As Radius To the Tangent of PO Then 10.0000000 9.9983442 10.2385606 GP PO=o0° 06', that is 6 Ita lian Miles, is the greatest Difference of Latitude the Ship made in that Course. PROBLEM IV. Admit a Ship fets fail from B in Latitude 18° 46', and fteers her Course away between E. and N. on the Arch of a Great Circle 40° 22', and then finds her Latitude at A to be 51° 32', I demand her Course, Difference of Longitude, and Departure. Practice. In the Oblique Triangle A B P, there is known the Side B P 71° 14', the Complement of the first Latitude B A= 40° 22′, the Distance failed; and the Side AP 38° 28', the Complement of the Latitude arrived at; to find the other Parts. This is done by Cafe 5 Of Oblique Triangles, and otherwise by the fifth Propofition of Chap. 9, which has been fo often exemplified, that I need not here repeat it. For thus the Angle at P will be found 30° 00' for the Difference of Longitude; and the Angle at B 28° 43', the Courfe at first fetting Sail; but the Angle at A= 46° 58', the Courfe at her Arrival; which is 18° 15' more to the East than the first. The Departure will be found (by the Analogies in the two foregoing Problems) to be 18° 32′ = 11 1 2 · Miles 370 Leagues. But if the Ship had failed from A to B, her Departure would have been 28° 22' 1702 Miles = 567 Leagues. Now Now notwithstanding there is fuch a Difference of the Departure, yet the Proportion by Plain and Mer.. cator's Sailing, or by Middle Latitude, gives none. PROBLEM V. By having the Latitudes of two Places given, with their Difference of Longitude; to calculate the Diftance in the Arch; the Courfe of the Ship at fetting Sail, and at her Arrival; the intermediate Latitudes and Longitudes by which the Arch of Diftance paf feth; the Courfe and Distances between those several intermediate Points of Latitude and Longitude. Practice. 1. Suppose the Summer Islands be defigned by E in Latitude N. 32° 25', from whence a Ship fets fail for the Lizard's Point at F, in Latitude 50° 00'; and admit the Difference of Longitude be 70° 00'; then, 2. In the Oblique Triangle F P E there is given the Side F P = 40° 00; the Side E P = 57° 35′; and the Angle at P = 70° cc′; to find E F, the nearest Distance between the two Places. Now this will be found 53° 24′ = 3204 Miles 1068 Leagues, by Cafe 4 Of Oblique Triangles. 3. Then by Cafe 3 Of Oblique Triangles, you find the Angle FEP 48° 48', i. e. N. E. and 3o 48' Easterly, for the Ship's Course at first setting Sail; and the Angle EFP 81o 10', i. e. E. by S. and 2o 25′ Easterly, for the Course of the Ship at her Arrival to the Lizard Point. 4. In the next Place, let fall the Perpendicular P a, then is the Oblique Triangle FP E reduced to two Right-angled Triangles a PF, and a PE; now the Perpendicular a P will be found 39° 26', whofe Com plement plement 50° 34' is the Greatest Latitude through which the Ship paffeth. 5. Having done this, next find the Angles E Pa, and a PF, by the common Analogy; the first will be 58° 31', and the latter 11° 29'; thefe Angles being obtained, you may proceed to find the Latitudes to as many Degrees of Longitude from a as you please. Thus, 6. Suppofe the Latitude at every 10 Degrees Diftance from a be required; P b. a Pb which Sides are the Complements of the feveral Latitudes the Ship paffeth by at each tenth Degree of Longitude from a. 7. You may then alfo find the Angles af P, a e P, ad P, ac P, a b P, which will fhew the Course to be fteer'd in each of the Latitudes before found. Alfo the Parts Ef, fe, ed, &c. may be found if defired, which are the Distances between the aforefaid Points of Latitude. Thus I have fhewn how the whole Problem may be folved in each Particular; but have left the Practice thereof to the ftudious young Practitioner. These Problems are general ones, I could not branch them out into particular Cafes; for that would be thought not only a mercenary Defign to stuff a Book, but would be holding a Candle to a Reader at Noon Day, which is very abfurd and affronting. I rather fear he who hath regularly proceeded thus far from the Beginning, will find I have been often obliged Actum agere, i. e. to do the fame thing over and over again, than been deficient in any one Part of this Undertaking. VOL. II. S s A Ta CHAP. V. The Method of Making and Sailing by the Globular Chart. A S I have given an Epitome of the feveral Kinds of Sailing, in feveral Parts of this Work, which depend more immediately on Trigonometry Plain or Spherical; fo in order that the young Student might have a compleat Idea of all Kinds of Navigation in ufe, I have here thought good to fubjoin a little Tract of the Nature, Conftruction, and Ufe of the Chart which is generally called the Globu lar Chart; although it hath not that direct Relation to Trigonometry, as all the other Parts of Navigation have. I am the more willing to do this, becaufe 'tis reckoned as a very celebrated modern Improvement of this Art, and is not to be found in any Mathematical Tracts that I have yet feen; and therefore that every one may have the Benefit or it, I fhall explain it in the following Particulars. 1. First therefore you must understand, that it is peculiar to this Chart to have all the Properties of the Surface of the Globe it felf; and from thence it acquires the Name of the Globular Chart; for as on the Globe all the Meridians incline to each other from the Equator to the Poles, where they all terminate, fo do they on this Chart; as on the Globe the Parallels are all equidiftant leffer Circles, fo are they in this Chart. The Rumbs here, as on the Globe, are inclining Spirals, or Helical Curves interfecting the VOL. II. S s 2 Meri |
