On a Scale for Guaging Cylindrical Measures of Capacity. 247 



for gill, 

 33-219 x log deptb 2 + 66*439 x log dianieter 2 - 14*6368 = 20; 



for half-pint, 

 33-219 x log depths + 66-439 x logdiameter 3 -14-6368 = 30; 

 and so on, the right-hand number rising to 100 for the bushel. 

 It will be remembered that the logarithms are common loga- 

 rithms corresponding to the number of inches in the depth 

 and in the diameter. 



The number —14*6368 may be divided into two parts, 

 — a and — b, subject only to the condition that a + 5 = 14*6368. 

 And as we propose to use two engraved series of numbers, 

 one relating to the depth and the other relating to the dia- 

 meter, we may attach —a to the first and — b to the second; 

 so that the numbers of the first column will be 

 33*219 x log (depth in inches) — a, 

 and those of the second column will be 



66*439 x log (diameter in inches) — b. 

 And we have now to consider the details of the two engraved 

 series which will represent these. 



When we plunge the material scale into the cylinder, to 

 touch its bottom, we obtain, in inch-measure on the scale, the 

 quantity of "depth." But the engraved numeral is to give 



"(33*219 x log depth) -a," 

 or " (33*219 x log inch-measure) — a." 



Therefore 



" 33*219 x log inch-measure " = " engraved numeral + a ;" 



auc * ,, , . , ,, engraved numeral + a 

 " log men-measure = — ,, ia ; 



and, taking the exponentials of both sides, 



"inch-measure" = , , 



, , . engraved numeral +a 

 number whose log is — - „, V) , 



= number whose log is -030103 x (engraved numeral +a). 

 In like manner it is found for the scale which is applied to 

 the diameter, 



" inch-measure " == 

 number whose log is "0150515 x (engraved numeral +b). 

 By these formulae the measures are given corresponding to 

 every engraved numeral. 



It will be remarked that the succession of engraved lines on 

 these scales differs strikingly from that on the common loga- 

 rithmic scale. With equal intervals of numerals, the intervals of 

 engraved lines on the scale become larger with increasing num- 

 bers. The scales may properly be termed " exponential scales." 

 On referring to the investigation, it will be seen that, if the 



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