Lieutenant Radler de Aquino, B. N.—I feel deeply honored by the Board of Control’s kind invitation to discuss Commander Marvell’s ably presented paper on the development of the method of finding a line of position at sea and I think he has rendered to all readers of the Proceedings and to all navigators a great service in showing by comparative study the advantages of the use of modern methods of navigation.
While I still believe my “Altitude and Azimuth Tables” to be the “simplest and readiest in solution ” I think all those interested in the advancement of the science of navigation will welcome his final suggestion asking navigators to “ speak out and tell the service at large their experiences in using the different methods of finding a line of position.”
Since the publication of my article in the December, 1908 issue of the Proceedings, describing my tables, many improvements were introduced and a second edition of them has just been published in London with further simplifications and improvements.
As Commander Marvell is on the Asiatic Station this new edition has not yet reached him and the development he uses on page 218 is still the one to be found in my first edition.
The small differences he found in the azimuth and in the longitude are easily accountable and are not due to the tables themselves.
First, when he states there is a difference of 16' between the two azimuths, I believe he overlooked the fact that the position used in the example on page 216 is not the same as used on page 218. The azimuth S 56° 35' 50" W is from Lat. 39^{0} 45' 00" N and Long. 52° 35' 09" W, while the azimuth S 56° 52' W taken from the tables corresponds to Lat.. 39^{0} 35' and Long. 52° 25' W.
The difference of one minute and a half of longitude, while not important in a latitude as high as 39^{0} 38', is due to the fact that he has rounded up all the data to the nearest minute of arc.
Of course, as I have stated in my tables, if greater accuracy is desired the data must be taken to the nearest tenth of a minute of arc, and curiously enough if we take the Greenwich hour angle, the declination, the hour angle, the latitude and the altitude to the nearest tenth of a minute of arc we will find exactly the same longitude as Commander Marvell found by the rigorous solution.
Thus with G. H. A. = 94^{0} 27'.7, d = 2° 4o'.3, t = 42^{0} 3'.2, L — 39° 35'.8 and h — 36° 58'.8 we will find by means of the tables for Lat: 39^{0} 38'.4, the Long. 52° 29'.S.
The writer fully agrees with Commander Marvell's statement on page 219 “that a navigator is justified in using, and can say that the position determined by the method shown above, is fully as accurate as the position determined by the rigorous method, and when time is taken into account the shorter and quicker the method the more valuable it is.”
A slightly improved method of finding the altitude and azimuth by means of my tables has been developed recently whereby it is not necessary to interpolate and find the true values of b and t for the true value of declination d, as explained on pages xv, xxi, and xxv of my tables. It is sufficient to find only in column a the values of b (generally a whole degree) and of t that correspond to an approximate value of d. This approximate value of d is always the tabular value nearest to the true value, b is combined in the usual way with La. (also generally the whole degree nearest the dead reckoning latitude) to find C, and h' and Z' arc found corresponding to the a and C. Now as h' and Z' are for the tabular value of d, we must correct them for the difference Ad between this tabular value and the true value of the declination.’ We know from page xvii that a change of altitude Ah for a given change of declination Ad is given by the formula: Ah = Ad cos M, where M is the parallactic angle. If we call, in Fig. 2 on page xii, the angle mMP : a and the angle mMZ: P, we have M — a + p.
The value of a is found on the same line with b, d, and t (a being practically the same for all three values of a). In the same way p is found on the same line opposite C, h, and Z.
However, instead of finding C with La. and b, it is better to find La. from b and C, as explained below.
The working out of Commander Marvell's example will show the great advantage of this improvement.
Pages no and in. 2d ed., 1912. a = 42° 0' G. A. T.= 6^{[1]}' 17"" 5i‘ = g4° 27/7
b =4°.................. d'= 2° 58' ...f_{A}.= 42 4 W.............. «=87.°3
d = 2 40.3 N Ga. = S2° 23/7W
C —36° ........  A d= — 17.7' ...... It' — 36° 57' ...Z' — S 56° 52'W_______________________  ..... P — 47.°1 
La. =40° N  ^{Alt}~ — 13.5  M = 40.° 2 
 h_{A}. = 36° 43.'5 ll = 36 58.8 

h — lu.= + 15/3
Note.—Numbers taken out of the tables by inspection are blackfaced in order to distinguish them from data given or found.
In addition to the formulae given on page xxviii for finding L_{A}. with b and C, we have added those for finding M with a and p.
t <90‘
d and La. same name
f La. < b : La. = b — C and M = a.J_{r}p
( La. > b : La. = b + C and M=a. — 3
.(> 90°....... : La. = C + b and M = P — a
d and La. contrary names....... : La. =C— b and M—a + p
When t > 90° the sum C + b > 90° also, and we must subtract it from 180^{0} to obtain L_{Am}.
A simple inspection of these formulas shows that no difficult rules are necessary with this new process. A knowledge of the approximate value of La. is always known by deadreckoning, and, therefore, we can immediately find, in view of the fact that b and La. are generally whole degrees, the value of C that combined with b will give us Ij. The tabular value h' nearest to the true altitude h shows us opposite it also the value of C.
The formulae show also that when we subtract b and C to find La. we must add a and p together to obtain M. When ,we add b and C to find La. we must subtract a and (5 from one another to obtain M.
The “altitude correction” A/i = I3'.5 is given immediately by our altitude correction table,^{[2]} where we enter at the top with Ad — 17'.7, and with M = 40°.2 on the left hand side. If M is greater than 90°, enter the table on the right hand side. The correction has the same sign as d — d' or Ad when M is smaller than 90° and the contrary sign to d — d' or Ad when greater than 90°. .
In this way altitude and azimuth are found by means of simple mnemon ical rules without interpolating.
Assuming now, as Commander Marvell did, that the true latitude is 39^{0} 38'.4 we would find the true Long. 52° 29'S W which is exactly the same found by the rigorous solution. These exact results are obtained in spite of the fact that the exact values of d', ts., and W are respectively 2° 58'.3, 42^{0} 04/.2 and 36° 57'.4, whereas the tables only give them to the nearest minute of arc.
If we had tabulated and used these exact values of d', t_{A}. and h' we would find Long. 52° 29'.i W instead of 52° 29'.5, a difference of fourtenths of a minute of arc from the rigorous solution. This shows it is doubtful whether it would be wise to give data in the Tables to the nearest tenth of a minute of arc.
Commander G. W. Logan, U. S. Navy.—Apropos of the contention that, by reason of the unavoidable existence of certain errors which prevent absolute accuracy in the determination of position at sea, it is admissible to open the door to further errors by resorting to approximate solutions, it will be appropriate to quote the following from Chapter XVI of Bowditch’s “American Practical Navigator” (revision of 19023) :
“ It is a wellrecognized fact that exact results are not attainable in navigation at sea; the chronometer error, sextant error, error of refraction, and error of observation are all uncertain; it is impossible to make absolutely correct allowance for them, and the uncertainty increases if the position is obtained by two observations taken at different times, in which case an exactly correct allowance for the intervening run of the ship is an essential to the correctness of the determination. No navigator should ever assume that his position is not liable to be in error to some extent, the precise amount depending upon various factors, such as the age of the ‘ chronometer rate, the quality of the various instruments, the reliability of the observer, and the conditions at the time the sight was taken; perhaps a fair allowance for this possible error, under favorable circumstances, will be 2 miles ; therefore, instead of plotting a position upon the chart, and proceeding with absolute confidence in the belief that the ship’s position is on the
exact point, one may describe, around the point as a center, a circle whose radius is 2 miles—if we accept that as the value of the possible error—and shape the future courses with the knowledge that the ship’s position may be anywhere within the circle.
“ It is on account of this recognized inexactness of the determination of position that some navigators assume that the odd seconds may be neglected in dealing with the different terms of a sight; the average possible error due to this course is probably about one minute, though under certain conditions it may be considerably more. It is possible that, in a particular case, the error thus introduced through one term would be offset by that from others, and the result would be the same as if the seconds had been taken into account; but that does not affect the general fact that the neglect of seconds as a regular thing renders any determination liable to be in error about one minute. Those that omit the seconds argue, however, that since, in the nature of things, any sight may be in error two minutes, it is immaterial if we introduce an additional possibility of error of one minute, because the new error is as liable to decrease the old one as to increase it; but the fallacy of the argument will be apparent when we return to the circle drawn around our plotted point. The eccentricity of the sextant may exactly offset the improper allowance for refraction, and the mistake in the chronometer error may offset the observer’s personal error, but unless we know that such is the case—which we never can—we have no justification for doing otherwise than assume that the ship may be any place within the 2mile circle. If, now, we increase the possible error by I mile, our radius of uncertainty must be increased to 3 miles, and the diameter of the circle, representing the range of uncertainty in any given direction, is thereby increased from 4 to 6 miles
“There is a more exact way of defining the area of the ship’s possible position than that of describing a circle around the most probable point, as mentioned in the preceding article, and that is to draw a line on each side of each of the Sumner lines by which the position is defined, and at a uniform distance therefrom equal to the possible error that the navigator believes it most reasonable to assume under existing conditions; the parallelogram formed by these four auxiliary lines marks the limit to be assigned for the ship’s position; this method takes account of the errors due to poor intersections, and warns the navigator of the direction in which his position is least clearly fixed and in which he must therefore make extra allowance for the uncertainty of his determination.”
It is not maintained that, by adhering to exact methods of computation, an absolutely certain position results; but it is maintained that the position thus determined is the most probable one and is surrounded by a minimum of doubtful area.
The degree of exactness with which the work is to be carried out may depend upon attendant conditions. At sea, making a long passage, the. daily position serves only to show that the ship is making good her course and to afford a departure for the next day’s reckoning; in this case a radius of uncertainty of two or three miles is negligible. When, however, the position is to be used for a landfall or for governing the movements of a
vessel in scouting work or in deepsea sounding, greater accuracy should be sought.
It would be a happy compromise on this question if the navigator were supplied with tables of trigonometric functions which gave values of logarithmic and natural sines, cosines, etc., for every quarter minute of arc (15 )• With such tables, by assuming values of angles to the nearest of the quarter minute divisions and omitting interpolation, an error of not more than 7" would be involved in any term of the computation; this would save the time as well as the chance of error incident to interpolation and the results would be sufficiently exact for. all ordinary purposes of navigation. Such tables are already in existence, and it is hoped that they will shortly be made available to our navigators by being incorporated in a publication of the Hydrographic Office.
In enumerating the various formulas available for the computation of altitude, Commander Marvell has omitted the following, which possesses many advantages:
sin h err sin L sin d + cos L cos d cos t.
To show the simplicity of computation by this formula, an example follows, wherein the data are the same as those in the example worked by Commander Marvell as an illustration of the Aquino method:
L  39° 35' ’ sin 9.80428  cos 9.88688 
d  +2 40 18 sin (+) 8.66850  cos 999953 
t  42 03  cos 9 87073 
A  + .02970 log ( + ) 8.47278 

B  + .57166  log 975714 
 TN. sin .60136 

li  LAng. 36° 58' 03" 

It will be seen that this formula is well adapted for expeditious and accurate computation; the form is simple and chances of error are slight.
In this example, taking the features tabulated by Commander Marvell in his comparison of other methods, we have for the computation of the altitude by this “ sine formula ” : time, 4^2 minutes; number of figures, 76; number of symbols or signs, 20; number of books, 1; number of openings, 6; chances of error, 11. This will be seen to compare most favorably with the other methods, even those in which the results rendered can be less depended upon for accuracy. To make the comparison complete, it is fair to take into consideration the determination of the azimuth, for which, if the diagram is used, one may add about half a minute of time, five figures, and four chances of error.
The selection of the method of solution will always be a matter of personal preference with the individual navigator, and will depend frequently upon what he has “ been brought up on ” and finds himself most accustomed to. If the new navigation is preferred to the old, an almost equal choice of solutions lies between the Aquino method, the haversine formula, and the sine formula to which reference has just been made. The lastnamed, for some reason, is not frequently found in works on navigation, and it is for the purpose of bringing attention to it as at least one of the best methods that this comment is submitted.
Lieut.Commander W. S. Turpin, U. S. Navy.—I have read with great interest Commander Marvell’s paper, “ A Study of the Development of the Method of Finding a Line of Position.”
The conclusions derived by the author agree with those that I have arrived at after duty for something over a year as navigator of the Delaware.
When I was first ordered to the ship I began to brush up my navigation and investigated the various methods that had come into use in the last few years. At that time I was not acquainted with Aquino’s tables and I decided to use a rigorous method that had been described in the Naval Institute about two years ago, in which the haversine was used to the exclusion of all other functions. This was facilitated by having on hoard Inman’s Tables of Haversines and shortly after joining, receiving the 1911 edition of Bow ditch’s LTseful Tables in which this haversine table had been incorporated.
In a conversation with the author, mention was made of the rapidity with which sights could be worked out by using Aquino’s tables, and I was so impressed by the advantages of their use which the author had pointed out to me that T at once began to investigate their merits. By working out several sights in the usual way and also by the tables I soon became convinced that the tables were accurate, as my results agreed, and the time required to work sights with the tables was very much shorter and the chances of error were, in my opinion, much less.
To show the rapidity with which sights can be taken and the line of position determined and plotted it is only necessary to cite the case where four sights of stars were taken and the lines plotted within half an hour. This included the determination of two of the stars’ names, about which I was uncertain. On an average you should be able to take a sight and work it out in from five to seven minutes.
I am firmly convinced that any navigator who once uses these tables will not use anything else in the present state of the development of the art of navigation. They are all the author claims for them, easy of application, rapid, and above all accurate.
Lieut.Commander W. R. Gherardi, U. S. Navy.—Those navigators with whom I am acquainted and who have used Radler de Aquino’s method are greatly in favor of it and work all their lines of position by the use of his tables.
I am of the opinion that its simplicity will appeal not only to the naval navigator but particularly to merchant officers. Having been an instructor of navigation in a nautical school ship, I am somewhat acquainted with the slim educational foundation upon which the merchant officer frequently must base his navigational structure.
There is in course of preparation at the Naval Observatory a book to assist the navigator in finding his position at sea. It is hoped to acquire the rights to publish Radler de Aquino’s tables and to give simple forms, some of which are now being tried out by our cruising ships.
It is believed that the part of practical navigation connected with finding the ship’s position at sea will be greatly simplified, that almost all the methods, except for meridian sights, now in Bowditch will eventually be dropped for the simpler methods, that the number of tables to be used with this book will be greatly reduced, and that the azimuth tables and the star identification tables will no longer be required.
Lieut.Commander H. G. Sparrow, U. S. Navy.—I have no new suggestion to make relative to the methods of finding lines of position set forth m Commander Marvell’s paper, further than to say that I confine myself to two types of sights, the Marcq St. Hilaire and the exmeridian. By so doing, greater familiarity is gained with the formulae, and the names of the different elements used need not be written down. I use seconds of arc and half seconds of time, in the data used, on the principle that in some cases all errors may be cumulative and would throw the result out more than the 1' tolerance.
A few expedients to reduce the mechanical work of solving sights ma> be of interest. In comparing watch with chronometer, I compare the watch, which is set to 75th meridian time (this being ship time always on this station), with 75th meridian time. The watches furnished navigators can be regulated to within 4s per day. The error of the watch is always kept less than 30s, and comparisons are made at daylight, 11.00 a. m., and sunset. The watch correction is easily remembered, and is applied mentally to the time of sight when entering same in the work book. This (with the addition of s hrs.) gives G. M. T. at once.
A table of combined altitude corrections for sun and stars, for each deck and bridge, is posted in front of the chart board. This is copied directly from Table 46 of Bowditch’s Useful Tables. By inspection the observed altitude is corrected and entered at once in the work book. This gives G. M. T., and true h without a single extra figure being written.
° At daylight each day a blank form giving all the needed arguments for sun sights, also R. A. M. S., is filled out and slipped in a holder in front of chart board. This makes reference to the N. A. for sun sights unnecessary. This form is printed aboard ship and is here shown:
Dafe........................................ ^{I}9^{1} ? ?
Dec. O’ ' H  H. D.  G. M. NOON Eq. Time to M. T. m s  H. D.  R. A. M. S. h m s 
Dec.  H. D.  G. A. NOON Eq. Time  H. D.  S. S. D. 
0  » 
 to A. T. m s 


The S. S. D. is unnecessary and is never filled in.
Lieut.Commandek E. B. Fenner, U. S. Navy.—In response to the Institute’s request for comment on the various methods of finding a line of position discussed in Commander Marvell’s article, I would like to submit the following points for consideration:
 As to the accuracy of the Aquino method : On page 215, the assumption is made that we have an observation error of ± 47" to start with and hence we might as well have an additional tabular error of ± 1' giving in some cases a total error of ± 1' 47". The observation error is unavoidable, but I do not consider that a valid reason for adding to it an avoidable error of one mile.
A second and more serious objection to the Aquino method lies in the plotting error. With the ordinary altitude formulae we use an assumed position as close to the true position of the ship as possible, very seldom in error as much as 10 miles. With the Aquino method we have to use whatever assumed position the problem gives us, often as much as 40 miles from the ship’s true position.
Every mechanical difficulty of plotting, as inaccurate charts, shrinkage, small rose, slipping parallel rulers and dividers will produce its error in either case; but in the average Aquino sight this error will be about four times as great, due to the greater distances to be plotted. Also a separate assumed position must be plotted for every Aquino sight, thus introducing still further chance of plotting error.
I believe another ± T would be a very moderate allowance for excess of plotting error in the Aquino method over the ordinary altitude formulae, although it would be difficult to prove the truth of this belief.
 As to speed: The actual working out of the Aquino sight is rather more rapid than any other 1 have tried, but I cannot find the difference shown in the table on page 219, if all classes of sights are worked out to the same degree of accuracy and for the same kind of results.
For example take the time sight on page 216, and work for a line of position by the tangent method, using azimuth tables for Z and dropping seconds of arc. The work in obtaining G. A. T., ?©, Dec., and Eq. t is exactly the same for both sights, for there is no reason why we should correct altitude differently in the two cases. In the remainder of the work there are about 87 figures in the time sight and about 73 in the Aquino, for one can easily pick an azimuth from the tables with ai^error of less than 2° without the help of figures and an error of 2° in Z when the assumed and true positions are ten miles apart, would give an error of about ± 20" in position.
The difference in number of symbols is of no importance to navigators who use printed forms, as most of them do.
Again, however, the greatest objection to the Aquino sight on the ground of speed lies in the plotting. If we take a cross of three stars and work them all out by the ordinary altitude formulae, we plot altitude differences and bearings from one fixed point which we assumed at the beginning. If we work these same sights by the Aquino method we must plot, in addition to the original D. R. or assumed position, a different assumed position for every one of the three sights, thereby losing most or all of the time gained in computation.
One other minor objection to the Aquino sight is the double entry tables in which we have to find an approximate coincidence of two arguments, always a longer process than the use of a single entry table.
Lieut.Commander C. C. Bloch, U. S. Navy.—The “ Newest” Navigation Altitude and Azimuth Tables by Lieutenant Aquino, of the Brazilian Navy, is all that the author claims for it—“ The Simplest and Readiest in Solution."
I have been using the Aquino tables for some time, and I have found them sufficiently accurate for all practical purposes, and preferable to all rigorous methods for the following reasons :
 The chances of error are very small.
 The facility of solution.
 The easiness with which error may be detected.
I have worked numbers of sights using the Aquino tables, and checked the solution by the haversine formula; in no case has there been a difference in result greater than 1.3'. In most cases the results have been identical.
In Lieutenant Aquino’s latest book^{8} he includes a “ Ready Reckoner which does away with the necessity of the two multiplications which Commander Marvell speaks of. However, this introduces a book opening, and I prefer making the two multiplications, usually mental, to making one book opening.
Recently, Lieutenant Aquino has devised a method whereby interpolation for the values of b and T ass. is no longer necessary. This brings into use the little used position angle M. While this method does away with interpolation, it necessitates picking out certain auxiliary angles a and P and the memory of certain precepts, all of which I consider more laborious than a simple interpolation.
Using the interpolations, I find it possible to work lines of position in about five minutes from the time of taking the sight.
Below is a form suggested for Aquino’s method; it is thought that such forms could be issued to ships for use, binding them in a loose leaf binder. A third column and back would be ruled for use in case of azimuths, meridian and exmeridian altitudes and for the navigator’s remarks, soundings, bearings, etc.
In case the navigator prefers a method of computation to the tables, I would suggest the use of Aquino’s “New Altitude Tables, where the formulae given below are used for the solution of the spherical triangle; solving for h only :
 log cosec ^{[3]}^{[4] [5] [6]} — P2 log sec L + V_{2} log sec d + log cosec —.
2 ^{2}
 versin (90°— /j)=:versin (L — cO + versin 6.
[Aquino’s new tables have all logs multiplied by io" so that no decimals appear. The tables can be used for these formulae only.]
I find these formulae, using Aquino’s tables, most simple and convenient.
I particularly like them on account of the absence of characteristics and decimal points.
The derivation of these formulae is explained in Aquino’s “ New Altitude Tables,” page ii *.
I. II.
 Dec. 30, 1912. P. M.  O sun sight.  Dec. 30  1, 1912. P. M.  9 Venus sight. 
 
I  Lat.D.R.  28° 50' 00" N 

 Lat. D.R.  29^{0} 07' 00" N 



2  Long.D. R.  78° 41' 00" W 

 Long.D. R.  78° 32' oo"W. 



3  Dec.  23° io' 58.5"  H.D.  9 73"  Dec.  16^{0} 14' 24" S  H.D.  16.6" 

4  Cor.  () 1' 28"  G.M.T.  9 h  Cor.  (—)io' 41"  G.M.T.  10.4b 

5  Dec.  23 ° 09' 30"  Corr.  i' 28"  Dec.  16^{0} 03' 43"  Corr.  io' 41" 

6  E.T.  2' 359"  H.D  1.21"  E.T. 

 
7  Corr.  (+) 10.9"  G.M.T.  , 9 h  Cor. 




8  E.T.  2m 46.8s  Corr.  10.9"  E.T. 




9  R. A. 

 R.A.  21 h 35m 46s  H.D.  11.5s 
 
10  Corr. 


 Corr.  2m 00s  G.'M.T.  10.4I1 

11  R. A. 


 R.A.  21 h 37m 46s  Corr.  119.6s 

12  W.T.  4I1 00m 30.5s 

 W.T.  5I1 23m 47s 

 
13  C.W.  4I1 58m 00s 

 C.W.  4h 58m 40.5s 



14  C. T.  8h 58m 30.5s 

 C.T.  10I1 22m 27.5s 



15  C.C.  (+)im 58.5s 

 C.C.  (I) im 58.5s 



16  G.M.T.  9h 00m 29s 

 G.M.T.  10I1 24m 26s 



17  E.T.  — 2m 46.8s 

 R.A.M. 0  i8h 34m 14s 



18  Table III. 

 Table 111.  (+) im 42.5s 


 
19  G.S.T. 


 G.S.T.  29I1 00m 22.5s 



20  R.A. 


 R.A.  21 h 37m 46s 



21  G.II.A.  8I1 57m 42s 

 G.II.A.  7I1 22m 36 5s 



22  arc.  134° 25' 30"  arc. :  134° 25' 30"  arc.  no° 39' 00"  arc.  110° 39'  00" 
23  A.D.R.  78° 41' 00"  T ass.  55° 48' 00"  A.D.R.  78° 32' 00"  T ass.  3i° 53'  00" 
24  T.D.R.  55° 44' 30"  A ass.  78° 37' 30"  T.D.R.  32^{0} 07' 00"  A ass.  78° 46'  00" 
25  a.  49^{0} 30' 00"  L ass.  28° 43' 00"  a.  30^{0} 30' 00"  L ass.  29 15'  00" 
26  b.  37° 17' 00"  h.  15° 16' 30"  b.  18^{0} 45' 00"  h.  35 30'  30" 
27  c.  66°  Corr.  4 6' 24"  c.  48°  corr.  — 8'  30" 
28  z.  232^{0} oo' 00"  h.  15^{0} 22.9'  z.  218° 24' 00"  h. /  35° 22'  00' 
29  h'.  15° ,i,9^{#} 00" 

 • h'.  35^{0} 12' 00" 



30  h'h.  3.9 Toward 

 h'h.  10 Toward 



Lieut.Commander W. K. Riddle, U. S. Navy.—Commander Marvell’s article brings out clearly the contrast between the “ time honored ” methods of finding a ship’s position at sea, and the improved facilities for solving the problem with modern tables and appliances. This is a matter of deep interest to all of us, and it is a pleasure to read a discussion of this important subject by one who knows the “ways and means” so thoroughly.
It may he safely said that every navigator is in favor of methods which economize time, provided that these “ short cuts ” involve no loss of accuracy. The assumption of a possible error of ± 47" in an observed altitude, combining as it does, personal error, error in refraction, and instrumental error due to graduation of arc, is surely a modest estimate: even an assumption of ± 1' error would be within reasonably small limits.
While the Atlantic Fleet is of course underway a good part of the time, most of the cruising is in piloting waters, so there has, of late, been comparatively little deep sea navigation (averaging, I should say, less than one day per month). Having made, however, during the past summer for purposes of comparison, a large number of simultaneous observations, on the bridge, the writer reached the conclusion that two careful observers will seldom differ more than one minute of arc, in their measured altitudes, when the horizon is clear. When conditions are less favorable, it is quite possible for them to differ a minute and a half, or even two minutes, with the ship perfectly steady, but the horizon, just a trifle indistinct. With a
poor horizon, there is a large element of uncertainty as to exact contact, and the difference, of course, becomes much greater.
One of the sources of inaccuracy mentioned by Commander Marvell, “ ^{sma,!} compass roses,” is completely eliminated by the use of the Court celluloid threearm protractor (ruling the meridians on it, as suggested by Lieutenant R. A. Koch), thus supplying a compass rose eleven and one half inches in diameter, for use with all charts— an idea which, incidentally, has proved of immense value for rapid plotting in piloting waters, especially in swift currents, and in localities where it is necessary to round a light at a specified distance.
Since reading some time ago a paper by Commander G. W. Logan, published in the Naval Institute Proceedings for December, 1908, the writer has used Captain Weir’s azimuth diagram almost exclusively, in all azimuth work, and found it more convenient than the azimuth tables, and much quicker: it is also well within the limits of accuracy (15' of arc) attainable with the navigational instruments in present use.
I have studied the paper by Lieutenant Radler de Aquino of the Brazilian Navy, in which he gives an excellent description of the “Simplest and Readiest Solution, but have had no practical experience with this method aboard ship: it is plain, however, that his “ Altitude and Azimuth Tables ” do facilitate the determination of the ship’s position, and mean a curtailment in the time and amount of work involved, so this method is, unquestionably, a step in advance. If the department should see fit to issue these tables to the service, I feel convinced that they would very soon be in general use.
Meanwhile, however, the haversine method for the Marcq St. Hilaire sight, is widely known among the navigators of the fleet and most of them seem to regard it as a great improvement upon the old cosine formula; the new tables are conveniently arranged, reducing the amount of mechanical work to a minimum, and the functions change so gradually that the interpolations are quickly and easily made. As a further saving of time, forms for this work (sun sight, star sight, azimuth, etc.), are printed aboard ship, and kept ready for use—to be afterwards pasted in the Navigators’ Work Book, table 46, Useful Tables, Bowditch, 1912, combining the corrections to the observed altitude (dip, refraction, parallax, and mean semidiameter), eliminates one “opening of books” and is, in itself, a good “time saver”— we no longer need refer to Muir, Table II. To use the nearest hour of G. M. T. in multiplying the H. D. to get the correction to declination, and then to use both d and h to the nearest minute of arc, as suggested, is an additional useful short cut ’; it falls well within the limits of accuracy possible to the observer, and so cannot affect the correctness of the result.
For purposes of comparison the problem worked out by the time sight formula, and again by the Aquino method in Commander Marvell’s interesting paper, has been worked out by the haversine formula for St. Hilaire sight below. In the first solution the “short cuts” mentioned above, are used; the second, is the rigorous solution :
MARCQ ST. HILAIRE
(i)
Date: March 27
W. 2I1 43m 15s
C.W. 3h 53m 28s C. 6h 36m 43s C.C. 12m i6sf
G.M.T. 6h 24m 27s
Eq. T. _ 5m 28s
G.A.T. 6h 18m 59s Long. 3° 30^ 00" L.A.T. 2h 48m 59s Lat. 39^{0} 45' dec. 2^{0} 40'
Sun: p.ni.
alt. 36° 48' 00" corr. 1i' 06" h'. 36° 59' 06"
Assumed Lat.: 39° 45' N Long.: 52° 30' W
I.C. +2' 00"
Cor. Ta. 4649' 06" corr. + 11'06"
dec. 2^{0} 34' 02.5" corr. 5' 52" dec. 2^{0} 40' —
58.71
________ 6
352.26
Eq. T. 5m 32.3s
corr. _ 4.6s
Eq. T. 5m 27.7s
767
4.602
Ld. 37^{0} 05'
90h 53° ^{I}5^{/}
n* 36 45
14'
— 52^{0} 30'W log hav. 9.11358 log cos. 9.88584 log cos. 999953 log hav. 8.99895
Nat. hav. 09976
Nat. hav. 10112
Nat. hav. 20088
or 14 miles nearer sun on line 236$
RIGOROUS METHOD
Sun
alt. 36° 48' 00' corr. + ii^{7} 04' h
Date: March 27
W. 2h 43m 15s
C.W. 3h 53m 28s
C. 6h 36m 43s
C.C. I2m_i6sj^
G.M.T. 6h 24m 27s
Eq. T. 5m 27.4s .
G.A.T. 6h 18m 59.6s
Long. 3° 30' 00" W
L.A.T. 2I1 48m 59.6s log hav.
Lat. 39^{0} 45' 00" log cos.
dec. 2^{0} 40' 17.5" log cos.
p.m.
(2)
Assumed Lat.: 39^{0} 45' N Long.: 52 dec. 2^{0} 34' 02.5" Eq. T 6' is''
36° 59' 04"
I.C.
D.
P.
R.
S. D.
30' W
5h 32m 3s
corr.
dec. 2° 40' 17.5" Eq. T. 5I1 27m 4s + 2' 00" Ta. 14 — 5' 48" Ta. 16 f o' 07" Ta. 20
9II358
988584
999953
1' 18" 416' 03" fix' 04"
log hav. 8.99895
.767
6.39
4.602
.230
.069
4.901
Ld.
90I1
h'
37^{0} 04' 42.5 53* 15' 10" 36° 34' 50
Nat. hav. 09976 Nat. hav. 10110 Nat. hav. 20086
14' 14", or 14! miles nearer sun on line 236° 36'.
Upon comparing the solutions, one is surprised to see how little they differ in the actual results obtained. 1 he first column of logs in the rigorous solution is identical with the first column in the “short cut ’ solution, and the second columns differ by two units in the last place (owing to the fact that the nearest minute of declination is used). A saving of 68 figures is effected by the “ short cut ” solution (which is certainly worth while), and the ultimate results differ less than a quarter of a mile.
’ As Ad is generally only a few minutes of arc, Z' does not need in practice any correction. The formula on page xvii: AZ = sin M sec li ? Ad shows us that AZ. is always smaller than sec/i • Ad. Under h=6o°, AZ is always smaller than 2Ad.
^{[2]} Also by our Plane Traverse Tables in Lat. column if we enter them with Ad as D and with M or 180° — M as Course.
^{[3]} The “Newest” Navigation Altitude and Azimuth Tables, by Lieutenant
Radler de Aquino, Brazilian Navy, second edition, enlarged and improved,
London, 1912. Published by J. D. Potter, 145 Minories, London, E. Price,
10s. 6d. net.