[+17] 
fion AED. And therefore the faid Portion AED is alio equal 
<^ to the Tfiangle ACL. 
" I am, Sir, Your &c. D. Gregorf. 
Mr C/j/uJf^had a fight of this Quadrature of Mr Terh ( be- 
fore Dr Gregorie or 1 had feen it \ ) And had given a Specimen ot 
Its bemg capable of further Improvement. But, without having 
Leifure, or giving himfelf the Trouble, of purfuing it through 
ali Its Appendages. I would ( with his leave ) have here in- 
ferted that Specimen : But he chofe rather to decline it ; faying, 
He thought it needlefs, becaule Dr Gregorie had, fince, done the 
like more fully. 
The Refult of it, is to this purpofe ; On the Center B, he 
draws by A, a Third Circle ; which forms another Lunula , 
than that Hippocrates : And he doth (very dextroufly) Square 
the Tortious of ihis Lunula. And doth thereby let us m, to a New 
Syftem, which may be purfued in like manner as Dr Gregorie 
hath done that Hippocrates. 
After thefe learned Difquifitions, on fo trite aSubje6l; it wili 
not be needful for me to fay much. I ihall but briefly Compare 
the Two Quadratures of Mr Tjchirnhauje and Mr Terk^^ (where- 
in they Agree or Differ with each other. ) And then fhew, How, 
by either of them, to Divide the Lunula in any Given Propartion, 
- 'Monfieur Tjchirnhauje \ Letting fall, from E (on AB) a Perpen- 
dicular EL, determines the Triangle ALC equal to the Portion 
ADE. 
Which being admitted ; We may 
thus Divide the Lunula in my 
Given Proportion. If we divide 
AB, atL, in fuch Given Proporti- 
on ; GL will, in the fame proper- -g 
tion (becaufeof the Common Alti- 
tude ) divide the Triangle ACB 
( which is equal to the Whole Lu- 
nula.) AndLE(ereaed at Right 
Angles on ALB ) will determine 
the Point E; from whence if we 
draw^ to C, the Streight line EC, this will, at DE, divide the 
Lunula in the fame Proportion. 
Mr Teris ; On EDC , drawing the Perpendicular AF , de- 
termines the Semi-quadrate AFE , equal to the propofed Por- 
tion ; 
