420 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, GDK, FGB, give 
the following proportions : 
HL or GB : FB :: DK : DC, and 
GF : FB CK : DC, which was the propofition 
to be demonftrated 
* When FB (fig. i£) is fo final! in companion of FA, that FG fhall be eva- 
nefcent in comparifon of FH, 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 
DK : DC ; for this reafon, and becaufe it has been proved, that as HL : FB :: 
DK : DC, it follows, that as HL : FB :: FH or BL : BC, that is, as the variation 
of the cofine is to cotemporary variation of the arc, fo 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 diminifhed that FG fhall bear a finite proportion to FH, and too 
great to be negleHed, BL will not be either to FH or to DK in a ratio of equa- 
lity : confequently, FH or BL muff no longer be fubftituted for DK : as BA 
becomes lefs, FB being ft ill fuppofed evanelcent, DK approximates to the fine 
of |FB to which it is ultimately -equal when B and F arecoinciding with A (fig. 1 6.) . 
In which cafe the proportion will become as HL or HA : FB or FA :: |FA : to 
CA, that is, as the verfed line of FA is to the arc FA fo is half the arc FA to radius, 
or fo is the arc FA to diameter. 
The proportions which have been demonftrated, comprehend the variation 
of the arc exprelfed in terms of the cotemporary variation of the line or cofine 
in every part of the quadrant without limitation, it being only allowed to fub- 
ftitute the arc FB inftead of its chord, thefe quantities approximating the 
tnore nearly to equality a3 FB is fmaller, and being ultimately equal in their eva- 
nefcent ftate. Moreover, it will be eafy from what has preceded to conftruft a 
plane right-lined triangle, which lliall 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 arc,' FA the chord, FH the fine, CH the cofine of the 
arc FA. Bife£t 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 vauilhing. 
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