VOL. II.] 



PHILOSOPHICAL TRANSACTIONS. 



233 



The Squaring of the Hyperbola. By Lord Viscount Brouncker.* 



N" 34, p. 645. 



Let A B be one asymptote of the hyperbola E d C, fig. 7- pi- 7 ; and let A E 

 and B C be parallel to the other ; also let A E be to B C as 2 to 1 ; and let the 

 parallelogram A B D E be equal to 1. And supposing that E A, a ^, K H, (3 n, 

 d9, yx, Sx, iju, CB, &c. are in an harmonic series, or the reciprocals of an 

 arithmetical progression. Then will 



1.1.1.1.1 



ABCdEA = 

 EdCDE = 



1 X 2 

 1 





3x4 

 1 



2x3 ' 4x5 



EdCyE=:^^,+ ' 



+ 



5x6 

 1 



+ 

 + 



7x8 

 1 



+ 



9 X 10 



1 



+ 



8X9 ' 10x11 



1 



&c. 

 &c. 



2x3x4 ' 4x5x6 ' 6x7x8 ' 8x9x10 



+ 



&c. 



For (in fig. 2 and 3) the 

 Parallelograms 



1 



l"x2 



1 



s'xi 



CA = 



dF = 



5x6 



fk = 

 ap = 



1 



7x8 

 1 



9x10 



^"™=riTT2 



el = 

 gh = 



13X14 

 1 



15 X l6 

 &c. 



EdC = 



Ebd = 



dfC = 



Eab = 



bcd = 



def = 



And (in fig. 4) the Triangles 



1 i=)dD — c=idF 



2x3x4 



1 



4x5x6 



1 



6x7x8 



1 cnaq— craap 



fgC = 



14x 15x 16 

 &c. 



Note. 



CA= dD + dF 

 -i-dD= br + bn 

 idF= £G+ fk 

 -Lbr = aq + ap 



^ b n = c s 4- cm 

 4- f G= e t + el 

 i-fk = gu + gh 

 &c. 



pFofessor of mathematics in the universities of St. Andrews and Edinburgh j where he was suddenly 

 struck with total blindness while employed in showing to his pupils the satellites of Jupiter, and died 

 a few days after, in l675, being only 36 years of age; to the great loss of the mathematical and phi- 

 losophical world. 



* William Brouncker, or Brounker, Lord Viscovmt of Castie Lyons in Ireland, was bom about 

 the year l620. He very early manifested a genius for mathematics, in which he afterwards became 

 very eminent. He was made M. D. at Oxford, l646. Afterwards he maintained an honourable cor- 



VOL. I. Gg 



