# Geometry 2: Maintaining integers using fractions ###### understanding the megalithic: circular structures: part 2

The megalithic sought integer lengths because they lacked the arithmetic of later millennia. So how did they deal with numbers? There is plenty of evidence in their early monuments that today’s inch and foot already existed and that these, and other units of measure, were used to count days or months. From this, numbers came to be known by their length in inches and later on as feet, and longer lengths like a fathom of five feet, the cubit of 3/2 feet and, larger still, furlongs and miles – to name only a few.

So megalithic numeracy was primarily associated with lengths, a system we call metrology. Having metrology but not arithmetic, the integer solutions to problems became a necessity. Incidentally, it was because of their metrological numeracy that the megalithic chanced upon a rich seam of astronomical meaning within the geocentric time world that surrounds us, a seam well-nigh invisible to modern science. Their storing of numbers as lengths also led to their application to the properties geometrical structures have, to replicate what arithmetic and trigonometry do, by using right triangles and a system of fractional measures of a foot (see later lesson – to come). In what follows, for both simplicity and veracity, we assume that π was too abstract for the megalithic, since they first used radius ropes to create circles, so that 2π was a more likely entity for them to have resolved.

It was Alexander Thom, who first drew attention to megalithic metrology by identifying a megalithic yard of 2.72 feet. He also identified some stone circle designs in which a circle had been systematically flattened, so that π would be closer to the integer 3, rather than an inconvenient fractional number (3.14). In part 1, we saw that forcing all radii to have 7 units ensured the perimeter of their circles would be an integer 44 of the same units in length. This was simple but somewhat inflexible, and a better method has been identified within the monuments. By remeasuring either the radius or the perimeter, with a unit only slightly longer or smaller, the value of 2π was effectively changed from 44/7 to either 25/4 or 63/10.

###### some other approximations of 2π

The 22/7 approximation to 2π is 6.285714, which is nearly 6.3 (or 63/10), so a radius of 10 units could give, to less accuracy, the expectation of 63 units on the perimeter. The number 6 and one quarter (6.25 or 2π equal to 25/4) was less than 2π but a simpler ratio of 25/4, so that 4 units in radius would give an expected perimeter of 25 units.

These approximations for 2π are in themselves less accurate than 44/7. But by cunningly blending the more accurate 44/7, the “good 2π”, with an inverted version of one of these “bad 2πs” (or visa versa, see later), integer results could be maintained between radii and their perimeters, to produce a more flexible alternative to using 44/7.

The first ratio was 176/175.

1. This used an integer radius rope, to predict the perimeter as being an integer number of feet that were slightly larger by 176/175, that is 6.25 times the radius rather than 22/7 times the radius.
2. 6.25 (25/4) is 175/176 smaller than 6.285714 so that,
3. assuming 4 feet as the radius and assuming 25 feet is the perimeter, one must count these 25 feet using feet that are 176/175 larger than the normal foot length.
4. 176/175 is the product of two fractions: the “good 2π” of 44/7 multiplied by an upside-down “bad 2π” of 4/25, making the required foot of 176/175, 1.00571428 feet.
5. Counting 25 of the longer feet, one can make a perimeter rope that is correct for a radius of 4 feet and reflecting the “good 2π” of 44/7 of the radius rope, rather than 25/4, a “bad 2π”.
6. Using this technique, one can make a perimeter rope of the correct length without using the radius rope to describe the circle.

By keeping the units in “root” form, another one longer by 176/175, and a third shorter by 175/176, the megalithic could accurately relate the radius to perimeter, and perimeter to radius of a circle (respectively) as if 2π was 25/4: while enabling integer values to be maintained.

The second ratio was 441/440.

Recalling that the second “bad 2π” was 6.3 (63/10), this was larger than the value of the “good 2π” of 44/7 and a correction was needed. This was arrived at in a similar method to above, this time multiplying the inverse of the original “good 2pi” of 44/7 by the second “bad 2π”: 7/44 x 63/10 = 441/440.

1. The integer radius rope of 10 feet could predict a perimeter rope as being 63 feet, an integer number of feet that were slightly smaller by 440/441, making the perimeter rope 44/7 times the radius rope.
2. In this case, 441/440 contains the “good 2π” as its reciprocal 7/44, multiplied by a “bad 2π” of 63/10, to give 441/440 (1.00227).
3. The geometry of a circle using a radius rope 20 feet long gives a perimeter rope close to 125.714285 (since 20 x 44/7 = 880/7), according to the “good 2π”. But this perimeter rope, divided by 440/441, returns a numerically larger perimeter rope of 126 such feet. Integers have been maintained.

It is very important to understand that, for the megalithic, nature was assumed to have formed a perimeter using the “good π”. All that the megalithic circle builder did was to apply one of these ratios to either the radius or perimeter rope, to restore that length to being an integer.

For this reason, the megalithic maintained their foot measures in slightly longer and shorter versions; to performed this “magic”. The micro-variations of the foot’s length were 441/440 or 176/175, larger or smaller, these forming the kernel of a grid pattern as seen in figure 2.

n.b. In recent history, John Michell mentioned 176/175 as present in ancient measures and 441/440 in the Great Pyramid uncapped height to base. By 2000, John Neal combined 176/175 and 441/440 as a grid unifying known versions of historical measures [Berriman, 1953]. (see figure 3 and Bibliography at end)

The variations of the same measure enabled both perimeter to radius and radius to perimeter calculations to maintain integer form. For example, an perimeter of 63 feet would yield a radius of 10 feet, of 441/440 feet or an integer radius of 10 feet would yield a perimeter of 63 feet, of 440/441 feet.

To the megalith builder,
this was the same as if the value of 2π actually was 25/4 and 63/10.

This lesson shows how the geometrical reality of the “good 2π”, shown in part 1, was usefully subsumed by the megalithic, into a system of metrological micro-variations that we would write as 176/175 & 441/440 but could only manifest in the megalithic as lengths adjusted by these fractions. The next part will show how the twin circles of the first part, diameter 11 and 14, also subsumed into metrology, by applying both types of transform so that 176/175 times 441/440 = 126/125 = 1.008.

For more on how twin fractions were used in this way by the megalithic, a technique I term “proximation” (<<click to see tag), or see Megalithic application of numeric time differences.

Bibliography

1. Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.