When the variation and lee-way are both to be allowed on a course, you may do it at once, by allowing their sum when they are both the same way, or their difference when the allowance is to be made in different ways, taking care to make the allowance in the same way as the greater quantity ought to be, whether it be the variation or lee-way. EXAMPLE I. EXAMPLE II. A ship steers W. by N. with her larboard tacks A ship steers E. S. E. with her starboard tacks aboard, and makes one point lee-way, there be-aboard, and makes two points lee-way, there be ing two points westerly variation; required the ing one point westerly variation; required the true course? true course? Lee-way to the right hand Variation to the left Difference allowed to the left Whence the course is west. 1 point Lee-way to the left In a violent gale, with a head wind and heavy sea, when it would be dangerous to carry sail, it is usual to lie to under sufficient sail to prevent the vessel from rolling so much as to endanger the masts and rigging. When a ship is lying-to, the tiller is put over to leeward, and when the ship has headway, the rudder acts upon her to bring her to the wind; the ship then loses her way in the water, which ceasing to act on the rudder, her head falls off from the wind, and the sail which is set fills and gives her fresh way through the water, which acting on the rudder, brings her head again to the wind. Thus the ship is kept continually falling off and coming to. In this case, you must observe the points on which she comes up and falls off, and take the middle between the two points for the apparent course, from which allow the variation and lee-way, and you will obtain the true course. EXAMPLE. · A ship lying-to under her mainsail, with her starboard tacks aboard, comes up E. by S. and falls off N. E. by E. there being one point westerly variation, and she makes 5 points lee-way-what course does she make good? The middle between E. by S. and N. E. by E. is E. by N.; and by allowing 6 points to the left hand (viz. 5 for lee-way and 1 for variation) the true course will be obtained N. by E. To exercise the learner we shall add the examples of correcting for variation and lee-way contained in the following Table. If the ship has been acted upon by a current Courses steered. 219. Having corrected the courses for lee-way and variation, and estimated the distances sailed, the latitude and longitude in at noon are to be found by either of the preceding methods of sailing. The latitude and longitude, thus calculated, are called the latitude and longitude by dead reckoning, and if the real course and distance made good by the ship could be estimated accurately by the compass and log, no N. W. W. S. N. N. E. S. S. E. S. E. S. S. W. W. W. N. THE TABLE. W. S. W. W. E. N. É. E. E. by S. S. N. by E. N. W. by N. S. W. E. by N. E. & S. S. E. N. W. N. W. by W. † W. W.AS. thing more would be necessary to determine the ship's place at any time; but by reason of the various accidents that attend a ship's way, such as heave of the sea, unknown currents, different rates of sailing between the times of heaving the log, sudden squalls, improper allowance for lee-way and variation, the latitude and longitude of the ship as deduced from the reckoning, will frequently differ from the latitude and longitude by observation. In this case it will be proper to re-examine the calculation to see whether a just allowance has been made for lee-way, variation, bad steerage, drift of the sea, error of the log-line and glass, &c. since it will sometimes be found that a different and more probable estimate of some of these quantities will make the dead-reckoning agree more nearly with the observations. Before the method of finding the longitude by lunar observations was introduced, the mariner had no other observation to be depended on except his latitude, and it was then usual to make allowances for supposed errors in the courses and distances, so as to make the latitude by observation and dead-reckoning agree. The method made use of by Robertson, Moore, and others, was divided into three cases, viz. CASE I. When the course was within three points of the meridian, the error was supposed to be wholly in the distance, on the principle that it would require a greater error in the course to cause the given error in the difference of latitude than could be supposed probable to have been committed. In this case the corrected departure, &c. were found, with the course, by deadreckoning, and the difference of latitude by observation, as in Case IV. of Middle Latitude or Mercator's Sailing. CASE II. When the course was between three and five points of the meridian, the error was supposed to be part on the course and part on the distance. In this case, the corrected departure was taken equal to the mean of the departure by dead-reckoning, and the departure which corresponds to the distance by dead-reckoning, and the difference of latitude by observation. With the corrected departure, and the difference of latitude by observation, the course, &c. were found as in Case II. of Middle Latitude or Mercator's Sailing. CASE III. When the course was more than five points from the meridian, the error was supposed to be wholly on the course, on the principle that it would require a greater variation in the distance to make the dead-reckoning and observation agree, than could be supposed probable, whereas it could require but a small change in the course to produce the sought effect. In this case, the corrected departure, &c. were found, with the distance, by dead reckoning, and the difference of latitude by observation, by Case V. of Middle Latitude or Mercator's Sailing. This method was given in the former editions of this work, in conformity to custom, though I was decidedly opposed to making such corrections, being convinced that the difference between the dead-reckoning and observation is more owing to unknown currents than to errors in the courses and distances given by the log. Even admitting the principle that an arbitrary correction of this kind is proper, the preceding method does by no means appear to be the most probable. To show this, let us take the following example. Suppose the course by dead-reckoning to be 330° 44′ 59′′, the distance by the log. 100 miles, and the difference of latitude by observation 73.1 miles. This comes under Case I. and the error must be placed wholly on the distance, which is to be found with the course 33° 44′ 59′′ (or 3 points nearly) and the difference of latitude 73.1, so that the corrected quantities are nearly by Table I. course 33° 44′ 59′′, distance 88, and departure 48.8. Now by altering the course two seconds, making it 33° 45′ 1", still retaining the distance by dead reckoning 100 miles, and the difference of latitude by observation 73.1, the example will come under Case II. and the corrected depar ture is the mean of the departure by dead-reckoning 55.6, and that corresponding to the distance 100, and the difference of latitude by observation 73.1; namely 68.2, so that this corrected departure is 61.9 miles, with which and the difference of latitude by observation 73.1, we obtain the corrected course 40° 15′ and the distance 96. Thus we see that by altering the course by dead reckoning only two seconds, the corrected course varies from 330 44′ 59′′ to 40° 15' or above 6 degrees, and the departure varies from 48.8 to 61.9, both of which are highly improbable. This defect of the rule evidently arises from the sudden change of the method when the course is near 3 or 5 points; it being much more probable that the variations take place by small degrees, in such manner that when the course by dead-reckoning is exactly on the meridian, the error ought to be in the distance, and when the course is 8 points from the meridian, the error ought to be in the course, and at intermediate courses the errors in distance ought to be greater, the nearer the course is to the meridian, and the errors in the course greater, the farther it is from the meridian. Both these objects are attained in a very simple manner, in the method proposed by the ingenious methematician, Dr. Adrian, late Professor of Mathematics and Natural Philosophy in Columbia College, New-York, which is somewhat similar to my method of correcting a survey. His method consists in finding, with the difference of latitude by observation, and the departure by account, the corrected course, distance, and difference of longitude by Case II. of Middle Latitude or Mercator's Sailing, so that no correction whatever is made in the departure. The propriety of this method will appear evident by observing that a change in the departure can have no tendency whatever in correcting an error in the latitude, and there can be no reason given why such change should be made to the eastward rather than to the westward, since it is supposed that all the allowances for heave of the sea, falling off the course, variation, error of the log, &c. have been previously taken into the calculation, and it seems to be contrary to sound reasoning to vary any of the elements when it will not serve to correct the known error of the latitude, particularly when there can be no reason given why the change should be made in one direction rather than another. In addition to this, the proposed method is not liable to the inconvenience of a sudden change in the rules when the course is near 3 or 5 points. It has also another advantage with respect to simplicity of calculation arising from the circumstance that the corrected difference of longitude is nearly the same as the difference of longitude by dead-reckoning. For the departure is not varied by the rule, and the middle latitude differs rarely more than a few minutes on account of the difference between the latitude by observation and account, so that in keeping a journal it will not be necessary to make any change in the longitude by dead-reckoning, even if you have not had an observation for several days. To illustrate this method I shall give the following EXAMPLE. Yesterday at noon we were in the latitude of 39° 18' N. and by an observation at noon this day are in the latitude of 37° 48′ N. our dead-reckoning gives 107 miles southing and 64 miles westing. Required the course, distance, and difference of longitude? With the difference of latitude by observation 90 miles (the difference of 37° 48' and 39° 18') and the departure by dead-reckoning 64 miles, I find by Case II. of Mid. Lat. Sailing, the course nearly 350, and the distance 110 miles; and with the middle latitude by observation 38° 33′, and the departure 64 miles, I find the difference of longitude to be 82 miles. If the middle latitude by dead-reckoning 38° 41' had been taken, the result would have been nearly the same. If you have not had an observation several days, and then find an error in the latitude by account, you may on these principles correct the latitude on the intermediate days, by saying, as the sum of all the distances sailed, since the first observation, is to the whole error in the latitude, so is the sum of the distances sailed from the time of taking the first observation, to the noon of any particular day, to the correction of the latitude by dead-reckoning on that day, southerly if the last latitude by observation is south of the latitude by deadreckoning, otherwise northerly. Thus, if the latitudes by dead-reckoning at noon, on four successive days, were 41° 0', 41° 30′, 42° 0′, 43° 0', the latitude by observation on the first day 41° 0', and on the last day 43° 15'; differing 15 miles from the latitude by account; the distances sailed by the log, on the three days respectively, 30, 90, and 105 miles; we must say, as the whole sum of the distances 225 miles, is to the error of the latitude 15 miles, so is the first distance 30, to the correction of the second latitude 2, and so is the sum of 30 and 90 (=120) to the correction of the third latitude 8', so that the corrected latitudes will be 410', 41° 30'+2′=41° 32′, 42° 0′+8′=42° 8' and 43° 15', and the corrected differences of latitude on the successive days will be 32, 36', and 67', with which and the departure by dead-reckoning, the corrected courses, distances, &c. on each day may be found, if thought necessary; but as the corrected longitude is not sensibly altered by any of these corrections, it appears to be in general wholly unnecessary to make any alteration in the Journal on this account. But if it be thought proper to notice these corrections in plotting off the track of a ship, it will be necessary first to plot off the courses by D. R. and then to place the points arrived at, at the end of each day, as much to the north or south of the places by D. R. as will make the latitudes of those points agree with the corrected latitudes found by the above rule. The latitude and longitude by dead-reckoning being found by the preceding methods; thence may be determined the bearing and distance of the place of destination; but when the mariner is fearful that his longitude by account is inaccurate, and he has no lunar observations to correct it, he must get into the latitude of the place, and (if possible) run east or west according to his situation, and the prevailing state of the winds. We have now given all the rules necessary for working a day's work and for the convenience of the learner (to enable him to refer to them easily) we have here collected them in the seven following articles. Rules for working a day's work. 1. Correct the several courses sailed for variation and lee-way, and enter them in a traverse table, and opposite to each course place the distance run on that course, found by summing up the knots and fathoms sailed by the ship on that course. Find in Table I. or II. the difference of latitude and departure corresponding to each course and distance, and set them in their respective columns: then the difference between the sums of the northings and southings will be the difference of latitude made good, of the same name with the greater; and the difference between the sums of the eastings and westings will be the departure made good, of the same name with the greater quantity. 2. Seek in Table I. or II. until the above difference of latitude and departure are found together in their respective columns; opposite to these will be the distance made good, and at the top or bottom of the page, according as the departure is less or greater than the difference of latitude, will be found the course. 3. If the latitude from which the ship's departure is taken, or yesterday's latitude, be of the same name as the difference of latitude, add them together; but if of different names, take their difference; the sum or remainder will be the present latitude, of the same name as the greater. 4. Find the middle latitude between the latitude of yesterday and this The set and drift of a current (if there be any) is to be reckoned as a course and distance, and on the first day after losing sight of the land the bearing and distance of it are to be taken into account, G% day, which take as a course in Table II. and seek for the departure in the column of Diff. Lat. then will the distance corresponding, be the difference of longitude, of the same name as the departure. 5. If the longitude in yesterday be of the same name as the difference of longitude, add them together; but if of different names, take their difference; the sum or remainder will be the long. in, of the same name as the greater. 6. If a lunar observation were taken at any time of the day, you must find, by the above method, the difference of longitude made since taking the observation for regulating the watch, and thence the longitude in at noon by that observation, and enter it in the Journal as the longitude by observation. 7. Find on a general chart the spot corresponding to the latitude and longitude by observation, and that place will represent the situation of the ship, whence the bearing and distance of the intended port may be found. The same may be obtained by middle latitude sailing, by inspection of Table II. thus: Find the middle latitude between the place of the ship and the proposed place, and seek for that latitude as a course in Table II. and find in the corresponding page of the Table, the difference of longitude (between the ship and the proposed place) in the distance column, opposite to which, in the latitude column, will be the departure. Seek in Table I. for this departure and the difference of latitude (between the ship and the proposed place) till they are found to agree, corresponding thereto will be the bearing and distance required. If the magnetic bearing be required, the variation must be allowed on the true bearing; to the right hand if the variation is westerly, or to the left hand if easterly. We shall now proceed to exemplify the above rules; first by a few examples of separate day's works, and then by a Journal from Boston to Madeira, kept in the usual form. |