A Rediscovered Unit of Length and Implications for the Neolithic

Following work which began in the 1970s and spanned nearly 40 years, Peter Harris and Norman Stockdale identified a “new” unit of length in 2015. They called it the Harris and Stockdale Megalithic Foot, or the HSMF for short, to distinguish it from other similar lengths. It is 14.142 inches (35.92 cm) or 1.1785 feet (35.92 cm), the length of the diagonal of a 10-inch (25.4 cm) square.

Illustration of the diameter of a 10-inch (25.4 cm) square. (Author provided)

In their analysis, they had surmised that celestial time periods of the Sun or Moon might have been used in conjunction with the HSMF to design and build megalithic structures. Their first breakthrough came when they tested the lengths of the sides and the diagonals of the four station stones at Stonehenge. These form back-to-back 13:12:5 Pythagorean triangles which create a slightly distorted rectangle, possibly due to being used as indicators for particular rising or setting positions of the Sun or Moon. They tested it in association with the time period of the “standstill” Moon (explained below) which is 18.61 years.

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Harris and Stockdale used whole number lengths converted to the HSMF and the value of about 14.15 inches (35.94 cm) was found each time.

Figure 1. Lengths (feet) by Richard Atkinson. [Note that the “rectangle” dimensions date from many years before the discovery of the HSMF.] (Author provided)

When precise values by Richard Atkinson (1978) are used we find:

Mean of the two short sides:

 91-92, 93-94 = 109.6(9) ft.

5x18.61 HSMF = 109.6(6) ft.

‘Error’ 0.03%

Mean of the two long sides:

          92-93, 94-91 = 262.7(8) ft.

    12x18.61 HSMF = 263.1(8)

‘Error’ 0.15%

Mean of Diagonals = 284.8 ft.

    13x18.61 HSMF = 285.1(1)

           ‘Error’ 0.11

The results of these calculations tell us three things:

1) The HSMF unit of length was used.

2) The time period of 18.61 (years) was known and used.

3) The near perfect result was achieved despite the apparently intentional distortion of the rectangle.

The Lunar Standstill. An Explanation

The Moon’s orbit round the Earth is inclined at about 5 degrees to the ecliptic – the plane of the Earth’s orbit round the Sun. The ‘nodes’ (N ₁ and N ₂) process slowly round taking 18.6 years.  Thus, the declination of the Moon varies from being about 5 degrees greater than that of the Sun (and 9 years later 5 degrees less that the Sun). When greater than the Sun, each month the Moon will swing from being high in the sky to low in the sky and back again. At these times the Moon returns to almost exactly the same maximum declination which changes very little for about a year and so it is called a ‘standstill’

The next standstill will be in 2024-25.

Figure 2. The Parthenon width Penrose measured. (Author provided)

The Origin of Ancient Units of Length

In 1888 Penrose measured with great care the width of the Parthenon platform finding it to be 101.34 feet (30.89 m). This is almost exactly 100 Greek feet which is 101.25 feet. The sexagesimal numerical system with sixty as its base, as opposed to the decimal (with 10 at its base), was commonly used in antiquity. This is the reason for why there are 360 degrees in a circle, 60 seconds in a minute and 60 minutes in an hour.

101.25 x 60 = 6,075 feet (1,851.66 m)

The Nautical Mile is 6,076 feet (1,852 m) and is the length of one minute of arc of the Earth’s polar circumference, i.e. One minute of latitude. Therefore, we learn from this that the Greek foot has a geodetic origin, meaning that it is based on the dimensions of the Earth.

There is a very useful book by A. E. Berriman (1953) which has information on ancient units of length, weights, volume and their origins. He details how nearly all of the units of length used in Greece, Rome (Italy) and Egypt had a geodetic origin, many of them being variations on other lengths.

Figure 3. The Fertile Crescent. (Author provided)

There is evidence for an advanced, very early society in the region of Mesopotamia - the Ubaid in Sumeria (c. 6500 – 3800 BC). The region was fertile owing to the regular flooding of the Tigris and the Euphrates. It seems likely that the dimensions of the Earth were known there. In principle it is not difficult.

To do so you need to measure the angle of a bright star above the horizontal when it is at culmination, its maximum height (angle) in the south, using a plumb line, water for a level, and an accurate scale on a large protractor. Move north or south so that the angle changes by a measurable amount perhaps a degree and measure in some way the distance. Call it 60 “miles”. In fact, it is probable that most, if not all ancient measurements were ultimately based on the dimensions of the Earth. This includes the meter: the Earth’s meridional circumference is 40,009 km, (i.e. via the poles.)

Figure 4. HSMF was used in the construction of the Great Bath of Mohenjo-Daro, now in modern-day Pakistan. (CC BY-SA 3.0)

Tracking the Spread of the HSMF Unit of Length

It is not known by what route the inch was developed, but it was known in Sumeria. Berriman discusses an ancient measure, the Sumerian foot of 13.2 inches (1.1 feet, or 33.43 cm). This became the unit of measure of the north German tribes in Roman times. When the Romans left Britain, the Saxons, one of the north German tribes, came to Britain bringing with them the Saxon foot.

This became the standard unit of length until one of the medieval kings, probably King Edward I, changed its length from 13.2 to 12 inches. So, we use a length that has an ancient origin. In another, more direct way the Sumerian foot is still with us. The Rod or Perch, a length of 16.5 feet (41.91 cm), is still used by surveyors. It is 15 times the Sumerian foot, and 14 times the HSMF.

The HSMF travelled east and was used in the construction of the Great Bath of Mohenjo-Daro in the third millennium BC located in modern-day Pakistan. Precisely made bricks were used. According to Berriman, the mean length is 471.5 inches (1,197.61 cm) which suggests bricks of 4.714 inches (11.97 cm), one third of the HSMF, were used.

Figure 5. Eridu Temple VII with 72cm grid. (Author provided)

During the early civilization in the Ubaid period (c. 6500 to 3800 BC) in Mesopotamia, many buildings including temples were built and the foundations and the lower parts of the walls of some of these exist, dating back to the 5th millennium BC. Grids drawn on tracing paper were overlaid on plans of the temples. It was found that the grid of 72 cm (28.35 in) was the one that fitted closely in nearly all cases. Kubba (1990) claims that the close fit is self-evident. Keep in mind that 72 cm is almost exactly 2*HSMF (71.84 cm), a difference of only 0.16 cm.

The HSMF was used in China and was known as the Kung Ch’ih. A length equal to one half of the Kung Ch’ih (7.07 inches, or 17.96 cm) was also in use. Confucius was 9.6 of the standard Ch’ih and was considered to be tall. Moving across the globe, it is not certain by what route the HSMF reached Britain but a recent paper by Schulz Paulsson (2019) suggests that lengths may have travelled west through the Mediterranean and then north up the Iberian Peninsula.

The HSMF in Britain and Northern Europe

It is believed that nearly all the stone rings in Britain were built between 3000 and 1500 BC. Alexander Thom, who was a very able mathematician and engineer, measured the dimensions of several hundred stone rings. The book BAR 81 (1984) contains drawings and information for most of them.

The dimensions of each ring, usually given in Thom MY (2.722 feet) and/or feet, were converted to the HSMF.  An analysis was begun using basic Excel tables. The row headings and the analysis of one site in the table was:-

              

Explanation: -  On each row the HSMF diameter and perimeter was divided by 18.61, 27.55 and 29.53 but then they were divided into 346.62 and 365.25. In most cases this leads to values greater than unity. The five values are examined for values close to an integer.

In the above example a multiple of 4 for 27.55 is indicated. This is assessed as follows:-

4 x 27.55 = 110.20   The percentage ‘error’ from 110.32 is found:-  0.11%

The results are recorded in three ranges:- 0 – 0.20%, 0.21 – 0.50% and 0.51 – 0.80%

The five row headings used were, the standstill period (18.61 years), the anomalistic moon (27.55 days), the synodic month (29.53 days), the eclipse year (346.62 days) and the calendar year.

Initial investigation seemed to show that in true rings, i.e. circles, the diameter or perimeter was a simple multiple of the period of the anomalistic moon or the eclipse period more often than would be expected by chance. Computerised analysis was required and so I approached Victor Reijs, a computational specialist in archaeoastronomy whom I had met at several SEAC conferences.

I sent to him the relevant details of the rings. There were 62 true rings and 42 ‘others’, e.g. Flattened rings, egg shaped rings.

The 62 true rings having only one value for each of the diameter and perimeter were used for analysis. Any unit length could be entered and the table and output was automatically updated. The ratios tested for were the integers 1-10 only.

The colors for ‘Hit codes’ are for the ranges of ‘errors’ shown.

There follows part of a larger Victor Reijs table.

Figure 6. Part of the table for the HSMF. (Author provided)

Explanation: - Part only of a table for an analysis for the HSMF (1.1785 ft.), 62 true rings; diameter and perimeter.  (The columns for each site name and details for been omitted.)

As explained earlier the dimension in HSMF is divided by the period 18.61, 27.55 & 29.53, but into 346.62 & 365.25. The five columns of figures are automatically revised for each of the four ‘error’ ranges as given in the ‘Hitcodes’ earlier. The column totals are given in B&W at bottom right.

A selection of ‘hits’ are highlighted to aid the reader..

My brother, J.W. Gough, carried out a full analysis using Z-score which is a statistical formula used to compare the score of what is being tested with the mean of a set used for comparison. Each of the 5 values was tested separately. The output from the excel table for the HSMF was tested against the mean output of 50 random unit lengths. The scores for 27.55 and 346.62 were markedly different from the mean of the 50 random values. This indicates that they were strongly associated with the HSMF. 

The analysis was checked by 3 independent statistical analysts.

None of the other three period values registered as being associated with the HSMF.

(Note: The association of 18.61 was using the pair of Stonehenge triangles, not stone circles.)

Figure 7. The variation of apparent size of the Moon. The anomalistic moon of 27.55 days is in a slightly elliptical orbit and so its apparent size varies slightly by about ± 3ʹ of arc. (Author provided)

With the exception of the Solar year of 365.25 days each of the other four values tested have periods which vary about a mean. In order to determine the mean period records would have to be kept for many cycles. Lunar eclipses have a mean period of 173.3 days. Since in some years there can be three eclipses, in others none at all, this would be difficult to determine.

Figure 8. Periods of the anomalistic moon. (Author provided)

The Anomalistic Moon

The variation of the anomalistic moon period is considerable, varying as it does by up to nearly 3 days. When a diagram is available then a pattern is seen but to have determined the mean of 27.55 days as was done in prehistory is very impressive. We do not know if the determination of the period was done in Britain or elsewhere, but it was built into a significant number of megalithic stone circles in the Neolithic. The earlier Z-score analysis showed that that 27.55 and 346.62 were used in the construction of many true stone rings. These rings were planned and for whatever reason these values recorded in them.

The eclipse year of 346.62 is a large number and only one ring records it as a diameter; Sanctuary A (345.7 HSMF). It is common as a simple fraction. Thus, there are four rings known where either the perimeter or the diameter is 288.4 – 288.7 HSMF. (346.62/1.2 = 288.85). The diameters of the two large rings of Brodgar, Orkney and the Great Circle at Newgrange, Ireland are 288.5 and 288.4 HSMF respectively. Other ratios 1.25, 1.333, 1.667, 2, 4, 5, and 6 are known. Multiples of 1, 2, 2.5, 3 and 4 are usually used with 27.55 and for diameters.

The Thom Megalithic Yard of 2.72(2) Feet

The unit was used in a limited manner for some rings and was used at the stone complex of Carnac in Brittany. The rings of dimension close to 288.85 HSMF discussed earlier, are given in Thom’s work as 125MY. This can be explained as follows:

The HSMF and the MY are related as: 1.1785*4/√3 = 2.7216 (feet) (Thus 288.85/125 = 2.31.  2.722/1.1785 = 2.31)

Figure 9. Ratios of Unit Lengths and the HSMF. (Author provided)

This may look complex, but triangles make the transition simple:

Figure 10. Conversion between the HSMF and the MY by use of triangles. (Author provided)

Since there is no evidence for the use of the Megalithic Yard other than in Europe, and the HSMF appears to predate the MY, it is suggested that the MY is derived from the HSMF.

The Stonehenge Sarsen Circle Uses the HSMF in its Construction

Figure 11 The Stonehenge Sarsen Circle. (Author provided)

The inner and outer circumferences of the Stonehenge sarsen circle are both simple ratios of the eclipse period (all lengths in HSMF):

Inner circumference, 82.65 x  = 259.65               346.62/1.333 = 260.03 (0.15% or 14m “error”).

Outer circumference, 88.65 x  = 278.50              346.62/1.25 = 277.23 (0.46% or 0.46m “error”).

Bluestone circle lies within the sarsen circle. Its circumference is a multiple of 346.62.

Diameter 28.65 Meg. Yards (Thom1978, p144).  28.65 x 2.72 = 77.93 ft. = 66.12 HSMF.

Circumference = 207.7 HSMF 346.62/1.667 = 208.0 (0.11% or 0.10m.).

 

From the HSMF we note that 14.142/3 = 4.714 inches and 1.1785 x 4 = 4.714 feet.

The bricks at Mohenjo-Daro apparently measured 4.714 inches.

Figure 12. Ellipse in Sweden. (A. S. Thom 1983 / Author provided)

Archie Thom, Alexander Thom’s son, measured rings in Scandinavia in 1980. One of these was an ellipse.

In the image, the ellipse is constructed on a 3,4,5 Pythagorean triangle. He expressed surprise on finding that the unit of length was 4.71 feet.

The Bush Barrow Gold Lozenge

This very thin gold plate was found in the Bush Barrow in 1808. The dimensions are very close to the lengths shown.

Figure 13. The Bush Barrow Gold Lozenge. (Author provided)

Peter Harris and Norman Stockdale examined rock carvings for possible evidence. They deduced a possible length of 1/56 of the HSMF and tested it on rock carvings. It gave positive results. The HSMi (Harris and Stockdale Megalithic Inch) is 0.2525 inches or 0.6414 cm. Rubbings were made of a number of rock carvings which had cup marks.

Figure 14. The Swastika stone on Ilkley Moor. (Author provided)

A fylfot known as the Swastika Stone is on a nearly flat slab on Ilkley Moor. The “square” outlines have been superimposed on a scale copy of the rubbing. The four sides of each “square” were measured to the nearest 0.1mm, summed, and converted to HSMi.

The inner “square” 81.7 HSMi

3 x 27.55 = 82.65

The outer ‘square’ 165.0 HSMi  (3x 27.55)

6 x 27.55 = 165.3

If the outlying cup mark is linked to the top RH corner of the outer square, then the total of all the lengths is 274.6 HSMi.

10 x 27.55 = 275.5

Figure 15. Kirkmuir 1. Rock Carving. (Author provided)

Known as Kirkmuir 1, there are two rock carvings on slabs in southwest Scotland, near Wigtown. If the cupmark centers are linked as shown, then the total length is 347.9 HSMi (‘Error’ from 346.62, 0.37%).

What Does the HSMF Mean When Analyzing Knowledge and Ability in the Neolithic

1) The HSMF is of length 14.142 inches (10√2) or 1.1785 feet.

2) It probably originated in Mesopotamia more than 7,000 years ago.

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3) Megalithic structures, especially true stone rings, were built using the HSMF in combination with celestial time periods. Analysis has shown that the time periods 27.55 days (the anomalistic moon) and 346.62 days (the eclipse year) were favored.

4) Determining the mean time periods of these cycles would have required extended observation and record keeping.

The above history of measurement and units of length indicates much greater ability and knowledge in the Neolithic than is currently accepted.

 

This article summarizes some of the descriptions and workings included in ‘A New Dimension to Ancient Measures’ by Dr Thomas Gough and Peter Harris available via Amazon (UK only) U.K., Europe and North America through the Megalithic Portal www.megalithic.co.uk. Go to - Books - Earth Mysteries £15 + p&p.

For more information, email: [email protected]

Top image: Collage of various measures of unit of length. Source: Andrey Armyagov / Adobe Stock

By Dr T. Gough and Peter Harris