Captain G. W. Logan, U. S. Navy.—The study of this subject made by Lieutenant Schuyler gives us some convenient formulas for the determination of the error due to disregarding seconds in nautical-astronomical computations. His deduction of the mean value of that error is also interesting and instructive. But I venture to take issue with him as to some of the conclusions which he draws therefrom.
To begin with, we will recall that, aside from the error due to disregarding seconds, the average line of position may be considered subject to a possible error of two nautical miles. This value is generally accepted as the result of experience, and is confirmed by Lieutenant Schuyler’s analysis of signaled positions of naval vessels sailing in company. This error is made up of: (i) the personal error of the observer; (2) the error in the tabulated dip and refraction due to existing atmospheric conditions; and (3) the instrumental errors of the sextant. Hence, when we lay down a line of position on the chart we cannot say that it positively includes the ship’s position, but must admit that the position may be anywhere within a region four miles wide extending two miles from the computed line in either direction.
Turning now to the effect of admitting a further error through the disregard of seconds, I'd us accept the mean value of that error deduced by Lieutenant Schuyler—namely 0.21 minute, or 13 seconds. It is to be assumed that the navigator who rejects the seconds in the data for his computation will also reject them in his computed result, and will pick out the term for which he is working to the nearest even minute of arc; this is understood to be one of the features of the plan of disregarding seconds, and is looked upon as having the same justification as any other part of that plan. Assuming a positive value of the mean error of 13 seconds which the approximate method of computation involves, it will be seen that, if the number of seconds disregarded in the resulting term lies between 17 and 29, both inclusive, the effect of the 13 rejected seconds would be to make the fraction of a minute greater than one-' half, and that the nearest minute would be really one greater than the apparent nearest minute of the approximate computation. A similar condition obtains in an opposite direction in the case of a negative value of the mean error, with fractions of a minute in the resulting term represented by a number of seconds between 42 and 30, both inclusive. Thus, there are 26 chances out of 60 that the result will not be correct to the nearest minute, provided that the sign of application of the mean error due to approximate computation is unfavorable; and there is one chance out of two that that sign will be unfavorable. Hence, the probability of occurrence of an error of one minute in the “nearest minute” of the final result is ½ x 26/60 or .217; that is, the effect of disregarding seconds is to introduce such an error more frequently than once in every five times. It is an accepted principle that, in order to insure accuracy to the nearest figure of a given denomination, a computation must be based upon terms correct to the next lower denomination; if we reject fractions of a minute in the data, it is not to be expected that the result will always be correct to the nearest minute.
So far, we have dealt with the mean error of the method under consideration, but really we are less interested in this than in the maximum error that may be considered reasonably possible. The penalties of a mistake in navigation are too serious for an officer to consider anything less than the worst that might happen with all the chances against him; in case of disaster it would not be a sufficient justification to claim that, in laying his course, he allowed for the mean error as mathematically deduced, instead of for the extreme error that might be expected.
Assume, therefore, a possibility of an error of one mile due to disregarding seconds, and add this to the possible two miles that we must admit from causes already enumerated, and the region of uncertainty changes from a strip of four miles to one of six miles in width.
Assume an error of three miles in a line of position and consider an intersection with another line which crosses at no sharper angle than 450; the point of intersection will be in error by the error of the line multiplied by the cosecant of the angle of intersection; in this case 3X14 = 4.2 miles. Just imagine that luck is against you and that the other line is three miles in error also, and in the unfavorable direction; the error of the point of intersection is nearly doubled. If you were making a landfall on a dangerous coast would not the elimination of one-third of this error have been worth the trouble of working to seconds?
Suppose we work to seconds in the result, if not in the given terms; it may still be worth pointing out that while the mean error that we are considering is less than a quarter of a minute, it takes no very unusual case to bring about values of much more considerable amount. Assume h = 60°, L= 40°, and d= 40°—a problem that might readily present itself. Solving, we find Z=M=28° 41', the cosine of this angle being .88; consider that any error in time is included in that of measuring altitude and therefore that it forms part of the personal error and is not to be taken account of in this connection. With errors in d and L of 24" each, operating to throw out the result in the same direction, we shall have a total error amounting to 2 (.88 X 24) or 42 seconds. With a 450 intersection, the resulting error in the fix would be a mile. Admitting that there is another and larger possible error that you cannot eliminate, does this fact lead you to believe that it is not worth a little trouble to avoid increasing the possible error from two miles to three miles?
When we have tables of trigonometric functions (especially the logarithmic ones) giving values for each quarter minute, we shall no doubt all admit that further interpolation may be neglected. Until then, whenever a navigator is in a position where accuracy is essential in the determination of position, he will place himself at a disadvantage if he rejects seconds in his computations. At sea, making a passage, with no landfall in prospect and with nothing more important depending upon his position than the accuracy of his next day’s dead reckoning, an error of one mile or of ten miles is of small consequence and he may cut corners in his work to his heart’s content.