l :.-' 



NAVIGATION. 



SAlLIXO. 



P 



6, Book V., or Prop. 10 of 



nd therefor. (Ku. 

 the Treatise on I'; 



4- .to., : A6' + be' + cJ' + *o. 

 But A6 + 6c + ed + &O., U the whole distance tailed on 



Flf. II. 



the course, and \f>' + be' + cd + <ta , is tlie difference 

 of ! \ C between A, the place left, and 1!, the 



place arrived at. Consequently, if a right-angled triangle, 

 similar to the little right-angled tri.i . l>e con- 



structed that is, a right-angled triangle, in which A, 

 rig. is. Fig. 15, is the angle of the course, and if 

 tlie hypotenuse AB be made to represent 

 the distance sailed that is, the tra k 

 A B on tho globe then, obviously, the 

 perpendicular A C will represent the 

 difference of latitude, while the base 

 < ' It the side opposite to the course 

 will represent the sum of all the minute 

 departure* which tho ship makes from 

 the successive meridians which it 

 for since 



A'. :l.h' :: AB :BC 



''' :: be : ee' :: ed : <W, <fec. 



.'. \'. : W/ :: A6 + &e + erf + to. : W + cc' + dd* + to. 

 and since by construction AB = A6- r -6r-)-cd- r - <tc., 

 tli.-refure B C ~ 66' + cc' + dd' + Ac. The length B C is 

 called the dejxtrburc made by the ship in sailing from A to 

 Ml it therefore follows that the distance sailed, tho 

 difference of latitude made, and the departure, may Jbe re- 

 presented by the sides of a right-angled plane triangle, the 

 angle opposite to the departure being the angle of the course. 



I. Radius 



II. Radius 



III. Cos. Course 



IV. Itadius 



V. Distance sailed 



VI Distance sailed 



VII. Sin. course 



VIII. Tan. course 

 IX. Dilf. Latitude 



Sin. Course 

 Cos. Course 

 Radius 

 Tan. Course 

 Diff. Latitude 

 Departure 

 Radius 

 Radius 

 De-part ure 



If tho table of natural sines, oosii es, ifeo., be used, 

 then Radius = 1 ; if the table of logarithmic sines, 

 cosines, <tc., be used and they are employed by seamen 

 too indiscriminately then log. Radius = 10. || 



In all books on Navigation, the latter tables, exclu- 

 sively, are referred to ; but, as already stated in the 

 INTRODUCTION, we would recommend a departure from 

 this practice ; we shall, therefore, in general, exhibit the 

 calculations by both tables. 



It is not con.sidi Ti-d necessary, in the examples that 

 follow, to introduce diagrams of tho several tri 

 but the learner should always roughly sketch the suitable 

 triangles himself, observing that, as is usual in maps, the 

 top of the page is to bo regarded as North, and the bot- 

 tom at South ; the right-hand side East, and the loft 

 West. In sketching his right-angled triangle, tli< 

 he should first draw the N. and IS. line, or the horizontal 

 decant, p. Ml. 



Sre an/. . p. 1037. 



t We harr i in ni (tit It u well toexpreu the rale* and formal v irivfn In 

 * ImoDVCZU*, In the funo of proportion* hen, u x*an are more 



, when any two of the four things Dictum, 

 Difference of Latitude, Departure, and Course are 

 known, the remaining two may be determined by the 

 solution of a right-angled plane triangle ; so that, as far 

 as these particulars are < is are the 



same as it the ship were sailing on a piano surface, tho 

 meridians In -iu^ replaced by parallel straight lines, and 

 the perpendiculars to these, taken for the parallels of 

 latitude. It is thus that that part of Navigation, win. h 

 is concerned only with the four things just mentioned, is 

 called PLANE SAILING. 



The line B C, or the side of the right-Angled triangle 

 opposite to the course, is not the representation of any 

 corresponding line on the globe : it is the entire sum of 

 all the minute departures made by the ship in passing 

 from meridian to meridian, from A up to It. 



The foregoing investigation comprehends the whole 

 mathematical theory of 1'lane Sailing, into which, it 

 will be observed, the consideration of lunyitutle does not 

 enter. 



The attentive reader will perceive, that in replacing the 

 spiral track of a ship's run, and the great circle arc which 

 measures the difference of latitude mado good in that 

 run, by tho hypotenuse and perpendicular of a i 

 angled triangle drawn upon a finite surface, we sai ; 

 not the slightest amount of accuracy, it is shown above, 

 that if this spiral track were unbent into a straight lino 

 A B, and at one extremity, A, of this straight line a plane 

 angle equal to that of the course were made by A C, and 

 the perpendicular B C drawn it is shown that A C must 

 accurately represent the difference of latitude, and B C 

 the departure. 



It cannot be any objection to this conclusion that we 

 have taken small triangles on the sphere to be plane 

 triangles ; for the reasoning fixes no limit to the d 

 of smallness of the sides ; nor must it be understood that 

 we have assumed the sphere, on which these trial 

 are assumed to be figured, to be itself a plane. The 

 assumption extends only to the length of supposing tho 

 sphere to present a succession of triangular plane faces ; 

 and as each face is contracted to any degree of minute- 

 ness, the error of this supposition ultimately disappears. 

 As a corollary to what is proved above, we may add 

 that In sailing upon a single rhumb, the differences of 

 latitude made, are proportional to the distances run. 

 And, from the theory of the right-angled triangle estab- 

 lished in the INTRODUCTION, + we have all the proportions 

 usually given on this part of the subject in books on 

 Navigation ; they are expressed as follows :J 



Distance sailed 

 Distance sailed 

 Diff. Latitude 

 Ditr. Latitude 

 Radius 

 Radius 

 Departure 

 Departure 

 Radius 



ture. 



Diff. Latitude. 

 Distance sailed. 

 Departure. 

 Cos. Course. 

 Sin. Course. 

 Distance sailed. 

 Ditf. Latitude. 

 Tan. course. 



meridian, and then take a portion of it for the difference 

 of latitude, drawing, from tho latitude reached, the I 

 of the triangle, to represent the departure to the n'y/if, 

 if the departure be east, to the li-ft if it be west. Tho 

 hypotenuse will then represent the distance sailed, and 

 the angle between it and the difference of latitude, the 

 course. It will be as well to regard the vertex of this 

 angle as at the centre of the compass-card, since it is the 

 centre of the sensible horizon at starting on tho c<> 

 and thus no mistake can be made as to which side of tlie 

 meridian line tho angle of the course is to lie on, or 

 whether its opening be upward or downward. 



Example*. 



1. A ship from latitude 47 3ff N., has sailed S.W. by 

 S., a distance of 98 miles: what latitude is she in, and 

 what departure has she made I 



accustomed to UM them in thu "h.-vpe ; bat the reactor of the Introduction 

 will nee that this rulc-of-lhree arrangement of tin- t>Tin employed U nut 

 ncef*arjr, and he may therefore adopt It or not, M he pleane*. 

 i SCO out*, p. 66A. H Sec ante, p. boo, rt tef. 



