used.
Tbe azimuth tables were therefore the second short cut or help.
The next step was the construction of diagrams and nomograms for the finding of the azimuth or Z, and thus do away with the azimuth tables. This has been done.
It would be of the greatest use if it were possible to construct tables so that in (1), (2), (3) and (4), having obtained tbe data, we could look in a table and pick out the corresponding answer t. or L, the local apparent time, or the latitude.
But stop and consider not only the tremendous amount of work that would be necessary to calculate such tables, but the size of the volumes when completed. About thirty million triangles would have to be solved for one latitude, d and h varying T between o° and 90°, and about three hundred thousand million triangles if L varies T between 0° and 90°. We can therefore say that these tables are impracticable and that this method must be one of calculation solely.
The next improvement to facilitate the working up a line would be to have tables to find h and Z in place of the formula (5). Z is already arranged for in the azimuth tables and diagrams.
Tables for which li could be exactly picked out would be large and unwieldy, therefore we must consider the advisability of using tables when the value of li can only be approximately obtained.
It is necessary to determine the nearest approximation that will answer the requirements of navigation. We have seen that ±15' are sufficient in the case of Z, owing to the crude instruments we use. In measuring the altitude of a celestial body at sea we use either a sextant reading to 10" (high grade) or one reading to 30" (surveying). The latter is more commonly used. Therefore in the first case we can have an error of ±3" and in the latter ± 15".
The correction for dip without refraction can of course be accurately determined, and for every change of 3 feet in the height of the eye of the observer, from 10 feet to 50, we have a change in dip of from 30" to 12". If the observer’s platform is absolutely steady then the dip, neglecting refraction, would be known, but as the ship is usually rolling and pitching more or less, thus giving an unsteady platform, it can be safely said that 6 feet would be an average change. This causes an error of from ±12" to ±30". Combining the least of these with the least of the sextant we have ±17". Assuming a personal error of ±15", which is very small, and an error in the refraction, which of course affects dip, of ±15", we find that the error of an observed altitude, no matter how carefully taken, will be ±47".
Therefore if tables can be constructed in which, when entered with L, d and t, the value of h can be picked out to the nearest five tenths (0/5) of a minute of arc, these tables will be well within the limit of error. Such tables have been constructed and both h and Z can be obtained with ease. There are a number of them. Ball’s tables are straight away, entering with the values of L, d and t given in the data, h is by interpolation picked out, and, by renaming the columns, Z is obtained.
Other tables are Guyou’s Nouvelles Tables de Navigation, Pes’ LeRette di Posizione, and Aquino’s Altitude and Azimuth Tables.
These tables now bring us to a new phase of the problem, viz.: “ How close must the data be to obtain results within the limit of error of observation ? ”
Navigators agree that when two observers take an observation of the same object at approximately the same time, i. e., within a few seconds, using their own observed altitudes and G. M. T. of their observation, their results may differ by at least ip of arc or 6^{s} of time.
Therefore if tables are devised that will permit the finding of It within ±1'; they will answer the purposes of navigation.
In finding h from Ball’s tables, interpolation, similar to the method we use in picking out Z from our azimuth tables, is employed, and the same for Z. This requires three multiplications and a combining of results in each case, making six operations in all.
In Guyou’s tables the effect of the curvature of the earth is taken into account, two volumes are necessary to work from, and the method is of too great a refinement for the navigator.
In Aquino’s tables two openings of the book, two multiplications and two simple formulas are necessary to get the result of It and Z.
From all this it is seen that the trend towards simplifying the work has been along the line of recognition of the fact that results cannot be any closer to the true result than the data issued is in error on the true data. 
Now let an examination be made of the error in results caused by the taking of data approximately true. To do this an observation of the sun will be worked out in the most rigorous manner in order to compare the other results with it.
h m s
W. 24315 G 36 48 00 C. — W. 35220 1104
C.6 35 35 0365904
C. E. +12 16 G. M.T.6 23 19 MCI127
Eq. t. 5 27.4
I.C.+ 200 D. 5 48 R.— 1 18 P.+ 7
S.D.+1603
+ 11 04
Deck 2 34 02.5 6 15.0
d 2 40 17.5N
G. A. T.6 17 51.6
Eq. t. 5^{m}3230 49° 5 27.40
H. D. .767 6.389
4.602
.230
61
II. D. 58.71 H.D. 58.57
U4
.006
.02
58.69
6.39
352.14
17.61
528
3750
On March 27, 1912, in D. R. Lat. 39^{0} 45' N., D. R. Long. 52^{0} 30' \V., observed the sextant altitude of O to be 36° 48'. I. C.+ 2' 00". Height of eye 35 feet. W. 2^{h}_{43}^{m} 15^{8}, C.W. 3^{1}' 52” 20^{s}. Chro. error fast or ( + ) I2^{m} 16^{s}.
7
4.90
 h  36°  59'  04" 



 sec.  .09757 
 L  39  45  O O 
 sec.  .11416 
 sec.  .11416 
 P  87  19  42.5 
 cosec.  .00047 



 2S  164  03  46.5 






 S  82  01  53.2 
 cos.  9.14181 
 cos.  9.14181 
5  h  45  02  49.2 
 sin.  9.84984 



S—pt  )  5  17  493 

 9.10628 
 cos.  9.99814 

 h  m  S 




 9.35168 
L. A.  • T.  2  47  31.0 
 sin.  9.55314 

 
G. A.  T.  6  17  516 






Long.  3  30  20.6  w  or 52'  ^{5} 35' 09"  W 

 
 4Z  6i°  42'  05" 



 cos  ;. 9.67584 
Z  N  123  24  10  w  or S 56  ^{0} 35' 50"  W 


With the same data, but correcting the altitude by the use of Table II, Muir, or Table Aquino, p. 5, we have:
I. C. + 2/0 + 8.8 + 10. 8
© 36° 48/0 Aquino 10.8 ©36 58.8 L. A. T. = 2h 47^{m} 33^{s}
© 36° 48' 00" I. C. + 2' 00" Corr. Muir 11 04 9 04
© 36 59 04 + 11 04
Which differs by 2s from the exact solution, or 0/5 of arc.
Which is the same as in exact solution.
L. A. T. = 2^47™ 31s
Therefore we can see at once that these tables simplify the work and do not affect the accuracy of the result.
With the same data and taking the nearest second of declination, calculating it as follows :
Deck 2^{0} 34' 02"5 H. D. 58.71
5 5^{2}3 6
d ^{2} 39 55 N 352.26
it is found that
L. A. T. = 2^{h} 47^{m} 30^{s}
This shows a difference of I^{s} in time or 15" in arc.
With the same data, taking d to the nearest minute, we find that
L. A. T. = 2^{h} 47^{111} 30^{s}
Now take h to the nearest minute, and
L. A. T. = 2^{h} 47^{m} 31“5
Now take the values of li and d to the nearest minute. The result is as follows :
L. A. T. = 2^{h} 47™ 30? 5
^=S 56° 35' 30" W.
From this it can be seen that the difference in the local apparent time in this example is but 05 of time or 7^5 in arc.
The books ordinarily supplied to a navigator in our service will not permit of closer work than that given in the first solution. It is evident then that a navigator need but take his G. M. T. to the nearest hour when multiplying the H. D. to get the correction to d, and then take d to the nearest minute of arc. It is also evident that h to the nearest minute of arc is sufficiently accurate.
Having now found that the results from the rigorous and approximate solutions differ so slightly, the method used by Lieu
tenant Radler de Aquino, of the Brazilian Navy, can be discussed. Briefly stated his method is to find the h and Z of a celestial body at some place within thirty miles of the D. R. position, and the problem is first to find the latitude of this spot, then the longitude, and h and Z any time after the latitude is found. This is explained in the United States Naval Institute Proceedings for December, 1908.
The example used above will now be solved by this method.
W. 2h 43m 15s  O 36° 48/0 I. C. + 2/0  Decl. 2° 34/0  H. D. 58.7  
C—W. 3 52  20  10.8 + 8.8  59  6. 
C. 6 35  35  © 36 59 + ^{I0}^{8}  d 2 40 N  352 
C.E. + 12  l6 



G.M.T. 6 23  19  Mcli. 27  Eq. t. 5^{m} 3^{2}^{s}3  H. D. .767 
Eq. t. S  28 
 4.6  6. 
G. A. T. 6 17  51 
 5 28  4.6 
G. H. A. 94° 27'  45' 



Long. 52 30  00  W 


t+ 41 57  45  a = 42°00' b= 3^{0} 35'  1.36 ^{1}  =42° 03' 

 90+6 = 93 35  26. G. H. A.  94 28 

 # L 39 35N  8.2 Long.  52 25W# 
comp. 0=36°  5/  Z = S 56° 52' W B = 54°  27.2 

obs.©= 36  59 
 354 

 2'  S. +w.  .03 

The true azimuth rigorously worked out is S. 56° 35' 50" W., a difference of 16' which is 1' greater than the 15' error, but well within reasonable limits, as azimuths laid off on charts can have a greater error than that and still give reasonably correct results.
Assuming that the true latitude is 393824 N. the position determined from the rigorous solution will be
Lat. 393824 N. Long. 522930 W.
The position determined from Aquino will be:
Lat. 393824 N. Long. 523100 W.
The results therefore of the Aquino method of finding a line of position are within the limits of error. The endeavor has been made to choose an example that when worked both ways would
have as great a difference as possible, and the writer has no doubt that there are other examples that would give a greater difference, but doubts whether they would ever occur in practice.
The element of time of solution has not been touched upon, nor have chances of error in picking out functions, logs, etc., or the amount of work; however, the following table gives an idea of these:
Method.  Time.  No. of Figures.  Symbols or Signs.  No. of Books.  Opening of Books.  Chances of Error. 
Time Sight Formula....  . . 8  150  28  2  7  23 
Altitude Cos Formula. .. .  ? ? 9  153  37  2  9  24 
Altitude Haver Formula.  7  139  38  2  8  24 
Aquino ....................  .. 4  IOO  25  I  2  13 
Common to all.........  •• 5  126  29  2  3  18 
These figures may vary slightly in different examples.
The question now arises, is it safe for navigators to use these short methods, which do not give exactly the same results as the rigorous solution. The writer considers that, taking into account every condition that surrounds an observer at sea, a moving platform, uncertainty in the time of the standard meridian, unknown refraction, inaccuracy of the charts on which plotting is done due to shrinkage of paper after printing and small compass roses, a navigator is justified in using the Aquino method, and can say that the position so determined is fully as accurate as the position determined by the rigorous method, and that when time is taken into account the shorter and quicker the method the more valuable it is.
An endeavor has been made in this article to show that, starting from the original formulas as used by earlier navigators which gave accurate mathematical results but inaccurate answers to the question of the ship’s position, the tendency has been to recognize this latter fact, and to introduce short cuts, which decrease the amount of work and yet give as accurate answers, and that these changes have been along the lines of, first, rearrangement of formulas ; second, construction of tables; third, construction of diagrams ; fourth, tables again; and fifth, graphic representation on charts, instead of mathematical work on paper. The method of Aquino shown is in the writer’s opinion the best of any that have been published.
The writer considers that this article is for navigators, and not for students learning the rudiments of navigation expecting to advance in knowledge of the subject. The student must be taught to be accurate, and all short methods, unless used carefully and fully understood by those who use them, are liable in time to make the users careless, and cause them to depend upon a rule and not upon their knowledge of the “ reason why.” It is hoped that other methods may be carefully examined and worked, to see if they are not better than the one finally worked in this article. The main things to be kept in mind are accuracy of tables, compactness and small number of books, and ease of solution.
Let the navigators speak out and tell the service at large their experiences in using the different methods of finding a line of position.