( 66 ?)

The fimc J{atio of t to t may be exprefled also by that of the Summ osehe Co-sisles of the two Latitudes, to the Sine of their Difterence : As like-wile by that of the Sine of the Summ of the two Latitudes, to the Diffe-rcnce of their Co-sines: Or by that of the Versed-sine of the Summ of theCo-Lätitudej , to the Difference of the Sines of the Latitudes Or as the fameDifference of the Sines of the Latitudes , to the Verfed-fine of the Difterenceof the Latitudes ; all which are in the tarne Ratio of the Co-line of the Mid-dle-Latitude , to die Sine of half the Difterence of the Latitudes. As it wereeasic to Demonstrate, if the Reader were not fupposed capable to do it him-self, upon a bare Inspection of a Scheine duly reprelenting diese Lines.

This Variety of Expression of the fame Ratio 1 thought not fit to be omit-ted, because by hdp of the Rationaliiy of the Sines of jo°, in all Caseswhere the Summ or Difference of the Latitudes is 6c°, 9 c 0 , 1 ic°, orryco-; some one of them will exhibit a Simple Series, wherein great Part ofthe Labour will be fäved. But the former seems for all Uses the most con-venient, whether we design to make the whole Meridian-Line, or any Part

thercof, vi%.

s r3 S’

s 77'S 7

<3c. wherein «,

9 S 9

is the Length of any Arch, which you design {hall be the Integer or Vnityin your Meridional Parts ( whether it be a Alinute, League or Degree, orany otherj ) S‘ the Co-fine of the Middle-Latitude , and s the Sine of half theDifference of Latitudes ; but the Seeants bring the. Rcdprocals of the Co-fines

~ will be equal to*^—

putting*/ för the Secant of the Middle-Latitude ; and <

i r

mto

4

S

will be —

r 4

This multiplied by

1l

? SS

that is by

sss s

1

3 r r r r-

will give die Second Step ; and that again by t fie third Step ; and *

y r r r r 1

so forward till you have compleated as many Plaees as you desire. But theStjuares of the Sines bring in the tarne Ratio withthe Vcßed-ßncs of the Dou-

ble Arche«, we may instead of assume-for our Multiplicator or the

Verfed-fine of the.Difference of the- Latitudes divided by thrice the Versed*sine of the Summ of the Co-Latitudes , &c. which is the utmost- Compendium

l can think- of for this pyrposey and the fime. Series willbeeome - into x

V 2. V $ 'J/4-

-4- — -4-- h —— *4- —r—. Hereby we areenabled to est>

3 V y V 2 7 v 3 9V+ 1

mate the Default of the* Method of making the Meridional Line, bythe continual Addition of the Seeants of Equidifferent Arches, which asthe Differences of thole Arches are (maller, does still nearer and nearer ap-

proacb-

/