[+14] 
I fhewed this Qiiadratnre of Mr. Terks to Dr. I>aviJ Greg(^y 
( our learned Profeflbr of Aftronomy at Oxfor^^) who gives his 
Opinion about it ( with his Improvement of it ) in a Letter of 
his to me ; which I (hall give you in his own words, 
Reverend Sir, The ^adrature of the Tarts of the Lunula of 
Hippocrates Ch'mSy by Mr. Terks ( which you fhewed me ) is 
" very Elegant. 
" I remember, the like was done, fome years fince, by MonGeur 
" Tchirnhaufe ; who afligns, as equal to the fame Portion, not the 
fame Triangle with that of Mr. Terks^ bnt another Equivalent 
" thereunto, (as I Ihall (hew by and by.) We have his Theorem, 
"jn tht y^Si^a Lipfia ^ for the Month of 4S'^//^'w/'<fr, 1687. But, 
without any "Demonftratwn. 
" But, both the One and the Other, feem not to have confidered 
" this affair in its full extent. 
" For, if you compleat the Two Circles, whole Arcs contain 
" the Lunula oi Hippocrates\ the fame is true, as well of the Points 
" in the other Semi circle ACB,as of thofe in the Semi-circle AEB; 
" and, for the fame Reafons. As appears in the Scheme annexed, 
" wherein I have mark'd the Points in the Semi circle ACB, ( cor- 
"refpondent to thofe of Mr. Terks in AEB,) with the correfpon- 
"dent /mall Letters of the Roman and Greek Alphabets. 
"If Mr. Terks had made his conftrudion univerfal; by ma- 
"king both E A and EB, meet with the Greater Circle, (which he 
" might have done by protradling thefe Lines and the Greater 
Circle 'till they meet ; ) he might have found that the Portions 
" of the Spaces A 2 CM, BHCN, (fuppofing MCN parallel to AB) 
" are Quadrable as well as thole of Hippocrates s Lunula : And 
" that E A > being a {freight Line, the Portion AED of Hip- 
^^pocrates's Lunula , is to A g <r ( the Correfpondent of A 8 C M ) 
" in the Duplicate Proportion of C 2 to A s. For E Rg (at Rthe 
" Center of the Lefler Circle ) is, in this cafe, a Right Angle. 
" Moreover ; If you take any Point s in the Semi circle ACB^ • 
" and proceed according to Mr. Ter}i% conftruftion Univerfalized 
"as above-faid; you will find, on the one fide, xht Trilineum 
"As-^" (contained by the Arcs A?, A^T, and the flreight line rcT) 
"equal to the Redlilineal Triangle A s?^. And, on the other fide, 
" the Triltneum contained by the Arc B « ( the Complement of e A 
"to the Semi- circumference,) and the Arc B d ( the Complement 
of A ^ to the Fourth part of the Circumference, )and the [freight 
"line 2 d, (that is, the Tr'tlmeum BHCd diminifhed by the Se- 
"gment 
