of the Angle fuhtended by Two ObjeSis, &c. 4 i p 
drant, the increment of the line becomes the veiled line 
of the arcs, kit increment, which gives this proportion : as 
cof. : 2 x rad. :: fin- : arc. And lince in this cafe arc — coline, 
we (hall have, arc — \/ 2-xfin. radius being 
i. 
In the other 
parts of the quadrant which are not very near its extremity,. 
- 1 — fin. 
arc = — ; Having given, therefore, the variation of the line 
or cofine of any arc, the line or coline being known, the 
cotemporary variation of the arc itfelf may be obtained,, when, 
it is either at the very extremities of the quadranty or at 
fome diftance from thofe extremities. The difficulty lies in 
afcertaining in what part of the quadrant the value of the 
arc 
fin. 
— cof. 
~ col. ~ 
fin. 
=V 2 : 
x fin. 
begin to fail, and the value expreffed by 
arc = v 2 x- tin. or — A^2 x cof. to take place. This leads to 
a general proposition comprehending both thefe values for thee 
arc’s variation, extended to every part of the quadrant- 
The propofition is this : the difference of the cofines is to 
the chord of the difference of any two arcs, as the fine of an 
arithmetical mean between them to radius ; and the difference of 
the fines is' to the chord of the difference, as the cofine of the 
fame arithmetical mean to radius. Let i\B, AF (fig. 15.) be 
the given arcs ; BF their difference ; BL, FH, the fines ; CL r 
CH, the cofines of the arcs AB, AF, refpe&ively join CA, CJB r 
GF, and FB ; FB will be the chord of the difference of the arcs 
AF, AB. Through B draw BG parallel toCA then HL==BG 
will be the difference of the colines, and FG the difference of 
the fines. Bifedt FBffn D, fo ffiall DA be an arithmetical mean 
between the arcs FA, BA ; join DC, which will interfeft FB at 
right angles in E ; through D and E draw DK, El, perpendicular 
to 
