# -: On Circles and Angles :-

Maths:Circles Infinity Parallels

It is interesting to note that mathematicians one and all, hold that the circle is measured by its radius, when in the real world, nothing of the kind happens. In practice, there are three kind of circle.

Full Circles
A circle, cylinder, disk, or sphere one might fully handle, is measured by its diameter.
Arcs of Circles
A horizon circle one stands in, and measures the distance between remote points by a measure of arc, or inclines over small angles (such as a road), are measured by the radius to arc-length measure, is in radians.
Circles in the Sky
A circle in the sky might be visualised as being mapped onto a wall-chart, the coordinates are found by the remainder against a day-cycle (right ascession) or year (degree), and that coordinate mapped onto the chart. Stylistic day-degrees, of which 3.00 make the circle are used to plot the point. Signs represent months, and the measure of right ascession is given by siderial time.

Scientists and mathematicians measure angle of the circle from the right-most point, (ie x=1), and measure anti-clockwise, so the top is 90, the left is 180, and the bottom is 270.

The normal convention, is to measure in clock-face notation, even where degrees are given. Top or north is 0, Right or East is 90, Bottom or South is 180, and West or Left is 270. Such are to be found on compasses.

## Angle

A good deal of time is spent trying to replicate the degree-like unit, such as the metric degree, 400 (twe: 340) make the circle. The angle system that comes down to us is the nominal sumerian year circle, the year divided to 360 (twe: 300) days, and then divided as time is, to 60 ghurries, of 60 pali of 60 vipali. (to use the hindoo terms).

In practice, a circle is a single thing to be divided. The normal divisions of anything by roman practice is into uncia or twelfts. Even to this day, one can give a coordinate of circle by a clock-face expression (six-oclock).

A circle is a single thing, a cycle, and is divided accordingly. A circle, is thus divided into (twe: 100) degrees of (twe: 100) minutes of (twe: 100) seconds.

While one might divide the sphere into angles of excess, a sphere being two circles of excess, the rule taken here is that all space is 1, and all lesser angles are written as fractions according to the twelfty-number. A cube-corner holds 0:15 sphere, and is thus 15 solid degrees. The rule of excess is that the angles between the faces, form a triangle of three right-angles, ie 30+30+30 gives 90, this excedes the euclidean triangle by 30 degrees.

Mathematicians use radians, and their prismatic powers. But the solid radian exceeds the sphere surface in nineteen dimensions. The tegmic power of the radian, is somewhat less than the solid angle of a regular simplex, the simplex lying somewhat less than sqrt(n/4) of this angle, but definitely greater than 1 (and 1:72 for dimensions greater than N=24).

In any case, Euclid did not rely on trignometry and radians in his classic 'Elements', and the Polygloss does not brook the elements of maffamatiks either.

## Of Tau and Eta

Unlike constants like Napier's exponent-constant e, Pi is not something that one is invariably lead to. The brief here, so to speak, is to set a constant to hive off the irrationalities, rather than something that all people are invariably lead to.

c/1 = 6:34 Tau, is something that mathematicians are invariably seemed to be lead to, by way of their use of radians, and arc-measures over small angles. The various taylor series etc for measures like cosines and sines, evaluate to a radian, when the number fed in is "1". Its downfall is that real people simply do not use circles by radius.

c/2 = 3:17 Pi. The ancient measure of the ratio between the diameter and circumference of a circle, stand in this ratio.

c/3 = 2:11.40 The diameter of a circle, as arc. Heath discusses a Sumerian measure of the circle, where the circle is taken to have a diameter of 60 ells, and the circumference is divided into 180 ells, each of 24 digits.

c/4 = 1:68.60 Eta=h, comes from the crind product, where the volume of a sphere of n dimensions, is given by the formula CPn = h^(n % 2) / n!!. The percentage sign, as in REXX, means to discard the remainder. The double exclains (!!), mean to multiply downwards in steps of 2, as long as the number remains positive.

In terms of the tegum product, CTn = (n-1)!! h^(n%2). Naturally CTn * PCn = PTn = n!. All three products are coherent, which means in a dimensional analys, one can freely associate L^n to mean Cn.L^n etc.

c/6 = 1:05.80 The ratio between the sextant and radian. The ratio of a hexagon to an inscribed circle (both of circumference and area), stand in this ratio.

c/8 = 0:94.30 The ratio between a circle inscribed in a square (circumference and area), and a cylinder in a cube. It is rather interesting that the density of wheat to water stand pretty much in this ratio: a cube holding a pound of wheat would hold also a cylinder of a pound of water.

## The values of Pi

The following table gives various approximates to the constant hait pi. Some irrational values might arise from the use of square roots, etc. For example, the equating of a quadrant arc of 10 to a quadrant chord of 9, produces a square root of 2.

Although the table gives an indicated value of a number typically referred to as (twe: 3:17), it does not mean that any real attempt is being tried to find a more exact value. The example above, known to the Egyptians, is mearly something one might find with a sewing tape, for example.

The values of Pi
Pi in
decimals
Description
3.000 000  000 000   3 This is the value of pi, assuming the hexagon represents the circle. In Sumerian measure, a circle of diameter 60 ells, has a circumference of 180 ells (Heath).
3.072 000 000 00 A five inch sphere has the same volume as a four inch cube.
3.125 000 000 00 This value of pi appears as one of the forms in Berriman. It was also known as an approximation to the Egyptians.
3.136 000 000 00 A five-inch sphere has a volume of 65 1/3 cu in (= 2744 / 42).
3.141 361 256 54 600/191. This is the value of pi, when one radian equals 57.3 degrees. As an expression of 1/pi = &38&24/120, is accidently equal to the solid angle of the twelftychoron x5o3o3o this being 38s 24f or &38&24/120 of the 4 sphere.
3.141 567 230 22 823543/262144: This appears to be the best approximation to pi, using numbers whose prime factors are 2, 3, 5 and 7. The value is 7**7/8**6. Written in octal, the value of pi is 3.1103 7552 52, and this number is 3.1103 67.
3.141 592 653 59 Pi to 11 decimals. (twe: 3:16E8.E212.7796)
3.141 592 920 35 355/113, a very good approximation to pi. If ye want to run an experiment, the result depends on pi, ye can fake the results using this approximation.
3.141 600 000 00 3.1416: This value of pi is known in India. Interistingly, the recripical of this written in twelfty is 0:38.23.78.23.78.23.78... Note: 3.1416 is 187×168.
3.141 640 786 50 1.2 phi², this is known from a number of different sources.
3.141 666 666 66 377/120. This version of Pi is known in the sexagesimal form 3&8&30/60. In the form 3&17/120, it reflects a version of 1.2 × phi². In this form, it can be written as 1.2×377/144, where 377 and 144 are fibanacci mumbers.
3.141 818 181 81 864/275. This equates a cubic foot with 2200 cylinder inches.
3.142 337 619 40 24/35 sqrt(21). This comes from 1 cyl ft = 2800 hoppus in = 36 Litre.
3.142 561 983 47 1521/484: This is 3&3&3/22. This ratio is a square (of 39/22), and thus a 44-inch circle has the same area as a 39-inch square.
3.142 696 805 27 20/9 sqrt(2): This is the square root of 800/81. This is the value of pi, implied by equating the quadrant to 10 parts, where the inscribed square has an edge of 9. This is known to the ancient Egyptians. This value also makes an f-unit equal to a cubic inch on the surface of an eighteen-inch four-sphere, there being 14400 f-units making full space.
3.142 857 142 85 22/7: 3&1/7. This is the most common approximation to pi, used in calculations. Metrologically, it is not at all common. The one place where it arises is the rating of the Wine Gallon as 294 cylinder inches, and 231 cubical inches.
3.146 108 329 53 6912/2197: This equates a 13 inch cylinder with a 12 inch cube.
3.150 000 000 00 63/20: This value of pi appears in Berriman, etc, as a form of the geographic pi.
The Swedish army mill uses this pi.
3.152 161 253 53 The geographic pi , defined as length of equator ÷ polar axis. Both of these measures have been extensively used in metrology.
3.160 493 827 16 256/81, a value known to the Egyptians. A nine-inch circle has the same area as an eight-inch square.
3.162 277 660 17 sqrt(10): At one time, the square root of 10 was put forward as a version of pi.
3.200 000 000 00 32/10: Four parts of Four The value pi used in the NATO army mil, since it gives the binary division of the circle demanded by the compass rose.
--------------------- End of table.

army mil
The circle is divided into an exact number of units, which are meant to correspond to milliradians. This generates a pi-approximate, eg 3.15 or 3.2
Cylinder measure
The cylinder of unit height, and unit volume.
Geograpic pi
The ratio of the polar axis to the circumference of the Earth. Both measures have been used extensively in metrology.