tween the latitude sailed from, and the latitude arrived at. This was probably a casual discovery; and when the places are near the equator, or when their parallels of latitude are not very far distant from each other; the nautical conclusions, drawn from a supposition that the departure is equal to the meridional distance, in the middle latitude, bétween the latitude sailed from, and the latitude arrived at, are very nearly the same as the conclusions derived from Mercator's sailing. But in high latitudes, or in a long run, when the course is not rear some of the four cardinal points, this method is not sufficiently accurate. And, if a ship sail upon several courses, she makes a less departure near the pole, and a greater departure near the equator, from one place to another, than if she were to sail in a direct course; yet in such small distances, as a day's run, the difference is almost insensible. PROPOSITION II. (Plate III. Fig. 2.) (Z) Straight lines equal in length to the distance run, difference of latitude, and departure, form a right-angled plane triangle; having the angle opposite to the departure equal to the ship's course. For it is shewn (U. 355.) that, ab : ai :: al : al; or ab aL :: Ai: Al; and by W. 356. we have ab: AL:: ib: ib+hc+kz+ te, &c. therefore ai: al :: ib : ib+hc+kz+te, &c. But the small elementary triangle is considered as straight lined, and is right-angled at i; therefore the triangle to which it is similar, may be considered as straight lined; AL will be the hypothenuse, Al the difference of latitude, and the departure ib+hc, kz, te, &c. (A) All problems solved by the preceding propositions are said to be in plane sailing; because the very same conclusions would be drawn, if the earth were a plane, and all the meridians parallel to each other. Hence it appears that plane sailing is true, so far as course, distance, difference of latitude, and departure, are concerned. (B) SCHOLIUM. Since, from the third proposition, the distance run, difference of latitude, and departure form a plane triangle; let CA in the annexed figure represent the distance, CB the difference of latitude, AB the departure, and the angle ACB the course (S. 42.), then will CAB be the complement of the course. Hence are deduced the following proportions for solving all the cases that can occur in the practice of plane sailing. W A B A IN -C E IS 1. Radius CHAPTER II. PARALLEL AND MIDDLE LATITUDE SAILING. (C) WHEN a ship sails directly east or west, upon any pa rallel of latitude, the method of finding her difference of longitude, distance, &c. is called parallel sailing. (D) Middle latitude sailing is founded on a supposition, that the meridional distance, half way between the latitude sailed from, and that bound to, is equal to the departure which the ship makes in sailing from the one latitude to the other. It is a compound of plane, and parallel sailing. PROPOSITION I. (Plate III. Fig. 2.) (E) In sailing upon any parallel of latitude, or directly east or west; the cosine of the latitude, is to radius, as the distance run, is to the difference of longitude. Let c be the centre of the sphere, P the north pole, woQE an arc of the equator; pmw, pno, meridians; min the distance run on the parallel mns, and wo the difference of longitude. In the plane of the parallel mns, draw mir, nr, meeting PC the axis of the sphere in r; and in the plane of the equator woQE, draw wc, oc, meeting the axis of the sphere in c: then the angle mrn is equal to the angle wco (R. 132.); therefore the arcs mn, and wo are similar. (Stone's Euclid. VI. and 33d, corol.) Hence, mn parallel of lat. :: wo equator. Or, mn wo parallel of lat. : equator. But the circumferences of circles are as their radii (Simpson's Geometry VIII. and 4th); therefore mr: wc: parallel of lat. : equator. Consequently mr : wc:: mn: wo. But mr is the sine of the arc Pm, or the cosine of the latitude wm, and wc is the radius of the sphere. Therefore, cosine lat. : radius :: dist. : ditf. long. Q. E. D. (F) COROLLARY I. Radius, is to the cosine of the latitude; as the difference of longitude is to the distance run. (G) COROL (G) COROLLARY II. The distance between any two meridians, on one parallel of latitude, is to the distance between the same two meridians on any other parallel, as the cosine of the latitude of the first parallel, is to the cosine of the latitude of the second. (H) SCHOLIUM. This proposition and Corollary I. will solve all the cases that can occur in parallel sailing. Examples ex ercising these cases have been given already at page 261. PROPOSITION II. (Plate III. Fig. 2.) (I) Cosine of the middle latitude (viz. the latitude half way between the latitude sailed from, and that bound to), is to radius, as the departure, is to the difference of longitude (nearly). Let A la represent the parallel of latitude sailed from, lL that bound to, Mkzqn the middle latitude between these parallels, and AbczeuL the rhumb line, or track of the ship, in sailing from A to L. Now whether the ship sail from A to L, or from 1 to A, it has been shewn in the scholium, X. 356. that the departure will be ib+he+kz+ te, &c. and (Y. 356.) this departure is nearly equal to the meridional distance мkzqN. But, Cosine wм radius: мkzqN: woQE. (E. 359.) Viz. Cosine mid. lat. ; rad.:: departure: diff. long. (nearly.) 2. E. D. PROPOSITION III. (Plate III. Fig. 2.) (K) Difference of latitude, is to the difference of longitude; as the cosine of middle latitude, is to the tangent of the course (nearly). For, diff. lat.: radius :: dep. : tang. course (B. 357.) And cos. mid. lat. : radius:: dep.: diff. long. (I. 360.) And, cos. mid. lat. x diff. long.=radius x dep. Consequently, diff. lat. x tang. course=cos. mid. lat. × diff. long. · Diff. lat. : diff. long. :: cos. mid. lat. : tang. course. Q.E.D. (L) COROLLARY. Cosine mid. lat. tang. course:: diff. lat. :diff. long. (nearly.) (M) Distance sailed, is to the difference of longitude, as cosine of the middle latitude, is to sine of the course (nearly). For, radius: distance :: sine of course: dep. (B. 357.) And, coș. mid. lat. : radius :: dep. : diff. long. (I. 360.) Hence, Hence, radius × dep.distance x sine of course. = Consequently, distance x sine of course cos. mid. lat. x diff. long. Distance: diff. long. :: cos. mid. lat.: sine of course. q.E.D. (N) SCHOLIUM. The second, third, and fourth, propositions, include all the cases that can occur in the practice of middle latitude sailing; the several proportions, for the sake of uniformity, are here collected. (0) MERCATOR's sailing is the art of finding on a plane surface, the motion of a ship upon any assigned course by the compass, which shall be true in latitude, longitude, and distance sailed *. (P) In a Mercator's chart, from which this method of sailing is derived, the degrees of longitude are every where equal, the degrees of latitude increase as you approach the poles, and the rhumb line, or track the ship describes, is represented by a straight line. (Q) On the globe the degrees of latitude are everywhere equal, and the degrees of longitude decrease as you approach the poles: that is, the distance between any two meridians in This includes the whole theory and practice of navigation, and if any method could be devised for measuring a ship's course and distance truly, nothing more would be wanted to complete the art. 3 A any |