Ad Quadratum is a convenient and profound technique in which continuous scaling of size can be given to square shapes, either from a centre or periphery. The differences in scale are multiples of the square root of two [sqrt(2)] between two types of square: cardinal (flat) and diamond (pointed).
Einar Palsson [1, at end] saw that the myths of foundation for Iceland’s settlement in 930 had Pythagorean roots. Since then Petur Halldorsson has identified patterns that could not have been influenced by Pythagoras (c. 600 BC) and Pythagoras was known to have adapted the existing number sciences found (according to his myth) from Egypt to China.
Such patterns, called Cosmic Images by Halldorsson , seek to establish a geometric connection between places on the landscape and on the horizon, here in the south-western region near Reykjavik, the only Icelandic city. The spirit of a region or island was integrated through organising space in this way, according to centers (Things) of circles and their radius and diameter as numbers of paces, circles punctuated with places and alignments to other places, horizon events or cardinal directions. John Michell provided a guide to some of the techniques in his books [2, at end].
In 1972 John Michell inferred an enormous ten-sided form nearly sixty three miles across, in which important historical and neolithic sites had been intended as ten vertices around an ancient centre, signified by a Whiteleafed Oak.
Michell had previously  developed the idea of the enchantment of the land as an actual practice; land areas were enchanted by using a geometrical pattern integrated with myths and ritual calendars, enacted within that framework. This was long before, around 930, such a pattern was being established of thing-places in Iceland. The idea of thing places is still find-able in English names such as Goring, the centre northeast of Stonehenge, where the summer solstice sun arose.
“Perpetual choirs were a Celtic institution, from pagan into early Christian times. In Iola Morganwg’s Triads of Britain, translated from Welsh, it is stated that ‘in each of these three choirs there were 24,000 saints; that is, there were a hundred for every hour of the day and the night in rotation, perpetuating the praise and service of God without rest or intermission.’ ” – The Measure of Albion
“Three of the choirs were located at Stonehenge, at Glastonbury, and near Llantwit Major in Wales. Others appear to have been at Goring-on- Thames and at Croft Hill in Leicestershire, a traditional site of ritual, legal, and popular assemblies.” The Dimensions of Paradise
Only two type-D stone circles (see figure 3) are
known to exist, called Roughtor (in Cornwall) and Seascale (in Cumbria). Seascale
is assessed below, for the potential this type of flattened circle had to
provide megalithic astronomers with a calendrical observatory. Seascale could also
have modelled the harmonic ratios of the visible outer planets relative to the
lunar year. Flattened to the north, Seascale now faces Sellafield nuclear
reprocessing plant (figure 1).
Stone Age astronomical monuments went through a
series of evolutionary phases: in Britain c. 3000 BC, stone circles became
widespread until the Late Bronze Age c. 1500 BC. These stone circles manifest
aspects of Late Stone Age art (10,000 – 4500 BC) seen in some of its geometrical
and symbolic forms, in particular as calendrical day tallies scored on bones.
In pre-literate societies, visual art takes on an objective technical function,
especially when focussed upon time and the cyclic phenomena observed within
time. The precedent for Britain’s stone circle culture is that of Brittany,
around Carnac in the south, from where Megalithic Ireland, England and Wales probably
got their own megalithic culture.
Around Carnac in Brittany the land is peppered with uniquely-formed megalithic designs. In contrast, Great Britain’s surviving monuments are largely standing stones and stone circles. One might explain this as early experimentation at Carnac followed by a well-organised set of methods and means in Britain. What these experiments near Carnac were concerned with is contentious, there being no appetite, in many parts of society, for a prehistory of high-achieving geometers and exact scientists. Part of the problem is that pioneers interpreting monuments are themselves hampered by their own preferences. Once Alexander Thom had found the megalithic yard as a likely building unit, he tended to use that measure to the exclusion of other known metrological systems (see A.E. Berriman’s Historical Metrology. Similarly, John Neal’s breakthrough in All Done With Mirrors, having found the foot we still use to be the cornerstone of ancient metrology, led to his ambivalent relationship to the megalithic yard. Neal’s interpretation of the Crucuno rectangle employs a highly variable set of megalithic yards, perhaps missing the simpler point, that his foot-based metrology is supported as present within the dimensions of the Crucuno rectangle; said by Thom to be a “symbolic observatory” of the sun: this monument was an educational device, in which Neal finds the geometry of “squaring the circle” which, as we see later, was probably the Rectangle’s main metrological meaning.