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 α Α α' 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), one 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 B B' 4 project as before the lines b and W; 3 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 b b in the other. Within this space 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 form—a 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 azimuthal 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 altitudes. Let us assume an error of a quarter of a point on the course, which produces an error of one mile in twenty ; l' 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 c 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 1t' 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 shows that an error of 1' in the altitudo 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 illustrate 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 6 -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 appermost 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 the objects near the prime vertical, the longitude will, ander such conditions, be but little affected. If you now turn the page top to the side, and look at the Fig. with its length trending to the right and left, you will see that the lines of position, B and A, indicate that the observations were made when the objects were on different sides of, and not far from, the meridian,—the azimuthal difference being as before 30°; in this case the longitude is much more likely to be affected than the latitude, and to a greater extent; but the quadrilateral still defines the space in some part of which is the ship, and it will continue to define it in whatever intermediate direction you tum the Fig., and though the latitude and longitude undergo change. (To be continued.) NOTES ON THE MARITIME DEVELOPMENT OF JAPAN. YTHOLOGICAL obscurity enshrouds the very early history of Japan. The writings of native authors are throughout tinged with the marvellous, containing many allusions to “ Divine ancestors " and mythical events and personages. It is, however, possible to gather here and there isolated facts which give some indication of the progress of Japanese shipping, and lead up to the period when European relations with the “ Land of the Rising Sun” became more intimate and settled. Japanese annals show that several centuries antecedent to the Christit an Era, ships arrived from distant southern countries and from the adjacent mainland of Asia. It is recorded that about 250 B.C., the Emperor of China sent over to Japan for the “ Elixir of Imp mortality,” and that subsequently a warlike expedition was desp atched from the mainland, and was destroyed by a typhoon. Nativa annalists also state that about 81 B.C., during the reign of Sujin, hips were first built in “ Dai Nipon.” In the early centuries of our Era the Asiatics were great travellers, the propagandists of Buddhism especially; and Japanese |