PHILOSOPHICAL TRANSACTIONS. 551 



series makes -5- or -f . But if out of this again another progression be culled out 

 by single skipping, as -^ -f tt + inrj ^^' ^^^ sum of that infinite series will be 

 the semicircle, the square of the diameter being 1 . Now, because by the same 

 means the arithmetical quadrature of the hyperbola is obtained, I thought it not 

 amiss to represent to view the whole harmony. 



1 23 4 5 6 7 8 9 10 11 12 13 14 15 l6 17 18 IQ 20, &c. 

 1 4 91625 36 49 64 81 100 121 144 169 196 225 256289 324 361 400, &c. 

 03 8 15 24 35 48 63 80 99 120 143 168 I95 224 255 288 323 360 399,&c. 



1 I 1 1 1 1 I I I , 1 1 I I I I 1 ! I i^ firr- — 3 . 



T -& -mnr TT Ts TT To TT TlTo" T 4 3 Tg T 1 6 i i aV ft i b ' i i i 3^3 3<6 Ti Ti*^^' — T» 



» 1 1 _. _ i_ 1 1 1 I 1 I Sim — « • 



T • TT • TX • TT • -^-gr • 1 i 3' • TTT * « 5 5 • TTT • 3'9 ij"-*^* — t* 



' 1 I 1 1 1 I 1 1 5lrr> — I . 



T • • • TT • • • TT • • • TTT • • • TTT * * OC*^' — v^ » 

 T • • • TT • • • Ti 6 • • • TTT • • • TTT * <*C. ^ II J 



where C is the circle ABCD, whose inscribed square is -|-, 

 and H is the hyperbola CBEHC, whose square ABCD is -f. 

 In the same figure, viz. fig. 5, pi. 15, to the asymptotes AF, AE, at right- 

 angles to each other, let there be described the curve-line of an hyperbola GCH, 

 whose vertex is C, and ABCD the power or square to which every rectangle 

 made of the ordinate as EH and the intercepted part AE, is always equal. 

 About this square let a circle be drawn, and let the hyperbola be continued from 

 C to H, so that AE be double of AB. Then putting AE to be 1, AB shall 

 be i, and its square ABCD will be 4-, and the circle whose power ABCD is 

 inscribed, will be -i. -f- ^ -f ^, &c. but the portion of the hyperbola CBEHC, 

 whose power inscribed is the same square 4-, which represents the logarithm of 

 the ratio of AE to AB, or of 2 to 1, will be ^ + Vr + ttt^ &c.* 



* The above series, for the quadrature of the circle and hyperbola, were before invented by M. 

 Mercator, and James Gregory. 



END OP THE PHILOSOPHICAL COLLECTIONS. 



A Description of Pen-park Hole in Gloucestershire. Communicated by Sir 



Robert Southwel. N*' 143,* p. 2. Fbl XIIL 



There is a place in Gloucestershire called Pen-park, about 3 miles from 



Bristol, and above 3 from the Severn, where some miners for lead discovering 



a large hole in the earth, one Captain Sturmy, a warm inquisitive seaman, who 



\ After being discontinued during four years, the Philosophical Transactions were begun again with 

 this N° 143, by Dr. Plot, who had succeeded Mr. Hooke, one of the secretaries, Nov, 30, 1 681; 

 and who again undertook to renew the publication, on the encouragement afforded by the Societ}'-, in 

 promising to purchase 60 copies of each number of the work. 



