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Mr. atwood’s Theory for the Menfuratlon 
to CA : DK will be the fine, and CK the cofine of the mean 
arithmetical DA : the fimilar triangles CEI, CDK, FGB, give 
the following proportions : 
GF : FB :: CK : DC, which was the proportion 
to be demonftrated *. 
% When FB (fig, 15) is fo fmall in comparifon of FA, that FG fliall be eva~ 
nefcent in comparifon of FIT, FH and BL will be in the ratio of equality, 
and confequently the ratio FH : FC equal to the ratio BL : BC, or to the ratio 
I)K ; DC ; for this reafon, and becaufe it has been proved, that as FIL : FB :: 
DK : DC, it follows, that as HL : FB :: FH or BL : BC, that is, as the variation 
of the cohne is to cotemporary variation of the arc, io is the line of the varying 
arc to radius ; and, for fimilar reafons, as the variation of the fine is to the 
cotemporary variation of the arc, fo is the cofine to radius. 
If BA be fo diminilhed that FG fliall bear a finite proportion to FH, and too 
great to be negletfted, BL will not be either to FH or to DK in a ratio of equa- 
lity : confequently, FH or BL muft no longer be fubftituted for DK : as BA 
becomes lefs, FB being ftiil fuppofed evanefcent, DK approximates to the fine 
of §FB to which it is ultimately equal when B and F a recoinciding with A (fig. 16.). 
In which cafe the proportion will become as HL or HA : FB or FA :: |FA : to 
C A, that is, as the verfed fine of FA is to the arc FA fo is half the arc FA to radius, 
or fo is the arc FA to diameter. 
The propofitions which have been demonflrated, comprehend the variation 
of the arc exprefled in terms of the cotemporary variation of the fine or cofine 
in every part of the quadrant without limitation, it being only allowed to fub- 
fiitute the arc FB inftead of its chord, thefe quantities approximating the 
more nearly fo equality as FB is fmaller, and being ultimately equal in their eva- 
nefcent flate. Moreover, it will be eafy from what has preceded to conftrubl a 
plane right-lined triangle, which fliall be fimilar to the mixtilinear triangle con- 
tained under an arc, its fine and verfed fine when they are diminifhed fine limite. 
Let FA (fig. 16.) be any are, FA the chord, FH the fine, CH the cofine of the 
arc FA. Bifedf FA in D, join CD, and draw the right fine DK : then will the 
plane right-lined triangle KDC continually approximate to fimilarity with the 
mixtilinear triangle FDAH as FA becomes fmaller, and the two triangles will be 
ultimately fimilar when FA is vaniflfing. 
HL or GB : FB :: DK : DC, and 
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