Then by Cafe I. find the Difference of Latitude and Departure to every Courfe, and fet them in their proper Columns; as where the Course is Northerly, fet the Difference of Latitude under Northing, or in the North Column; and where the Courfe is Southerly, fet the Difference of Latitude in the South Column.

Again, when the Courfe is Eafterly, fet the Departure in the East Column, and when the Course is Wefterly, fet the Departure in the Weft Column; then adding up each Column by itself fubtract the North and South Columns, the leffer from the greater, the Remainder is the Northing or Southing made good. Alfo fubtract the East and West Columns, the leffer from the bigger, the Remainder is the Eafting or Wefting made good; then have you Difference of Latitude and Departure given, to find the Course and Distance, by Cafe VI.

In this Example, the firft Courfe is E.S.E. 30 Miles; or two Points 30 Miles; for which, by Cafe I. I find the Difference of Latitude 27: 7. Now the Course being between South and Eaft, I place my Difference of Latitude in the South Column, and my Departure 11: 5, in the Eaft Column, leaving the North and Weft Columns blank,

Then for the fecond Courfe, N.E. by N. or 3 Points, 40 Miles, here my Difference of Latitude 33: 3, is to be placed in the North Column, and the Departure 22: 2, in the Eaft Column, because the Course is between the North and Eaft.

Then the third Course being E. by N. or 7 Points, 25 Miles, I place my Difference of Latitude, 49, in the North Column, and Departure, 24: 5, in the East Column.

And fo for the fourth Courfe, N.N.E. or two Points, 44 Miles, I place my Difference of Latitude, 40: 6, in the North Column, and my Departure, 16: 8, in the East Column, and then adding up each Column, the Sum of the Northing Column is 78: 8, and the Sum of the Southing Column is 277, which fubtracted from the Northing 78 8, the Remainder 51 I, is the Difference of Latitude made good, which is Northing, because the Northing was the greater Number.

Again, the Sum of the Eafting Column is 75: 0, which (because there is no Wefting to fubtract from it) is the Eafting made good. Thus you have the Northing, 51: I, and the Eafting, 750, given, to find Courfe and Diftance, by Cafe VI. and although you cannot find in the Table the exact Numbers of 51: 1, and 75: 0, together, yet find the nearest you can, which is 75 4, and 50: 9, over which at the Top, you find 34 Degrees for the Course, which is N.E. by N. od. 15m. Easterly, and the Distance 91 Miles.

But if you have a Place propofed that you are bound for, whofe Courfe and Distance from the Place failed from is given, find thereby the Difference of Latitude and Departure to the Place bound for, from which fubtract the Difference of Latitude and Departure, made good, (if it be the leffer Number, or the other from it,

if

if one of them be the greater Number) the Remainder is the Difference of Latitude and Departure from the Ship to the Place bound for, by which you may find the Course and Distance by Cafe VI.

Example. In the first Question in Traverse Sailing Geometrical, there is a Place proposed to be fail'd to, diftant 129 Miles N.E. half E. the Difference of Latitude for that Courfe and Distance is found by Cafe I. to be 76: 2, and the Departure, 92 8. Now the Difference of Latitude made good by the Ship being but 51: 1, if fubtracted from the whole Difference of Latitude to the Place bound for, 76: 2, it is evident that the Remainder, 25: 1, must be the Difference of Latitude from the Ship to the Place bound for. Alfo the Departure made good by the Ship, 75: 0, fubtracted from the whole Departure, 92: 8, the Remainder, 17 8, is the Departure from the Ship to the Port bound for. And thus you have the Difference of Latitude, 25: 1, and the Departure, 17: 8, to find the Courfe and Distance by Cafe VI. and the neareft to thefe two Numbers that can be found is 25: 4, and 17: 8, over which you have 35 Deg. for the Courfe, and againft it 31 Miles for the Diftance, viz. N.E. by N. id. 15m. Eafterly, 31 Miles is the Courfe and Diftance to the Place bound for, &c.

But fuppofe the Difference of Latitude made good by the Ship were more than the Difference of Latitude to the Port bound for, you must fubtract the leffer from the greater, and the Remainder is the Difference of Latitude from the Ship to the Port, but of a contrary Denomination, which feems to be fo plain, that it needs no Example; for if a Man intends to travel to a Place that bears due North from him 12 Miles; if a Man travels 15 Miles due North, his Courfe from thence to the Place bound for is South 3 Miles; even fo if a Ship intending for the Port aforefaid, whofe Difference of Latitude is 762 North, and Departure 92: 8 Eaft, fhould fail between the North and Eaft, till her Difference of Latitude be 100 North, and Departure 750 Eaft, tis plain fhe is to the Northward of her Port; because she has got more Difference of Latitude Northerly; and the

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is alfo to the Weftward of her Port, because she hath not got fo much Departure Eafterly; and therefore fubtracting the leffer Difference of Latitude from the greater, the Remainder, 23 8, is the Difference of Latitude Southerly from the Ship to the Port bound for; and the Departure 57 0, fubtracted from the whole Departure, 92: 8, the Remainder, 17: 8, is the Departure Easterly from the Ship to the Port bound for; and in this Cafe the Course and Distance would be found by Cafe VI. to be S.E. by S. & S. 29: 05 Miles.

SECT. III.

How to find the Difference of Longitude by the Table of Difference of Latitude and Departure.

HAVING one Latitude, Course and Distance given, or both Latitudes and Courfe, &c. you may find the reft according to Plane Sailing, by the foregoing Rules, which being done, find the Middle Latitude, (always obferving that Caution, to take a Latitude rather too big than too little, where the Middle Latitude cannot exactly be had without a Fraction, then the Rule is :

Find the Complement of Middle Latitude amongst the Degrees at the Top or Bottom of the Table, and under it, (if you find it at the Top, or above it, if you find it at the Bottom) in the Column of Departure, find your Departure, and right against it, in the Column of Distance, is the Difference of Longitude.

And although this Method is not fo practicable in great Numbers, yet it is very useful at Sea, for working a Day's Run, or the like, which feldom exceeds 50 or 60 Leagues; and therefore I fhall recommend it to the Learner, as the moft expeditious Method, and of fufficient exactness for fhort Distances; here being Methods fufficient laid down in this Treatife for more exactly correcting the Reckoning once

or

or twice in a Week, of which more when we come to that Part.

Note, If you think it too much Trouble to fubtract the Middle Latitude from 90, to find its Complement; you may find the Middle Latitude amongst the Degrees, as before; and find the Departure in the Column of Difference of Latitude, and againft it, in the Column of Distance, you have the Difference of Longitude as before.

I shall give an Example or two, and refer the Reader for further Practice of this Method to the Examples laid down. for keeping a Reckoning in the latter Part of this Book, where I fhall illuftrate this Method by Example, and fhew its general Agreement with the Longitude, as found by Mercator's Sailing.

Example 1. A Ship in Latitude 55: 0 North, fails in the North Eaft Quarter, into Latitude 56: 10, her Departure 50; I demand Course, Distance, and Difference of Longitude.

See the Question in Cafe IV. of Middle Latitude failing Trigonometrical.

Difference of Latitude 70, Departure 50, the Course will be found by Cafe VI. hereof to be 36 Degrees neareft; omitting the Minutes, and the Distance 86.

The Middle Latitude is 55 35, but because we have only whole Degrees, we muft ufe either 55 Degrees, or 56; and because, as I faid before, it is beft to take a Degree too great rather than too little, we shall call it 56, whole Complement 34 found in the Degrees, look under it for the Departure 50, the nearest Number to which is 49: 8, against which in the Column of Distance, I find 89 for the Difference of Longitude required.

Example 2. In Cafe V. of Middle Latitude Sailing Trigonometrical.

I demand the Course, Distance and Difference of Longitude between two Places, one in Latitude 56: 15 North, the other in Latitude 58: 35 North, their Meridian Diftance 81.

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