961 



RECKONING AT SEA. 



BECKONING AT SEA. 



therefore if A and g be joined, and yp be drawn perpendicularly to Ap, 

 the angle p\y is the resulting course, &g the resulting distance, Ap the 



difference of latitude between A and y, and p<j is what in called the 

 departure, which, in plane sailing, is identical with the difference of 

 longitude between the same points A and ,'/. 



By drawing lines perpendicular and parallel to Ap, as in the above 

 diagram, there will be formed the several right-angled plane triangles 

 Aim, ben, &c., in each of which there are given the hypothenuse and the 

 angles; and consequently by the rules of plane trigonometry the 

 several sides AM, km, en, Im, &c., may be computed. Now, let these 

 computed values be placed in the third and succeeding columns of the 

 above table in the following order : those which are parallel to Ap in 

 the column N. or S., according as the lines which represent them lie 

 towards the north or towards the south of that extremity which is 

 first, in order of sailing, on the corresponding hypothenuse ; and those 

 which are perpendicular to \]> in the column E. or W., according as 

 the lines ,vhich represent them lie towards the east or west of that 

 same extremity. Then the sum of the numbers in the column N. 

 being subtracted from the sum of those in S. will be found to leave 

 76-24, and this will be the value of Ap in geographical miles (or equa- 

 torial minutes) ; consequently 1 16' 14" will express the extent in 

 latitude to which, on the whole, the ship has sailed southwards during 

 the day. Again the number in W. being subtracted from the sum of 

 those in E. will leave 95'68 ( = 1 35' 41" ), and this will be the value otpij, 

 or the extent in longitude which, on the whole, the ship has sailed east- 

 ward during the day. Thus the position of A being known, we have 

 that of g. In the right-angled plane triangle Apg, having Ap and m 

 in miles, as above, we may compute Ay and the angle pAg, that is, the 

 resulting distance and course. The former will be found to be = 122'35 

 miles, and the latter S. 51 27' E. The series of zig-zag lines which a 

 ship may describe is called a traverse ; the preceding table is called a 

 trarene table, and the whole operation of finding the resulting course 

 and distance is called traverse tailing. 



In practice, both the construction and calculation above indicated 

 are superseded by the use of the table of difference of latitude and 

 departure, which is given in treatises on navigation, and is called a 

 traverse table. The numbers in the table are nothing more than the 

 computed values of the sides of right-angled triangles ; the hypothenuse, 

 or the distance, and the adjacent angle, or the course, being given. 

 Thus, by referring to such a table, the courses and distances being 

 used as arguments, the numbers in the columns N. S. E. W. above, 

 might have been found sufficiently near the truth. And, conversely, 

 seeking in the table the difference of latitude ( = 76) and the departure 

 ( = 96), the corresponding distance ( = 122) would be seen in its proper 

 column, and the angle or course ( = 51) at the bottom of the page. 



The logarithmic or Gunter's scale [SCALE] was formerly, for the sake 

 of expedition, much used in the resolution both of plain and spherical 

 triangles for the purposes of navigation. If, for example, it were re- 

 quired by that instrument to find the values of cq and yd in the triangle 

 cqd, the following proportions 



Rad. : in. gcd (60 34') : : cd (60'5): yd ( = 44) 

 Had. : sin. cdq (29 26') : : cd : cq ( = 25) 



might be worked by taking in the compasses the distance from 90" to 

 60 34' on the Une of sines, and applying that distance on the line of 

 numbers from 60'5 towards zero ; the other foot of the compasses 

 would fall on 44, which is the value of qd ; again by taking the dis- 

 tance from 90 to 29' 26' on the line of sines, and applying it on the 

 ABT8 A!CD SCI. MV. VOL. vr. 



line of numbers, from 50'5, as before, the other foot of the compasses 

 would fall on 25, which is the value of cq. But it is evident that when 

 the angle is small, or nearly a right angle, the instrument must be 

 very inaccurate. 



Should a ship, on any part of the eai-th's surface, sail for a short time 

 iu a direction either due east or due west, so that during that time it 

 might be considered, without sensible error, as sailing on the circum- 

 .ference of a parallel of latitude, the determination of its place is 

 obtained by a different process. Thus, the earth being supposed to be 

 a sphere, the length, in miles, of any arc of the equator between the 

 meridian circles passing through its extremities is to the length, in 

 miles, of the arc between the same meridians on any given parallel of 

 latitude as radius is to the cosine of the latitude < f the parallel. 

 Therefore, when the number of geographical miles passed over on any 

 parallel of latitude is known by the log (all due corrections being 

 supposed to be made), the difference of longitude corresponding to 

 that distance may be found at once by the above proportion. Evidently 

 also, if any three whatever of the terms are given, the fourth can be 

 found ; and thus every variation of the case may be resolved. This is 

 called parallel sailing. 



But the tables of difference of latitude and departure may be ren- 

 dered available for finding the required term if we consider the 

 latitude of the parallel on which the ship is sailing to represent what is 

 called the course in those tables ; the distance in miles on the parallel 

 as the difference of latitude, and the difference of longitude in geo- 

 graphical miles as the distance in the tables ; and then, by inspection 

 as before, the required term may be found. 



The third method of operating, which is called middle latitude sailiny, 

 has been de6ued under LONGITUDE AND LATITUDE, METHODS OF FINDING, 

 and we have here only to point out its application. Let A E be a 



portion of the rhunib-liue which a ship describes while her motion 

 continues to coincide with the direction of one point of the compass, 

 that is to say, while it makes a constant angle with the meridians of 

 her successive places. Let this curve be divided into any parts, A B, 

 BC, &c., of small extent, so that each part may, without sensible error, 

 be considered as a straight line ; and imagine both meridians and 

 parallels of latitude to be drawn through A, B, c, &c. Then the several 

 triangles BAA, o B c, &c., being considered as plane triangles, if the con- 

 stant angle B A b, c B o, &c., be represented by A, we shall have 



A B COS. A = A& , B COS. A=BC , &C. J also 



A B sin. A = B 6 , B c sin. A = c c , &c. ; whence, by addition, 

 (AB + BC + &C.) cos. A = A6 + Bc + &c., and 

 (AB + BC + &C.) sin. A = B6H-cc + &c. 



It is evident therefore that the sum of all the distances multiplied 

 by the cosine of the course will be equal to E M, the difference between 

 the latitudes of A and E as in plane sailing ; but the sum of all the 

 distances multiplied by the sine of the course (that is, the sum of all 

 the departures, B 6, c c, &c.) will be less than A H and greater than N E. 

 Therefore, as an approximation to the truth, we may consider the sum 

 of all these departures as the length, in miles, of the arc X 1 (between 

 A N and M E) of a parallel of latitude equally distant from A M and N E ; 

 that is, of a parallel whose latitude is an arithmetical mean between 

 the latitudes of A and E. Consequently, as in the theorem for parallel 

 sailing, the difference between the longitudes of A and E will be 

 obtained from the proportion 



Cos. mid. lot. ( = lat. of x Y) : Rad. : : (A B + B c + &c.) sin. A : diff. of long. ; 



and, instead of working the proportion, the tables of difference of 

 latitude and departure may be used as before. 



If we imagine the earth's surface to be developed on a plane so that 

 the meridians and parallels of latitude may be respectively parallel to 

 themselves as in the plane chart ; and if the lengths of infinitely small 

 portions of the circumference of the equator have to the lengths of 

 corresponding portions of a meridian, in any latitude, the ratio that 

 the radius bears to the secant of that latitude [MEBOATOR'S PIIOJECTION ; 

 RHUMB-LINE], there may be formed a species of chart which will afford 

 a general and at the same time a sufficiently easy method of deter- 

 mining the elements relating to a ship's place. This method is called 

 Mercator's sailing. In the chart just alluded to, the length of a degree, 

 minute, &c., of longitude, on any parallel of latitude, is constant, being 



