184 PHILOSOPHICAL TRANSACTIONS. [aNNO 1715. 



In the foliate the equation is vv = ~ — From these last two 



equations it seems that these curves differ no more from each other, than the 

 circle from the ellipsis. 



The quadrature of the curve here described has something of simplicity with 

 which I was well pleased. With the radius ba, and centre b, describe a circle 

 AKG, let the square hpst circumscribe it, so that hp be parallel to ag : prolong 

 FE till it meet the circumference of the circle in m, and through m draw lmgi 

 parallel to hp. The area bfe is equal to the area khlm, comprehended by kh, 

 HL, LM and the arc km. And the area sfe is equal to the area KmLH or kmpq. 

 Therefore if bf and Bf are equal, the two areas bfe, Bfe taken together are 

 equal to the rectangle hq, and therefore the whole space comprehended by 

 BEAXBCYGZ (supposiug Y and z to be at an infinite distance) is equal to the cir- 

 cumscribed square hs. 



N. B. This quadrature is easily demonstrated from the equation : for by it, 

 a J^ z'.a — z :: zz : vVy that is af : ef :: mf : fb ; and so (pF the fluxion of af 

 to l/ the fluxion of mf. Hence the areola E¥<pe will be always equal to the 

 areola ml/jia, and therefore the area aef always equal to the area mal. 



Hence it appears that this curve requires the quadrature of the circle to square 

 it ; whereas the foliate is exactly quadrable, the whole leaf of it being only one 

 third of the square of ab, which in this is above three sevenths of the same. 

 Again, in our curve, the greatest breadth, is when the point p divides the line 

 AB in extreme and mean proportion : whereas in the foliate, it is when ab is 

 triple in power to bf. And the greatest ef or ordinate in the foliate, is to that 

 of our curve, nearly as 3 to 4, or exactly as ^ ^y/^ — i to ^ b^\ — 54-. 



But still these differences are not enough to make them two distinct species, 

 being both defined by a like equation, if the asymptote sgp be taken for the 

 diameter. And they are both comprehended under the 40th kind of the curves 

 of the third order, as they stand enumerated by Sir Isaac Newton, in his incom- 

 parable treatise on that subject. 



j^n easy Mechanical Way to divide the Nautical Meridian Line in Mercators 

 Projection; loith an Account of the Relation of the same Meridian Line to the 

 Curva Catenaria. By J. Perks, M. A. N° 345, p. 331. 



The most useful projection of the spherical surface of the earth and sea for 

 navigation, is that commonly called Mercator's ; though its true nature and 

 construction is said to be first demonstrated by our countryman Mr. Wright, in 

 his Correction of the Errors in Navigation. In this projection the meridians 



