andGEAj tSLchofthcra Haifa Square. And AG to AE, as 
to I ( proportional to the Refpedtive £adu of the Two Circles. ) 
And the Like Segments ADG, AE, in their Refpedlive Circles 
( as the Squares of their Refpe&ve i?^^^// ) as 2 to r. And there- 
fore the Semi-fegment AFD, equal to the Segment AE. And con- 
fequently ( one taking from the Triangle as much as the other 
addestoit) xh^ T or tion of the Lunula equal to the Trian- 
gle AFE. Which was to be Demonflrated. 
( I take the liberty ( both in this and the things that follow ) to 
vary fomewhat from the Authors Words, (but to the fame fenfe, 
and without any difadvantage to Them,) fo as to Defign the lame 
jRefpedive Points ( in all the Figures ) by the fame Letters* 
Which makes it fomewhat Shorter ( without Repeating the iarae 
Conftrudtion anew for every Figure ; ) and prevents the Confu- 
lion which might arift to the Fanfy, if the fame Refpeilive 
Points, in feveral Figures, were defigned by different Letters ; 
and the fame Letters, in the different Figures, deOgn different 
Points. ) 
If the Point E chance to be in K ( the middle of the Arc AEB) 
there will be no Interfedion at G ( the Points G5B being then 
coincident, but without any difturbance to the Demonllration : ) 
If it happen beyond it, toward B ; then G will be on the other Ode ; 
and what is here fayd of EGB, mutt be accommodated to EGA : 
which things are fo obvious, as not to need any long difcourfe. 
The whole proceeds upon the fame general notion with that 
of fquariug the whole Lunula (and fome other Curve-lined Fi- 
gures; ) that, if as much be added to the one fide, as is taken from 
the other, the Equality remains. 
And theftrefs of the Deraonftration, is, to prove the fegments 
ADG and AE, to be Like Segments ; and therefore Proportional to 
their Refpe6live Circles; the Whole of one,equal to Half the other. 
The Ground of the whole Proceft is plainly this. The Angle 
ACE, being an Angle at the Center of the Greater Circle, but at 
the Circumference of the Lefler, the line CDE ( as it pafleth from 
CA to CB ) doth, in the fame proportion, divide the Quadra ntal 
Arc ADB, and the Semicircular AEB : whence all the reft doth 
naturally follow. 
And this is Applicable to other Lunulas ( befide that Hif' 
pocrates) if ( by altering the Angle at F, or otherwife,) we take-in 
fuch a Portion of the common Segment ABD on the one fide 
(inftead of AE cut-off on the other fide) as the Proportion of the 
two Circles requires. 
R r r 2 I (hewed 
