"A true hereditary and traditional measure is something almost as vital as a language, sometimes more so; and to stamp it out utterly and instantly is not within the province, or the power, of any government of men for the time being.”
Charles Piazzi Smyth
How John Michell first established certain rules that governed ancient metrology
John Michell (9 February 1933 – 24 April 2009) the great philosopher-antiquarian was the first man to establish definite values for units of measurement. Prior to this, due to variations in linear measurement, virtually any close value could be proposed as an intended module. Through many comparisons of recorded modules he stated that examples were found in Roman, “Polar” feet, Greek, royal Egyptian and Ancient Jewish that all existed in two distinct variants that related as 175 parts to 176. This inferred that the modules of these different nations all referenced an earlier canon, which in turn implies an earlier and now forgotten culture that had been universal.
These are the exactly expressed modules given in English feet:
The two variants are under the headings “Tropical” and “Northern” because it was believed at that time that it was the variations in the length of the geographic meridian degree at 10º latitude and latitude 51º that dictated the lengths. These variants enabled him to calculate two specific earth radii from the “sacred” cubit of the Jews, that of 20,854,491ft for the polar radius and 20,901,888ft as the mean radius of the earth and that these radii differed as 440 to 441.
Having these exactly expressed values enabled the foundation of the more complex structure of ancient metrology to be understood because these data points were now in place. Firstly it was realized that that the fraction 175 to 176 was also the difference between the anciently used pi ratios of 25/8 (or 3.125) and 22/7 (or 3.142857), thus establishing a purely mathematical as well as the geographic explanation as to its presence in metrological values. Another focusing was the observation that the values of the “polar” feet at six parts to seven of the royal Egyptian, were in fact the universally known common Egyptian feet.
Also pertinent to the structure was the recognition that the term “sacred Jewish” feet were far too long to be termed “feet” – they were arrived at through the division of the sacred cubit by 1 ½, when in point of fact the sacred Jewish cubit is a two feet cubit. Two important factors were brought to the fore when the correction was made. Firstly take the value 2.0736ft as the sacred cubit, this divided by two is 1.0368ft, it is therefore a two-feet cubit of the common Greek foot. Furthermore, if the value 1.3824ft is divided by 1½ it is .9216ft and this is a value of the pie or Spanish foot, one third of the vara; it is an Iberian 1½ feet cubit. Thus was revealed an exact integration of metrological values of far greater complexity than John had first proposed that goes much further than merely two variants, or this limited number of foot modules.
Just this small exercise is a perfect illustration of the total amalgamation of metrological values that underscores the point that it is a singular system. The sacred Jewish is common Greek and relates to Iberian as 9 to 8; the common Greek is 21 to 20 of the common Egyptian and 9 to 10 of the royal; the Iberian is 14 to 15 of the common Egyptian and 4 to 5 of the royal. In this fashion, all of the foot values of the world so interrelate– by unit fractions. Having knowledge of this connectedness and more that one value to maintain accuracy (by comparisons) it enables any module to be identified and classified.
Prior to the world’s adoption of the Système International d'Unités, there was already an international system of measurement. It had become fragmented into what superficially appeared to be an unconnected collection of quite separate methods of mensuration. On close inspection of these, from all periods of history and from all quarters of the world, it is manifestly obvious that there was only one “system” of measurement. The greatest impediment to an understanding of this extremely ancient science is the metric system itself.
No traditional unit that was replaced by the metre is expressible by an exact exchange; since it is so utterly divorced from the system that it supplanted no historic module of measurement is an exact number of millimetres.
Further to this, in the classification of modules, whatever the multiple lengths (stadia, miles, leagues etc.) that are encountered they should first be reduced to their constituent foot length and the comparisons made at that level, but the metre has no subdivision that approximates to a foot length. It is often difficult to identify what is any of the ancient multiple feet modules under consideration when they are expressed in metres because the prefixing numbers are lost. This will be become obvious in due course.
The algorithm or axiomatic rules of the structure of metrology, is that all of the foot modules of measurement are connected through a series of unit fractions, and all of these foot lengths have identical variations in their length. It is these variations that have occluded the detection of the whole-number interconnectedness of the various feet – because the wrong variants are most often compared, but the comparison must be made like for like.
The following list is of the core variants of the Greek feet; it is more extensive, but these are the variants most commonly encountered. The Root or number one is the English foot.
These are the fractions that govern these variations in the same feet
The significance of these unit fractions, that of 441/440 and 176/175, is that they both have the property of yielding whole-number solutions in diameters and perimeters of circles in differing modules of measurement. If diameters are multiples of four the rationalisation of pi that was used historically is 3.125 as opposed to the more accurate 3.142857 and 25/8 is 175 to 176 of 22/7. This means that a diameter of four Roman feet would have a perimeter of twelve Greek feet; the ratio of the Roman foot to the Greek is 24 to 25.
It is the 440th fraction that is used on the diagram to the right; the royal Egyptian foot relates to the Greek as eight to seven, this is 1:1.142857 but the addition 440th fraction increases the foot of the diameter to 1.145ft. This is how these fractions – both those relating the module lengths and those of the module variations – were utilised to give integer solutions in designs.
Listed above are the core (most commonly found) variations of the Roman feet; .96 Root differs from the Greek Root of 1ft at .96ft as 24 to 25. It is because we are using the English/Greek foot to express these values that the Greek feet may also be regarded as the formulae by which all other foot lengths are classified. Other modules that also relate as 24 to 25; the Sumerian and royal Egyptian for example, would also share the same relationship of integer radius and perimeter.
There are nineteen potential Mathematical feet and they all evidence the identical variations. The term “mathematical foot” distinguishes the module from the “natural foot” – these natural or anatomical feet are half cubits. The list of the mathematical feet of all nations is shown below at their “Root” classification expressed in English feet. The list shows one of the unit fraction linkages between the modules in the offset column; these all relate through square numbers that are shown in bold type.
Three of the foot lengths are marked “unidentified” this does not mean that they are never encountered, but are too rarely found to give a positive nomination. They should be regarded as “potential” but must be there to give the connective unit fraction links; additionally, the canonical height of a man is six feet and in one or another of the mathematical feet – every man is six feet tall. The relationship of the anatomical foot to the height of a man (or woman) is of one seventh; these non-conforming feet would be legitimate on this basis alone.
The complexity of ancient metrology is further compounded by the fact that many of the Root values have a greater or lesser identification that vary by 1.008 (125/124), which is a combination of both fractions, – 176/175, plus 441/440. The most commonly encountered example of this reduction of the “Root” classification occurs in the Roman foot; there is no doubt that the unit faction linkage between the Greek and Roman feet is that of 25 to 24. The Greek feet also relate to the Egyptian foot as 7 to 8 but he Roman foot of .96ft has no unit fractional link with the Egyptian; the Roman foot must be reduced by its 1.008th part to equal .9523809ft and is then 5 to 6 of the Root royal Egyptian foot of 1.142857ft. At this reduced value it maintains its unit fraction link with the Greek but at the ratio 20 to 21.
Another module that often displays this property is the Belgic foot; it has been accepted since antiquity that the Belgic foot is 9 to 8 of the Roman. At a Root value of 1.08ft this displays no unit connection with the Greek, yet reduced by its 1.008th part to 1.071428ft it is 15 to 14 of the Greek. At this reduced value when encountered it is referred to as the “Doric” foot. This brings us on to the identification terminology that is met with in metrological research.
Identification of the units of metrology
It has been demonstrated that that the term Doric is also a variant of what is termed Belgic. This practice of naming feet after the location, architectural style or nation in which they are found leads people to assume that there are a far greater number of quite different foot modules than actually exist. This is clearly evidenced by the spread of terminology that is applied to what is termed here as the “Assyrian,” or the least mathematical foot at .9ft. It and its variants have been termed Lesser, Archaic, Italic, Oscan-Italic, Oscan-Umbrian, Campanian, neo Babylonian, Lydian, Assyrian, Mycenaean, Basque, Geometric – and many more.
The convention here is to collect all of the variants of the same basic foot into a common terminology by calling them after the variant that relates to the English/Greek foot by a unit fraction; in this case it is the Assyrian foot that is .9ft; then all of the variants as encountered in the above list would then be “Assyrian” followed by their classification term – Root Geographic for example (this example would be the so called Lydian foot which is .9 x 1.0114612). All foot lengths that are encountered may be precisely identified in a similar fashion. Roughly speaking, a module may be classified by dividing the example in question by the closest Root value, but care must be exercised, for the following reasons.
Because of the closeness in value of certain of the listed foot lengths, in the spread of their differences – a greater variant of a lesser foot can sometimes exceed the length of a greater foot at its lesser variation. Caution must therefore be exercised in matters of identification; most often the intended module can be identified from the context in which it is found. One such example would be the foot termed Archaic English; it is the foot of the “yard-and full-hand,” at 10 to 9 of the English it is 1.111ft and at its greater variations has overlaps with the Nippur foot that is 9 to 8 of the English at 1.125ft.
The Root Geographic variant of the Archaic English foot is 1.12385ft and it is easy to see how this may be likened to the Nippur Root foot of 1.125ft. But the situation in which this variant was encountered was in calculating the foot of a geographic degree recorded in China as being 180 li in length. The convention is that the geographic degree contains 360,000 geographic feet. Therefore the variety of foot may be calculated by dividing 360,000 by, in this case, 180 and this will give the number of feet in the li; in this case it is 2,000 Root English feet. A li consists of either 1,500 or 1,800 “feet” so this must be an 1,800 feet li of 1.111 feet; then adjusted in length for the latitude in question (ca. 40º) it is 1.1239ft. This is how this particular module is identified, and illustrates the point that the Root foot that it resembles is not necessarily the correct interpretation. All other lengths, wherever they are encountered can be similarly classified, largely through the context of their use.
It has been broached that in certain circumstances, module variations are governed by geographic considerations; certain of the variants accommodate the lengthening degrees of the earth’s meridian in order to maintain the same number of feet to the degree at different latitudes. The geographic foot variations that are common to all of the different “systems” are 1.008, this is termed Standard Canonical, 1.011461, this is Root Geographic and 1.01376 is Standard Geographic. In terms of the Greek feet they are the lengths of the geographic foot at 10º, at 38º and at 51º. There are other localized variations that are met depending on the latitude, but the above variants are found universally and are the lynch pins of the geographic structure.
The geometrical nature of “cubits”
As with all modules of measurement, the descriptive terminology often carries a wealth of interpretation; mile, league, stadia, furlong, iteru, parasang etc. cover a wide range of measured distance that are termed the same unit in different localities. So it is with the cubit; often enough it merely means the “covid” or “module.” Halves or doubles of any measure often have the same term applied to them. Anywhere, from the Far East to Europe to the Americas, in ancient texts one finds references to the “step” of 2½ feet and this is sometimes termed cubit, half this length, termed remen or palimpes may also be referred to as cubit.
The basic cubit is of 1½ feet and is the length of the forearm and outstretched hand, the antebrachium or ulna, whence comes the term ell – is also a very general module term. The Aztec cemmolicipitl has the same meaning – one elbow. This is termed a short cubit and the long cubit is of two feet and these two are the true cubits. Short cubit, long cubit and step relate as 3 – 4 – 5 and are therefore geometrical ratios. It is through the basic geometry of the human form and the shape and dimension of the earth that the canon of measure stems.
The geometry of cubits:
The structure of ancient metrology was devised to express integers in designs. To this end, incommensurable ratios particularly pi and √2 are rationalised through close approximations. These two ratios are intimately interconnected through the following geometry.
The commonly used approximations of √2 – 99/70 and 140/99
As illustrated, the two fractions 99/70 and 140/99 are divisible by 7 as one factor and by 11 as the other. As both factors are common to the most often-used pi ratio of 22/7 it is not surprising that that these two approximations prove to be related through the medium of measurement – squares and circles and how they are reconciled as being basic to metrology. The most obvious relationship is that a square inscribed in the circle shares the diagonal of the square as the diameter of the circle.
99/70 equals 1.412857 and this number exhibits the recurring 7th fraction that is constantly found in metrological values; 140/99 is 1.414141 and exhibits the recurring 11th fraction also commonly encountered in metrological values. These two approximations of √2 when multiplied together are exactly 2. Additionally, as so often happens, the numbers expressed in ratios are often measurements when expressed in English feet and the number/length, 1.4142857 is the Samian 1½ft cubit that stems from the foot of .9428571ft. Just as with the varying values of pi being selected to produce whole numbers in squares and circles, so it is with these values of the square root of two. It becomes obvious which value is being applied, because it will give an exact number of the modules in question in the solutions.
The basic module of all methods of mensuration is the foot. The basic factor that governs the dimension of a circle is the radius. The nature of the relationship between feet and cubits is exemplified in the nature of the circle and circumscribed square. If the radius is one foot (in any of the feet) the diameter is a two-feet cubit and the quadrant chord of the circumscribing circle is a one and a half feet cubit of a different foot. It is also true that most often, the quadrant arc is also a cubit module.
The above diagram to the right illustrates how the English/Greek foot radius propagates differing modules in the other dimensions; the Samian cubit in terms of the English foot is 99/70 as a number.
The diagrams above illustrate how integer modules arise from the same geometry, the resultant integers being maintained by values that differ from the Root classification radii by using the 176/175 or 441/440 fractions. Which of the exact approximations for √2 that are used (140/99 or 99/70) is dictated by the integer solution being a proven module. The feet that are used in the above radii are English feet and greater. If lesser feet are used, shown below, the quadrant chord may reduce to a 20 digit remen cubit.
The diagrams above illustrate how these lesser feet, namely the Iberian and Samian radii, do not form 24 digits cubits but are 20 digits remen as the quadrant chord integer. It is fair to term a remen as a cubit because it is the forearm and clenched fist; the fist reduces the long-palm length (10 digits) by one short palm (4 digits) thereby reducing the 24 digits true cubit to the 20 digits remen cubit. (This point will be made clear in the section on the human canon as the source of measurement.)
Another pertinent fact in the above geometry concerning the Samian foot radius, is that the value one uses for pi in order for the quadrant arc to be a common Egyptian cubit is that of 864/275, also known as the “Fibonacci pi” of 3.1418 (and this point will be clarified in the section concerning the dimensions of the world as the secondary source of measurement.) Both exemplars, Man and the World, illustrate how the ratio one uses, either for pi or √2, must be selected to produce a correct solution in the final module.
The best example of this point concerning the √2 ratios is the dimensions of the Great Pyramid as illustrated by the following diagrams.
For many reasons, it is the south side of the Great Pyramid of 756 English feet that is the datum of the visible structure. The King’s Chamber level has the ratio of 1 to √2 with the base of the pyramid. If the diagonal at that height would be the 756 feet of the base side, then the side of the reduced pyramid would be a five hundred feet stadium in terms of the Persepolitan foot, at 534.5454ft – if the ratio used is 99/70. Then the diagonal of the pyramid base is twice that at 1000 Persepolitan feet but the value for √2 that is used to achieve this precise number is that of the alternative – 140/99.
The diagram above is Flinders Petrie’s rendition; it is not only illustrative of the √2 relationship regarding the base of the pyramid to the King’s Chamber level, but it twice demonstrates the pi ratio as the quadrature of the circle, whereas the previous diagram depicting √2 uses the inscribed square. It is the most often quoted geometrical basis of the pyramid design, that of the height corresponding to the radius of a circle of which the base is the length of its circumference. In order for the geometrical functions of the Great Pyramid to be demonstrated, it is of paramount importance that the correct dimensions are used in calculation; these are height – 481.0909ft and datum base side 756ft, or 7 to 11. All modules that are used in pyramid are of the Standard variation – Root plus the 440th part.
Another example of how the pi ratio governs the modules of antiquity is the straightforward way that the radius of a circle relates to the 60º arc of the perimeter, rendering both as related modules:
Once again, if this occurrence is shown to be regular throughout the range of feet it is more clearly evidenced if the module of the radius is given the extra 440th part that transforms it to the “Standard” classification; it makes things a little clearer by the resultant arc being elevated to the “Root” classification. If the Root Greek foot (which is the English foot) as depicted in the diagram above, is extended by its 440th fraction to be 1.002272ft then the Persian foot of the arc is 1.05ft. This is then a Root value in the solution, this is not true across the board of all of the different feet; sometimes a Root radius will yield a Root arc as so: if the Root Assyrian of .9ft is the radius, the arc is the Root Samian of .942857ft.
It is through these comparisons of radii to the 60º of arc of the resultant circles that the manipulation of modules by the fractions that govern the module variants is most obvious.
Assyrian .9ft –
Iberian .9163636ft –
Roman .96ft –
Greek 1.008ft –
common Egyptian .981818ft –
common Greek 1.030909ft –
Persian 1.05ft –
Persepolitan 1.060606ft –
Belgic (Doric) 1.07386ft –
Sumerian 1.090909ft –
Saxon* 1.1ft –
Archaic English 1.11363ft –
Nippur 1.125ft –
Royal Egyptian 1.14545ft –
common Greek 1.028571ft
Archaic English 1.111111ft
royal Egyptian 1.142857ft
royal Egyptian 1.152ft
Samian remen 1.178571ft (5 to 4 of .942857ft)
Roman remen 1.2ft (5 to 4 of .96ft)
It is informative that in both the √2 and the π relationships that the remen (or palimpes) replaces the true cubit in the chord as the radius module diminishes, and conversely increases from the foot to the remen in the arc.
The whole number solutions are often maintained in the above results through the manipulation of the modules by the fractions 441 and 176 – meaning they are not all Root to Root solutions; the majority of results as shown have just one side of the equation as “Root.” *The Saxon foot is marked with an asterisk and indicates that it is the only solution that does not use 22/7 as the pi ratio, but uses 864/275 to produce the correct module in the result. (This practice will be dealt with below concerning both the geographic measurements and canonical man). The fact that emerges from the above comparisons, that of both the Sumerian and Saxon yielding a royal Egyptian solution, indicates that the Saxon foot is truly a modified Sumerian measurement.
These same radii and the 60º arc relationships also hold good on greater modules, Schwaller noted the same connections as in the diagram below:
He gives the correct ratio of 21 radius to 22 of the 60º arc only in the bottom right section of the hexagon. If the six fathoms of the arc were six English feet, he may have noticed that the radius is then exactly the length of the King’s Chamber in the Great Pyramid (34.3636ft exactly 20 royal cubits). He uses “Denderah” cubits, a slightly longer version because his fathom is of one of the longer Greek feet.
The diagram below is of the feet and potential feet modules as shown in the original table above. It shows their relative lengths.
The numbers to the left are the unit fractions by which they all vary from the English foot. Lesser modules than the English foot, such as the common Egyptian are 48 to 49 English, and modules greater than the English foot, such as the common Greek are 36 to 35 English – and so forth. Two modules, the Samian and the Iberian, have been reduced from their true Root by a factor of 176/175 to produce this correspondence; this is the purpose of these fractions in the field of metrology – the maintenance of in integers in designs or numerical series.
The Shape of the World
If not the sole reason, then at least one of the principle reasons for the regularly found variants of the modules, is the shape of the earth and how its oblateness affects the lengths of the meridian degrees. If the mass of the earth were spherical it would have the following geometric relationship with its satellite:
The method of squaring a circle to a ratio of 22/7 from the departure point of a 3-4-5 triangle
It is not spherical and there are therefore three radii that determine its oblateness. The polar, the mean and the equatorial and their whole number relationships are as follows:
These lengths correspond to one of the modern acceptances of this magnitude as so:
Earth’s Radii in English feet
At the integer ratios
The ancient estimates are as accurate as are the modern, this is because there is no current common agreement on an exact definition and diverse geoids, of which WGS84 is merely one, give different magnitudes within which these ancient estimates comfortably fall.
The oblateness of the geoid is due to its rotation, is not arbitrary in its numerical composition, it evidences a numeric-geometric regularity. This regularity could push the credibility dangerously close to creationism, but in conjunction with the specific patterns of metrology is better regarded as glimpses of the underlying symmetry of natural phenomena.
The meridian degrees of the Polar Regions and the degrees near the equator vary very little progressively from degree to degree – only about one metre per degree for three of the degrees at either end of the scale. From either end, in a regular fashion, this variation increases as they approach the median of 45º where the difference becomes ca. 20 metres between successive degrees. This allows the variable modules of antiquity to be assigned to very specific degrees on the surface of the spheroid. These diagrams of the quadrant arc of the earth explain certain of the variations that are universally found in the measurement system.
What has been shown in the various tables are the “core” values that for the reasons of neatness, if nothing else, merely to make them more comprehensible. The additions to this core that are included in the above scale, is the northernmost degree of Egypt, this fits the pattern as being 441 to 440 of the least degree at 10º who’s foot of 1.008ft is the first that achieves a geographic length, then at 1.01029ft is a localised Egyptian length and is not universally found. Also the northernmost degree at the latitude of the Arctic Circle where the foot length has increased to the maximum length and is twice the fraction of the least of 1.008ft, it is at 1.016064ft – 1.008.² This precise extended value is undoubtedly found in practise, but not as commonly as the core variants; it is the Greek foot of the Pantheon design for example. (The full range of these variations will be demonstrated at a later point.)
Pi and the World
As was stated in the first diagram of this series – if the world were a true sphere it would be 7920 statute miles in diameter. This is a radius of 20,908,800 feet and the mean radius of the oblate earth is 20,901,888 feet. The difference between these measurements is 3024 to 3025, the difference between the pi ratios of 864/275 or 3.14181 and 22/7 or 3.142857 is 3024 to 3025. In practice this means that the same meridian circumference number would be expressed if the pi value used on the perfect sphere were 864/275 and the universally used 22/7 were applied to the mean diameter, (here we show the radius): -
864/275 x 20,908,800 = 131,383,296 (from the radius of the true sphere)
22/7 x 20,901,888 = 131,383,296 (from the mean radius of the oblate at 30º)
The number 131,383,296 is the number of English feet in the meridian circumference that would be calculated from the mean radius. There are 129,600,000 geographic feet in this circumference; divide the former by the latter it equals 1.01376ft and this is the geographic foot calculated from the mean radius of the earth. Nowadays usually given as 309mm it is one of the two most widely accepted values of a Greek foot, the other is 308.3mm and this is 1.0114612ft and both are sometimes termed “Olympic.” The Parthenon was termed Hecatompedon which means hundred–footer, the stylobate width is 100 of these 309mm feet.
The Meridian Circumference
Although there are many reckonings of the meridian circumference from the ancient world, the two most often referred to are in terms of the Greek 5,000 feet itinerary mile and the 6,000 feet sea mile. The former begins as two remen - the step, two steps - the pace, one thousand paces - the itinerary mile; and the latter stems from two cubits - the yard, two yards - the fathom, one thousand fathoms - the geographic mile. The mile of the former is 5,068.8ft or 5,000 feet of 1.01376ft at 25,920 to the circumference; the mile of the latter is 6,082.56ft or 6,000 feet of 1.01376ft at 21,600 to the circumference.
Some other classical estimates of the circumference are:
Eratosthenes 252,000 common Greek 500 feet stadia (36 to 35 of the “Olympic” Greek)
Posidonius 240,000 Belgic 500ft feet stadia
Ptolemy 24,000 Belgic 5,000 feet miles (66 ⅔ miles to the degree)
Numerous sources give 27,000 Roman miles (75 miles to the degree)
Habash 20,160 these are 6,000ft miles of the lesser Belgic feet
Ibn Yunus 20,250 these are 6,000ft miles of the Persepolitan foot
Other Arabian sources state the circumference is 66,000 farsakh (three miles to the farsakh or parasang) in which case this would be 19,800 miles of 6,000 Sumerian feet.
All of these estimates are accurate to the datum of 129,600,000 Greek geographic feet circumference.
The Human Canon as the Source of Measurement
The modules of measurement stem from the human form; wherever the skeleton articulates it is a module of measurement. Modern antiquarians and archaeologists regard this human proportion as crude, “rough and ready” as it is often described. However, it would be better described prosaically as intelligent and convenient, or romantically as inspired and sublime. Undoubtedly, the best-known portrayal of the canon is Da Vinci’s “Vitruvian Man” as shown below.
Based upon the Roman architect Vitruvius’ account of the principles of Greek architecture, in “The Ten Books on Architecture” Book III – “On Symmetry: in Temples and in the Human Body” he gives a description of the canonical scale between the various parts of the human anatomy. Although correct in every other detail, Da Vinci made one obvious correction to Vitruvius’ description; this is the length of the anatomical foot in proportion to the stature. Given by Vitruvius as one sixth of the height, Da Vinci knew from his own draughtsmanship of the human anatomy that the correct proportion was one seventh and drew it accordingly.
Because canonical man is drawn in a square, then the circle of the same perimeter was superimposed upon Da Vinci’s rendition; with the additional, previously described, means to draw the circle. Every aspect of the design proves to be a module of measurement from the datum of six Greek/ English feet as the height and reach of the man. The scale to the left is 144 half-inches and shows that the centre is taken to be the half-pi ratio 56 to 88 and not the phi ratio of 55 to 89. The perimeter of the square is 24 English feet and Da Vinci’s circle is 24 Roman feet, the pi ratio is 864/275 in order to make the perimeter of the circle 24 × .96ft.
If the datum of six feet were any of the feet of the tables then comparable related modules would result. If similar data is applied the scale of the woman one gets the same results from the underlying geometry. The basic rendition below is by the Australian artist, Susan Dorothea White that she has entitled “Sex Change for Vitruvian Man” and the identical geometry has been superimposed upon it.
The rationale of the basic measurement is that in societies where nutrition is adequate and there is no sex discrimination, the ratio in stature between man and woman is 12 to 11. Therefore if the canonical man is six feet then canonical woman is 5½ feet. Through the application the fractions 441 and 3025 as additional to the Root, then the woman is six Iberian feet in stature. It is gratifying to see the previously mentioned “cubits” of the ratios 3,4 and 5 in practice. This how the units of measurements arise from the human anatomy.
The “Iberian” foot of that arises from the female canon is no fudge of number, it is quite spontaneous and has (relatively) recent historical precedent. At .9166ft this is naturally 11 to 12 of the English foot and three such feet are 2.75ft. The Root Iberian foot is .9142875ft and if it is increased to the Standard classification by the additional 440th part it becomes .916364ft; it reaches the female canonical value of .9166ft by the addition of the 3024th part it then becomes rational to the English foot at eleven to twelve.
Shown above are the core values of the Iberian foot that are subject to the same variations as the English/Greek. (It is the Standard value plus the 3024th part that is the basis of the female canon.)
In relatively modern times this identical adjustment was made to the vara after the annexation of the states of Florida, Texas, New Mexico and California by the United States. In Texas it was rounded to be 33⅓ inches but in New Mexico and California by 1854 it was rounded to be exactly 33 inches and this is 2.75ft, which divided by three, is the basic foot measure of canonical woman. Coincidentally, the measures used by the Spanish in Mexico were identical to the Aztec whom they supplanted, that were also based upon these feminine anthropometric values.