Now to discover how the Meridians are expanded from the Equator in Proportion to the Degrees of Longitude decreasing towards the Poles; Let A B D in the annexed Scheme, reprefent the Quarter of the Meridian; F B, the Radius of a Parallel of Latitude; now C G, which is equal to F B, is to CD, as a Degree on the Parallel is to a Degree on the Meridian, or any great Circle; and as CG is to CD, fo is CB to CE, the Secant of the Latitude of the Parallel. F E G D Therefore in a Projection of the Globe where the Meridians are kept parallel, it is evident, a Degree on the Meridian at any Parallel must be equal to the Secant of the Latitude of that Parallel; and the Distance of any Point, upon the Meridian, from the Equator, is equal to the Sum of the Secants contained between it and the Equator. Hence it is evident, that by a continual Addition of Secants, beginning at the Equator, a Table of Meridional Parts may be compofed for every Degree and Minute in the Quadrant. Therefore the Meridional Difference of Latitude between any two Places may be eafily found, by finding the Meridional Parts answering to both Latitudes, and either adding or fubtracting, according as the Cafe requires; that is, if both North or both South, fubtracted; but if one North and the other South, added gives the Meridional Difference of Latitude between them: but the Meridional Difference of Latitude between any Place and the Equator is found, by taking the Meridional Parts belonging to the Latitude of that Place. To find the Meridional Parts belonging to any Number of Degrees and Minutes of Latitude required. In the Table of Meridional Parts, feek the Degrees on the upper Part of the Table, and in the left or Right Hand Column the Minutes marked on the Top with M. oppofite to which, and under the Degrees, are the Meridional Parts required. Suppofe the Meridional Parts belonging to 57° 18′ were required? Look in the Table under 57°, and oppofite to 18 ftands 4216, the Meridional Parts of 57° 18'. The fame may be observed of any Degrees and Minutes required. The Solution of the following Problems, as F G well as all other Trigonometrical Operations in Navigation, depend upon the fourth Propofition B of the fixth Book of Euclid; where it is demonftrated, that Triangles which are fimilar or alike their like Sides are Proportional: Therefore in the annexed Triangles A B C, and AFG, the Radius, Sine, and Sine Complement, or the Radius Tangent, and Secant, form a right-angled Triangle; and the Sine Tangent and Secant of A PLAN LANE SAILING, as has been before obferved, fuppofes the Earth and Sea to be in the Form of a Bowling-Green, on which the Meridians are parallel, and the Degrees of Latitude and Longitude equal in all Places; but the Earth and Sea include a round Body or Globe, on which the Degrees of Latitude are equal in all Places, and the Degrees of Longitude decrease from the Equator in Proportion to the Sine Complement of the Latitude. Though the Meridians all meet at the Poles, and the Parallels to the Equator continually decreafe, and that in Proportion to the Co-Sines of their Latitudes; yet in old Sea Charts the Meridians were drawn parallel to each other, and confequently the Parallels of Latitude made equal to the Equator, and fo a Degree of Longitude on any Parallel, as large as a Degree on the Equator Alfo in thefe Charts the Degrees of Latitude were ftill reprefented (as they are in themselves, equal to each other, and to thofe of the Equator; by thefe Means the Degrees of Longitude being increased beyond their juft Proportion, and the more fo the nearer they approach the Pole, the Degrees of Latitude at the fame Time remaining the fame, it is evident, Places must be very erroneously marked down upon thofe Charts, with respect to their Latitude and Longitude, and confequently their Bearing from one another must be very false, To remedy this Inconvenience, fo as ftill to keep the Meridians parallel, it is plain we must protract or lengthen the Degrees of Latitude in the fame Proportion as thofe of Longitude are, that fo the Proportion in Eafting or Wefting may be the fame with that of Northing or Southing; and confequently the Bearing of Places from sach other to be the fame upon the Chart as upon the Globe itself. Now to discover how the Meridians are expanded from the Equator in Proportion to the Degrees of Longitude decreasing towards the Poles; Let A B D in the annexed Scheme, reprefent the Quarter of the Meridian; F B, the Radius of a Parallel of Latitude; now C G, which is equal to F B, is to CD, as a Degree on the Parallel is to a Degree on the Meridian, or any great Circle; and as C G is to CD, fo is CB to ČE, the Secant of the Latitude of the Parallel. F E G D Therefore in a Projection of the Globe where the Meridians are kept parallel, it is evident, a Degree on the Meridian at any Parallel must be equal to the Secant of the Latitude of that Parallel; and the Distance of any Point, upon the Meridian, from the Equator, is equal to the Sum of the Secants contained between it and the Equator. Hence it is evident, that by a continual Addition of Secants, beginning at the Equator, a Table of Meridional Parts may be compofed for every Degree and Minute in the Quadrant. Therefore the Meridional Difference of Latitude between any two Places may be eafily found, by finding the Meridional Parts answering to both Latitudes, and either adding or fubtracting, according as the Cafe requires; that is, if both North or both South, fubtracted; but if one North and the other South, added gives the Meridional Difference of Latitude between them: but the Meridional Difference of Latitude between any Place and the Equator is found, by taking the Meridional Parts belonging to the Latitude of that Place. To find the Meridional Parts belonging to any Number of Degrees and Minutes of Latitude required. In the Table of Meridional Parts, feek the Degrees on the upper Part of the Table, and in the left or Right Hand Column the Minutes marked on the Top with M. oppofite to which, and under the Degrees, are the Meridional Parts required. Suppofe the Meridional Parts belonging to 57° 18′ were required? Look in the Table under 57°, and oppofite to 18 ftands 4216, the Meridional Parts of 57° 18′. The fame may be observed of any Degrees and Minutes required. F c G The Solution of the following Problems, as well as all other Trigonometrical Operations in Navigation, depend upon the fourth Propofition B of the fixth Book of Euclid; where it is demonftrated, that Triangles which are fimilar or alike their like Sides are Proportional: Therefore in the annexed Triangles ABC, and AFG, the Radius, Sine, and Sine Complement, or the Radius Tangent, and Secant, form a right-angled Triangle; and the Sine Tangent and Secant of A any Arch in one Circle, is in Proportion to the Sine Tangent and Secant of the fame Arch in another Circle, as the Radius of the one is to the Radius of the other. Let A B reprefent the proper Difference of Latitude; B C the Departure; A C the Distance; and the Angle B A C the Ship's Courfe. Produce A B to F, to reprefent the Meridional or enlarged Difference of Latitude; and parallel to B C draw FG, to reprefent the Difference of Longitude. It is plain that A B is in proportion to B C, as AF is to FG; and that the Sine Tangent and Secant of the Triangle FAG, is as the Radius AC is to the Radius AG: Wherefore, as the Sine Complement of the Courfe is to the Meridional Difference of Latitude, fo is the Sine of the Courfe to the Difference of Longitude, and the contrary. And as the proper Difference of Latitude is to the Departure, fo is the Meridional Difference of Latitude to the Difference of Longitude. Hence it will be eafy to reduce Departure into Difference of Longitude, and Difference of Longitude into Departure. Therefore, all Cafes in Mercator's Sailing are worked by Geometry, Trigonometry, Gunter's Scale, Inspection, and the Tables exactly the fame as in Plane Sailing, by only confidering the Meridional Difference of Latitude as proper Difference of Latitude; and the Difference of Longitude as Departure: For it is no more than enlarging the Difference of Latitude that the Difference of Longitude may be in Proportion to the Departure, as the Meridional Difference of Latitude is to the proper Difference of Latitude, the Courfe continuing the fame; and the Sine Complement, or Tangent Complement of the Course bears the fame Proportion to the Meridional Difference of Latitude that the Sine or Tangent of the Courfe does to the Difference of Longitude, and therefore is found in the fame Manner as if they were Difference of Latitude and Departure in Plane Sailing. CASE I. The Latitudes and Longitudes of two Places given, to find the direct Courfe and Distance between them. What is the Course and Distance from the Lizard, to the Eaft Part of Barbadoes? Lizard's Lat. 49° 57'N. Mer. Parts 3470 Long. 5° 14'W. Difference 36 452205 M. Diff. 2671 Diff. 54 23=3632 M. 1 MERCATOR's SAILING. By PROJECTION. ift. Draw the Meridian of the Lizard A B, which make equal to the Meridional Difference of Latitude 2671 Miles, on B erect the Perpendicular BC, upon which fet off the Diff. of Longitude 3263 Miles. Draw the Line CA; then will the Angle BAC be the Course 50° 42'. 2dly. Set off the proper Diff. of Latitude 2205, upon the Me- C, E 93 Prop 2205 A Cou 50 42 Dep 2694 D B ridian, from A to D, and ED parallel to BC, to cut A CinE; then will E A be the Distance 3481 Miles. To find the fame By CALCULATION. To find the Distance. To find the Course it will be, As Mer. Diff. of Lat. 2671 3,42667 As Co-Sine Courfe 50.42 9,80166 Is to Radius 90 10.00000 Is to P. Diff. of Lat. 2205 3,34341 So is Diff. of Long. 3263 3,51362 So is Radius 90° 10.00000 13,51362 To the Tan. Cou. 50° 42′ 10,08695 To the Distance 13,34341 9,80166 3481 3,54175 Whence the direct Courfe from the Lizard to Barbadoes, is S. 50,42 W. or nearly S. W. by W. W. Distance 3481 Miles. By GUNTER. Firft. The Extent from the Meridional Diff. of Lat. 2671 to the Diff. of Long. 3263 on the Line of Numbers, will reach from Radius, or 45° to 50° 42' on the Line of Tangents." Secondly. The Extent from Co-Course 39° 18' to Radius or 90° on the Line of Sines, will reach from the Proper Diff. of Lat. 2205 to 3481, the Distance, on the Line of Numbers.' By INSPECTION. RULE. Look for the Meridional Difference of Latitude, and Difference of Longitude, as if they were really Difference of Latitude and Departure in Plane Sailing, and the Courfe will be found among the Points or Degrees at the Top or Bottom; then, inftead of the Meridional Difference of Latitude, look for the proper Difference of Latitude, in the Column marked Lat, and in the Column marked Dift. will be the Distance required. Thus, with the 20th Part of the Mer. Diff. of Lat. 2671, and Diff. of Long. 3263, viz. 133 and 163.1 the nearest Numbers are 163.1 and 133.8 ftanding over 4 Points, which is the Courfe, over 1 |