## Tuesday, September 19, 2006

### Dring & Fage

Recently I managed to acquire an old slide rule, made in England by the firm of Dring & Fage sometime around 1820 or 1830. Its construction is interesting — it’s made of boxwood, with brass fittings to hold it all together. The scales on the slide rule are hand-engraved; it must have taken dozens and dozens of hours to make them all. The one I bought is still in great condition, and works just fine.

But what has proven to be the most interesting aspect of this slide rule is its purpose. Most (though not all) modern slide rules are purely mathematical instruments — you can use them to multiply, divide, obtain logarithms, make trigonometric calculations, etc. The Dring & Fage slide rule can be used to multiply and divide, but its real purpose is to compute taxes and duties on alcoholic beverages stored in barrels. It was used by people who worked in ports, warehouses, and other facilities where barrels of booze were bought and sold.

The taxes were based on the amount of alcohol, which is a function of the total volume of liquid in the barrel (they were often only partially full) and the “proof", or percentage of alcohol in the liquor. So the tax assessors carried around a little kit, with a long ruler to measure the depth of the liquid in the barrel, a hygrometer (a set of glass balls of calibrated specific gravity) to measure the percentage of alcohol in the liquid, and a slide rule like the one I have. The slide rule was used to compute the volume of liquid in the barrel, then multiply that times the percentage of alcohol to get the total alcohol content — which gave you the tax basis for the barrel. These kits were called “proofing kits", and the slide rules in them were called “proofing rules”.

Calculating the volume of liquid in a partially full barrel is a non-trivial exercise in three-dimensional geometry. The barrels are not cylinders (that would be easy!) as their sides are curved outwards. It turns out that the barrels used back then all had sides that were, in cross-section, shaped like a small section of a circle. There were only a few basic shapes of barrels used, and they could all be characterized by the radius of these circularly shaped sides and the diameter of the barrel’s top and bottom. With a proofing rule, you could calculate the volume of liquid in a barrel with just a few quick movements of the slides, so long as you knew the diameter of the barrel’s top, the height of the barrel, the radius of curvature of its sides, and the height of the liquid in the barrel. The only challenging thing there is the radius of curvature, and this was handled by the barrel industry standardizing on just a few (about four or five) “varieties” of barrels, known as “Variety A", “Variety B", and so on. Each variety had a fixed radius of curvature. And the proofing rule has a separate scale on it for each curvature.

I was surprised how widespread such a sophisticated tax basis calculation was, so very long ago. So far as I know, the tax folks today don’t have to deal with things like this (though this could just be my ignorance)…