The design of the C.O.F. system

The basic rules follow those provided here: https://www.tapatalk.com/groups/twelftyonline/on-designing-new-systems-t47.html

Getting density and velocity right in any given system provides more 'true to scale' units.  But in place of a time unit, we select gravity as the additional base unit. 

The measure \( v/g\) velocity/gravity = time, is chosen to keep gravity as unity, and the velocity as in the human range.  This table gives the present common velocities, the velocity and the derived length as (velocity)^2 / gravity.

Unit A	g / unit	Length (mm)
m/s	9.80665	101.971
mph	21.936851	20.3784
knot	19.050394	20.378494
ft/s	32.174	9.4735022
KINE	34.56	8.210568
kph	35.303940	7.868155

Although the units are small, the system does not produce large numbers.  The length is shorter than the cm (of CGS), but the velocity is some 30 times faster.

The unit of time here is a day divided to 12^6.  In the usual clock face, the clock is divided to 12 hours, of 144 minutes of 144 seconds, but this unit of time is 1/12 second, or an instant. 

In the second table, we give the day divided into powers of 12, and the distance travelled at the speed of one kine, with names in the 'yard' style metric, and as common names.

day	24 h			24.516625 km
hour	2 h	kiloyard	mile	2.043052 km
	10 min	hectoyard	furlong	170.254340 m
minute	50 s			14.187861 m
	4.166 s	yard	yard	1.1823218 m
second	1/2.88 s		hand	98.526817 mm
instant	1/34.56 s	centiyard	corn	8.210568 mm

One can have a foot-like unit, so a rod=3 yards, a foot = 300 my, an inch = 30 my, and a line = 3 my.

Area

Although the metric system makes use of 'square length', the present system will use a comma-unit, making the acre = 1000 sq yards.

The acre is derived in the traditional English way, being a rectangle of width 4 rods, and a length of 40 rods (or a furlong).  But a rod = 3 yards make these 10z yards wide and 100z long, or 1000 sq yards.  A square yard is then a milliacre, a square mile is a kiloacre.

Volume

Volume refers to geometrically derived measure.  In metric, this was the Stere, or solid measure for firewood, landfill, and irrigation water. 

Capacity

Capacity is poured measure: that is, what one derives against a graded vessel.  The quantity can be measured by weight or by volume, the former practice with grains, fruit and coals, was by measure, but the advent of effective scales made weight the option.  One sees the weight of a bushel of this or a bushel of that.

Liquid capacity is still by volume, it is easier to measure by flow or graded vessel than to weigh it.  Note that aeroplane fuel is weighed.

The unit of capacity is taken as a cubic hand, or Litre.

Weight

Weight means measured that are compared by swinging on a balance.  The ruling formula here is that of a torque balance: \(m_1 g_1 l_1 = m_2 g_2 l_2 \).  Gravity needs to be present, but apart from the assumption \(g_1 = g_2\), it plays no further part in the weighing.  The acts specify an equal arm balance, where \(l_1 = l_2\), but unequal arms can also be used.

The units of weights are derived in the first instance as a cube of water.  This specifies that the specific gravity of water is 1, or alternately the density of water is 1 spig.

Currency

The original weights and measures specify currency, against the value attached to a weight of silver or gold.  This was the order of the day well to the second world war.

Temperature

The idea of setting constants like \(g\), \(\rho\), and \(j\) to one, is that the same unit functions for different scales of measure.  These already form name bridges without invoking a coherent system.  That is, something like a pound of force, or a foot of pressure are already with meaning, the first \( \mbox{lb}g\), the second \( \mbox{ft}\rho g\).

The third of these measures, from what I can trace, occurs in Pendlebury's TGM.  It sets the unit of heat \(jM\Theta\) equal to mechanical and gravitational energy.  When James Prescott Joule first measured the mechanical equivalent of heat, it was that to raise a pound of water by a degree Fahrenheit, is equivalent to raising it 778 feet against gravity.  That's about a furlong.

This equates to \(j/g = 778 \mbox{ ft/°F }\), but we plan to use a corn of heat, which is much smaller.  Before we do, we should look at another interesting unit.  If one supposes that from freezing to boiling consists of 116.3 degrees H, then this converts the joule constant 4186.8 J/kg.K, to 3600 J/kg°H.  This makes 1 kg.degH/hr = 1 kg.secH / sec = 1 Watt.

Heat, then is something acquired mechanically over hours. 

Since we are setting \(j=1\), we get 1 corn =  T² g²/j, or 1 kelvin = 1 second². j/g²

Putting g=34.56, j=4186.8, and g=9.80665, we get 1 kelvin = 51998.2 corn.  Bearing in mind that we note this unit represents a 'second' of temperature, we find that a degree is this number divided by \(12^4\), or 2.507629 degH = 1 degC.

Since the degree is a division of a much larger unit (hectokelvin, where 0=freezing, 1=boiling), this larger unit is 250.7629 degrees, or 18V,9 doz. degrees.  This makes the hectokelvin = 1.741 of the dozenal equivalent (144 degH), and it's a closer fit to supposing 100 F = 100degF.

With fahrenheit, the scale amounts to putting freezer-cold = 0, boiling = '4', and dividing each interval to 60.  Römer's degrees (of 4 fahr ea), overestimated boiling, but the rest of his estimates were pretty close.  Water freezes at 32 in this scale.  Now 32 is close to 0.3333 hundred, so we can set this to 40z.  Water boils at 18E+40, or 210, say.

Electricity

The usual formulae to derive electricity, is either Ampere's relation or Coulomb's equation.  The prototype definitions suppose a number N, whereby

Biot wrote:The (current), is that current, that when flowing in each of two wires of negligable cross-section, and seperated 1 (length) apart, creates a force of 2/N (forces) per (length) of wire.

Franklin wrote:The (charge) is that charge, when placed at each of two points, one (length) apart, exert a force of N (forces) on each other.

The '2' in the first equation is a result of an integration, that the surface area the curved wall of a cylinder is twice the enclosed sphere.

These have a dimensional analysis of an biot = \(F^{\frac 12} \), and the franklin = \(F^{\frac 12} L\).  The relation of Fr / Bi.s  has the dimensions of velocity, this is one of the velocities that Maxwell compared the speed of light to.  It represents the electrostatic charge delivered by a magneticly measured current over a period of time.  It equates to C=N (Fr/Bi.s), where C is the speed of light in L/T.

These constants are generally not base-free, because to get suitable-size units, one must set C/N to be a smallish number, and thus N by preference is a very large power of the base, eg 12^8 or 10^7.  Even so, eliminating this measure by setting c=1, brings similar problems into the electro-dynamic theory favoured when setting c=1.

Instead, we plan to set \(Z_0 = \mu_0 c = 1/\epsilon_0 c = 1\).  This is the essence of a symmetric electric + magnetic system. 

Ampere's law becomes \( F/l =  I_1I_2 / 2\pi c R\)

Coulomb's law becomes \( F = c Q_1 Q_2 / 4\pi R^2\).

Setting c=1 removes the constants here, the resulting system being the Heaviside-Lorentz-theory as an electro-dynamic theory.

The unit of resistance is an electrical resistance of 376.730313462 Ohms.  You can never create an electrical resistance from \(Z_0 = E/H\), because there are hidden factors that do not resolve in SI.  This is one of the \(4\pi \) values: the MKSA system that SI replaced, has this at 29.9792458 Ohms.

One of the interesting things here is that the capacitance of parallel plates is \( C=\epsilon_r A/c D\), where A is the area and D the distance between them.  If these are measured in light units, (ie ct), then C=A/D is the formula.  Since light-units are also something that people think in terms of for how long a signal takes to move a distance, this means the same scale of HLU-like formulae works here too.

Stroud-Wallot Rules

These are rules that were taught to engineers at the turn of last century.  The two concentrated on that one could only cancel out units of the same name, but they were implemented differently. 

In one case, a unit was thought to be a numeric and a base unit, eg '1 ft = (30.48 cm), one could replace 1 ft with this expressinn, eg g = 32.174 ft/s² = 32.174 (30.48 cm)/s² = 980.663 cm/s².

The other case is that one could apply 'identities' to the equation, which were fractions of unit measure (eg \( \frac {1 \mbox{ ft }}{30.48 \mbox{ cm}}\), so 32.174 ft/s would become \( 32.174 \frac{\mbox{ft}}{\mbox {s}^2} \times \frac{30.48 \mbox{ cm}}{\mbox {ft}} \). 

The results are the same, but in the second calculation, there are things to cancel.

Applied to COF. each unit would equate to a number of COF units.  Because things like \(g\) are set to unity, the units are proportionally scaled.  A pound might be '600'z while an inch is '3'z.  So 1 psi = 600/3² = 80z.  Atmospheric pressure is 890z, so this equates to 11.16 psi, or 2E feet of water, or whatever.

The main advantage of this is that it is easier to get an estimate of a size in an appropriate unit, than some hundreds of smaller units.  One would measure distances on foot in a known 'hectometre', than to multiply 'metre' 100 times in the ground. 

Given that the usual practice is to select base units from the acts of legislation, it is highly unlikely that an instant be included in this, given that most people don't need it.  So the most probable system of units from COF, would be the dm.kg.s (units of 10, 1000, 10 COF, dozenally).  1 kg/sq dm = 1000/10² = 10.  Atmospheric units would be 89 kgf/sq dm, but because this needs a further 'g', (10 dm/s/s), it comes to 890 Pa.

wendy.krieger wrote:Instead, we plan to set \(Z_0 = \mu_0 c = 1/\epsilon_0 c = 1\).  This is the essence of a symmetric electric + magnetic system.

There are a number of procedures concerning the setting or normalisation of proportionality constants that might affect symmetry.
1. Coherence
A system is said to be coherent if the magnitude of a proportionality constant in some physical equation is set to one. It may be possible, provided some previous choices for the constants, that setting some constants to one in the electromagnetic equations could produce symmetry in the electric and magnetic aspects, such that the setting of the constants to one may not be the exclusive or primary cause of the symmetry. That is, there could be a less stringent condition leading to symmetry than setting of the magnitudes of the constants to one. The setting of the magnitudes of fundamental constants to one is often associated with Planck units, though those are more about setting the smallest magnitudes to one.

2. Rationalisation
The setting of a proportionality constant of some force law to involve a term containing the mathematical constant pi to correct for the surface area of the region through which the field is passing is called rationalisation. It is associated with the name of Heaviside. A rationalised system need not be symmetrical in the electric and magnetic aspects.

3. Symmetry

https://en.wikipedia.org/wiki/Centimetre–gram–second_system_of_units wrote:Note that of all these variants, only in Gaussian and Heaviside–Lorentz systems \(\alpha _{L}\) equals \(c^{-1}\) rather than 1. As a result, vectors \(\vec {E}\) and \(\vec {B}\) of an electromagnetic wave propagating in vacuum have the same units and are equal in magnitude in these two variants of CGS.

Symmetry of the electric and magnetic aspects is associated with the name of Lorentz and may be linked to relativity.
It seems that a simpler version of the system-independent versions of the equations of Maxwell would happen when \(k_C\), the Coulombic force constant, and \(α_B\), the Biot–Savart force constant, are equal to the reciprocal of 4\(\pi\), though such versions may not be the most realistic for conditions such as for media that are not empty free space or vacuum, in which case properties such as the dielectric of the medium could be included, or the velocity of the current if it is not near the speed of light. Could it be that a ratio of the constants \(k_C\) and \(α_B\) should more realistically be the velocity of the current rather than one distance unit per time unit? Might the magnitude of \(k_C\) be set to \(c/4\pi\) and the magnitude of \(α_B\) to \(1/4\pi\), or rather \(k_C\) as \(1/4\pi\) and \(α_B\) as \(1/4\pi c\)?

The formulae used at the Wikipedia are unnecessarily complicated, in reality, there are simply two variables you add to SI to produce all other systems.  I use three, but this shows the use of the rationalised and unrationalised systems fairly.

\( \qquad \nabla\cdot\vec D = \rho/\beta \qquad \kappa\nabla\times\vec H = \tau \vec D + \vec j/\beta\)

\(\qquad \nabla\cdot\vec B = 0 \qquad \kappa\nabla\times\vec E = -\tau\vec B\)

Where \(Q = I\kappa t\), and  \(\kappa\vec E = c\vec B \).  The relation \( \kappa E = zcD = \kappa zH = cB\) applies throughout.

The ampere constant is \(k_A = \frac {2\mu}{\gamma}\) and the coulomb constant \(k_C=\frac 1 {\gamma\epsilon}\), the relation \(\gamma R^2 = \beta S\), where S is the surface of a sphere of radius R.

Setting \(Z_0=1\) simply amounts to setting \(\epsilon=\mu\), since \(Z_0 = \sqrt{\frac{\mu}{\epsilon}}\).  We set \(\kappa=1\), which means that unit electric charge is delivered by unit current in unit time.

Ampere's law looks like \(F = \frac{zcQ_1Q_2}{4\pi\epsilon_r R^2}\), and ampere's law looks like \(F/l = \frac{zI_1i_2}{2\pi c R}\).  

1.  Coherence.

There is no such thing.  Coherence is a relation between a set of units and a body of formulae.  There are books on engineering to which the gravitational systems are coherent, and SI isn't.  Of course, the theory that we're setting here does not affect coherence.

2.  Rationalised

This can be treated in dimensional analysis, for which SI treats as a dimensionless group.  

3.  Symmetry

The proposed system is double-symmetric, compared to the CGS, which is singly, and SI, which is singly in a different axis.

wendy.krieger wrote:Setting \(Z_0=1\) simply amounts to setting \(\epsilon=\mu\), since \(Z_0 = \sqrt{\frac{\mu}{\epsilon}}\).  We set \(\kappa=1\), which means that unit electric charge is delivered by unit current in unit time.

wendy.krieger on Tue Sep 17, 2019, https://dozenal.forumotion.com/t23-pendulum-system#90 wrote:I still have not figured out a worded definition for Z_0, since it is not an electrical resistance.

Bill Hall has used this symbol Z_0 for what was called impedance.

Bill Hall, May 20, 2012, #1, https://www.tapatalk.com/groups/dozensonline/new-system-idus-t629.html#p22010826 wrote:The base unit for electrical units is impedance. This is unlike SI which uses current as its base unit. The unit of impedance is called the Heaviside which equals the impedance of free space (Z0). Due to speed of light being set to one (by a factor of twelve), the permittivity of free space (ε0) and the permeability of free space (µ0) equal one (by a factor of twelve).

wendy.krieger wrote:3.  Symmetry

The proposed system is double-symmetric, compared to the CGS, which is singly, and SI, which is singly in a different axis.

If the SI is only singly symmetric yet there is not much of a problem with it its units being used in the main metrological system in the world, then being doubly symmetric may not be crucial for the design of electromagnetic units.

The practical electrical units date from the 1860s, and replaced various units like "1 mile of #6 copper wire", or a "German mile (8236 yd) of 6 mm Iron wire" or a 'foot of #26 silver wire'. They were chosen to be decade-multiples of the metric e.m.u., nearest Wm. Siemens unit of 1 metre of section 1 mm², of mercury, and the voltage nearest the Daniells cell.

Various national units were implemented to get as close to these stated units, so you have the BA volt (UK), and the Legal volt (Germany), and various other volts, until they decided to replace the different standards by an international one. Such is the international units.

The only reason that the ampere came to be the base unit, is that it was the first one that one could write it in a way suited for legislation. That is, it is the description of an experimental state, without the underlying theory attached. The actual definition defines a constant that I write \(\mu/\gamma = 10^{-7} H/m\), but you can't say that in an act.

The measure \(Z_0\) is indeed called 'vacuum impedence' for no better reason that it has the units of electrical impedence in SI units. However, in SI, it's 376.730 Ohms, whereas in the pre-SI system called the MKSA, it's 29.979 Ohms. It involves the same 4pi that appears in the value of \(\mu\) in SI units. You can't measure it in a circuit with it acting as an impedence or resistance.

The reason for implementing the various constants in COF, is that you allow calculations where they are equal to 1. The design of C.O.F, by setting \(\mu=\epsilon=1/c\), does not bring a huge advantage that setting g=1 does, but it becomes more apparent when you set c=1, then these two constants also disappear.

Of course, you don't have to set c=1, but simply use V = v/c, as an exponent. By placing the units at separate powers of V, the things like c=1 arises because length = V1, T=V0, and velocity is therefore V1, or (v/c). So something like a foot is 1v1, and a light foot is also 1v1 seconds.

Bill Hall gets to Z_0=1 by virtue of setting \(1/\mu = c = 12^8\), and hence \(\mu c=1\). I don't do that.