RHU 



R H U 



point or inftant of the progrefs may be efteemed the be- 

 ginning, the veffel always makes the fame angle with the 

 meridian of the place where it is each moment, or in each 

 point of its courfe, which the wind makes. 



Now a wind, e. gr. that is north-eaft, and which, of con- 

 fequence, makes an angle of 45 with the meridian, is 

 equally north-eaft, wherever it blows, and makes the fame 

 angle of 45 with all the meridians it meets. A veffel, there- 

 fore, driven by the fame wind, always makes the fame angle 

 with all the meridians it meets with on the farface of the 

 earth. 



If the veffel fail north and fouth, it makes an angle mii- 

 nitely "acute with the meridian, i. e. it is parallel to it ; or 

 rather fails in it. If it run eaft and weft, it cuts all the 

 meridians at right angles. 



In the firft cafe, it defcribes a great circle ; in the fecond, 

 either a great circle, viz. the equator, or parallel to it. If 

 its courfe be between the two, it does not then defcribe a 

 circle ; fince a circle, drawn in fuch a manner, would cut 

 all the meridians at unequal angles, which the veffel can- 

 not do. 



It defcribes, therefore, another curve, the effeatial pro- 

 perty of which is, that it cuts all the meridians under the 

 fame angle. This curve is what we call the loxodromlc curve, 

 rhumb-line, or loxodromy. 



It is a kind of fpiral, which, like the logarithmic fpiral, 

 makes an infinity of circumvolutions without ever arriving 

 at a certain point, to which it yet ftill tends, and towards 

 which it approaches at every ftep. 



This afymptotic point of the rhumb-line is the pole : at 

 which, were it poffible for it to arrive, it would find all the 

 meridians conjoined, and be loft in them. 



The courfe of a veffel then, except in the two firft cafes, 

 is always a rhumb-line ; which line is the hypothenufe of 

 a right-angled-triangle, whofe two other fides are the (hip's 

 way or diftance run in longitude and latitude. Now, the 

 latitude is ufually had by obfervation, and the angle of the 

 rhumb, with one or other of the two fides, by the compafs. 

 See Latitude, and Compass. 



All, therefore, that is required by calculation in failing, 

 is the value of the length of the rhumb-line, or the diftance 

 run. 



But as fuch curve line would prove very perplexing in the 

 «alculation, it was neceffary to have the (hip's way in a right 

 line ; which right line, however, mull have the effential por- 

 perty of the curve line, viz. to cut all the meridians at 

 right angles. See Chart. 



If PA, PF, PG, &c. (P/a/,rII. Navigation, Jig. 5.) 

 be fuppofed meridians, A I the equator, and E B, K L, 

 M N, parallels ; A O will reprefent a rhumb-line, which 

 makes equal angles with the meridians, and confequently 

 different from thofe made by a great circle, which cuts the 

 meridians at unequal angles ; whence it follows that the 

 rhumb is not a great circle of the fphere. If a (hip, there- 

 fore, be at firft directed towards E, and conftantly perfift 

 in the fame rhumb, it will never arrive at the place E, but 

 at the place O, which is farther from the equator A I. 



Hence, as on the furface of a fphere the (horteft way 

 between A and O is an arc of a great circle between A 

 and O ; the rhumb-line is not the (horteft way, or lead 

 diftance from one place to another. 



R.HUMB-/,<Wr, Ufe of the. I. If the meridians PA, 

 PB, PC, PD, &c. (Jig. 6.) be not very far apart, the 

 rhumb-line A I H G is divided by the equidiftant parallels 

 L E, M F, N G, &c. into equal parts. 



Hence, 1. The parts of the rhumb A I and A G are as 



the latitudes A L and A N of the places A and G. 

 2. Since the arcs A B, IK, H F, are equal in magnitude, 

 and therefore unequal in number of degrees ; the fum of the 

 arcs, called the latus mecodynamicum, or miles of longitude, is 

 not equal to the difference of longitude A D of the places 

 A and G. 



2. The length of the rhumb-line A G is to the change or 

 difference of latitude G D, in the fame ratio as the whole fine 

 to the cc-fine of the angle of the rhumb. 



Hence, I. The rhumb failed on being given, together 

 with the difference or change of latitude, turned into miles, 

 the length of the rhumb-line, or the diftance from the place 

 A to the place G upon the lame rhumb, is had by the rule 

 of three. 2. The rhumb being given, together with the 

 quantity of the (hip's way on the fame rhumb, /'. e. the length 

 of the rhumb A G ; the difference of latitude D G is had, 

 by the rule of three, in miles, to be converted into degrees 

 of a great circle. 3. The difference of latitude D G bein^ 

 given in miles, as alfo the length of the rhumb-line A G, 

 the angle of the rhumb, and confequently the rhumb failed 

 on, is had by the rule of three. 4. Since the co-fine of an 

 angle is to the whole fine, as the whole fine to the fecant of 

 the fame angle ; the difference of latitude G D is to the 

 length of the rhumb-line A G, as the whole fine to the 

 fecant of the angle of the rhumb. 



3. The length of the rhumb-line, or of the (hip's way in 

 the fame rhumb, A G, is to the latus mecodynamicum or 

 mecodymanic fide A B -f 1 K ■+ H F, as the whole fine to 

 the fine of the loxodromic angle GAP. 



Hence, 1. The rhumb, or angle of the rhumb, being 

 given ; as alfo the (hip's way in the fame rhumb-line A G, 

 the mecodynamic fide is had, by the rule of three, in miles, 

 «'. e. in the fame meafure in which the length of the rhumb 

 is given. 2. In like manner, the mecodynamic fide A B 

 ■f I K -f HF being given, as alfo the rhumb-line or (hip's 

 way A G, the rhumb failed is found by the rule of three. 



4. The change of latitude G D is to the mecodynamic 

 fide AB -f IK + HF; as the whole fine to the tangent 

 of the loxodromic angle P A G or A I B. 



Hence the rhumb or loxodromic angle P A G, and the 

 change of latitude G D being given, the mecodynamic fide: 

 is found by the rule of three. 



5. The mecodynamic fide AB+IK+HFisa mean 

 proportional between the aggregate of the rhumb A G, and 

 the change of latitude G D, and their difference. 



Hence the change of latitude G D, and the rhumb-line 

 A G being given in miles, the mecodynamic fide is found 

 in the fame meafure. 



6. The mecodynamic fide AB+ IK + HF being given, 

 to find the longitude A D. 



Multiply the change or difference of latitude G D by fix, 

 which reduces it into parts of ten minutes each ; divide the 

 mecodynamic fide by this product, and the quotient gives 

 the miles of longitude, anfwering to the difference of 

 latitude in ten minutes ; reduce thefe miles of longitude in 

 each parallel into differences of longitude from a loxodromic 

 table, and the fum of thefe is the longitude required. 



7. If a (hip fail on a north or a fouth rhumb, it defcribes 

 a meridian ; if on an eaft or %veft rhumb, it defcribes either 

 the equinodtial, or a parallel to it. Wolfii Elem. Math, 

 torn. iv. cap. 1 1. 



1 . Tojind the rhumb between two places by calculation, or 

 geometrically, we have two canons, or proportions : the firft, 

 as the radius is to the co-fine of the middle latitude, fo 

 is the difference of longitude to the whole departure from 

 the meridian, in the courfe between the two places propofed. 

 7 The 



