Useful tricks for building containers according to the The Carpenter's and Joiner's Pocket Companion (1888):

• A box \(8 \times 8\) inches square, and \(4 \frac{1}{8}\) inches deep, will contain approximately one gallon.
• A box \(4 \times 8\) inches square, and \(4 \frac{1}{8}\) inches deep, will contain approximately half a gallon.
• A box \(4 \times 4\) inches square, and \(4 \frac{1}{8}\) inches deep, will contain approximately one quart.
• A box \(4 \times 4\) inches square, and \(2 \frac{1}{8}\)inches deep, will contain approximately one pint.

An alert visitor of this page pointed out that these numbers appear to be wrong. One gallon = 231 cubic inches and you get much more than 231 if you do the multiplication for the first one (you get 264). It should be more like \(3 \frac{3}{4} \) inches deep than \(4 \frac{1}{8} \). What's going on here? Below are theories proposed by Curious and Useful Math visitors:

Back in 1888 lumber was much larger. Even finished lumber usually exceeded the dimensions it was sold at. So a 2x4 was actually more like 2.5 by 4.5 - not smaller, as it is now. That could explain the loss of volume inside the box.

Having noticed the dilemma of a "gallon" not quite matching an 8 x 8 x 4.125 inch box, I realized that an Imperial gallon (being 4.546 litres) IS QUITE CLOSE, compared to a US gallon, being less at approx 3.9. litres. This is probably the reason for the "error", that is, no error but different standards.

Your speculation about the measurement of the wood is exactly right. A carpenter will ALWAYS use the measurements of the outside of the wood. Building a box 8" x 8" x 4.125" using 1/4" plywood (which actually measures 3/16") will yield very close to a gallon because the inside dimensions of the box will be 7.625" x 7.625" x 3.9375" for a total volume of approximately 229 cubic inches. I am assuming no top on the box.

I am not a mathematician either (I am an engineer) but there is a reason the box is much bigger than a gallon. The only reason to build a box that holds a gallon would be to move water from one place to another. That being the case you can't carry a box of water filled to the brim because it will spill. Also since it is made of wood, the wood will soak up some. I believe that this box will give you a much more accurate gallon at the point of delivery. You wouldn't have a gallon after picking up the box if it were built to hold exactly a gallon. I hope that makes sense to you, too.

If you assume the carpenter was using finished lumber, a square box using an "8 inch" plank would actually be 7.5 inches square. This comes very close to the right volume.

The given measurements are accurate for the appropriate measure. There are two types of measure that use the terms pints, quarts and gallons - liquid and dry. These instructions refer to dry units. For example, strawberries and other small berries are sold in dry units.
1 dry gallon = 4 dry quarts = 268.8 ci, rule #1 gives 264 ci.
1 dry quart = 67.2 ci, rule #3 gives 66 ci
Since this was taken from a carpentry book, it seems very logical that if a carpenter is building boxes for harvesting fruit, then this information would be very useful.