The Geometric Origins Of Our Contemporary World’s
MEASURES OF LAND
An Analysis of Lengths and Areas
This next chapter should be of particular interest to both surveyors and architects. None of them have ever been taught the true roots of the measuring units handed down to them from the past, and with which they work every day. Neither they, nor the rest of humanity, have any idea that the inch, foot, yard, and mile; the acre and township; even the grain and both ounces and pounds (troy and Avoirdupois) all are inextricably connected not only to one another, but to America’s system of gold and silver coinage. The clear mathematical evidence in support of the above allegations has been hidden from the public for thousands of years!
When Edmund Gunter introduced his newly invented land measuring chain at the beginning of the 1600’s, in all probability he was releasing some very old and secretive knowledge along with it. Whether or not Gunter himself actually knew what I am about to disclose to you, the first readers of this manuscript, no one can say with any certainty. Personally, I think he knew everything I am about to disclose. What I do know, and can prove, is that the system of land measures designed by man, for man, corresponds to the eternal geometric principles governing surface area to a better than 99.999%. . . approach to perfection! To the best of my knowledge, these simple mathematical and special relationships (that I am revealing here and now) never have been taught in any “traditional” classroom, public or private.
The Geometric Origins of the
Surveyor’s Chain, Acre, and Mile
By the time The United States Of America was established back in the late 18th century, the science of land surveying had already evolved to a relatively sophisticated and disciplined art-form capable of producing very accurate results for their times. Surveyors were even using their own unique measuring rule, called the “surveyor’s chain”, complete with its own unique set of measures. All of this science had been established centuries earlier in England, and was brought to America by the British colonials.
To get at the true roots of our land measurement system it is important to understand the chronology of key events leading up to the creation of “townships” in America. The first was the codification of the “mile” (equal to 5,280 feet) into British law in 1593 under the first Queen Elizabeth. The second event involved a man named Edmund Gunter. History records that in 1607 Gunter’s work resulted in Parliament standardizing the “rod” as a survey measure of 16.5 feet. A few years later (ca. 1620) he introduced a land survey devise that quickly became known as “Gunter’s chain”.
In Gunter’s vision of land measurement, the unit known to the general public as a mile of 5280 feet was instead equal to 80.0 chains. Since there was no metric system in general use at the time, the natural division of the mile was into halves, quarters, eighths, and so on. Gunter’s mile consisting of 80.0 chains can be seen to first divide into 8.0 parts; and then, each part divided into 10.0 chains.
Edmund Gunter was a mathematician and astronomer and was certainly aware of a growing science that the general public had no knowledge. Scientific technicians of his time were aware of the “base-ten” system, and it is obvious Gunter was no exception. When he chose to divide the mile into “tenths”, after its first division into 8 parts, Gunter had purposefully broken with what until then had been tradition. This division into ten was out of context for that time in history. Nonetheless, soon these one-tenth parts of an eighth of a mile increment became customary to land ownership units, just as the “foot” measure was to the mile.
The “chain” that Gunter designed as a physical measuring tool was literally in the form of a chain. But this chain is of a very unusual and unique design. It bears little resemblance to what was recognized then and now as a common chain of uniform links. Overtly, the chain is 66 feet long. It is comprised of 100 “links” with every 10.0 links marked by a brass ring. Property lines and distances in general were recorded in numbers of “chains and links”. It was a simple scale based on 1.00 (chain) and 100 (links).
Since the chain is 66 feet long and there are 100 links, each link is then .66 of a foot, which is also 7.92 inches. Once again these are the overt measures. To Gunter, the chain’s creator, and to his behind the scenes unnamed colleagues and collaborators, the true occulted symbolic measures lay hidden beneath the surface. For the fundamental unit chosen to be the designed micro-measure (on a scale where the chain of 66 feet is the macro-measure) is .99 inch! In this way, each link of 7.92 inches is really the sum of 8.0 units each .99 inch long. This makes the chain a unit of measure totaling exactly 800 “survey”, or “chain” inches, that are .99 the length of the standard inch.
I know this sounds crazy, but it even gets crazier. Let’s look at the “mile” measurement again. To Gunter, and for the rest of Europe it measured 80.0 chains. Again, it was divided “traditionally” into 8.0 parts, and then each part further divided into 10.0 (chains). But only Gunter and his illuminous colleagues knew that the true length of his chain was 800 units of .99 inch. Now for them, the mile measured an even 64,000 (“survey-chain” or illuminati) inches. The rest of Europe had no idea (until today) that their mile, measuring 63,360 inches, had just been “covertly” altered; nor why?
Now, when Gunter overtly divided his chain of 100 links into increments of 10.0 links (marked by brass rings) he was also concealing its more important covert division into 8.0 parts. For to the illuminated ones, just as each 7.92 inch link became 8.0 units .99 inch long, the chain itself naturally divides into 8.0 units, making each of the eight segments 8.25 feet; or, 99 inches in length! So from this vantage point, the chain is sub-divided into eight 99 inch units, with each one containing one-hundred .99 inch units.
It doesn’t stop here. No, as we continue, it just gets more and more unbelievable. But it will also all start making sense. For example, a length of 5280 feet becomes 640 units, each exactly 99” long. This means that a “square-mile” (which is one “township” section) measures 640 of these 99 inch “units” per side. Moreover, 640 squares measuring 99” per side, laid out in a single line edge-to-edge across the earth’s surface, not only equals one mile in length, but also covers an area exactly equal to 1.0 acre. Since there is 640 acres in 1.0 square mile you can repeat this process 640 times perfectly covering every square inch contained within that square mile.
Let’s take a closer look at this very unique 99 inch square. First, sub-divide it in the same manner that a standard “township section”, one square mile, is typically sub-divided. This is into “quarter-sections”. The 99 inch edge-measure, sub-divided, creates four “quarter-sections” with 49.5 inch long edges. But to Gunter and his collaborators, and to the contemporary custodial illuminati in charge today, the occulted expression of this “quarter-section” square’s edge-measure is twice 24.75 inches; and/or equally 4.125 feet. Readers of the previous chapters will recognize these quantities as ( powers of) respectively the pure gold content (in grains) in the $10 “eagle”, and the gross weight of America’s foundational monetary unit, the $1.0 silver coin.
Forget it! It’s NOT a coincidence! It gets better. Remember, America’s coinage system, and its system of land division, were enacted into law at about the same time in our history; generally, the early 1790’s. So, when the occulting veil is pulled still further back, and we see deeper into the mathematical and geometric composition of this 99 inch square we also find that each quarter-section contains an area of either:
66 times 37.125 square inches
99 times 24.75 square inches
Again, both quantities are (powers of) the foundational monetary measures of weight right out of the 1792 Coinage Act, where it specified there be precisely 371.25 grains of pure silver in the dollar coin; and, 247.5 grains of pure gold in the ten dollar “eagle”. And those behind the scenes, who made “damn certain” that these were the measures America use for her new monetary system, also were fully aware that the two coins they had designed contained exactly 99 troy ounces of pure silver in every two times 64 silver dollar coins; and, that there is 66 troy ounces of pure gold in every two times 64 gold eagles. Of course, we just saw the 99, 66, and 64, to be the occulted foundational quantities of Gunter’s system of land division.
Look at these monetary measures of weight, and the inter-play between them, when instead of weight these “magical quantities” are units of length or area:
41.25 square feet = 371.25 square inches times 16
41.25 square feet = 247.5 square inches times 24
Now watch how these quantities as area morph into actual physical replicas of the monetary weight measures:
66 sq.’ / 16 = 4.125 sq.’ = 594 sq.”
594 sq.” = 24.75 sq.” (X) 24; or, 37.125 sq.” (X) 16
594 (27milligram “cubets”) = 247.5041632… grains
Once again we look to the “cuboids” in the previous photograph that model the weights of the pure silver (371.25 grains) and pure gold (247.5 grains) that was specified for each coin in America’s 1792 Coinage Act. The $10 gold Eagle is the cuboid behind the larger cuboid in front. Its 594 cubets, comprising its allocation of pure gold, is the aggregation of the white 27mg cubets. The remaining 54 red cubets is the amount of alloy in this coin’s 270 grain gross weight. Note that:
594 white cubets + 54 red cubets = 648 cubets;
648 X .027 gram = 17.496 grams;
17.496 grams = 270.0045417… grains
( i.e., the gross weight of the $10 gold eagle)
There are 891 white 27mg cubets comprising the cuboid in the foreground of the above photo. This is the 371.25 grains of pure silver contained in every $1.0 coin. The remaining 99 red cubets is the amount of copper alloy specified for this coin. This cuboids’ 990 “cubets” in all, add up to this coin’s gross weight of 412.5 grains. Now, morphing back into measures of area, we see that:
(99 sq.’ /16) = (24.75sq.’ /4) = 891sq.” = (4.125sq.” (X) 216) = (37.125 sq.” (X) 24)
(110sq.’ / 16) = (41.25sq.’ / 6) = 990sq.” = (24.75sq.” (X) 40) = (41.25sq.” (X) 24)
990 (27 milligram “cubets”) = 412.5069387…grains
Clearly, the coinage weight measures, specified in America’s 1792 Coinage Act, are quantities arising from the same occulted math and geometry as that which created Europe’s much earlier terrestrial measures of length and area. We can see this in the number of increments of 41.25 lineal feet in one mile: two times 64. Half this length is 20.625 lineal feet. If this length is used as one edge of a rectangular tile and 20 lineal feet the other, then the area of this tile is 412.5 square feet. Using [(2)10 X 66] tiles of this size an entire square mile, 1.0 township section, can be perfectly covered. Tiles 20.625 inches by 20 inches measure 412.5 square inches in area; and precisely [3(2)15 X 99] of these will likewise perfectly cover 1.0 square mile.
Ten silver dollars contain 3712.5 grains of pure silver. And, 3712.5 square feet equals 412.5 square yards. Tiles of this size won’t perfectly cover a township “section”, but precisely [(2)12 X 66] of them fit perfectly on one entire township of 36 square miles (quite fitting for a yard equaling 36 inches). The edges of this tile, like the other tiles mirroring the silver dollars gross weight of 412.5 grains, are in a 20.625 by 20 proportion. This can be converted to a simple 66/64, or a 33/32 ratio between the two edges. This shape is very close to being a perfect square. And if we scale this form down to “the fundamental micro-unit of land measures”, the square inch, so that the shorter edge equals one inch (1.0”), then the other edge is exactly 1.03125”, which is 1/32nd of an inch longer. One side is 32 thirty-seconds of an inch, and the other 33. This “difference” is key to understanding the reason “land” and “monetary” measures share powers of the same quantitative units.
Also at this point, it may be important that we take note of the following excerpts from “The Great Pyramid of Giza” by Eckhart R. Schmit regarding our ancestral “Royal Cubit”:
The units of measure used in the construction of the Great Pyramid are the Royal Cubit and the Pyramid Inch. There is some evidence that the measurement unit known as the Royal Cubit was already in use as much as one hundred years prior to the building of the Great Pyramid, and perhaps even somewhat earlier. It was employed as a unit of measure for the construction of buildings, for the measure of land, grain quantities etc. The Royal Cubit was of central importance as a standardized unit of measure at the time of unification of upper and lower Egypt(2). One can infer that the dimension of this unit of measure could be geodetic in nature, that is to say, having some specific relationship with the physical attributes of the Earth’s shape. Generally the Royal cubit is understood to have been 524 millimeters +/- 2mm (20.63 inches) in length. Professor Flinders Petrie, who is regarded as the founder of Egyptology, and had studied the Temples and Buildings of Ancient Egypt with utmost exactitude in the 1880’s assigned a value for the Royal Cubit of 20.632 +/- .004 Imperial British Inches based for the most part on the dimensions found within the King’s Chamber in the Great Pyramid(3). (and) In review of the dimensions of the King’s Chamber, Sir Isaac Newton ascribed a value of 20.63 Imperial British Inches for the Royal Cubit(4).
If the “Royal Cubit” is in fact “524 millimeters +/− 2mm”, then the 20.625 inch measurement also being 523.875mm, certainly must now be a strong candidate for assumption to that Royal honor.
Measures of Land and Coinage Measures
Are Powers of the Same Units
Generally, “land measures” are units of length and area, whereas “coinage measures” describe weight and volume. But in the light of occulted geometry, we find both land and coinage measures derive from the same geometric properties of form.
Euclid showed us that the fundamental units of both length and area are inherent to a simple “square”. An area of 1.02 (one square unit) in the form of a square has an edgelength equal to 1.01 (one lineal unit). All of “geometry” ultimately is scaled to this basic form, and from it, so too do our measures of land and money.
Geometry shows us that there is a maximum amount of volume that 1.0 unit of surface is capable of completely enclosing within the confines of its surface. This amount is 0.094031597… of 1.0 volume unit; and this volume will be in the form of a perfect sphere (these quantities are depicted in the image at the beginning of this chapter). The mathematical relationship between one square unit of surface and this ideal volume quantity is expressed by this ratio: 1.0 ∕ 0.094031… i.e., 1.0 surface “unit” to 0.094031… “unit” of volume. But there is another equal mathematical description of this same relationship since:
1.0 ∕ 0.094031… = 10.63472310… ∕ 1.0
These portions, these quantities are etched into the foundational bedrock of geometry, and as we will see, lead directly to the measures of land and coinage.
In its pure form, geometry refers only to the unit, not to “The Unit”. For to do so, requires assigning a “name” to that original square “unit” of surface area. It could be (1.02) square inch, foot, yard, meter, acre, hectare, . . . or any other named unit, and an entirely unique geometry would unfold from each one. Some time in the past, someone made that name choice, at least for measures here on earth. You be the judge after reading the following.
The Fundamental Unit
Of Terrestrial Measurement
By assigning the name “one foot” to geometry’s originating square of one surface unit, a scale was imparted to these forms. But why choose one “foot”? Well, remember the ancient maxim “Man is the Measure of All Things”? Maybe there really is something to it?
Now, 1.0 ∕ 0.094031597…means 1.0 “square foot”, and a 0.094031597… portion of 1.0 “cubic foot”. This means that 10.63472310… ∕ 1.0, its equivalent mathematical description, is also scaled in “feet”. And it is here, at this strange quantity of area, where we have finally arrived. We are now in possession of the mathematical key to a doorway, which when opened, will reveal some of the world’s best kept secrets. Let’s start by looking at the following equation:
10.63472310… sq. ft. = 1531.4001… sq. in.
and then this equation:
41.25 in. X 37.125 in. = 1531.4062… sq. in.
The results of these equations are identical to a 99.9996001…% approach to perfection. And in them, we can clearly see the quantities 41.25 and 37.125 “inches” to be the essential (sub-unit) building blocks, at what is literally the very birth of geometry itself. And remember, all of this was gleaned from the geometric properties of a simple square equal to one square unit. It’s also worth noting here that these lengths are in a 9/10 ratio, just like the pure silver alloy in their counterpart coinage weight measure. Geometry is telling us that this square area, equal to 10.63472310… square feet, is something unique, something very special. And indeed it is! For it is quite literally the seed of the acre; and from it are derived all of the other standard measures of land we (in America and much of the world) have come to know today. Here is how it works.
Traditionally, from America’s birth in 1789 through today (2014), the “township” has been the “macro-unit” of land area sub-division. A “township” is simply a square area measuring 6 miles per edge and encompasses 36 square miles. Each square mile is called a township “section” and is the first sub-division of the macro-unit.
This 1.0 square mile traditionally subdivides first into “¼ sections”, then “¼, ¼, sections”, and finally into “¼, ¼, ¼ sections” subdividing the edge of the square mile into 8 parts. Now, like the classic “checkerboard”, the area of the square mile has been subdivided into 64 sub-unit squares. The area of each one of these squares is 10 “acres”. No further subdividing of the mile’s edge is possible because the resulting smaller parcels won’t contain whole increment acre units.
So far it works like this. The “township” square sub-divides once into 36 sub-squares, each a square mile. The square “mile” first subdivides into 4–160 acre parcels. Next, by quartering each quarter, there then are 16–40 acre parcels. One more quartering brings us to 64–10 acre square parcels. That’s it for the square mile, leaving only the 10 acre parcels for further sub-division.
Here arises a geometric “problem” in that a square cannot be easily sub-divided into 10 acres. That is, unless you have the correct measuring rod. Gunter shows us the solution to this problem in the early 1600’s with his special “chain” measuring 4 “rods”. . . i.e., 66 feet. Since a 10 acre square measures 660 feet per edge, or 10 chains, the 10 acre square naturally sub-divides into 100 square chains. 1.0 acre then is simply 10 square chains. Each square chain is 66 feet per edge. And it is this “square chain”, 100 in all, that constitutes the 10 acre square’s next sub-divisional unit.
The most efficient geometrical arrangement of 10 square chains into 1.0 acre is a rectangle measuring 2 chains by 5 chains. Each 1.0 of the 10 square chains is 1/10th acre.
Thus far, beginning with the “township” unit, down through the sub-divisioning all the way to the “square chain”, there really is no mystery. The “traditional information” just summarized in these last 5 paragraphs is all out in the open for anyone interested in the subject of surveying. But, what now follows here, you will find nowhere else but here.
If one sub-divides the “square chain” exactly in the manner as a square mile, i.e. into ¼ sections; ¼, ¼ sections; and ¼, ¼, ¼ sections, we’ll have divided it into 64 smaller squares. This means that the “chain” itself has just been covertly divided into 8 parts. I say “covertly” because, as was stated earlier in this chapter, Gunter’s “overt” divisioning of the chain is into 100 links marked off with brass rings every 10 links.
The division into 64 sub-unit squares makes each square’s edges measure 99” X 99” (twice 4.125’) enclosing 9801 square inches. Now, sub-divide this square into 64 more sub-unit squares. The edge-length of each one of these squares is 12.375” (which is 33/32 of a “foot”); and each encloses an area of 153.140625 square inches (which is 4.125” X 37.125”) . . . or, 1.063476563… square feet. The following simple arithmetic shows just how deeply the three powers of America’s foundational coinage weight measures combine together to form this basic unit of length for land measurements:
(.2475) + (.37125) + (.4125) = 1.03125
If the measures highlighted in red in the above equation are portions of 1.0 “foot”, then their sum is 1.03125 “foot” (or 12.375 “inches”). And again:
(24.75” / 2) = 12.375” = (37.125” / 3)
Ten of these 12.375” squares arranged in one rectangle measuring 2 squares by 5 squares is itself a micro-model of the 10 combined to form one acre. This rectangle measures 10.63476563… square feet.
Therefore, it must be concluded that this 10.63476563… square foot rectangle is to the acre as the 1.063476563… foot square is to the square chain. And, it is this 1.063476563… foot square that is, in practice, The Fundamental Unit at the very basis of all land measures predicated on the mile, yard, foot, and inch.
Readers don’t need to be “rocket scientists” to conclude for themselves that all of the quantities above already had been exposed a few pages back, having been distilled directly from the properties of geometry’s “Fundamental Unit of Surface Area” . . . a simple square with its edge 1.0 unit in length. From this square area, by following a path well marked by simple geometry, we were led to conclude that a surface area equal to 10.63472310… square “units” is an equivalent mathematical expression describing geometry’s Fundamental Unit of Surface Area.
Now remember, the 10.63476563… foot rectangle that we just discovered three paragraphs back began with an investigation of the largest square “unit” called a “township” comprised of 36 square miles. Its sub-unit, the “square mile”, was then sub-divided into 64 smaller squares of 10 acres each. Each of these 10 acre parcels are further sub-divided into 100 “square chains”. The “square chain” divides into 64 squares 99” per edge, which are in turn finally divided into the 64 squares. As stated just above, these last 64 Fundamental Units of land measurement “square chains” each contain 1.063476563… square feet and measure 12.375 inches per edge.
Now, let’s see how they compare; “they” being “man’s measures”, compared to “geometry’s measures”, which existed before time itself.
When geometry’s special area of 10.63472310… square feet is sub-divided into 10 squares and arranged like the 10 “square chains” comprising an acre, the exact edgelength of each one of these geometric squares is 12.37497526…inches. This compares to the 12.375 inch square arrived at working back from the “township” as:
12.37497526… ∕ 12.375 = 0.999998001…
These two measures are literally 99.9998% identical! But how can this be? Surely “they” should have taught this to all of us in school; why isn’t this in our history or science textbooks?
Even today’s surveys don’t approach this degree of accuracy. For example, the statute mile of 5280 feet contains 5120 units, each 12.375 inches in length. This is a total of 63,360 inches. And 12.37497526… inches 5120 times equal 63,359.87333… inches (which I propose be called “The Geometric Mile”). The difference between these two measures is a mere 0.126670000… inch, exact! It is .00167” over 1/8th inch.
It should be noted here that one can lay out a perfect rectangular acre using U.S. silver dollar coins. One side of this acre will measure 5280 dollar coins and the other 528 (Note: the coin’s diameter is 1.5”; 528 X 1.5” = 792” or 66’, or 1.0 chain.). It is also important to note here, that by using the long edge of the old United States issued “large” paper “bills”, printed from 1862 through 1928, precisely 51200 of these laid out (small) edge to edge will perfectly delineate the six mile edgelength of a “township” (see chapter titled “The Geometry of Paper Money”). And in keeping with the land measures schematics thus far exposed above, these quantities 528 and 512 are related as
1.03125 / 1.0; which we also know to be 33/32.
A Tale Of Two Squares
Lets return to the early 1600’s and revisit Gunter’s chain and the “mile” measure that was already well established and written into English law. Since the mile is equal to 80 chains, a “square mile” contains 6400 square chains. An since the chain of 66 feet is actually 8.0 units each measuring 99”, then a single square chain contains 64 of these 99”squares; again, just like the classic chessboard. Using these measures, a square mile becomes an assemblage of 99” squares; 409,600 of them in all; 640 per edge.
Now, remember earlier in this chapter it was demonstrated that the relationship between 1.0 unit of surface and the maximum amount of volume it is capable of containing is expressed by the ratio 1.0 / .09403…. and equally expressed by its reciprocal 10.6347… / 1.0. These are two different mathematical expressions meaning exactly the same thing. They are equal (=) to one another.
Apply this same logic to the edge–length of the square mile with its 640 measures of 99”. This is to say, look at the same mile measure as consisting of 99 units, each 640” in length. In this way, the square mile is subdivided into a grid 99 units by 99 units mirroring its interior sub-division into 99” by 99” squares. The white squares in the following grid number 99 per edge. Each of these squares measure 640” by 640”. And the sum of their individual areas equals the 640 acres comprising 1.0 square mile.
With the addition of the indicated red squares to the grid, we have a slightly larger square; it now has an even 100 squares per edge, rather than 99. It also measures an even 64,000 inches compared to the mile’s 63,360 inches.
If each of the 100 links in Günter’s chain was 8.0 inches rather than its overt measure of 7.92 inches, then the mile would measure 64,000 inches . . . but it doesn’t. It’s obvious that one line of (640”) squares was removed from each of two adjacent edges to create the remaining perfect 640 acre square area. So how many acres were subtracted from the larger square to create the smaller? Or equally, how many acres were added to the smaller square to achieve the larger?
Since there are 100 red squares on one edge, and 99 left on the other, there is a difference altogether amounting to 199 squares with edges measuring 640”. Each of these individual square’s areas measures 409,600 square inches (and is itself a reflection of the greater square mile unit and its division into 409,600 squares with 99” edges). Next, we multiply this area by 199 units and arrive at a total of 81,510,400 square inches. If we divide this by 144 we have the number of square feet: 566,044.444…; and if we divide this by 43,560 (the number of sq. ft. in 1.0 acre) we will have found the number of acres: 12.99459239…
So by subtracting 12.99459239… “acres” from the large square we arrive at a little smaller square containing precisely 640 acres; 1.0 square mile. To be consistent with all that has been revealed thus far, one would expect this 12.99459239… quantity to be a very special unit. Let’s see how this quantity relates to measures of area with respect to The Geometry of Form. This investigation will bring us full circle, back to the “beginning”, and the 1.0 Surface Unit in the form of a sphere’s surface. . . and from there, the geometric origin of the “township”.
Once again, look at the image on the first page of this chapter. The spherical volume rests on top of its surface area, which is laid out flat in the form of a square. This construct is quanta-sized by the sphere’s surface area being equal to 1.0 areal or square unit. In the previously uncharted dynamic transformational geometric system, that I have been independently exploring now for nearly forty years, and which I have long referred to as The Geometry of Form, changes in surface and volume are strictly accountable. The following example is typical of this accounting principle, and at the same time will show the significance of this 12.99459239… quantity with respect to measures of surface area.
What happens when geometry divides the volume of this sphere into two new spheres? Remember, its volume is .09403159… so the volume of each new sphere (made from the original’s volume) is .047015799… In this sphere’s transformation by division, one-half of the original sphere’s surface area (0.5000…) is likewise imparted to each new sphere. But that’s not enough. A sphere with a .047015799… volume has a .629960526…surface area. Thus each of the two new spheres is missing a .129960526 portion of 1.0 unit of surface.
This specific quantity of missing surface area is a geometric constant. We know this because regardless of whatever shape or form that the 1.0 surface unit assumes, if it divides its volume into two new forms identical to the original form each will be in need of an additional .129960526… surface unit.
Look at the data thus far. We have two components of surface area: one has been distilled from terrestrial land measures, and the other from a spherical geometric transformation. Though differing by a magnitude of 100, the essence or root unit of these two quantities is 99.99% the same:
12.99459239… / .129960526… = .009998876…
At the heart of this quantity’s essence is a prime numeric expression: 1299. This is a magnitude, used in conjunction with various powers, to form proportions defining geometric objects and their surface, volume, and lineal transformations through distortion, division, and fusion.
Here are some examples of how The Geometry of Form uses this very special 1299 quantity. In order for the sphere with a surface area equal to 1.02 unit to be able to “distort” into a cube (with a surface area equal to 1.02 unit) it must first eject a quantity of what is now excess volume amounting to two times .012995108… volume unit. Another example is if two times .12995108… surface unit is added to an octahedron with a 1.02 unit surface area, its volume can then “distort” into a tetrahedronal form. And another simple (2-dimensional) example begins with a circle with a 1.0 lineal unit radius and an inscribed equilateral triangle; 1.2990381… units is the amount of surface area enclosed by the triangle. Or if instead, the 1.0 lineal unit is the edge of a tetrahedron and one imagines it as sitting flat on a table top, then one of the tetrahedron’s four sides is this pyramid’s base and is unexposed; the amount of remaining exposed surface area is 1.2990381… units.
The correspondence between what I’ve termed the Illuminati’s system of weight measures based on the 27 milligram cube and their system of land measures was exposed earlier is this chapter and previous chapters. In the same way that the two cuboids constructed from these 27mg cubets (depicted in the earlier photo) replicated the weights and exact alloy proportioning for America’s flagship gold and silver coins, so too is the much older Avoirdupois ounce of 437.5 grains represented by 1050 of these 27mg cubets. And we find that 1050 surface units, each measuring 1.2990381… units, equals the surface area of (what in The Geometry of Form is called) the equilibrious icosahedron, which is so scaled that its surface area is equal to its volume. Then again, we shouldn’t be surprised since 1.2990381… units is also:
(27)1/2 ∕ 4
Now, I’ll remind the reader that the silver dollar coin’s gross weight of 412.5 grains, which is modeled as the larger cuboid in the foreground of the previous photograph, consists of 990 (27mg) cuboids. And we just saw how important the 640 inch measure is with respect to deriving the square mile from a 100 X 100 grid. So too may we derive the mile measure by performing any of the operations below:
640” ∕ [(1.0 / .990) −1.0] = 1 mile
640” ∕ [(1.0 − .990) ∕ .990] = 1 mile
640 X 99” = 1 mile
64.0” X 990 = 1 mile
(two times) 64’ X 41.25 = 1 mile
Earlier in this manuscript, it was shown how the medieval 42 gallon “tierce” later became the 20th century’s adopted “petro-barrel”; and how geometry determined why the modern gallon is a measure of 231 cubic inches, and why there is 42 gallons in the petro-barrel. It was also shown how 42 gallons disassembles into its constituent 9,702 cubic inch sub-units, and how they can be neatly arrange 2-dimensionally into a near perfect square measuring 99 cubes by 98 . . . and how, in practice, additional capacity within the barrel is required for fluid expansion due to changes in temperature and pressure. The “completed” square, measuring 99 X 99 cubic inches, accomplishes that function. Also, it is recognized as the same 99 X 99 inch square that was shown to be a fundamental sub-unit of the acre, with its sub-divisioning into quarter-sections measuring 4.125 feet per edge.
With this in mind, let’s now transfer some of the previous quantitative units from geometry into some of humanity’s actual measures of volume and area, and then see how they relate among one another.
For example, one petro-barrel of 42 gallons equals (4.125/3) barrel dry; 1.333 barrel fluid; and is 9702 cubic inches. But here is the covert “quanta-sization” of some of our common units of volume based on the cubic inch.
A .37125 portion of one petro-barrel equals:
2 X .2475 barrel fluid; as well as
120 X .129937… gallons
.4125 barrel oil = 1.333… X .129937… gallons
.4125 barrel fluid = 12.9937… gallons
When we look at the quantities immediately above we see the familiar monetary weight measures (in red), which are also, as we have just seen in this chapter, measures of land (i.e., length and area). Obviously, they are standards in our volume accounting system as well. And here too, they appear working in conjunction with powers of the geometric constant governing surface transitions: .129937…
Now, let’s look at the volume of the originating 1.0 unit surface area in the form of a sphere and some more of man’s measures deriving from it.
This spherical, geometrically special volume measure is .094031597… cubic “unit”. It was discovered by “naming” its surface area one square foot; and from it, we’ve been able to derive all of our measures of land (and the entire system of dry measures culminating in the Winchester bushel). Now we find that this volume measure, as a portion of one cubic foot, also has left its mark in some of our other measures of volume. For example, a volume of .094031597… cubic foot, which is also [1.0 / (36pi)1/2] cubic foot, is at the same time 1.5708990… Scottish pints, which quantity is simply (pi / 2); again at the same time, this quantity is 4.8359497… pint dry measures. In geometry, this quantity 4.8359758… is the surface area of a sphere having a 1.0 cubic unit volume; and it is also related to the pi portion: (36pi)1/3. And this quantity 1.06347572…, so special to measures of land, as volume of a barrel dry can be re-written (0.1 / .094031295…) and is equal to 33/32 barrel fluid. It is also a (371.25 / 480) portion of one barrel oil, which portion is at the same time describing the number of grains of pure silver in the American dollar coin (as a portion of the 480 grain troy ounce). And 480 chains is the length of one township. This fraction (371.25 / 480) equals .7734375, and as a portion of one barrel oil measures 259.86159… pints. But 259.86159… is (2 X 129.9307…), once again a power of the (previously described) geometric constant governing surface transformations of a volume Unit.
There is something else of great importance that should be noted here. It will be picked up again, and expanded upon in a later chapter but should at least be mentioned now in the context of the last few paragraphs. Remember, “the Unit” acquires a physically defined actuality only after being named. If in this example we name the Unit “1.0 meter” then .094031… describes a portion of one cubic meter and its surface area as a sphere is exactly one square meter.
Now, only when we go further and assign a name to the material substance comprising that sphere’s volume is it then possible to calculate a weight. If it is made of pure gold, then it must weigh 2.0 tons; 4000 pounds: (gold weighs 19,300 kg/m3, and .094031… X 19,300 kg/m3 = 1,814.8098… kg. Since there is 2.204624… pounds/kg we multiply that by 1,814.8098… kg which equals 4000.974… pounds). So are we to believe it is just coincidental that 1.0 square meter comprising the surface of a sphere made of pure gold just happens to weigh precisely 2.0 tons? That is, at least as precise as the “pure gold” in America’s coinage which is refined to a .999 purity:
4000 / 4000.974… = .999756…
When applied to gold, the above geometry uses commensurate whole units of meters, with whole units of pounds and tons in the same system. Modeling weight using the (illuminati’s) system of 27 milligram cubets shows the very same commensuration. We can easily see this by forming a cuboid out of 27 milligram cubets: make this cuboid’s base layer of cuboids a square measuring 480 cubets by 480 cubets. Stack them 437.5 layers high. Calculate its weight: 480 X 480 X 437.5 X .027 gram = 2,721,600 grams. Given 15.43235835… grains/gram equals 42,000,706.49…grains. Given 7000 grains/lb equals 6000.100927… pounds…i.e., exactly 3.0 tons to a .999983… degree of perfection! Again, this shows a natural affinity, a geometric commensuration uniting, in this example, both the troy ounce (480 grains) and avoirdupois ounce (437.5 grains) working in conjunction with the gram based 27 milligram cubets to create a greater unit obviously denominated in whole unit measures named “ton”.
If we continue to follow the geometry (embedded in the DNA, so to speak) of the .094031… volume unit, it will lead us straight to the source of the township’s unique configuration . . . and beyond. I promise the reader some very interesting geometry, and I am reasonably confident that, in all probability, it has never been seen in any public classroom or textbook.
In the beginning of this chapter, we saw how (1.0 / .094031…) described the relationship between geometry’s fundamental unit of surface and it’s maximum potential to contain volume. And, we saw how this ratio was perfectly equal to (10.63472310… / 1.0).
Now we’re going to see another view of this very special volume unit:
.094031597… / (1.0)1/2 = (10.63472310…) / 36
This equation reads: “the .0940… spherical volume unit is to the square root of its 1.0 unit surface area . . . as 10.6347… is the square root of the surface area of its 36pi spherical volume”. Of course this is a special case, since the right hand side of the above equation is a description of geometry’s unique “equilibrious sphere” with its surface area being equal to its volume: neither has precedence over the other.
The left hand side of this equation is pictured in the photograph on the first page of this chapter. The graphic to the left illustrates these relations. On the right side of the sketch there is a 6 by 6 grid above the line. This represents the surface area of the sphere having a volume equal to 36. This surface is also equal to 36pi, and is naturally subdivisible into 36 separate areal units.
In this sketch, the square above the sphere on the right side describes the same quantitative information as the sphere above the square on the left side.
What this is showing us is the primordial geometric relationships between surface and volume. Clearly, at the very inception of geometry’s organizational structuring we discover a grid subdivided into 36 sub-units, with each one’s area equal to pi. This leads directly to, and is inseparable from, the one surface unit in the form of a sphere; and from there, as was previously demonstrated, humanity’s measures of length and area. The fact that geometry’s equilibrious sphere has a diameter equal to 6.0 units (6 X 1.0), and that its surface naturally sub-divides into a 6.0 unit grid, is strong evidence (in light of everything else thus far presented) that the macro land unit measure named “township” derived directly from this specific geometric arrangement. In all probability, so to did the “foot’s” sub-division into 2 X 6 “inches”. And its radius of three units is also (27)1/3 and models as the edge of a cube which is identical in structure to the 27 milligram cubet at the heart of the illuminati’s systems of weight measures.
Before moving on we should take note of another significant correlation between this quantity of 36pi and the exact (.12667″) difference between the statute mile and what I’ve previously referred to as the geometric mile:
.12667 inch = (7 feet, 4 times, times 36pi) / 3(105)
and for future reference
.12667 inch = pi(1008.0 feet) / 3(105)
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