yOL. VI.'} PHILOSOPHICAL TRANSACTIONS. 623 



— z 



A 1,1 r a* + ab + za + zb 



And therefore = — = x, 



z 



Wherefore ^ a = b. 



And ^/:*-"^2.i+.^ + 4.r:H:t-.^^_ 



And therefore a and b being given, x is known by the first equation; and a 

 and X being given, b is known by the second; and b and x being given, a is 

 known by the third. 



The horizontal line BFBAF is substituted for GABEFb, when the close 

 end of the tube is not even with the surface of the water, to avoid the breach 



cC = bB = — ; in the length of the tube. 



Note. That the perpendicular immersion of the tube or cylinder, spoken of 

 in N° 73, is not to be understood of the depth of the bottom or open end in 

 the water, but of the depth of the air within the tube or cylinder from the 

 surface of the water, viz. FC, not FD. 



An Answer to four Papers of Air. Hobbes, lately published in the 



Months of August and this present September, \6jl. N" 75, p. 2241 . * 



In the former Part of his Jirst Paper. 



By reason of a proposition of Dr. Wallis (prop. 1, cap. 5, De Motu) to this 



purpose, for he does not repeat it verbatim ; if there be supposed a row of 



quantities infinitely many, increasing according to the natural order of numbers, 



1, 2, 3, &c. or their squares, 1, 4, Q, &c. or their cubes, 1, 8, 27, &c. whereof 



the last is given ; it will be to a row of as many, equal to the last, in the first 



case, as 1 to 2; in the second case, as 1 to 3; in the third, as 1 to 4, &c.; 



where all that is affirmed is but. If we suppose that, this will follow. Which 



consequence Mr. Hobbes does not deny ; and therefore all that he says to it is 



but cavilling. 



Mr. Hobbes moves these questions, and proposes them to the Royal Societ}' 

 to pass a judgment on them. 1. Whether there can be understood (he should 

 rather have said, supposed) an infinite row of quantities, whereof the last can 

 be given? 2. Whether a finite quantity can be divided into an infinite number 

 of lesser quantities, or a finite quantity consist of an infinite number of parts.? 

 3. Whether there be any quantity greater than infinite.? 4. Whether there be 



* We think it necessary to give this paper (evidently ccmposed by Dr. Wallis) at full length, as it 

 contains a good deal of pleasing information on a very dry subject} containing the sum and substance, 

 and indeed closing the long and tedious dispute with Mr. Hobbes, relating to his pretended quadra- 

 tures of the circle, which he so obstinately and perv^ersely defended to the last. • 



