( 44 * ) 
E D the Tangent of the Angle DCE = i the Arch 
E F, and C Dis the Secant of the fame— the Diameter 
of the Circle CED F, and therefore its P^adius is half 
that Secant. 
Now from the Figure ’tis plain, that in very fmall 
Arches the Radius of their circular Place will be 
half the Radius of the Quadrant, that is, putting 
this Radius= io, the other will be And the Radius 
for the Arch of 5)0, the higheflto be ufedonthe Qua- 
drant will be the Square Root of half the Square of 
the Radius = Sine of 45: Degrees = 7.071, and the 
Arches at the Center drawn by thefe two Radij are 
the Extreams, the Medium of which is 6.03 55. And 
if a circular Arch be drawn with this Radius Ath 
Part of the Length of it, that is, in an Inftrument of 
20 Inches Radius, the Length of one Inch on each 
Side of the Center affording two Inches in the whole, 
to catch the Coincidence of the Rays on, which 
muff be own'd is abundantly fufiicient, the Error at 
the greateft Variation of the Arches, and at the Ex- 
tremity of thefe 2 Inches, will not much exceed one 
Minute. 
But in fixing the Curvature or Radius of this Cen- 
tral Arch, fomething farther 
than a Medium between the 
Extreams in the Radius is to 
be confidered : For in fmall 
Arches the Variation is very 
fmall, but in greater it equal- 
ly encreafes, as in the Figure 
where it appears, the Differ- 
ence between the Angles ABC 
and ADC is much greater than the Difference be- 
tween 
