62 4715 4807 4840 4872 6914 Degrees 1 The Use of the Table of Meridional Parts. Latitude 26.00 Deg- 1616 Merid. Parts. 63.00 Deg- 4905 M. P. &c. 2. What Degrees and Cetesins do answer to. For the Refolving of the following Problems, which (in kind) Fig.. will be the fame with those in Plain Sailing as to the Trigonometri. LXIX. bal Work. But in the performance of them, I will fet the places proposed down upon a Sea-Chart, made according to the Projection of Mercator, where the Degrees of Latitude are enlarged according as they tend nearer and nearer to the Pole: And on the out-fide of fucha Chart, I have described a Plain Sea-Chart also, where the Meridians and Parallels of Latitude are every where of an equal distance; and fo wrought the Questions according to both Charts, by which the difference will more plainly appear: And the Chart which I have here made to work the following Examples upon, begins (at the bottom of it) at about 49.50 deg. of Latitude, and extends upwards to 55. 50 deg. of Latitude; and at the Top and Bottom to Six Degrees Difference of Longitude: But the Parallels of Latitude in the Plain Chart, (which are diftinguished by Pricked Lines) extend to above 59 deg. of Latitude within the fame Bounds. PROB. І. The Latitudes of two Places, and their difference of Longitude being known; To find, (1) The Rumb leading from one to the other: And (2) Their Distance upon that Rumb: according to Mercators Chart. By Trigonometrical Calculation ET one Place be at A L in the Latitude of 50 deg. and the other at C, in the Latitude of 55 deg. but differing in Longitude Eastward from B, 6. 50 deg. which two places thus fet downin the : Fig. the Chart, draw the Line A C, which is the Rumb (or Course) LXIX. from A to C: And thus have you upon your Chart conftituted a Right-angled Plain Triangle ABC, in which you have given, (1) B C, the Difference of Longitude 6.50 deg. which you muft Reduce into Miles (or Minutes) of Longitude by multiplying them by 60, (as in Plain Sailing) and they make 330.00 Miles. — (2) АВ, the Difference of Latitude 5.00 deg. which must be Reduced into Meridional Parts thus : The Meridional Parts for 50. 00 d. are For 55.00 Their Difference 3475 2968 493 Which are the Meridional Parts answering to the 5.00 deg. Difference of Latitude of the two Places A and C: And by these you may find; (1.) 'The Rumb BAC. (2) The Meridional Distance upon the Rumb A C. By Trigonometrical Calculation. (1.) For the Rumb BAC As the Merid. Dif. of Latitude AB, 493 M. P. So is the Radius, Tangent 45 deg. 2.692847 12.518514 IO. 9.825667 To the Tangent of BAC, 33.80 degrees. Which is the Rumb, leading from A to C: whose Complement 56. 20 deg. is the Angle BCA, or the Complement of the Rumb: (2.) For the Meridional Distance on the Rumb AC. As the Co-fine of the Rumb: BCA, 56. 20 d. 9.919592 Is to the Merid. Differ. of Latitude AB 493 M. P. 12.692847 So is Radius Sine 90 deg. 10. 2.773255 To the Merid. Distance A C, 593.27 M. P. onal Parts, you must Firft, Substract the MPO of AB, from the MP of A C, their Difference is ICO. 27 M.P. the half whereof is 50.13 M. P. Secondly, Add this half Difference 50. 13, to the Meridional Parts of the Greater Latitude 55.00 deg. viz. 3968, and it makes 4018. 13, which are the Merid. Parts answering to 55.50 deg. of Latitude. Alfo, fubftratt this half Difference 50. 13, from the Merid. Parts of the Leffer Latitude 50.00 deg. viz. 3475, and the Remainder Remainder will be_3424.87, which are the Merid. Parts answering Fig. to 49.38 deg. of Latitude. Thirdly, The Difference of these two Latitudes last found, viza, 55. 50 de. and 49. 38 deg. is 6.12 deg. And that is the true Distance upon the Rumb between A and C, in Degrees. And according to this Method, may all the other Problems (before wrought by the Plain Sea-Chart) be performed by Tri. Calculation; by the fame Canons: Only remember, to Reduce the Difference of Latitudes and Distance on the Rumb, into Meridional Parts, but the Difference of Longitude into Miles, as, in the other. Now that you may fee the Difference between these two ways; See the Figure of the Chart, wherein the two Places are laid down by the Plain Chart; as in the Triangle AOS, which Triangle be ing refolved (as is before shewed) you will find (1) The Rumb OAS, to be 47. 73 deg. Differing from the truth 13. 93 deg.. (2) The Distance upon the Rumb AS, to be 7. 43 de. Differing from the other 1. 31. And let thus much fuffice for Sailing according to Mercator. There is a third way of Sailing, which differs not much from this way of Mercator's, but may be performed without Reductien or use of Meridional Parts; which is called Sailing by the Middle Latitude: of which a little. III. Of Sailing by the Middle Latitude.. Shall exemplefie this way of Sailing by Four of the most usual Irablexem Navigation: And they shall be the fame as in the foregoing: whereby the Difference will the better appear.... PROB. I. The Longitude and Latitude of two Places, A and C, Given (1) The Rumb BAC. (2.) The Distance upon the Rumb: AC. 1.ΧΙΧ.. LET L ET one Place be in 50. 00 de. the other in 55.00 de. of North Latitude: And 6. 50 deg. difference in Longitude: Then the Middle Latitude between 50. 00 deg. and 55.00 deg. is 52. 50 de. And its Complement 37. 50 deg. Then (1) For the Rumb: The Proportion is, As the Difference of Lat. 300.00 Miles. 2.477121 Is to the Dif. of Longit. 330. 00 Miles. 2.518524 So is the Co-fine of Middle Latitude, 37. 50 de. 9.784447 12.302971 THE Latitudes of two two Places, A and C, and theRumb, BAC, (or Course) between them Given : to Find, (1) Their Distance AC. (2) Their Difference of Longitude, В С. For the Distance it may be found as in the former Problem. But To the Difference of Longitude BC, 330.01 Μ. 2.518550 PROB. |