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XXVII. On the Construction and Use of a Scale for Guaging 

 Cylindrical Measures of Capacity. By Sir Gr. B. Airy, 

 K.C.B., Astronomer Royal*. 



SEVERAL years ago, when I was officially engaged on the 

 national measures. I arranged a simple scale for rapidly 

 guaging the cylindrical measures of capacity which are sanc- 

 tioned by law ; and which are in fact the only measures in 

 ordinary use. I understand that this scale is now sometimes 

 employed for its proposed purpose. I do not remember that 

 the principle of its construction has been published; and I think 

 that there may be advantage in making it known. 



The cubical capacity of a cylinder, whose axial depth in 

 inches is called " depth/' and whose diameter in inches is called 

 "diameter/' is = depth x (diameter) 2 x '785398. As the 

 cubical capacity of the gallon is 277-2738 cubic inches, that 

 of the half-gill (the smallest of our measures), or -^ of the 

 gallon, is 4*332403 cubic inches. Hence we have for the half- 



depth x x (diameterO'x -785398 = 4-332403, 

 or depthi x (diameter:) 2 = 5-51619. 



As the cubical capacities of our measures (half-gill, gill, half- 

 pint, pint, quart, half-gallon, gallon, peck, half-bushel, bushel) 

 proceed in continued binary progression, we have 



for gill, depth 2 x (diameter 2 ) 2 = 2 x 5-51619; 



for half-pint, depth 3 x (diameter 3 ) 2 = 4 x 5-51619; 

 and so on, the factors of 5*51619 being the successive powers 

 of 2. 



Taking the logarithms of both sides, 

 log depthx + 2 x log diameter! = -74164, 

 log depth 2 + 2 x log diameter 2 = -74164 + -30103, 

 log depth 3 + 2 x logdiameter 3 = -74164 + 2 x -30103, 

 log depth 4 + 2 x log diameter 4 = -74164 + 3 x -30103 ; 



and so on, the numbers on the second side increasing in arith- 

 metical progression, with the common difference '30103. We 

 are at liberty to multiply these equations by any arbitrary 



number ; and we shall adopt the multiplier- ^- l/ . „ r0 or 



J 'oUlOoOO 



33-21928. And we shall add to both sides the number 



_^ x . 7 4164 + 10=-14-6368. 



Thus we find, f or half-gill, 



33-219 x log depthi + 66*439 x log diamete^- 14-6368 = 10 : 

 * Communicated bv the Author. 



