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project these points on the chart, each pair of points being joined by a straight line, the intersection of the two lines is very nearly the ship's position. If the intersection does not happen to fall between the two assumed parallels, then, for greater accuracy, assume another latitude, such that it shall do 80; compute and project again. If one person observe both altitudes, it will be necessary, as they are not exactly simultaneous sighte, to take the Greenwich time for each observation ; if quickly done the small change in the ship's position in the interval will not greatly effect the result.
2. The altitude of the same object—as in the case of the sunmay be taken at two different times, and the circles laid down as before. When the ship has changed her position in the interval between the two observations, either the usual reduction of the first altitude for change of place must be applied, or—as is more practical—the circles of position for each observation having been projected, the first must be moved parallel to itself in the direction of the course, made good, and by a quantity equal to the distance run; the intersection of its new position will give the place of the ship at the second observation.
In Fig. 7, if we suppose the lines A and B
A to represent lines plotted on the chart to their respective latitudes and
7 longitudes as derived from simultaneous altitudes of two celestial objects, then the ship being somewhere on A,
Fig. 7. and also somewhere on B, the intersection of the two lines at the point P at once determines the position.
But in Fig. 7 if, as in the case of the sun, where there has been an earlier and later observation, with a course and distance run in the interval, we, as before, project a line A as that on some part of which the ship is supposed to be at the time of the first observation; and B the line as derived from the second observation ;
then, from any part of the line A we must also project the line C equal to the course and distance; from the extremity of C draw a line A' parallel with A ; and the intersection at P' of the two lines A' and B determines the position of the ship at the time of taking the second observation.
A single altitude of a celestial object at any given Greenwich time, with its polar distance and two assumed latitudes, determines the elements for a line of position A, which is plotted on the chart according to the respective latitudes and longitudes; if the data are correct, A. is unquestionably a line on some part of which is the ship; if the altitude is assumed to be doubtful to the extent of 2' or 3', in one direction or the other, this can also be shown. When the 3
a A e altitude is too small, the hour angle is too great ; when the altitude is too great, the hour angle is too small. Hence, by projecting the lines a and a' (Fig. 3), onė on each side of A, and parallel with it, and to the extent of
FIG. 3. the error of altitude, we get a zone, or linear space, bounded by the lines a and a', within which it will be safe to assume the ship's position to be.
If the altitudes of two objects have been taken at the same time, then, assuming the data to be correct, we at once determine the point by the intersection of the lines of position A and B (Fig. 4); but if, as in the case of A, the altitude which gives B is also doubtful, we
A project as before the lines b and b'; we thus get a space, indicated in the figure by the shaded quadrilateral, and which is determined by a a' in one direction, and by bb in the other. Within this
is the ship's position, and the area of the space is naturally more circum
Fig. 4. scribed than either zone. If we now assume a small error in the chronometer, we can delineate it around the quadrilateral; but as
this gives no error in latitude we get a figure of a different forma hexagon, which determines the limit of error of the point, and gives an area or surface of certitude within which lies the ship’s position.
When the azimutbal angle between the lines of position is 90° the form of the quadrilateral will be that given in Fig. 4; it will change its outline considerably for smaller or greater angles ; its area, nevertheless, defines the limit of error, though the exact position of the point within it is unknown (see also Fig. 6, p. 115).
A position obtained by two altitudes, with an interval of time between the observations, is affected to the extent of the errors in the “dead reckoning " during the interval, and by errors in the altitades. Let us assume an error of a quarter of a point on the course, which produces an error of one mile in twenty ; 1' on each altitude ; and the chronometer doubtful to about 5 or 6 seconds. In Fig. 5 we have the line of position A (together with a and a')
as the result of the first altitude, and B (together with b and b') as the position line for the second altitude. If the course and distance be correct, A transferred to A' gives, by its intereection of B, the point where the second observation was made ; but A and B have each an error, which, developed near the point of intersection, gives a small quadrilateral, easily recognised by the reader.
The distance run may, however, be 19 or 21 miles, due to the
error on the course; this one mile projected on each side of A' gives the zone contained between c and d' ; but, since c and care the representatives of A, we must reproduce a and a' outside of a and c'. Thus the quadrilateral (the light shaded portion of Fig. 5) defined by the zone b l' in the one direction, and by the zone a a' in the other, becomes the space or area within which the ship's position may possibly be. If we now carry the quadrilateral bodily to the east and west 1}' for the error of the chronometer, we, as before, obtain a hexagonal space definitively limiting the position of the point. Each of the geometrical figures will be small or great, in proportion to the errors in the data, concurrently with the azimuthal angle between the lines of position. In Fig. 5 the ship may be at the intersection of A' and B; it may also be at any part of the quadrilateral, or of the hexagon. But where there are so many errors of different kinds as those we have now taken into account, since some are likely to be of one character, and some of another, it is just possible, but very improbable, that they could be accumulative in any one direction; hence Fig. 5, which, from the various errors delineated, looks so very formidable, may be considered to define the ship's position, not within eight or nine, but within four miles.
The position determined by simultaneous altitudes of two stars, if the angle at the vertical is good—and this is a mere matter of selection-can only be affected to the extent of the errors of altitudes and those of the chronometer; and the navigator should never lose an opportunity of observing them.
It is evident from the nature of the projection that the most favourable case for the accurate determination of the intersection is that in which the lines of position intersect at right angles. Hence the two objects observed, or the two positions of the same object, should, if possible, differ about 90° in azimuth.
We give below, in miles, the greatest errors likely to arise on the point, for different values of the errors of altitude at different angles of the intersection of the lines of position.
A reference to the Table sbows that an error of l' in the altitude will produce an error of position on the earth's surface equal to at least 1.4 miles, even when the azimuthal difference of the lines is at its best (90°). When the angle is very small or very large, the error is proportionally greater; and the latitude and longitude will be more or less affected accordingly, the latitude most by observations made when the object is near the prime vertical, the longitude most by observations taken near the meridian. Fig. 6 will illastrate this: B and A are lines of
B position projected for observations on each side of the prime vertical, with an azimuthal angle between them of about 30°; if both altitudes are correct, the intersection of B and A gives the correct position. For altitudes with equal errors -each too great and too small—the shaded quadrilateral defines the space within which must be the ship's position. If both altitudes are equally too great or too small, the ship may be at the outermost part of the quadrilateral, to the right or left. If one altitude is too
Fig. 6. great, and the other equally too small, the ship's position may be at the uppermost or lowermost part of the quadrilateral; in which case the latitude will be most in error, and is likely to be so, for the observations having been made with