454 PHILOSOPHICAL TRANSACTIONS, [aNNO lOgg. 



Monsieur Tchirnhause ; who assigns as equal to the same portion, not the same 

 triangle with that of Mr, Perks, but another equivalent to it. His theorem is 

 in the Acta Lipsiae, for the month of September 1687; but, without any de- 

 monstration. But neither of them have considered this affair in its full extent. 

 For if we complete the two circles, whose arcs contain the lunula of Hippocrates ; 

 the same is true, as well of the points in the other semicircle acb, as of those 

 in the semicircle aeb ; and, for the same reasons. As appears in the scheme, 

 fig, 3, wherein I have marked the points in the semicircle acb, correspondent 

 to those of Mr. Perks in aeb with the correspondent small letters of the Roman 

 and Greek alphabets. 



" If Mr. Perks had made his construction universal, by making both ea and 

 EB meet the greater circle, which he might have done by protracting these lines 

 and the greater circle until they meet, he might have found that the portions 

 of the spaces AeCm, bhcm, supposing mcn parallel to ab, are quadrible, as well 

 as those of Hippocrates's lunula: and that, EAy being a straight line, the portion 

 aed of Hippocrates's lunula, is to AeJ, the correspondent of AeCm, in the duplicate 

 proportion of ct to ae. For ere, at r the centre of the lesser circle, is in this 

 case a right angle. 



" Moreover, if you take any point t in the semicircle acb, and proceed ac- 

 cording to Mr. Perks's construction universalized as above-mentioned ; you will 

 find on one side, the trilineum AiS (contained by the arcs Ae, aS, and the 

 straight line £<?) equal to the rectilineal triangle ae^. And, on the other side, 

 the trilineum contained by the arc be (the complement of ea to the semi- 

 circumference,) and the arc b d (the complement of aS to the fourth part of the 

 circumference,) and the straight line id, (that is, the trilineum BHcd diminished 

 by the segment ce) to be equal to the rectilineal triangle Bsf, And that those 

 two spaces AtS, and the difference of BHcd from the segment Ce (parts of the 

 lunula ACBgyA) taken together, are equal to the triangle acb ; as well as the 

 two spaces aed and bed, parts of the lunula of Hippocrates. 



" So that, on the whole it appears, that the two circles, containing the 

 lunula of Hippocrates being completed; this lunula aebga, and the other 

 ACBgyA, make up one system, and are conjugate figures. For (drawing a 

 straight line cde, or ce^, or Cid, at pleasure through c the centre of the 

 greater circle, and cutting tiiose two circles,) the space contained within two 

 arcs of these two circles and part of the said straight line, (as aed, or AeS, or 

 BHEd,) is equal to the rectilineal triangle aef, or A£<p, or BEf respectively. And 

 it so happens that, if this line going out from c be on the same side of the dia- 

 meter MN with the lunula of Hippocrates ; the aforesaid space (which receives 

 a perfect quadrature) is solitary ; such as are the parts of Hippocrates's lunula ; 

 and of the two spaces AeCm, bhcn, which therefore are parts of the lunula 



