NAVIGATION 271 screw till the moment the contact is formed between the sun s reilected image and the edge of the horizon, when the eye is dropped from the telescope of the sextant to the face of the watch, and the second or even the half second noted ; the minutes should be written before commencing, and seen to be correct on finishing the set. Take bearings again of the sun as before, noting the time. Make the mean of the sun s bearings correspond with the mean of the time of sights (i.e., the sun s altitudes). On October 17, 1882, about 8.24 A.M. apparent time, in 88 15 N. and 6 50 E. , observations were taken to ascertain the longitude and variation of the compass. The mean of the times by watch when the altitudes were taken was 8 h 25 m 35 s 5. The result of comparison between the watch and chronometer was - 7 m 14 s , the negative sign meaning that the watch was faster than the chrono meter. The latter was known to have been fast on that day 36 m 3 s . Therefore, both those amounts being subtracted, the mean time at Greenwich was 7* 1 42 m 18 s 5. The mean of five observations of the sun s lower limb was 20 42 30". The corrections were index error + 2 ; eccentric error (in proportion to altitude) - 15" ; dip of the horizon corresponding to the height of eye, 24 feet, - 4 50" ; san s semidiameter + 16 5" ; refraction, minus parallax, - 2 22" ; hence the true altitude of sun s centre was 20 53 8" and zenith distance 69 6 52". The sun s declination is more conveniently taken from the Nautical Almanac for mean noon, as the mean Greenwich time is always known by the chronometer, but the hourly variation is given on the opposite page. Thus, on the day in question, 9 19 29" S., having been corrected for 4 h 18 m before mean noon, gave 9 15 33" as the sun s declination at the time of observation. We have now to find the hour angle and azimuth from three sides : viz., co-latitude 51 45 , co-declination or sun s polar distance 99 15 33", and the zenith distance 69 6 52" (fig. 18). For convenience let these be placed under each other as mentioned, take their sum and half sum (110 3 42") ; also the difference between the half sum and the side opposite the required angle, the zenith distance, = 40 56 50". Add to gether the logarithm cosecant of the first two and logarithm sine of the last two, remembering that when the degrees exceed ninety the supplement must be used, or the amount over 90 with reverse term. Half the sum of those four logarithms, rejecting twenty from the index, will be the cosine of half the angle required (3 h 35 m 48 s 4), the time from noon. This is the principle upon which the angle is found, but it is rather quicker to use the latitude and declination direct, tude, declination, and zenith distance under each other, marking the declination + if on the contrary side of the equator to the latitude, and -if they are on the same side. In the present in stance the latitude being N. and declination S. the latter is addi tive. Take the difference between the half sum and the zenith distance, that still being the side opposite the required angle. Add together the secant of latitude, secant of declination, sine of half sum, and sine of difference ; the sum of those four logarithms, rejecting twenty from the index, will be the sine of half the hour angle, l h 47 m 54 S 2. That being doubled and taken from twelve hours shows that the apparent time at ship was 8 h 24 m 11 s 6. The equation at mean noon was 14 m 35 s 1, the correction for four hours and eighteen minutes before noon was -2 s ; leaving 14 m 33 S- 1 to be subtracted from apparent time (as the sun was then in advance of a mean clock). The result for ship mean time is 8 h 9 m 38 s "5, the difference between which and the Greenwich mean time as found by the chronometer, 27 m 20 s , is the longitude in time = 6 50 , and it is east, because ship time is the greater! The azimuth would in practice be partly sought at the same time as the hour angle ; as the sine of the polar distance or cosine of the declination can be taken at the same opening of the tables where the cosecant or secant was found. We have then sin P x sin PS
- ^ = sm ,
sm ZS the sun s azimuth or angle from north, observing that when the angle is more than 90 the supplement is found opposite the log. sine, and the amount over 90 opposite the cosine. Thus sin P 3 h 35 48 s x sin 80 45 (or cos 9 15 ) = ,70 gno , - = 8111 58 40 , sin ZS 69 7 which is the supplement of PZS, and therefore reckoned from the south, which in this instance is most convenient, as the mean of bearings taken of the sun, by the standard compass, immediately before and after the altitudes for time, was S. 43 30 E. , which taken from the sun s true bearing S. 58 40 E. showed an error of 15 10 18. Write the lati- westerly, that is, that the north point of the compass was so much to the westward of the true north. It is unnecessary to reckon seconds of angle in an azimuth. It is always necessary to note the direction of the ship s head at the time of observation, as there is in general an amount of local deviation caused by the iron in the ship. In the instance given the ship s head was W.N.W., on which point, as previously ascertained, the local deviation was 2 westerly ; consequently the variation of the compass when freed from the ship s influence was 13 10 westerly. When it is desirable to take an azimuth without finding the hour angle as above, there appears no clearer or better mode than that of treating it as a spherical triangle in which the three sides are given to find the angle opposite the sun s polar distance. From the above example take the zenith distance, co-latitude, and polar distance ; place the degrees and minutes only under each other in the above order ; take the difference between their half sum and the polar distance, which will be 10 48 . Add together the logarithm cosecant of the first two terms and the logarithm sine of the last two ; half the sum of those four logarithms will be the cosine of JPZS. Thus PZS = 60 40 x 2 = 121 20 . This is evidently from north, consequently the supplement is S. 58 40 E. as before. It is some times necessary to have recourse to this mode of finding the sun s bearing even when the hour angle is known, as it may be difficult to decide which side of the east or west line the sun may be upon when the angle is near 90. As the last method brings out the angle in two halves there can be no mistake. Position ~by Cross-Bearings of the Sun. The most practical method of obtaining the greatest amount of information and immediate benefit from observations similar to those just described is by laying off the position on the chart (especially if it be on a large scale), using the estimated latitude by which the sights were worked, and the longitude found by the chronometer, the accuracy of which depends on that of the latitude. Through that position draw a line at right angles with the sun s bearing ; in the above instance it would run N. 31 20 E. and S. 31 20 W. Though the latitude were wrong by many miles, the ship would, if all else were correct, be somewhere on that line. For the line thus drawn cor responds to a small arc tl in fig. 18, drawn at right angles to ZS, and all positions for which the sun s zenith distance = ZS lie on that are. As n illustration, suppose that the above sights were worked by a dead-reckoning latitude, which the next observations about 10 h 14 m will prove to be 17 miles too far south. The ship sailed during the interval W.N.W. 15 miles, wind south, leeway half a point, variation + 13 W., and local deviation -2 W. Hence the true course was N. 77 W., which by inspection, under distance 15, gives diff. lat. 3 4 + , and dep. 14 6. The latter opposite the comple ment of the latitude (38) gives 18 5 in the distance column, which is the difference of longitude, subtractive as it was westing and the longitude was east. The latitude was then assumed to be 38 18 24" and longitude 6 31 30". Draw the bearing taken at gh 24 through the newly found point, which will represent the line the ship was on at 10 h 14 m . Five observations of the sun were then taken, and the mean of the times by watch was 10 h 17 m 39 s ; the comparison showed - 7 m 16 s , and the error of chronometer (fast) - 36 3 s . Greenwich mean time was 9 h 34 m 20 s . The equa tion was at that time 14 m 34 s + to mean time, and the corrected declination was 9 17 15" S. The mean of the altitudes when corrected gave a true zenith distance of the sun s centre 53 39 52". All the data are thus given to find the hour angle as before described = l h 46 20 s , and apparent time at ship, 10 h 13 40 s ; subtract the apparent time at Greenwich, 9 h 48 m 54 s ; the differ ence, 24 46 s , is the longitude in time, =6 11 30" E. As the longitude carefully worked up from the former sights appeared to be 6 31 30", it is evident they have both been worked by an erroneous latitude, which is easily discovered. The sun s bearing was not taken at the time of the second set of observations, as it was too high for accuracy, but the true bearing was calculated as be fore described and found to be S. 33 15 E., a right angle to which was E. 33 15 E". If a chart on a sufficiently large scale be in use (or if one of similar latitude can be sub stituted) the most simple mode is to lay down the two positions (as A and B, fig. 19) on the FIG. 19. Cross bearings of the sun. same parallel and draw the two lines through them, as A: and Ex. The point of intersection will be the ship s true position, which can be proved by working the second set of observations again with the corrected latitude. If the chart be too small,
