Discussions and school lessons on Roman arithmetic are solely about integers. In reality the Romans, and the medieval merchants who continued the tradition, handled fractions all the time using a sophisticated notation.
Integers operated on a pair of 5s, not a simple base 10. Fractions operated on a pair of 6s, not a simple base 12. The separate bases ruled out modern tricks like moving the decimal point for convenience in entering numbers. Parts and wholes were distinct universes.
Non-decimal money was formerly familiar in pounds and shillings, long lost in other money. After Britain converted to decimal in 1971, the only remnant was no longer within common practice. The distinction is still strong in other measurements, but only in the US. England and all of Europe went totally decimal long ago. (This is one of the few areas where US retains sanity. We’ve been leading the crazies in all other ways!)
Counting whole numbers was based on the two hands. Counting fractional parts was based on folding or distributing into containers.
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The basic integer units were suggested by two hands:
V I I I I from one thumb and four fingers.
X was suggested by two thumbs crossed, so two Vs became X.
The next places C and M obviously came from Centum and Mille, though the M might also be suggested by another variation, as we’ll see later.
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Fractions were made by factors of 2 and 3, forming 6 and 12. Base 12 has a genuine advantage over 10 in the real world of commerce and carpentry and cooking. Folding a letter or towel by two then three is familiar. You can accurately divide a cup or fold a string or saw a board by 2 or 3 repeatedly but partitions by 5 are rare in the real world. The Romans recognized the advantage and chose the better base for each type of number.
Fraction notation followed the same pattern as whole numbers, using dots for single items instead of I, and S (semis) for half instead of V.
Mixed numbers looked like this:
17 = XVII
17 and 4/12 = XVII ⋅ ⋅ ⋅ ⋅
17 and 6/12 = XVII S
17 and 9/12 = XVII S ⋅ ⋅ ⋅
Zero was unneeded for most purposes, but around 700AD some astronomers and apothecaries started using N for Nulla to mark the starting point of a table or measurement.
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The British money system remained in use during the mechanical age and part of the digital age, so a wide variety of machines were built to calculate money the Roman way. The smallest one was the Addiator. Looking through some old books on business math in pounds/shillings/pence, noticed a description of an Addiator specialized for LSD. It was made in Germany from the 20s through the 70s. Most Addiators were decimal; the LSD version was popular in England.
Addiator history, mainly from this website:
The older version used the same sliders for adding and subtracting, with opposite scales on opposite sides. The later form had separate add and subtract sections on one side so you could see both operations at the same time. The older flipover version was sized to fit a suitcoat pocket, clearly meant for use by salesmen or insurance agents.
Checked Ebay, and amazingly one of the LSD addiators was available from a charity thrift shop in England. Immediately bid on it and won the auction. Only $30. I would have paid twice as much.
Here the Addiator is posing behind my existing calculators.
The wide black one is an Addometer, about the same vintage as the Addiator. White plastic is an IBM hex calculator from 1957, designed for use by techs. The little circle is a Russian watch-type slide rule, and the regular slide rule needs no introduction. All of them work properly. The wide Addometer is the most pleasing to the hands, with detents and carries that feel just right.
The Addiator belongs to the slider type of simple machine, which is not automatic. As with an abacus, you have to unset one row and carry to the next row when the row goes beyond 9. The two rotary machines carry automatically with internal cams that flip the next digit for you.
Here’s an ad for decimal addiators from 1923:
And an ad from Popular Science in 1959.
The Addiator came in a neat book-like wallet, with openable covers on both sides and a strap for the stylus. The middle part was partly open to expose the sliders, and the reset lever was on top. Each cover had photo corners on the inside so you could slip in a table of prices or a scratch pad. Unfortunately someone had inserted the Addiator upside-down so the reset lever was hidden at the bottom, and the reset lever was firmly lodged inside the wallet. I had to break the wallet to get the Addiator itself out. No great loss; the wallet was cheaply made and already coming apart.
Closeup of the two sides.
Flipping it top to bottom shows the Subtract side, with the same sliders working upside down.
The carries are indicated by the boundary between red and white in each slider. If you’ve placed your stylus in a red number, it’s time to slide up and over.
It has two Shillings sliders so you can count up to 19 before flipping the first Pounds slider.
The Pence slider goes up to 12 instead of 10. There’s an extra slider for Hapennies and Farthings.
The gold bar on top is the Reset lever, which pulls all the sliders to zero.
Why was an LSD adding machine needed? Ordinary adding machines and cash registers had two decimal places for cents, and all computers have a limited number of decimal places. Some of the 12ths don’t work in decimal form.
For instance, one Pence is 0.0041666… pounds. If you tried to represent this on a two-place adding machine it would give zero. Adding machines and cash registers multiplied by repeated adding, so every multiplication would still yield zero. Even on a computer with a finite amount of precision, multiplying would give an answer less than 1, so it would never compare correctly in a statement like
if (TotalPence == OnePound): Pay(TotalPence)
When the machine counts each unit separately, all answers** are accurate. Electric adding machines and cash registers sold in Britain also recorded the three units separately. UK mainframes didn’t have special registers. The programs simply multiplied Pounds by 240 and Shillings by 12, so everything was expressed in Pence as integers. Then did the calculation in decimal on Pence, then reconverted back to L/S/D.
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Continued with a much older device in Part II.
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** Fussy footnote: To be exact, all entered amounts are accurate. In any unit of any type, decimal or practical, some divisions will yield answers that can’t be shown by a whole number. We’re dealing here with comparative advantages of one system over another.
Polistra's Mill 