C ^03 ) 
The firft that demotiftrated the faid Analogy y was the ex- 
cellent Mr. James Gregory in his Exercitationes Geometrica^ 
publiftied Anno 1668. which he did, not without a long train 
of Conlequences and Complication of Proportions, where- 
by the evidence of the Demonftration is in a great mealiire 
loft, and the Reader wearied before he attain it. Nor with 
lefs work and apparatus hath the celebrated Dr. BarroTi/ in his 
Geometrical Le^ures ( Led. XI. App. 1. ) proved, that the 
Sum of all the Secants of any arch is analogous to the Loga- 
rithm of thQ ratio of Radius -^^ Sine to RaJ, — Sine, or^whick 
is all one, that the Meridional parts anfwering to any degree 
of Latitude, are as the Logarithms of the rationes of the Fer- 
fid Sines of the diftances from both the Poles. Since which 
the incomparable Dr. WaSis ( on occafion of a paralogifm 
committed by one Mr. Norris in fhis matter ) has more ful- 
ly and clearly handled this Argument, as may be feen in 
Num. 176. of thefe TranjaBions. But neither Dv.Watiis 
tior Dv. Barrow in their faid Treaties have any where touch- 
ed upon the aforefaid relation of the Meridian-line to the 
Logarithmick Tangent I nor hath any one, that I know of, yet 
difcovered the Rule tor computing independently the inter- 
val of the Meridional parts anfwering to any two given La- 
•titudes. 
Wherefore having attained, as I conceive, a very facile 
and natural demonftration of the faid Analogy, and having 
found out the Rule for exhibiting the difference of^Merldional 
farts^ between any two parallels of Latitude^ without finding 
'both the Numbers whereof they are the difference : I hope 
I may be entituled toa ihare in the emprovements of this ufe- 
iiil part of ^^^G^ Defiring no other favour of fome 
Mathematical Pretenders, than that they think fit to be lb 
fuft, as neither to attribute my defire to pleafe the Honou- 
nb\Q thQ Royal Society in thefe Exerciies,to any kind of Vanity 
or Love of Applaufe in me, ( who too well know how very 
few thefe things oblige, and how fmall revv^ard they procure) 
^nor yQt to'compl^h coram non jtidice^ that I arrogate to my 
^If the inventions of others^ and\upon that pretest to de- 
•■^retiate wha^ 1 do, unlels at the fame time, they can pro- 
duce ithe Aiithor 1 wrong, to prove their-affertions. Such 
'^^genuicy ^ Lhava always moft carefully avoided, iol 
■^ifh l had not top niuch experience of it in the very fame 
L ' perlbns 
