310 DESCRIPTION OF GARNETT S 



The principles of the Chart together with the application to 

 sailing by the Arc of a Gi eat Circle, can also be deduced from 

 the following propositions. 



PROPOSITION I. 



The ansfle of convergence, or inclination of two meridians to 

 eacli other, at any given latitude, is == the difference oj longitude 

 x by the sine of the latitude. 



Let P A, P B, be two meridians, on which let a, b, be two 

 places in the same latitude; draw the two tangents a T, AT, 

 meeting the axis CPT in T, and a c, be, perpendiculars to it; 

 also c d, T d perpendiculars to a b, and draw a 

 C to the center C; then will the angle «T4 

 represent the inclination of the meridians P a, 

 P b, to each other at the points a and b. 



From the right angled triangles a dT, ad c 

 and the similar triangles acT, acC. 



a T : rad.:: a d : sine 7 inclination of merid. 

 and rad.: a c:: sine (dca=) i diff. of Ion. : ad. 

 By composition, (a T : a c=.) a C : C c :: sine 7 

 difference of longitude : sine 7 the inclination of meridians; 

 that is, radius : sine oj the latitude :: sine 7 difference of longitude : 

 sine f the inclination of meridians. Or taking the arcs themselves 

 for their sines, which is sufficiently accurate in small arcs, and 

 agrees with the construction of the Chart; the angle oj converg- 

 ence of any hvo meridians at a given latitude, is = to the sine oj the 

 latitude x by the difference oj longitude. Q. E. D. 



REMARK. 



If the meridians be considered as great circles of the sphere, 

 and their inclination to the central meridian (or that which 

 bisects the angle at the Pole) as the complement of the angle 

 made by a great circle passing through them at any given lati- 

 tude, then in the spher. triangle Pad we have rad.: cosine 

 Pa:: tang, a P d: cotan. Pad; that is, rad.: sine oj the latitude:: 

 tangent of half the difference of longitude : tangent of the inclination 

 to the central meridian; which seems more correct. 



See the Table of the Inclination of Meridians, deduced from this Proposition. 



