MIDDLE LAtitude.

The MIDDLE LATITUDE is the parallel of latitude midway between two places; hence it is half the sum of the two latitudes when they have the same name (N. or S.); half the difference when they have different names-one N. and the other S.:

We use the middle latitude when obtaining the longitude by Dead Reckoning, and proceed as follows to find it :

Lat. left... 49° 9' N.

Diff. Lat.... 2 56 N.

Lat. in...... 52 5 N.

2)101 14 sum of Lat. left and Lat. in.

Mid. Lat.... 50 37 N.

DEPARTURE.

DEPARTURE, which you saw we obtained in working the Traverse, receives its name from being taken to (approximately) measure how far the ship has departed from the meridian: it must not be confounded with taking a departure from a place (which has yet to be explained), nor with longitude, although it is the basis whence longitude by Dead Reckoning is got. I said (p. 28) it was the east or west distance that a ship made when sailing on any oblique course. It is often called the meridian distance (in naut. miles) between the meridians of two places, but it is not strictly such, unless the two places are on the same parallel.

I will explain the difference between departure and longitude. If you refer to Fig. 2, p. 11, you will notice that all the meridians, however far apart at (EQ) the equator, converge (come together) at the poles; consequently on any parallel, as a b, the distance along that parallel from one meridian to another must be much less than between the same meridians on the equator. Table, p. 34, is drawn up to show the distance (in nautical miles), in any latitude, of two meridians one degree (60′) apart; look to it: if you are on the equator (Lat. 0) and sail due east or due west for 60 miles, you change your longitude 1 degree; but if you wish to change your meridian by 1 degree on the parallel of Lat. 40°, the table shows that you would do so by sailing due east or west only 46 miles; and in Lat. 60°, the same change would be effected by sailing 30 miles. You know that the nautical mile is everywhere of the same length-viz., 6087 feetbut nevertheless the 60, 46, and 30 of such miles, which are so much departure from one meridian to another, represent for the respective parallels the same thing, viz., one degree of longitude; and a continually decreasing departure or meridian distance (on successively higher parallels) measures the convergence of the meridians as they approach the poles. You can now understand that difference of Longitude (expressed as arc) always exceeds the departure, except on the equator; also that there must be some ratio by which the distance between the meridians decreases in their convergence towards the poles, and that it has

Length of a Degree (60′) of Longitude in different Latitudes, expressed in English Statute Miles, and in Nautical Miles.

some connection with the parallels on which the meridian distance or departure is measured. Hence we want rules which, by Inspection, shall enable us

TO CONVERT DEPARTURE INTO DIFFERENCE OF LONGITUDE. Sailing on a parallel we make no northing nor southing, only east or west Departure, which, to convert into Diff Long., by the method of Parallel SailingRule. Enter Trav. Tab. II. with the Lat. as if it were a course, seek in the Lat. column for the Dep., opposite to which, from the Dist. column, take out the required Diff. Long.

Ex.-A ship makes 123 miles of easting (departure or distance) in Lat. 41°, required the Diff. Long.

In Trav. Tab. II., under 41°, Dep. 123 in Lat. col., gives in Dist. col. 163'; which is the Diff. Long. on parallel of 41° corresponding to 123 miles; you would write down 163' as 2° 43′ E.

But we do not always sail on a parallel, therefore we want a rule for finding the Diff, Long. in cases where we change the latitude in an interval of 24 hours

or less. You can understand that the difference of longitude cannot be inferred either from the departure, considered as a meridian distance in the latitude left, or in the latitude arrived at; in the greater latitude it would give the difference of longitude too much, and in the less latitude too little; but we may generally infer that the departure is nearly, if not quite, equal to the distance of the meridians between which the ship's track lies, in the Middle Latitude, i.e., the mean of the two latitudes; at least near enough for practical purposes; hence— Rule FOR DIFF. LONG. BY MIDDLE LATITUDE SAILING;-Enter Trav. Tab. II. with the Mid. Lat. as a course, seek in the Lat. col. for the Dep., opposite to which, from the Dist. col., take out the required Diff. Long.

Ex.-A ship from Lat. 51° 3′ N. arrives at Lat. 53° 11′ N.: having made west Dep. 97.3 miles; required the Diff. Long.

The following Table may also be used in the conversion of Dep. into Diff. Long., since it shows the relation between the Nautical Mile and a minute (') of Long. on different parallels of Lat.

Thus, last Ex., Dep. 97.3 × 1.62 (the length of the' of Long. in. Lat. 52°) gives 157-6 Diff Long. 2° 37'6 W.

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Length of a Minute (') of Longitude, in different Latitudes, corresponding to 1 Nautical Mile.

Another example of finding the Diff. Long. by the method of Mid. Lat.will show how to use the Traverse Tables, when the quantities exceed those to be found in them*; and we will also project the figure.

When using Traverse Tables I. or II., the quantity or quantities given may be too great, in which case you can enter the Tables by dividing by 2. 3, 4, &c.; but the divisor, If applied to one, must be applied to all, except the course; then, whatever quantity or quantities you take out, you must multiply by what you divided, for a correct result.

Ez.-A ship from Lat. 42° 25′ N., on a N.E by E. course, runs a distance of 423 miles: what is her Lat. in, and the Diff. Long. made.

In Trav. Tab. I.; Course 5 pts. and Dist. 141 (one-third of 423) give Diff. Lat. 78.3 and Dep. 117-2, which multiplied by 3 (as we took only a third of the Dist.) make Diff. Lat. 234.9 N. and Dep. 351·6 E.; then—

In Trav. Tab. II, Mid. Lat. 44° and Dep. 175-8 (of 351-6) in Lat. col., give 244 in Dist. col.: double this gives 488 for Diff. Long.

Also, Mid. Lat. 45° and Dep. 175-8 in Lat. col., give 249 in Dist. col.: double this gives 498 for Diff. Long.

498-488-10, of which we must take and add it to Diff. Long. found by Mid. Lat. 44°, since 44° 22' is towards 45°.

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Hence, 488+4=492 Diff, Long.

Projection of the Diff. Long. by Mid. Lat. Sailing.

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Draw the line A D of any length, and from ▲ lay off the angle BAC 5 points; from a lay off (from scale of equal parts) the Dist. ▲ c= 423; from c drop a perpendicular on the line ▲ D, meeting it at B; then will A B be the Diff. Lat. 235, and B C the Dep. 352: from c draw the line C D making an angle with B C equal to the Mid. Lat. 443°, then C D (measured off the scale of equal parts) will be 492, and this is the Diff. Long. corresponding to

Dep. 352 in Mid. Lat. 444°.

D.ofLat.235m.

D

Rule FOR DIFF. LONG. BY MERCATOR'S SAILING.-Another method of finding the Diff. Long. is by Mercator's Sailing, which requires the Table called Meridional Parts* (Tab. III. of Norie's Epitome).

Taking the last example given, we proceed as follows:

Explanation.-Enter Tab. III. with the degrees of Lat. at the top, and the minutes of Lat. at the side; under the degrees and opposite the minutes take out the Meridional Parts: do this for both latitudes: in this case, since both latitudes have the same name (N.) the difference of the Mer. Parts is taken for the Mer. Diff. of Lat.-When of different names (one N., the other S.) we take the sum of the Mer. Parts for the Mer. Diff. of Lat.

Meridional Parts are certain numbers formed on such a basis that we can project the earth's globular surface into a plane, such that while the parallels of latitude and also the meridians are parallel straight lines, every place is in its proper latitude and longitude, and the rhumb curve of a ship's course cuts every meridian it crosses at the same angle.

Then (in Trav. Tab. I.), with Course 5 points, and half the Mer. Diff. Lat. 164.5 (found in Lat. col.), stands 246 1 in the Dep. col.; this multiplied by 2 gives the Diff. Long. 492.2 as before.

As we have projected the Diff. Long. by Mid. Lat. Sailing, so we may do the same by Mercator's Sailing, that you may see the difference in the two methods.

Projection of the Diff. Long. by Mercator's Sailing.

Draw the line A D of any length, and from A to B lay off the Diff. Lat. 235; on в erect the perpendicular в c, and make the angle BAC equal to the Course 5 points; draw a line A E; then lay off the Mer. Diff. Lat. 329 from A to D, and through D draw D E parallel with B c; the Dist. A c will measure 423, and the Diff. Long. D E will be 492.

D

Diff. Long.

E

Departure

Mer. diff. Lat. 329
D. Lat. 235

Dist
Course 5 Pts.

When the Course is less than four points the method by Mercator's Sailing,and when the Course is greater than four points, the method by Middle Latitude Sailing, is considered preferable. But for the ordinary purposes of navigation, when making up the Dead Reckoning, and the Latitude is less than 55°, the method of Mid. Lat. for finding the Diff. Long. is sufficiently accurate.

By whichever method you obtain the Diff. Long., an unknown error in the course will generally produce as largè an error in the result as that arising out of the difference between the methods.

RAPER observes that "the Diff. Long. found by the method of Middle Latitude is true at the Equator, and very nearly true for short distances in all latitudes, especially when the course is nearly East or West.

"In high latitudes, when the distance is great and the course oblique, the error becomes considerable.

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When, however, the course is near 6 or 8 points, the Diff. Long. should be found by Middle Latitude in preference to Mercator's Sailing, since a small error in the Course may then produce a considerable error in the Diff. Long.,— notwithstanding that the method by Mercator's Sailing is mathematically cor

rect."

In high latitudes, it is quite safe to work by Mid. Lat. if the Diff. Long. is found on each course and distance at short intervals.

LONGITUDE LEFT AND LONGITUDE IN.

LONG. LEFT is the longitude from which the ship has departed.

DIFF. LONG. is the change of longitude in any interval.

LONG. IN is the longitude at which the ship arrives, and is determined either by observation, or by means of the Diff. Long. derived from the Departure.