i n 



NAVIGATION. 



[XIUDLI LATITUDE 8AILI5O. 



1 ..-. Iv 



and by thU the problem is sol Ted upon strict mathe- 

 matical principles. Middle latitude Bailing is a combina- 

 tion of plane sailing and parallel sailing, which are 

 united in the following way : 



It has been seen, in the theory of plane sailing, that 

 the line called the departure, is a lino equal to the sum <>f 

 all the elementary departures made by a ship in sailing 

 on an oblique rhumb. Thus, if A B, in Fig. 18, be the 

 distance sailed, the departure is made 

 u). of all the elementary portions of 

 the parallels of latitude lying between 

 A D and C B ; it is plain, therefore, 

 that the departure in less than AD, 

 and greater than C B, since the 

 meridians approach closer together, 

 the nearer the parallel is to the pole : 

 there is, therefore, some parallel, M L, 

 between A and C, such that the portion 

 M I, would be exactly equal to the 

 departure ; and that, in latitudes near 

 o the equator, this parallel ML cannot 

 differ materially from the middle 

 parallel. It is on the supposition that 

 the departure is equal to the distance 

 between the meridians left and arrived at, measured on 

 the middle parallel, that middle latitude sailing is founded. 

 From a mere inspection of the diagram or, better 

 still, of a common globe it is obvious that this sup- 

 position can differ but very little from the truth for low 

 latitudes, and for such short distances, A B, as a day or 

 two's run ; and more especially if tho angle of the course 

 be large, so that but little difference of latitude is made, 

 and therefore the parallels AD, C B, are pretty close 

 her. In such favourable cases, the method of 

 miilille latitude sailing though, if uncorrectod, only an 

 ai'pmxiination to the truth is preferable to the method 

 of Mercator's sailing, though this is theoretically accurate, 

 for reasons that will be hereafter shown. In high 

 latitudes, however, this method is not to be depended 

 on, at least for more than a single day's run, if the 

 angle of the course be small, and the middle latitude 

 be uncorrected ; because the interval between the latitude 

 left and that reached, may be too wide to warrant the 

 supposition that the departure is equal to the middle 

 parallel, between the meridians left and arrived at. But 

 if the middle latitude be corrected by Workman's table, 

 given at page 1060, all objection will be obviated. 



Investigation of the fiulcs for M iddle Latitude Sailing. 

 Let A B in the preceding diagram be the track of the ship, 

 then the difference of longitude made will be tho same as 

 if the ship had sailed from M to L, along the middle 

 parallel M L. By the present hypothesis, the distance 

 on this middle parallel is the departure made in running 

 from A to B ; hence, the departure being known by plane 

 sailing, we know the length of the parallel M L : and we 

 know, also, the latitude of that 

 parallel Consequently, the dif- 

 ference of longitude may be found 

 as in parallel sailing. Tims, if in 

 the right-angled triangle BCD, 

 Fig. 19, tho base represent the 

 departure, that is, M L, and the 

 angle at the base be made equal to 

 ' the latitude of M L, then the 

 D hypotenuse will represent the dif- 

 ference of longitude between M 

 and L ; that is, between A and B. 

 As tho base of the right-angled 

 triangle represents tho departure 

 made, wo may connect with it, 

 as in tho annexed diagram, the 

 difference of latitude A 0, and the 

 i auce A B, as in plane sailing. 

 Wo shall thus have a sort of double 

 triangle ; in one triangle (the lower one here) will be 

 represented the diff. lat. A C, the dist. A B, the angle A 

 r>f the course, and the departure C U, equal in length to 

 M L in tho preceding diagram. In the other triangle 

 will be represented by C U, tho distance M L in tho 



i ls> 



preceding diagram, the iliti long. B D, of M and L, and 

 tho angle CBD of tho mill. lat. ; tho lower triangles 

 being constructed conformably to tho principles of piano 

 sailing, and the upper conformably to the principles of 

 parallel sailing. In tho hitter the perpendicular C D is 

 of no significance. 



1. lu the triangle DCB we have 



cos. DEC : radius :: BC : DB, 

 that is, cos. mid. lat : radius -. : depart. : diff. long. ...(!) 



2. In the triangle D B A we have, 



sin. D : sin. A : : A B : B D (ItrrRODOcriow, p. 1044). 

 that is, cos. mid. lat. : sin. course : : dist. : diff. long. . . (2)- 



a Also, in the two triangles A B C, D B C, we have 



AC tan. A-BC, BD cos.D BC - B C, 

 .'. A C tan. A - B D cos. D B C, consequently 



AC:BD::oos.DBC:tau. A; that U, 

 diff. lat. : diff. long. : : cos. mid. lat. : tan. course. . . (3). 



The proportions marked (1) (2) (3) comprehend the 

 entire theory of middle latitude sailing. It is scarcely 

 necessary to mention that the middle latitude is half tho 

 difference of the extreme latitudes when both are north 

 or both south, and half their sum when one is north and 

 the other south. 



The three proportions given above may be united in 

 one set of equations, by which mode of expressing them, 

 they will, perhaps, appear in a form more convenient for 

 memory. They are as follows : 



diff. long. = - p?r - departure X sec. mid. lat 



dist X sin- course 



cos. mid. lat. 



dist. sin. course sec. mid. lat 

 diff. lat. Xjtan. course = 



cos. mid. lat. 



diff. lat. X tan. course X sec. mid. lat 

 But by imprinting upon the mind the two connected 

 triangles in tho preceding diagram, any recurrence t< 

 formulas will not bo necessary on the part of any o 

 familiar with what is taught in the INTRODUCTION to this 

 Section ; and this is one advantage of making tho plane 

 triangle subservient to the purposes of Navigation. How- 

 ever, without reference to the triangle, the single equa- 



tion, diff long. =^ 



liar in middle latitude sailing ; the other equations are, 

 in fact, implied in this ; instead of departure, the equiva- 

 lent to it is substituted. 



1. A ship from latitude 51 18' N., longitude 9 50 W. 

 steers S. 33 8' W., till she has run 1024 miles : required 

 the latitude and longitude in. 



1. To find the difference of latitude. 

 By logarithms. 



As radius . . 10 



: cos. course, 33 8' 

 : : dist. = 1024 . . . 



: diff. lat - 857-5 . . 2"9332 



Without logarithms. 

 diffi. lat. ~ dist X cos. course 

 cos. 33 8' 

 1024 reversed . 



857-6 

 /. the difference of latitude is 857$ miles. 



