This section is from the book "The English And American Mechanic", by B. Frank Van Cleve. Also available from Amazon: The English And American Mechanic.

To find the Contents in Gallons of a Vessel, whose diameter is larger at one end than the other, such as a Bowl, Pail, Firkin, Tub, Coffee-pot, etc.
Multiply the larger diameter by the smaller, and to the product add one-third of the square of their difference, multiply by the height, and multiply that product by .0034 for Wine Gallons, and by .002785 for Beer.
Required the contents of a Coffee-pot 6 inches diameter at the top, 9 inches at the bottom, and 18 inches high.
Large diameter | 9 | Brought up | 1026 | ||
Small do. | 6 | •0034 | |||
54 | 4104 | ||||
1/3 of the square | 3 | 3078 | |||
57 | 3.4884 wine gallons, or nearly 3½ gallons. | ||||
Height | 18 | ||||
456 | |||||
57 | |||||
Carried up 1026 | |||||
1020 multiplied by .002785 equal 2.8574 Beer Gallons. | |||||
 
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