VOL. XXI.] PHILOSOPHICAL TKANSACTIONS. 45S 



more nearly relating to each other. But if that line going out from c, be on 

 the other side of mn ; then the space which is equal to the rectilineal triangle, 

 is the difference of two mixtilineal figures, (the one a trilineum, the other a 

 segment of the lesser circle,) as is abovesaid ; neither of which can be squared 

 severally. 



"All these particulars are plain from Mr. Perk's demonstration; which, 

 with a little variation (such as is usual in the different cases of the same theorem), 

 is applicable to all of them : though perhaps he was not aware of it. 



" In the dimension of the parts of Hippocrates's lunula, it might perhaps be 

 expected, that the triangle assigned equal to a portion of the lunula should be 

 part of the triangle to which that whole lunula is wont to be assigned equal ; 

 that is, that the triangle assigned equal to the portion ade, should be the re- 

 spective part of ACB which is equal to the whole lunula, which in that of Mr. 

 Perks is not. 



" But in that of Mr. Tschirnhause, above-mentioned, it is so, which is to 

 this purpose. If from any point e, in the circumference of the lesser circle, 

 we let fall on ab, a perpendicular cutting it in t, and draw the line cl ; the 

 triangle cal is equal to the portion of the lunula aed. And consequently the 

 triangle cbl, equal to the portion bed. Which, because Mr. Tschirnhause 

 has not at all done it, I shall briefly demonstrate, so as the demonstration may 

 reach the portions of the conjugate space ACBgyA. For the triangles acb, 

 AEF, are like triangles, each being the half of a square: and therefore by IQ 

 El. 6, the triangle ace is to the triangle aef in the duplicate proportion of ba 

 to AE, that is, by 8 El. 6, as ba is to ..l. But, by J El. 6, the triangle acb 

 is to the triangle acl as ba is to al. Therefore by Q El. 5, the triangles acl 

 and AEF are equal. But the triangle aep is (by Mr. Perks) proved equal to the 

 portion aed. And therefore the said portion aed is also equal to the triangle 

 ACL. " D. Gregory." 



Mr. Caswell had a sight of this quadrature of Mr. Perks, and had given a 

 specimen of its being capable of further improvement, as follows : — On the 

 centre b, he draws by a a third circle, which forms another lunula than that of 

 Hippocrates ; and thus he very dexterously squares the portions of this lunula ; 

 and thence lets us into a new system, which may be pursued in like manner as 

 Dr. Gregory has done that of Hippocrates. 



After these learned disquisitions on so trite a subject, it will not be needful 

 for me to say much. I shall but briefly compare the two quadratures of Mr. 

 Tschirnhause and Mr. Perks, wherein they agree or differ with each other. And 

 then show how, by either of them, to divide the lunula in any given proportion. 

 Monsieur Tschirnhause, letting fall from e, on ab, a perpendicular el, fig. 4, 



