Friday, 5 February 2016

Fictional Measurement System.


As is the tradition, the Friday blog will be on something more abstract or light hearted than the usual topics.

Over the years I have read a number of sci-fi and fantasy books where the author has given a world its own measurement systems. Most of these seem to be fairly arbitrary rather than based on aspects of that world. It is with this in mind that I have pursued the following train of thought.

While doing some background reading for the last series of posts I came across a description of a duelling sword as being of “four palms length”. Seemed rather short for what was described as an excellent weapon for the duel! I did a little research and discovered that “palm” was a term sometimes used for a measurement based on hand length. Like many old measurement standards it varied from city to city but was between 8 and 10 inches long. 32 to 36” is about right for a handy duelling sword.

I began to read about anthropic measures. Something that occurred to me was that a “foot”, despite its name, was not an anthropic measurement. Very few people have feet twelve inches long. According to Vitruvian proportions a foot should be a seventh of your actual height. I’m about 71” tall so my feet should be over 10” long. The 15.3% value on the wiki article would give me 11” feet. Actually they are 9” long.  Someone with a 12” should be seven feet tall if Vitruvian proportions are correct! Vitruvian proportions don’t work for feet. Many people my height have feet larger or smaller than me. Many smaller people have larger feet. It has been suggested that “foot” as a measurement is a synonym for “shoe”. Coincidentally my favourite boots are exactly 12” long. My ancestors probably did not have moulded rubber soles so their shoes were probably shorter than this.


A fact I did come across was that the Romans used a measurement based on the pace. The Roman passus was a “double pace”, measured from the heelprint of a foot to the next heelprint of the same foot. It was standardized at 1.48m and also described as being five Roman “pes”. One pes or “roman foot” is 11.6 modern inches. The Romans used a decimal system for large measurements so one thousand passus or five thousand pes was a Roman mile. It seems likely that the pes was created as an increment of a passus rather than anything to do with the size of a human foot.

My reading also turned up the interesting fact that the metre was originally derived as being a ten-millionth of the distance between the equator and the north pole of the Earth. While logical there are a number of objections to this. One is that a measurement of the Earth will vary with how it is done. If we fly an aircraft from the equator to the north pole the actual distance it travels will depend on its altitude. It may occasionally have to fly higher to cross a mountain range so the distance travelled will be different to that on a map. If it was practical to walk this distance the walker would have to move up hills and down across valleys that the aircraft simply fly over. Again, the total distance travelled will be different. Perhaps our walker takes a little dog with him. Poor Fido will have to clamber across rocks and negotiate furrows and thus walk a different distance to the human beside him. Another objection is that this derivation does not make the measurement unit particularly tangible:-

      “Fred, do you think it is less than a hundred metres to the pub?”

     “Well, Bert, do you think that looks like a hundred-thousandth of a quadrant?”

Suppose, for discussion's sake, we had a measurement system based on something else. I am going to use the Roman passus (1.48 m) for this example. A pace is quite a handy, easily visualized measure. There are obvious useful real world applications for working out ranges or travel distances in paces. It is probably more useful to know something is twelve paces away than twenty yards. Pacing a distance gives a fairly accurate approximation even if your pace-length is not exactly 1.48 m.

           For reasons that will become clear in a moment I am going to suggest decimalizing the passus. Each passus is divided into ten deci-passus or “desips”, each equivalent to 14.8 cm or 5.8”. Interestingly, three desips equals 17.4”/44.4 cm, a shade under a standard (Biblical) cubit and nearly identical to the Roman cubitus. Vitruvian proportions gives the height of a male as four cubits. Twelve decips works out as 177.6cm or 5’ 9.6”, a fairly good estimation for the height of an average man. We can therefore create a second measurement called a “kupit” and use the length between a middle finger and elbow tip as an approximate measure. A cubit is quite a handy measure. It is readily to hand and in Vitruvian proportions is equal to a shoulder width. A man’s height or armspan is meant to be four cubits. The Wikpedia page on Vitruvian man defines a pace as four cubits, but this is obviously incorrect since men do not walk in bounds equal to their height. Cloth or rope can be easily measured out in “double cubits” by stretching it from centreline/nose to fingertip. This is a traditional way to measure yards but our system does not need yards (or feet) when we can use passus, kupits and double kupits.

One passus = Ten desips
One kupit = Three desips
Three passus = Ten kupit

So, our standardized measures of length are the passus, kupit and desip, two of which can be easily estimated by pacing or comparison to a body part. For smaller increments we can have the centi-passus, milli-passus  (centip and millip?) etc. For longer distances we have the kilo-passus (kilop?) equal to ten-thousand desips or 1.48 km/ 0.925 miles. A league, the distance that can be marched in an hour, is 3.25 kilo-passus. 200 passus is a shade under 300m, which is about the limits of the effective range of a rifle for many shooters. 150 passus around the battlefield range of a bow.
 
Kupits to Passus = n x 3/10
Passus to Kupits = n x 10/3
 
 A hektopassus is an area measurement equivalent to a square ten passus to each side. Ten thousand hektopassus make a square kilop.
 
Once we have a linear measurement we can move to volumes. A cubic desip works out to be 3.24 litres, not a very useful aliquot to base a system on. Instead, I am going to suggest a cubic half-desip, which is a cube of 5 centi-passus to a side. Eight of these will fit in a cubic desip and each is about 405mls. This is just under an American pint so is a handy amount to deal with. We will call this a “skulp”. Skulps can be divided into milli-skulps and micro-sculps, each milli-sculp being equivalent to a cube of 5 milli-passus sides. For larger volumes there is the kilo-skulp, eight of which make a cubic passus. Whereas the liner measurements are based on 3s and 10s the volumes are related by 5s and 8s!


For a unit of mass we might as well use the mass of a single skulp of water. We will call this a “skud” so one skulp weighs one skud or 405 gms. A milli-skud is 405 mg and a kilo-skud is 405 kg.

One milli-skulp of water masses one milli-skud and is a cube of 5 milli-passus per side.
One centi-passus cube of water masses eight milli-skud and contains eight milli-skulp

A cubical cup with internal dimensions of half a desip (2.9”) provides a handy standard. For a skulp, fill it up to the line. To weigh a skud measure a cup filled with water against its empty twin. The internal sides of this standard cup are half a desip across. The single standard cup is therefore a useful standard for mass, volume and length! All merchants should have a set! Beer mugs will hold a skulp so an improvised standard will seldom be far away. One skulp is a handy size for a scoop or ladle.
There you have it. A fairly simple but useful measurement system for a fictional realm. 

Update.
The derivation of a skulp suggests that a passus might be more logically subdivided into twenty parts rather than ten. I will call this unit a “vingt”. This is approximately a handbreadth. The desip becomes redundant. There are five centi-passus to a vingt. For convenience the centi-passus will be renamed “ort”.
 A two kupit (twelve vingt) pendulum allows the derivation of a time increment of 0.945 seconds at Earth standard gravity. This is only about 1/20th difference from an actual second so is effectively the same for most purposes. Using the system of counting to 144 on the fingers allows the use of seconds in multiples of twelve. 144 seconds is 2.4 minutes. Twelve such units (a zagier) is 28.8 minutes and 50 zaiger-seconds is 24 hours. 144 “short” seconds is just over two and a quarter minutes. A zagier of short seconds is just over 27 minutes.

One passus is 20 vingts. One kupit is 6 vingts. A skulp is a cubic vingt. A skud is the same weight as a skulp of water.


8.5 to 10 vingts would probably be a good length for the blade of a handy sword. Half a passus the height of a battle shield. A typical spear might be 5 kupits, a light lance 2 passus and a pike 12 kupits.
 
Summary.
  • A passus is a standardized double pace. One passus = 1.48 m, 58”, 20 vingt or 100 orts.
  • One kupit is a standardized cubit. One kupit = 44.4 cm, 17.4”, 6 vingts or 30 orts.
  • One vingt = 7.4 cm, 2.9” or 5 orts.
  • One ort = 14.8 mm or 0.58”.
  • One skulp = 405 mls or 14.25 fl oz. A cubic vingt.
  • One skud = 405 gms or 14.25 oz/ 0.89 lb. Weight of a skulp of water.