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-metical Quadrature exact ia whole Numbers, which wehave here done inFractions.

There are several things relating to thisQuadrature,which might be takenNoticcof, elpecially one, uns. That the Terms of this our Series y, y> y, y, y,Gc-

are of HarmonicalProgression, or in a continued Harmonical Progression,as will be evident to any one that (hall examine them. And a Series made by

Skipping, as y, y, y , - , -y, Gc. is also of Harmonical Progresiion.And y, y , , yy , yy, Gc. is also a Series of Harmonical Pro-

portional;. Wherefore since the Circle -i- 4 - y 4 - ■— 4 - -yy -+• yy-xFc. — - - -y--, Gc. by sobstracting the latter partial

Series from the former partial Series, the Circle will be the Difference of twoSeries in Harmonical Progression. And bccause the Summ of any Numberof Terms'in Harmonical Progression, how many soever, may by some Com-pendium be obtained; hence Compendious Approaches (if aster Van Ceulenthere be any need of them) may be be deduc’d , if one would in this ourSeries take out the Terms aflfected with the Sign —, by adding the two

next into one, 4 - \ — y, and 4 - y — y, and -+ y-‘y, and -+■

— 1 - x —, and 4 - — - 1 and so onward, hewillhave a new Series

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for the Magnitude of the Circle, namely y st hat is y-y) 4- y ( that

is --—) ■+ ~~ (that is y- — ) Gc. Wherefore

The Square inscribed being y

The Area of the Circle (hall be — 4 - — 4 - — 4 - —H- l -, Gc.

3 35 99 t?s 323

But the Numbers 3, 35, 99, 195, 32.3, Gc. by skipping are taken out ofthe Series of Square Numbers, 4, 9, 16, iy, Gc. diminimed by an Unite,and so made the Series 3, 8, 1 y, 24, Gc. out of the Members of which Se-ries every fourth aster the sirst, is a Number of this our Series. But I have

found (which is worth noting) the Summ of this Infinite Series — 4- —

4* y 4 * Gc. to be y. Nay, and by culling out by singlc Skipping, as

x 4 —— 4 * 7-, Gc. the Summ of this Infinite Series maketh — or —. But3 *5 35 42

it out of this again another Progression be culled out by single Skipping, as

~—b 3 j 99 * & c ' £ ^ e Summ of that Infinite Series (hall be die Semicir-

cle, the Square of the Diameter being 1. Now becaule by the fame MeansVoll. D the