( 668 )

when it is leis than 45". And by the fame Doctrine putting T for theTangent of any Arch, and t for the Difference thereoffrom the Tangent of

1 I r f f

another Arch, the Loiarithm of their Ratio will be — into — -i-—

m 1 z 1 r

s 3 t 4

+ 7tT ■ + 7T7' t '

I . * t r

— mto-=,--

m 1 i T 2

is the leiser Term.

t s

TtT*

, 3

Öfe. when T is the greater Tcrm; or,

-tVt H -W ' when T

4 1 4 y T3

And if m be suppolcd; o, 0002908881, (3c. = a, its reciprocal — will

4

be 3437, 7467707849392526, (Fe. whieh Mukiplied into the afordaid Se-rieif , Ihail give precisely the Difference of the Meridional Parts, between thetwo Latitudes to whose hals (Komplements the alTumed Tangente belong. Noris itmaterial from whether Pole you estimate the Complements, whether tlicElevated or Depreffed ; the Tangents being to one another in the fame Ra-tio as their Complements, but Inverted.

In the sime Difcourle I allö Ihewed that the Series might be made to Con-verge twice as fwift, all the Even Powers being omitted ; and that putting T

for the Summ of the two Tangents,., the sime Logarithm would be. — or —-

m 4

into

t 3

ll'-

t 9

5.7:

, &c. but the Ratio of t to t, or of

the Summ of two Tangents to their Difference, is the sime as that of theSine of the Summ of die Arches, to the Sine of their Difference. YVhereforetf S be. put for the Sine Complement of the Middle Latitudo , and s for the»

Sine of half the Difference of. Latitudes, the. sime Series will be-^- into —

4 S

s 3

s 7

$SS ■** 7 S 7

/ 99 89

, f 3 c- wherein as the Difference»

of’ Latitude are smaller, fewer Steps will suffice. And if- the Equator be put 1for the M iddte Latitude, and consequently S = R, and .s to the Sine of the-

Latitude, the. Meridional Parts reclconed from the Equator will be —

s 3 s S s 7

-----H- T—whieh is Co-incident with D t.lVal-

3 r r a, y 7 4 4 71 % ’

niJJ&XXXVh Hs’s Solutiori. And this sime Series, being half the Logarithm of the Ratio of/R .-4 / to R — /, that is of. the Verjed-fines of the Distances from both Poles,doe* agree with what Dr . Barr ow haddhewn in his XI Leäure.

\

The.-