Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/284

Rh 272 NAVIGATION the diagram can be made on a piece of paper, the difference between the two longitudes being laid off from any scale of equal parts ; e.g., BA is made to measure 20 and becomes the scale for the remainder of the figure. The lines at right angles from the sun s bearing intersect at x, whence a perpendicular drawn with the protractor will give y on the base line. Then By is the total error in longitude to be applied to that found by the second obser vation, in this case 33 3, which added to 6 11 30&quot; gives 6 44 48&quot;. The perpendicular xy represents the error in latitude, and would be correctly measured on the side of the chart as 17 2. But from the plain paper the numerical measurement would be too great in the ratio of the difference of longitude to the departure. The measurement of xy, on the scale BA = 20, would be 21 &quot;8. By inspection (in traverse table) distance 21 - 8 opposite co-latitude 38 is found to give departure 17 2, which in this instance will represent the miles of latitude increased in size to suit the miles of longitude upon which the diagram is formed. That is the required correction to be added to the latitude assumed at the time of the second observation, and will give 38 35 36&quot; as the true latitude at that time. The same corrections may be -found arithmetically by plane trigonometry, in which case it will be desirable to draw a rough figure showing how the lines cross. This mode of correcting the latitude admits of easy proof ; by working the second set of sights with the latitude increased by 17 12&quot;, the longitude will come out 6 45. Another good practical method is to work each set of sights by two latitudes, 10 or 20 miles apart. The logarithms can be taken out for the two with little more trouble than one, and they form a check on each other. When the two sets of sights are not worked by the same latitudes, but by latitudes in accord with each other by allowing for the run of the ship during the interval, and when the change of longitude due to 10 miles of latitude is known, the true position can be found by placing equiva lent proportions under the first longitude brought forward and the second as found, till they agree. By such double observations H. M.S. &quot;Devastation,&quot; in January 1847, was enabled, after having been set by current 28 miles past Goree, to which she was bound, to alter course at 11 A.M. instead of increasing her distance by another hour s steaming. By this method the greatest accuracy of position may be main tained throughout the day and night, if the horizon be clear. The one line is very useful though the other may never be obtained, which is not so with the double altitude, where nothing is known till the whole calculation is completed. In high latitudes during the winter the sun is of little use in finding the longitude; bright stars and planets must be used, taken near twilight when the horizon is clear, and also when they are near east or west if longitude only is required, and with sixty to ninety degrees difference in bearing at other times. The only difference in the treatment of stars and planets from the treatment of the sun when finding the longitude consists in ap plying the difference of their right ascensions and that of the sun, reduced to the time of observation in order to obtain apparent time at ship, and from it mean time, to compare with that shown by the chronometer as Greenwich mean time. A mistake in the manner of applying the difference in right ascension to the hour angle found by a star is less likely to occur in actual practice than in a problem taken for exercise. The stars may be considered as marks in the heavens which pass the meridian so many hours and minutes after the sun, until it becomes more convenient to reckon how much they are before the sun, remembering that, as the sun moves from west to east among the stars at a mean rate of nearly one degree or nearly four minutes of time daily, its right ascension must be carefully noted. It has been attempted to find the longitude by marking the time of sunset, without the aid of any instrument except the chronometer, but this process is very rough on account of the uncertain amount of refraction ; and if the lower limb is allowed to touch the bright reflexion which appears to rise to meet it, the correct angle will be lost, as the centre of the sun will then be about 18 minutes below the horizon, in addition to the &quot;dip of the horizon.&quot; Lunar Observation. The moon is the least serviceable of all the heavenly bodies for the purpose of finding apparent time at ship and longitude by chronometer, in consequence of its rapid motion entailing more care in the corrections, though the Nautical Almanac gives its place with great accuracy for every hour, and the variation in ten minutes. That rapid and uniform motion rendered it the most valuable and at sea the only means of ascertaining the longitude after the construction of almanacs, and before the present perfection of chronometers. In presence of the latter lunar observations have fallen into the shade, and like double altitudes are found principally in examination papers. As the moon passes the stars at the mean rate of 33&quot; of angle in one minute of time, it is obvious that an error to that amount in measuring the distance from a star would produce an error of 15 miles in longitude. As the moon s motion with regard to the sun is nearly one degree a day less, a similar error in the distance would produce still more effect. The Nautical Almanac gives the true distance between the centre of the moon and the sun while within range by the sextant, as it would be seen from the centre of the earth ; and in like manner the distance from some of the principal fixed stars and planets, for every three hours of Greenwich mean time, with the proportional logarithm of the change in that time. It is essential that the star from which the lunar distance is measured should be near the ecliptic, in order to obtain the greatest amount of change. If the star is quite in the path of the moon the distance may be small, when the measurement will require less steadiness. We take a star lunar as an example, which will introduce the problem of finding the time at ship by a star (fig. 20). On November 18, 1882, about 9 P.M. apparent time, the ship was in 38 10 N. and 45 15 W. by dead-reckoning ; height of the eye 20 feet. There was not a chronometer on board. An assistant was employed to show a light on the arc of the sextant, to take time, and to write down ; consequently the same sextant was used. An altitude of the moon s lower limb was taken before and another after the other observations, and the mean was reduced by propor tion to what it would have been if observed at the same instant as the mean of the distances, namely 35 15. Two or more altitudes of Aldebaran were taken before the distances and as many after, the mean of which when reduced to the same instant was 35 6. The mean of several measurements of distance from the moon s farthest limb was 94 40 18&quot;. The index correction was +1 20&quot;, eccentric error - 2 9&quot;, sernidiameter at midnight was 16 9&quot;, which was augmented by 9&quot; (taken from a table) in consequence of the moon being nearer the observer as it approaches the zenith. As the farthest side of the moon was used, the augmented semidiameter was subtracted. The corrected observed distance was then 94 23 11&quot;. The observed altitude of Aldebaran was 35 6, index correction + 1 20&quot;, eccentric -25&quot;, dip of the horizon -4 24&quot;; giving ap parent altitude 35 2 31&quot; and Z.D. 54 57 29&quot;. Apply refraction - 1 23&quot;; then the true altitude is 35 1 8&quot; and true Z.I). 54 58 52&quot;. Add the difference between latitude 38 10 N. and declination (from Nautical Almanac,_ p. 327) 16 16 21&quot; N. to true Z.D. 54 58 52&quot;, and note the difference betweeen the half sum and the true Z.D. Add together the logarithm secants of latitude and declination and logarithm sines of half sum and difference ; half the sum of the four logarithms will be the logarithm sine of half angle SPZ = 3 h 51 m 43 S&amp;gt; 8, which is the time the star was east of the meridian. By adding the longitude in time to the time at ship it will appear that Greenwich apparent time was about mid night, when, the sun s right ascension was 15 h 37 m 6 s . That of the star being 4 h 29 m 14 s, it is evident that the star would pass the meridian H h 7 m 52 s before the sun, and 12 h 52 m 8 s after it. The latter is more suitable at present. Taking the time the star is from the meridian from the time it is after the sun, it is seen that the sun had passed the meridian 9 h O m 24 s - 2, therefore that is the true apparent time at ship, provided the latitude was correct by which it was found. Hence the necessity of choosing a star when nearly east or west. The equation of time taken from the A Imanac and reduced to apparent midnight was 14 m 32 s 5, and the result was ship mean time = 8 h 45 m 51 s- 7. The altitude of the moon s lower limb was 35 15, index cor rection + 1 20&quot;, eccentric error - 25&quot;, dip - 4 24&quot;, semidiameter + 16 9&quot;, augmentation 9&quot; ; giving apparent altitude 35 27 49&quot; and apparent Z.D. 54 32 11&quot;. To the latter add the refraction 1 22&quot;, and subtract the parallax due to -altitude, 48 12&quot;. The result is true Z.D. =53 45 21&quot;. The horizontal parallax is the angle at the centre of the moon subtended by the semidiameter of the earth, and the parallax due to any altitude of the moon (that is, the amount by which the apparent altitude would increase if the observer were sunk to the centre of the earth) is given by the equation hor. parallax in seconds x cos of ap. alt. . . . -, = parallax in alt. rad. It must always be added to the apparent altitude. With all navi gation tables there is one for reducing the parallax, sometimes to every two minutes in altitude. The refraction is then included in the correction. The most important part of this problem- is finding the true distance between the star and centre of the moon at the moment of observation, from the data already obtained ; it is confined to the triangle ZSM (fig. 20). The two apparent zenith distances 54 57 29&quot; and 54&quot; 32 11&quot; with the apparent distance (that is, as measured) are drawn to scale, and from the three sides we find the angle MZS = 12753 54&quot;. As the correction applied to the altitudes of moon and star on account of pa rallax and refraction were in a direction to or from the zenith, the angle between the two zenith distances will be unaffected by any change in the length of those legs. We have therefore the true zenith distances Zm 53 45 21&quot; and Zs 54 58 52&quot; with the included angle just found, to find the third side a pro blem which has been already exemplified (see fig. 15 and the