of the Angle fuhtentied by Two Objefifs, &c.. 
drant, the increment of the fine becomes the verfed fine 
©f the arcs laid increment, which gives this proportion: as 
* ° o 
eofi : 2 x rad, :: fimrarc, And fince in this cafe arc — cofine, 
we fhall have, arc — V 2 x fin radius being — 1. In the other 
parts of the quadrant which are not very near its extremity,, 
— — fin. , ^ 
arc — — ; having given, therefore, the variation of the fine 
or cofine of any arc,, the fine or cofine being known^ the 
cotemporary variation of the arc itfelf may be obtained* when 
it is either at the very extremities of the quadrant, or at: 
iome diftance from thofe extremities* The difficulty lies ire 
afcertaining in what part of the quadrant the value of the 
1 • 
— — _A n * — c °f* . 
arc ~ cof. ~ iThT De g' m to- fail, and the value expreffed by 
arc = V^2 x fin. or — sf 2 x cof. to take place. This leads to 
a general propofition comprehending both thefe values for the 
arc’s variation, extended to every part of the quadrant. 
The propofitior 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 AB, AF (fig. 1 5.) be 
the given arcs ; BF their difference ;■ BL, FH, the fines ; CL,. 
CH, the cofines of the arcs AB, AF,.refpedHvely join CA,.CB,. 
€F, 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 cofines, and FG the difference of 
the fines. Bifedt FBin Dj fo (hall DA be an arithmetical mean 
between the arcs FA, BA ; join DC, which will interfedl FB at 
right angles in E r through D and E draw DK*EI, perpendicular 
