NAVIGATION CHART, &€. 311 



PROPOSITION II. 



The Angle of Position at the middle longitude between 

 two places is very nearly equal to the loxodromic angle or course 

 between them; and differs at each place from the loxodromic 

 course, by the inclination of the meridian at each place to the 

 central meridiem. 



Let a, b, be two places on the arc of a great 

 circle, extended into a right line which is 

 crossed by the central meridian T d, and the 

 meridians T a, I b, at their respective angles 

 of position, then the angles a T d, d I b will 

 be the inclination of the meridians at a and b, 

 to the central meridian, respectively. Draw 

 T t, dt perpendiculars to «T, ab, and draw 

 a t; then because the loxodromic course, cuts 

 the meridians at equal angles, the course near 

 a, may be considered as a portion of the logarithmic spiral ; 

 by the known property of which, t a, will be the radius of 

 curvature at the point a, and n a perpendicular to t a, its tang- 

 ent, making the angle dan=di a^=dT a, (being on the same 

 segment a d of the circle passing through a d t T;) and there- 

 fore, the exterior angle T d b is=T a d+(d T «=) d a n=zT a n, 

 the course; and the course must be parallel to ab wherever it cros~ 

 ses the central meridian T d, othenmse it cannot make an angle 

 with it=T a n. 



In the same manner 1 b m may be proved =</ b I +d I b 

 ==I d a. Q. E. D. 



SCHOLIUM. 



From these propositions may be deduced an easy practical 

 method of Sailing by the Arc of a Great Circle, by means of a 

 SPECIAL CHART, on a convenientscale.of the intended tract, 

 constructed in the following manner. (See the Chart.) 



Having calculated by spherical trigonometry, as in the fol- 

 lowing example, the angles of position ba T, ab I, the dis- 

 tance a b, and also the latitude of the point d at the middle 

 longitude ; draw on a sheet of paper from a moderate scale 



s 



