NAYIGAT1 



SAILIVO. 



dnT<nr-dt X wn- oouwe-diff. long. X cos. m 



.'. MIL ooune : cos. corrected niid. lat. : : diff. 



long. : distance. 



00!CST*CCYIO* Or A TABL1 OF MKRIMOXAL PARTS BY 



HOD. In Mr. \\ n-hfs method of 

 cowtructing table of meridional parts, every tabular 

 number, after the firrt, depend, upon the number pre- 

 viously calculated, as already shown at page 1001 Dr. 

 HalTey propoxl Mother plan, by which the nu-nd.t,al 

 p^ooVwiponding to any given latitude may be com- 

 puted independently of previous calculations. A* th 

 matter U of 10 much practical interest, we shall here show 

 ! ,> !; ii, v\ b <:!'. Otad . . . 



To transcribe Dr. Halley's investigation (PMoiophital 

 TVttfuorfum*, No. 21) would be to occupy more _ro< 

 than we can spare : we must therefore employ a different 



oceH and, if the reader be unacquainted with the 

 fcrrtpriiciptes of the Differential and integral Calculus, 

 be may pass it over, and omit this article altogether. 



U is shown at page 1061, that if a ship in latitude x 

 vary her latitude by a small portion of the meridian 

 A m, and that she continue her course till her departure 

 becomes equal to the difference of longitude due to the 

 differenoeof latitude A x, then the increased difference 

 of latitude A V, necessary to produce this effect, will 



It thus appears, that if the log. tan. <,f half the 

 nleim-nt of anv latitude IK- subtracted from 10, and the 

 remainder be multiplied l.y T'.'l.VTOl IC.7'., tho product 

 will be the meridional parts, in nautical miles, cor- 

 responding to that latitude ; and, therefore, as observed 

 already, the meridional parts for any latitude may 

 be computed independently of previous results 



The logarithm of the constant multiplier , lUBTQt . . . 

 is 3-8984895; so that from (2) we have 



A y-sec. x A*. 



A*" 60 -*- 



This equation is rigorously true only when A *, wl 

 consequently A V diminished to tlio last degree of 

 smallueM ; so that, employing the notation of the dif- 

 .ial calculus, we strictly have 



-/-sec. * .'. dy^sec. x dx 

 cut 



and the integral of this is the equation 



y-log. tan. (45 + $*) ... (1) 

 The logarithm here implied, is the Napierian loga- 

 rithm. To change it into Brigg's, or a common loga- 

 rithm, we must multiply it by the modulus .> 3^2585 . . . 



(see the article on Logarithms, page 519) ; we shall then 



.,'arithm of the natural tangent of 45 + J x, 



Ming to Brigg's system. But in the table* it is not 



the l"g. of the natural Ungent of 45 + i x, but this log. 



increased by 10, that is inserted : hence, for the common 



uhms, the equation (1) is 



=log.tan.(45' + i*)-10 

 .'. y-2-302585 [ log. tan. (45 + J x) -10 j 



This is the correct expression for the increased meridian 



li, measured in miles, in reference to a globe whose 



1 nnlo ; to adapt it to the globe of the earth, we 



ni'ist multiply the expression by the radius of the earth, 



or by 3437-74679 nautical miles; for in every circle the 



radius U equal to 343774G79 minutes of that circle, 



tlius : 



Rod. earth X 3-14159 . 

 nautical miles. 



10800 

 Iud - earth =:;!. iv,... 



Hence, for the number of miles in the lengthened 

 inrridinii y, from the equator to the latitude x, we have 

 the formula 



y. 70157044679 [ log. tan. (45 + i *) -30 j 

 or, since tan. (45 + J*)- cot (45 -$1) and that 



n 



cot. jj-, and .'. log. cot 20 log. tan. 



the preceding formula may be otherwise written, as 

 follows : 



Meridional lat. - 7 ' ' [10 -log. tan. (45 



| proper lat)}. ..(2) 



-length of .180=- 10,800 

 3437 -74679 nautical miles. 



log. merid. lat - 3'8984895 + log. 1 10-log. tan. } 



comp. lat I . . . (3) 



from which formula, the tnie meridional parts for all 



latitudes may be calculated. 



We shall conclude this part of our subject with a tei 

 remarks upon the peculiar character of the Rhumb une, 

 or, as it is sometimes called, the Loxodramic cunt. 



Ox THB CONTINUED RHUMB LINE. From the prin- 

 ciples of Mercator's sailing, or from the diagram at 

 pace 1061, which connects the enlarged meridian will 

 the difference of longitude, it is clear that if a ship set 

 out from any point on the globe, and sail on the same 

 oblique rhumb towards the pole, it can reach it only 

 after circulating an infinite number of times round i 

 for, from any point to the pole, the enlarged meridian is 

 infinite in length, and so, therefore, is the difference of 

 lon"itde due to this advance in latitude, provided lon- 

 gitude be measured round the globe in one uniform 

 direction : the longitude, thus measured, being infinite, 

 the ship must wind round the pole an infinite number ol 

 times. 



But paradoxical as the statement may appear- 

 nevertheless true, that the infinite number of revolutiu 

 about the pole are performed in a finite time, and that 

 the entire length of the spiral track of the ship is a finite 

 line. However strange this may seem, it follows, as a 

 necessary consequence, from the principles of platie tad- 

 ing ; for these principles correctly give, 



diff. lat 

 length of track oog course 



which is finite ; and, therefore, the rate being uniform, 

 : ibed in a finite time. 



The matter may be explained as follows : NN haw 

 be the progressive rate of the ship along its undeviat in<< 

 course, the time* of performing the successive revolutions 

 about the pole continually diminish as the latitude in- 

 creases ; both the extent of circuit, and the time of per- 

 forming it, evidently tending to zero tho limit is ac- 

 tually attained, only at the pole itself. Consequently, 

 an infinite number of circuits must ultimately be per- 

 formed to occupy a finite portion of time. 



The case is somewhat analogous to those infimt 

 descending series so frequently met with in arithmeti 

 and algebra ; they are infinite not in their aggregate 

 amount, but only in the number of tho continually- 

 diniinishing parts into which that amount is divided, 

 these parts becoming less and less, and ultimately 

 vanishing altogether. 



In like manner, every additional circuit the ship makes 

 round the pole, increases the length of the previously- 

 described track by a quantity less and less, the suca 

 sive increments continually diminishing, and ultimately 

 vanishing altogether ; so that, just as in a common 

 decreasing geometrical series, the sum of the incremer 

 thus continually tending to, and ultimately terminating 

 in zero i., finite ; and hence tho time of describing the 

 track must be finite too. 



TRAVERSES BY MID-LATITODB AND MBRCATOR s 

 INO. In order to work a traverse with a vi, w t>. I: 

 the ship's place, and tho direct course and distance to 

 it, we may proceed in either of tho two ways fol- 

 lowing : . 



1. Form a traverse table, in the first two columns pi 

 which insert the several courses and di nd m 



the remaining columns put the corresponding OUMNMM 

 of latitude and departures, found either by computation 

 or by reference to tho already computed traverse table, 



