The method commonly used by navigators to test the accuracy of a series of observations is to compute the rate of change of altitude from the watch times of the observations. If the computed rates are fairly uniform, it is assumed that the observations are satisfactory and the mean of all the sights is used with the mean of all the watch times. On the other hand, a wide variation in the computed rates is an indication of poor observations or gross errors in certain ones of the series. Under some conditions of observation it is difficult to get really good agreement among the computed rates of change of altitude. In such circumstances it would be desirable to have a separate test for the rate of change of altitude which will indicate which observations are reasonably accurate; a test which is independent of the measured altitudes. Such a test is available and is one which can be quickly and easily applied.
Spherical trigonometry provides general equations for the solution of the spherical triangle. Referring to Fig. 1, which is a drawing of the general triangle, the two equations which are most useful in this particular case are:
Figure 2 is the celestial counterpart of the general triangle in which P is the elevated pole, M is the observed body, and Z is the zenith of the observer. For the various parts of this triangle we may write:
short periods of time, such as are consumed in making a series of observations, the derivative of H with respect to time (t) will result in an equation for the rate of change of altitude with time. There is one further consideration: we must introduce the constant which expresses the relation between minutes of arc and minutes of time in order that the equation shall be dimensionally consistent. Since one minute of time is equal to 15 minutes of arc, the constant is 15. Introducing this factor into equation (4) and differentiating
from which it is evident that, if the bearing of the observed body and latitude are known, we have a means of testing the rate of change of altitude which does not depend upon the actual measurements of altitude nor the watch times of the observations.
We are so prone to think in terms of absolute values of altitude, in which declination plays an important part, that it is somewhat startling to find that declination does not enter into the equation for
constant according to our original premise. Consequently, declination does not affect the rate of change of altitude during short periods of time; the only factors involved are the values of bearing and latitude.
In order to avoid making the computation for rate of change of altitude each time we make a series of observations, it is possible to precompute the values of dH/dt for various values of bearing and latitude, and to arrange them in tabular form as in Table I. Such a table was prepared by Commander L. V. Kielhorn, U. S. Coast Guard, and is included as an appendix to his Stereographic Projections in the Plane of the Horizon, published by the U. S. Coast Guard, Treasury Department. Unfortunately this tabular form does not lend itself to ready interpolation; the values of rate of change do not come out in even figures.
A more convenient arrangement is to plot the tabular values as a series of curves in which interpolation can be made more readily by inspection. Figure 4 is such a series of curves prepared by the writer from his own computations which were checked against Kielhorn’s tables. Incidentally this writer discovered an error in Kielhorn’s table; probably an error in copying the table for reproduction. While these curves are more satisfactory than the tabular form, there is still some interpolation. In addition, it is well to note that, while a curve is drawn for a bearing of 5° from the meridian, the limit of usefulness of these curves is at about 7° bearing. When the body is within 7° of the meridian it is better to resort to method of reduction to the meridian because it can be shown that the altitude change, in this region, is proportional to the square of the time.
Probably the best method of solution is to construct a nomogram for equation (7). This device eliminates all interpolation and reduces the computation of dH/dt to one very simple operation. Such a nomogram was constructed by Charles E. Manierre and was published in the U. S. Naval Institute Proceedings, October, 1921. This nomogram is reproduced in Fig. 5.
To construct this nomogram, select a piece of good grade bond drawing paper, 11 by 17 inches. Draw two parallel lines the length of the paper, the first line to be ½ inch from the bottom of the sheet and the second exactly 10 inches above the first. About 2 inches in from the left-hand edge erect a perpendicular line; this line is to be the base-line for the scale of bearings and is marked off by spacing the scale proportional to the sines of the angles using a table of natural functions. The sine of 0° is 0.00 and is the origin of the scale at the top. Then, for example, sine 30° = 0.50 so that the mark for 30° will be exactly 5 inches from the top of the scale. Similarly, the mark for 60° will be 8.66 inches from the top of the line. While these distances are proportional to the sines of the angles, the scale markings are numbered simply with the degree numbers.
From the foot of the sine scale measure along the bottom horizontal a distance of about 13 inches; exact distance is not material. Draw a line connecting this point with top of sine scale. Next draw a line parallel to the sine scale and to the right so spaced that it is about 2 inches away from the sine scale; the important thing is to have the line of some even length, say 8 inches. This latter line is called the inside parallel. Using the intersection of the diagonal and the bottom line as a pivot point, transfer the marking of the sine scale to the inside parallel. The markings less than 30° need not be transferred. Number these points with the complementary numbers; the 30° mark becomes the 60° mark of this new scale. These transferred points are only temporary, so do not make them too heavy. Now, using the intersection of the sine scale with the bottom line as a pivot point, transfer the temporary marking to the diagonal and give them the same markings as the temporary scale. This is the latitude scale and the process just described is a projective transformation.
Erase the temporary marking on the inside parallel and then divide its length into 15 equal parts with suitable subdivisions as desired. This is the change of altitude per minute scale. Ink in all the scales with India ink, to make them permanent, and the nomogram will be complete. The accuracy of the diagram depends on the accuracy of the draftsmanship, so keep the pencil sharp and draw fine lines. A pin stuck into the drafting board through the pivot points is a great help when making the projections. The finished diagram can be bound in a standard notebook by using the 11-in. side as a binding edge.
To use the nomogram, lay a straightedge across the scales so that it cuts the bearing scale in the observed bearing and the latitude scale in the dead-reckoning latitude. Where the straightedge cuts the rate of change scale will be found the value of the rate of change of altitude per minute of time. The page must be kept flat while using the nomogram.
To illustrate the use of this method for testing a series of observations, let us assume that we have a series as follows:
Inspection of the calculated rates of change shows two which are in fair agreement, between obs. (1) and (2) and obs. (3) and (4); whereas some error appears to have come between (2) and (3) and also between (4) and (5) which would cast some doubt on the whole series. Taking the rate of change from our diagram, using the average bearing during the interval and the dead-reckoning latitude, we get 7.9 minutes per minute of time, so that observations (4) and (5) are probably the best. It must be realized that these data were purposely made discordant so as to bring out the method more forcefully. It is more than likely that an experienced navigator would obtain much more concordant results than shown here, but the test would still show him which were the better observations.
The test is not strictly rigorous for two reasons: (a) the latitude of a moving ship is not often constant, and (b) the bearing of a body from the meridian is not constant but varies with time. These two facts, however, do not invalidate the method. The latitude even of fast moving
an apparent error of only 0.5 mile. If we had computed dH/dt we would have found it to be 8.433 which would reduce the apparent error to 0.2 mile but the difference does not justify the additional labor.
There is nothing particularly new in the ideas presented here since the principles have been used for years and are well understood by all thorough students of navigation. The usual works on navigation almost never include the proof of such propositions nor do they make any attempt to investigate the probable accuracy when a given method contains certain assumptions or approximations. This is true of the at2 method of reduction to the meridian and the writer is willing to wager that very few navigators have ever seen the mathematical proof of the statement (as in Dutton)*, “It may be proved by mathematics that if a body changes its altitude a seconds while changing its hour angle one minute of time from the meridian, then, in any small period, say t minutes of time, within the limits of slow movement in altitude it will change its altitude at2 seconds of arc.” When one knows the proof for a given method, he is in a much better position to know its limitations and to judge of its accuracy in a given application. This particular proof has been presented in the hope that it will inspire confidence in its use.
* Navigation and Nautical Astronomy, 1932 Edition, page 187.
THE PILOT OF THE INTREPID
A native of Palermo, Sicily, Salvadore Catalano served as pilot of the ketch Intrepid when Decatur entered the harbor of Tripoli on the night of February 16, 1804, and recaptured and burned the Philadelphia under the guns of the enemy. It is interesting to note that Catalano was not forgotten by the United States. He came to America on one of our ships of war returning from the Mediterranean and was promoted to sailing master in the Navy. For many years he served in various capacities at the Washington Navy Yard, where he died March 4, 1846.