# Deriving a system of scientific units

The dominant society on my conworld is at a level of technological and scientific development not unlike our own, with minor differences here and there. Scientific standards are decided by Academy of Science (for want of a better translated term), and this includes the system of measurements, which in this world is similar to the Terrestrial Imperial system, but with derived units in a manner similar to SI derived units. The native numeral system is dozenal (or duodecimal, or base-12), so a lot of the unwieldy aspects of an Imperial system of measurement can be handwaved away (12" to a foot is nice and round, for example).

So far, I've got units of length, area, time, temperature, and mass. Following the example set by the SI derived units, I've also come up with equivalents for force, pressure, and energy (newton, pascal, and joule), though I haven't thought of appropriate names for these units.

How can I derive units for electromagentism (e.g. volt, ohm, watt, ampere), as the SI definitions seem to be a bit confusing in this regard? The ampere is currently the base SI unit for current, but how did they decide exactly what one ampere is?

For more context, I've listed all units in this system of measurements as they're currently defined below. Note that whilst I'm using the same names as common Imperial units, the actual definitions vary quite considerably.

## Base Units

• Length: foot = 267 mm
• Mass: pound = 528.72675 g
• Time: second = 0.8544 s
• Temperature: Originally defined by assigning 0° to the freezing point of water, and 144° to the boiling point of water, but supplanted in physics and chemistry with a unit using the same graduations but measured from absolute zero instead, analogous to degrees Celsius and kelvins.
• Molar weight is the amount of substance which contains as many elementary entities as there are atoms in one ounce of carbon-12, which makes it slightly less than one SI mol. The analogue of Avogadro's constant is correspondingly adjusted.
• Ampere equivalent?
• Candela equivalent?

## Length

• Point = 0.309 mm ($\frac{1}{72}$ of an inch)
• Line = 6 points = 1.854 mm ($\frac{1}{12}$ of an inch)
• Inch = 12 lines = 22.25 mm ($\frac{1}{12}$ of a foot)
• Foot = 12 inches = 267 mm (Base unit of distance)
• Yard = 4 feet = 1.068 m (Less commonly used instead of feet)
• Chain = 18 yards = 19.224 m (Commonly used in surveying (miles-chains))
• Mile = 72 chains = 1384.128 m (Common unit of large distances)
• League = 4 miles = 5536.512 m (4 miles, or the distance a person can walk in an hour)
• Pica = $\frac{1}{6}$ in = 3.7083 mm ($\frac{1}{72}$ of a foot)
• Furlong = 9 chains = 173.016 m ($\frac{1}{8}$ of a mile)
• Link = $\frac{1}{144}$ chain = 133.5 mm (Surveying unit)
• Rod = 24 links = 3.204 m (Surveying unit equal to $\frac{1}{6}$ of a chain)

## Time

• Second = 0.8544 Earth seconds
• Minute = 72 seconds = 1.02528 Earth minutes
• Hour = 72 minutes = 1.230336 Earth hours
• Day = 24 hours = 29.528064 Earth hours
• Year = 241 days ≈ 297 Earth days

## Mass

The base unit of mass is the pound, but the modern ounce was originally defined as the mass of one cubic inch of water.

• Grain = = $\frac{1}{6912}$ lb ≈ 0.019 g
• Dram = 48 grains = $\frac{1}{144}$ lb ≈ 0.918 g
• Ounce = 12 drams = $\frac{1}{48}$ lb = 11.015140625 g
• Quarter(-pound) = 12 ounces = $\frac{1}{4}$ lb = 132.1816875 g
• Pound = 4 quarters = 1 lb = 528.72675 g
• Stone = 12 pounds = 12 lb = 6.344721 kg
• Hundredweight = 12 stones = 144 lb = 76.136652 kg
• Ton = 12 hundredweights = 1728 lb = 913.639824 kg

## Derived Units

So far, I haven't thought of what names to give these derived units

• Force: poundal (pdl) = 1 lb⋅ft / s ≈ 0.193 N (probably only used with prefixes much like our kilogram?)
• Pressure: poundal per square foot ≈ 2.713 Pa (could also use pounds per square inch: 1 psi = 390.625 Pa)
• Energy: poundal⋅foot ≈ 0.052 J (possibly also only used with prefixes)
• SI defines ampere as follows: The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to 2×10−7 newtons per metre of length. Commented Dec 3, 2017 at 8:23
• Volt, ohm and ampere are no power units.
– L.Dutch
Commented Dec 3, 2017 at 9:00
• Your "pound force" has a misleading name. Most people understand by pound-force the weight of a mass of one pound; the unit described is properly called a "poundal". Commented Dec 3, 2017 at 11:30
• @L.Dutch The "slug" doesn't really sound like a scientific unit either but it turns up all the time in engineering textbooks by American publishers (who for some bizarre reason don't do SI-only versions of their books, at least in my university library). Commented Dec 3, 2017 at 11:58
• That's a fair amount of text. I added some boldface in an attempt to emphasize the actual question, making it easier to find. Feel free to roll back or edit further if you disagree.
– user
Commented Dec 3, 2017 at 13:38

Volts, ohms, and amperes are not units of power. Volts are units of electric potential, ohms are units of electric resistance and amperes are units of electric current.

The unit of power is the watt, with the straightforward definition of joules per second. In your system, the unit of power would be "pound" times "foot" squared over "second" cubed; one such unit would be equal to 0.060432434082031 watt.

To see why, you must consider the dimensionality of power. Power is energy over time; energy is force times length; force is mass times acceleration; and acceleration is length over time squared: so that overall power is mass times length squared over time cubed.

## Electromagnetic units

The quirky SI definition of the ampere comes from its history.

The basic problem is that there is no obvious way of relating electromagnetic units to mechanical units. Historically, electromagnetic units were defined in terms of forces, using either:

• Coulomb's law, resulting in the so-called CGS electrostatic units (statvolt, statampere, statcoulomb, etc.). In this system, the fundamental electromagnetic unit is the unit of charge, called statcoulomb or franklin or esu (electrostatic unit of charge), defined as the charge of an object which, when placed at 1 cm distance from another identically charged object, will repel it with a force of 1 dyne.

• Ampère's law, resulting on one hand in the CGS electromagnetic units (abvolt, abampere, etc.), and on the other hand in the SI electromagnetic units (volt, ampere, etc.) In this system, the fundamental electromagnetic unit is the unit of electric current.

Historically, it all began with the abampere (CGS electromagnetic unit of electric current), defined as the current which, when passing through two infinitely long conductors of negligible cross section placed 1 cm apart in a vacuum, produces between those conductors a force of 2 dynes per centimeter of length. Nice definition, with simple numbers. (1 cm is 0.01 meters; 1 dyne is 1 gram times 1 centimeter over 1 second squared, that is, 1/100,000 newtons.) Once you have a definition for the units of electric current, electric charge is current times time, and electric potential is work over charge, so you can derive immediately units of charge and potential.

When engineers came into the picture they found that scientists used absurdly small or absurdly large electromagnetic units, so in 1873 the International Electrotechnical Commission (which at that time was called the International Electrical Congress) voted to adopt as the engineering unit of electric potential the volt equal to 100,000,000 scientists' abvolt, and the ampere defined as 1/10 of the scientists' abampere.

To sum it all up, first you must choose if you want to have a system of electromagnetic units similar in spirit with one of the various CGS systems (electrostatic, electromagnetic, or Gaussian) or with the SI. Then, if you choose to have a system of electromagntic units similar in spirit with the SI (that is, a system where the electric permittivity and magnetic permeability of the vacuum are dimensional constants, and the 1/4π factor is not rationalized out), and you want to have a unit of electric current about the same size as the ampere, you can define it as the current which, when passing through two infinitely long conductors of negligible cross section placed in a vacuum at one unit of length apart, produces a force of 2×10−6 units of force per unit of length; this would result in a unit of current equal to 1.389 amperes.

A unit of length is 0.267 m, and a unit of force is 0.193 N. 1 ampere of current would produce 1/0.267 * 2E-7 N per meter at the distance of a "foot", or 2E-7 N per "foot", or 1.036E-6 units of force. We want to have nice numbers, so we need 2E-6 units of force, which is 1.93 times greater; take the square root because force is proportional to the product of the currents and you get 1.389 amperes.

## Photometric units

Unlike other units of measurement, the photometric units don't measure physical quantities but sensations in the brain of a "standard observer", which is more-or-less an average person with average vision ("average" taken over persons with no visual impairments such as color blindness); they work only for humans, and would baffle a dog or a cat. As a consequence, the photometric units on your world depend on the physiology and neurophysiology of the inhabitants.

In particular, the fundamental photometric unit, the candela, is defined so that a light source of one candela appears to a human to be about as luminous as an ordinary candle. The SI definition is that a candela is the luminous intensity of a light source which emits monochromatic green light with a radiometric intensity of 1/683 watts per steradian. For monochromatic yellow light human physiology makes the radiometric power needed lower than 1/683 watts per steradian; for red monochromatic light, the power would be greater; for blue light, even greater; and for infrared or ultraviolet light no amount of power is big enough to produce one candela, because we cannot see infrared or ultraviolet light at all.

If the inhabitants of your world have the same physiology as humans and you want to keep the same luminous intensity for the candela, the corresponding definition would require 1/41.275 units of power per steradian.

• Scientists: approximating everything to spheres with no friction since 1000 b.c. :)
– L.Dutch
Commented Dec 3, 2017 at 9:34
• Thanks. I'd been going over the Wikipedia entries to both CGS and SI systems of units, and I'd been struggling to relate the two and work out exactly how to define the analogous system of electromagnetic units in this world. Although incorrect, I used the word "power" in my question in the layman's sense as a synonym for electricity. Commented Dec 3, 2017 at 11:45
• @Robbie: The various CGS systems of electromagnetic units and the SI system are not directly relatable. The physics formulas have different forms in the various CGS systems and the SI system. For example, an electric potential of 100,000,000 abvolts is the same as 1 V, but you cannot convert volts in abvolts and use the same formulas, because the CGS systems abstract away the electric and magnetic properties of the vacuum, and some of them (called "rationalized") dispense with the 1/4π factor. Commented Dec 3, 2017 at 11:55
• I'm marking this as the answer, as it describes how to create an analogous system of electromagnetic units, which is what my poorly-worded brain-dump of a question was trying to ask. Commented Dec 4, 2017 at 2:04
• @AlexP How did you calculate the 1.378… value for the unit of current? Commented Dec 6, 2017 at 7:08

are you looking for names, or relationships?

Volt, ampere, watt, and ohm all take their names from the people who discovered the fundamental concept and the basic realtionship between these aspects of electricity.

Many of the names of SI units came from the people who discovered the aspect or relationship involved. Standardization into an SI unit came much, much later.

The reason I bring this up is that most of the units of measure we use today are spectacularly arbitrary. They may be universally defined, but they're hardly universal at all. Evidence of this are the wild numbers we must adopt as constants (e.g., the Planck constant) to make rational relationships that are irrational due to the arbitrary nature of our "standards."

Who says that time, even time associated with Earth, need be defined by 24 hour days? Do we even need minutes? What, really, is a second? For example, the ancient Babylonians divided the day into 24 hour segments, but the ancient Chinese divided the day into 100 segments. How would the formulas we use today have changed had the word adopted the Chinese solution rather than the Babylonian solution? It was an arbitrary choice, one being no more right from the point of view of physics than the other.

As I look through your list you appear to be assigning words in a one-to-one fashion with what we use today. You're providing basic relational differences (referenced to Earth), but in reality... it's 100% the same system.

So, use the names for the above four units of measure as an example and move forward. Create four inventors, and assign the their names to the unit name to honor them as we did.

Defining units begins with understanding relationships

Physicists of the past weren't looking to define a standard. They were searching for relationships in our physical world. Three of the four units are defined with one remarkably simple equation: Ohm's law.

R = V/I

A similar calculation defines power:

P = V*I

In other words, whatever definition you use for volt will impose itself on the definitions of the other three units because the relationships must never fail. They're actually what's important.

No standard unit sprang into existence. The relationship between two or more units was mathematically established first. A rock-solid, written-in-stone means of "defining" exactly what something is (like a second,) came much, much later.

Your choice of how to define any SI unit is as arbitrary as the methods used on Earth. However, if you want WB:SE to help you with alternative methods of defining standard units, you should ask for each definition as a separate question, or this becomes too broad.

However, how important is this? As an author, are you straining at a gnat? How important is mimicking the body of standardized measurement in your story? Better still, how important is it that the unit definitions exist? Without a bit more insight, it appears to be a ton of work with little value.

• Well done for pointing out that this may not be required at all for a story to work... Commented Dec 3, 2017 at 16:14
• I'm not writing a story (yet), I'm creating a world. The extremely long-term goal is to eventually write some kind of in-universe encyclopaedia, written in their language, that doesn't have references to terrestrial things at all, like a Codex Seraphinianus that actually means something. Commented Dec 4, 2017 at 1:58
• It's worth noting that the Codex does mean something, just not what you anticipate. Art is painful that way. While it's difficult to comment without knowing much more about your project, my guess is that you've put the cart before the horse in that you're replacing one-for-one the names/values of things with possibly arbitrary changes that do not relate to the origins of your world as they should. E.G., a duodecimal system could mean your peoples have six digits per hand/foot, if they have hands/feet. Frank Herbert uses it in Battlefield Earth for remarkably unique reasons. Just wondering.
– JBH
Commented Dec 4, 2017 at 5:26

There are many ways to define units. Even the SI unit for length has been derived in different ways over our history-- it initially used a seconds pendulum; now it is light/wavelength based. The definitions change as science allows more precision.

This means that your alternate history of science will generate many different bases. Ultimately, there are underlying physical constants, like the charge of an electron.

Units systems are just oversimplifications abstractions that engineers use to avoid doing real simplify the math.

• As an engineer I'd be offended by that last sentence if it wasn't 100% true. Oh, don't get me wrong, we love the math, but when all is said an done, our purpose in life is to solve practical problems, not mathematical problems. (It's all in how you see it, you see!)
– JBH
Commented Dec 3, 2017 at 14:52