One morning early in June, some eight years ago, while en route to San Juan, Porto Rico, and when about sixty miles north of that port, I, as navigating officer, obtained observations of two stars just before sunrise, both bodies being on the same side of the meridian and near the prime vertical, differing in azimuth not more than fifteen or twenty degrees. The conditions were apparently normal and there seemed to be no reason for doubting the result as a good determination of longitude ; nevertheless, as a matter of routine, I obtained a sun line about two hours later and discovered to my surprise that, while this line was almost parallel to that of one of the stars, there existed a discrepancy of nearly eleven miles for which I could not account. When landfall was made shortly after, a check back proved the sun sight correct and the two stars badly out. Ordinarily a difference of this amount would be attributed to personal error but it so happened that an assistant, using another sextant, his own watch, and his own tables, had observed these same stars and had agreed with me, a circumstance which led to the belief that some external and unusual condition had caused the variation. Since that time, I have compared a large number of star sights with those of the sun and have taken many observations from known positions, frequently finding larger differences than are usually expected in navigation. It is the purpose of this article to explain and enlarge upon a method that I discovered in an Italian textbook* about five years ago, a method which I have used with considerable success in guarding against errors like the one described, and in assisting me in properly interpreting lines of position in general.

*Nuova Navigazioiie, Astronomica, di G. Pes, Professore di Astronomia Nautica nel R. Instituto Nautico di Genova.

If nearly simultaneous sextant observations of two stars of about the same altitude and having an azimuth difference of one hundred and eighty degrees be made by an observer who uses the same instrument for each observation, two parallel lines of position will result, separated by an amount equal to approximately twice the personal, instrumental and other errors. It is the common practice in such cases to eliminate most of these errors by regarding a third line parallel to and midway between the two observed lines as representing the mean of the two results and as the correct interpretation of the two Sumner lines. This is but a special application of a general principle, the development of which leads to a number of interpretations of Sumner line intersection, of great interest and importance as they affect directly the result, the plotted position.

In the beginning of this discussion it will be necessary to divide into two general classes the numerous errors affecting in varying degree every observation. The most important class, because its errors are usually larger and easier to detect, is that which causes errors of a systematic type, that is, errors constant in sign and amount and which affect all sights alike. For example, an error in taking out the tabulated refraction, or an error in applying incorrectly the index correction, would affect each of a number of sights similarly. Abnormal conditions which alter refraction, the personal equation, and instrumental errors also come under this class. Perhaps the largest error of all is that which is generally peculiar to star observations—the false horizon that is so often encountered during morning and evening twilight, a condition which results sometimes from a low lying mist or fog, sometimes from the observer misjudging the horizon because of light lines filtering through the clouds or cats- paws moving over a calm sea, and frequently because of the optical effect caused by two different strata of air displacing the visible horizon.

The other class belongs to the accidental type, errors which may change both in sign and value from one observation to another. Such are, for instance, the mistakes due to bringing the body to an improper contact with the horizon, the reading of the vernier, errors of dip caused by the rolling of the ship, discrepancies in plotting and carrying forward lines of position, and the excess of error (where more than one observation is made) in those conditions causing systematic error.

We may properly infer from the foregoing that the systematic error is usually much larger than the accidental error, and deductions from numerous groups of observations confirm this opinion. A practiced observer with a high grade sextant should find that his accidental error under favorable conditions does not exceed 30", and it is likely that the average error will be found not larger than 20". The systematic error, however, may be very large, especially where abnormal dip or false horizon is encountered, or where the index error of the sextant is applied in the wrong direction. Where two observations are made there is also a probability of one-half that the accidental error may affect each line of position alike, in which case part of the error will have a systematic effect and only the difference of the two accidental errors need be considered as accidental.

In Figure 1 let ab and cd be two simultaneous lines of position as laid down on the chart, and let a'b' and c'd' represent the two corresponding accurate lines of position, the error of ab and cd being the same in amount and in the same direction. Let the arrows (at right angles to the lines of position) be the direction of the bodies from which the Sumner lines were derived. It is evident that o' and not o is the true position of the vessel.

In the foregoing we have assumed that we knew the direction and the amount of the error. In practice, using but two lines of position, simultaneously obtained, we could not know either the amount or direction of the error.

In Figure 2 let ab and cd be two simultaneous lines of position as laid down on the chart, and the bands inclosing hatched lines on either side represent an unknown error due to abnormal dip, the arrows showing the direction of the bodies observed. Since the error is due to dip both sights will be affected alike and the position of the ship must be either within the parallelogram otpv or mros, and cannot be within either the parallelogram rnto or sovq, for then the error would have affected each sight differently, which is contrary to the supposition. As the amount of the error is unknown it may not be said that the position of the vessel is at m, o, or p, but rather on a line connecting these three points, the bisector of the angle tov. In other words, because of the systematic error, two lines of position do not give the ship’s position, but do give a geometrical locus of the ship’s position which reduces to a new line of position, the bisector of the two angles at the vertex supplementary to the difference of bearing of the two bodies observed. This line, which may be called the Sumner Bisectrix, possesses two valuable characteristics; first, it is a new line of position free from all constant or systematic errors, being affected by the accidental error only; second, it is limited in length by the angle between the Sumner lines and the amount of the probable error.

If two or more observations are made within a brief interval it is probable that the error, if any, will be largely systematic. Thus, under favorable conditions, with clear sky and smooth sea, all the errors due to incorrect application of index correction, instrumental errors, errors due to dip, to refraction (where bodies are of nearly the same altitude), to the effect of the personal equation and conditions of sea horizon, will affect each sight alike, and the correction will be constant and systematic. While it is difficult to assign any average value to the systematic error, which may vary from zero to ten or fifteen minutes of arc, or even more, the Sumner Bisectrix is free from this error and it is not important to determine its amount.

Ride for determining direction of Sumner Bisectrix. The bisector of that angle made by the intersection of two Sumner lines which is equal to 180 degrees minus the azimuth difference of the two observed bodies is the Sumner Bisectrix. The bisectrix will also bisect the angle made by the two bearings of the bodies. Thus, in Figure 3 where ab and cd represent the two plotted position lines, and the dotted lines represent the bearings of the stars it will be seen that the Sumner Bisectrix bisects both the angle bod and the angle s'os".

In considering the combined effect of systematic and accidental errors refer to Figure 4 where ab and cd are two lines of position as laid down on the chart. Let a'b' and c'd' be two corresponding lines of position having been corrected for systematic error. Also let a"b" and c"d" be the two lines corrected for both systematic and accidental errors. Then the “fix” of the two lines ab and cd will be at o. The Sumner Bisectrix will pass through o and o', but the exact position of the ship will be at o" a distance of y from the Sumner Bisectrix. This distance y will vary with the amount of accidental error, the proportion of the systematic error to the accidental error, and also with the difference of azimuth between the two bodies. It can readily be seen that the smaller the azimuth difference the greater will be the value of y, and, conversely, the greater the difference the less will be the value of y and the more nearly will the Sumner Bisectrix be a true line of position. Hence it may be pointed out that the two stars selected should have a bearing difference of at least thirty degrees, and preferably more.

In Figure 5 let the point o be the accurately known position of an observer who takes observations of the three stars S_{1} S_{2}, and *S _{3}*. If there arc no errors in his sights, three Sumner lines will result, each passing through the point 0. However, assume that there is a systematic error present of value e. If the sign of the error is positive the three lines of position will form the triangle abc; if negative, the triangle a'b'c'. Whether the error be positive or negative it will be noted that the Sumner Bisectrices of either triangle, being the bisectors of the angles of the triangles, will pass through the point o, and that this point will be outside the triangle as long as the sum of the angles between the stars –S

_{1}

*o*.

*S*, and S

_{2}_{2}oS

_{3 }is less than 180 degrees, and inside the triangle when the sum of the angles is more than 180 degrees.

Referring to Figure 6, the triangle abc represents three lines of position as plotted on the chart, the directions of the stars for each line being shown by the small arrows. The Sumner Bisectrices are shown by dotted lines and intersect at the point o, which, neglecting the accidental error, is the true position sought. In Figure 7 the triangle abc represents the Sumner lines of three stars, the sum of the azimuth angles of which is greater than 180 degrees. It is seen that in this case the Sumner Bisectrices intersect within the triangle.

There are only two ways in which the accidental error may affect a fix by three lines of position. First, the error may be applied in the same direction to each line, in which case all but the excess of error becomes systematic, and second, the error may be applied in one direction to one line, and in the opposite direction to the other two lines, in which case the error becomes systematic with respect to two of the lines of position, only one being wholly affected by accidental error. Thus we may treat three lines of position as though but one were displaced by the accidental error.

In Figure 8, abc represents three lines of position, resulting in the fix o. Now, assuming that the line ac had an accidental error and should be plotted as a'c', then the true position of the vessel would be at o’ instead of o. Similarly any other line may be considered in error and the resulting difference found.

Experience seems to indicate that observations made in broad daylight, whether of sun, moon, or planet, are seldom in error more than two minutes of arc and, accordingly, a single line so obtained may be treated as fairly reliable; while those made at night or during twilight are doubtful and uncertain, and a single sight then should be accepted only with caution. The foregoing discussion, while general in scope, applies more particularly to star observations, especially to those cases where only two lines are obtained or where three lines form other than a very small triangle. Where Sumner Bisectrices are used it will be sufficient ordinarily to estimate them and their intersections, rather than drawing them, remembering that groups selected in the direction of the best horizon will not only give the best results but a fix which will fall outside the triangle.