186 PHILOSOPHICAL TRANSACTIONS. [aNNO 1777. 



has to B such a multiplicate ratio of a to b, as is expressed by the 

 number /j. 



In the reasoning above, I fixed on b as the magnitude to which the rest were to 

 be referred; but I might as well have fixed on a, or any of the other magnitudes. 



rni r • i. 1 P— 1 B— A /)— 1 p — 2 (B — a) , . (b — a),"-' 



Thus, for mstance b + - — . b . \- — . . b . ^ + &c. b. — 



1 A ' 1 2 A A 



has to A, such a multiplicate ratio of b to a, as is expressed by the number /j; 



U X. iP— • B — A,p—lp—2 (b - a)' , o (B— a)A-' ,, 



or A has to b + . b . 1- . '— -— . b . ^ + &c. b. ^ , the 



I a I 2 A A 



r 1 P — ' ^ — ^ I P — ' P — 2 (a — b)^ , (a — b)p - I , 

 ratio of A + '—-—.A. \- '—■— . '— -— .A.^ + &C. A. to b; 



'1 B ■ 1 2 B ' B 



that is, such a multiplicate ratio of a to b, as is expressed by the number p. 

 Each of these indeed is demonstrated separately from the same sort of geome- 

 trical reasoning; but for the sake of brevity these separate demonstrations are 

 omitted, as they are both contained in the general reasoning above, which fur- 

 nishes likewise a great variety of other expressions, according as certain numbers 

 of the ratios c to d, e to f, g to h, &c. are supposed to be respectively equal 

 to, greater or less than, the ratio of a to b. 



XXI F^. The Case of yinn Davenport. By Mr. Fie/ding Best Fynney, Surgeon, 



at Leek, Staffordshire, p. 458. s 



May ]6, 1775, Mr. F. being desired to visit Ann Davenport, a native of Leek, 

 he beheld a truly miserable object, with the most cadaverous countenance he had 

 ever seen, emaciated to the last degree by a hectic fever, and profuse colliquative 

 sweats. She had a continual thirst, her appetite was totally gone, and she was 

 always in the extremes of being too loose or too bound. Her mother said, she 

 was then in her '2 1 st year ; and that she had been a strong and sprightly child 

 from her birth, till she was about 5 years of age, from which time she had been 

 a stranger to health, and every now and. then had been seized with excruciating 

 fits of the colic, especially whenever she ate or drank any thing the least acid. 

 The young woman said, that about a year before she had first perceived a 

 swelling on the right side of her belly just above the groin ; which, if at any 

 time she attempted to stretch out her thigh, gave her inexpressible pain, as if 

 something stabbed her in that part: that therefore she was alw.iys obliged to 

 keep up her knees, more or less, towards her breast, by which means slie had, 

 in some degree, lost the power of extending her limbs. 



Mr. F. ordered her to take -i-dr. of powdered bark in a little port wine every 

 4 hours; and, as matter had already formed within the tumor, he desired that a 

 maturating poultice might be applied every night and morning; for he imagined 

 that nature, without such assistance, could never bring the abscess to a head in 

 her weak condition. July 10th, the matter pointing at the upper end of the 

 tumor very near the os ilium, he made a large opening, from which was dis- 



