Howard Crowhurst lives near Carnac and has been studying Brittany's megalithic remains for over twenty years. Last year he published his second book called Carnac, The Alignments, When Art and Science were one. This book studies a means used for laying out monuments, specifically appropriate at Carnac, and linking monuments there into a larger coherent and intentional arrangement using the properties of a relatively simple geometrical idea, that of multiple squares to form rectangles that have a most fortunate mathematical property.

We know that Pythagorean triangles were popular 3500 years before Pythagoras lived because the megalith builder's at Carnac had moved to a latitude upon the spherical Earth where the solstice sunrises and sunsets occured along the 5 side of a 3-4-5 triangle, the first such whole number triangle which has very low numbers and these all adjacent. The hypotenuse of 5 can be achieved without using a triangular construction. If instead a recangle is made out of 4 squares by 3 squares, the rectangle will contain 12 squares of the same size. Squares are easy to build since the sides are the same length and two diagonals between each opposed corner should have the same length, actually the square root of two of the side lengths, but that is irrelevant to the practicality of construction. It is therefore easier to build a right angle out of a square than it is to build a Pythagorean triangle that always has a right angle at one corner.

However, Crowhurst finally got the phenomonological message that the lunar maximum standstill and minimum standstill formed near single square and double square diagonals as alignments within the landscapes. These lunar extremes reoccur every 18.6 solar years due to a retrograde motion of the lunar nodes at which nodes the angle of the moon's orbit cuts across the solar ecliptic wherupon eclipses can occur of both the sun or (more easily observed) the moon when opposite the sun.

The geometry of multiple squares therefore becomes very important at carnac where the moon and the sun, in their extremes, can align to diagonals across rectangles made up of one or more, same sized, squares. When Robin found a possible lunation triangle at Le Manio we followed up with a survey (see report on site) and this reframed the origins of this solar-lunar invariant triangle from being 12 units in its base and 3 units in its shortest side to being a rectangle of four squares with its solar year hypotenuse being the diagonal. This was made especially clear during the survey when 36 lunar months along the base equated to 27 metres to reveal that the metre contains within 3/4 of its length a perfect day-inch count for the lunar month, that is 29.53 inches. It was already discovered by Robin that type B flattened circles had a three square geometry in them.

I therefore found it possible to define Carnac's sightlines and some key astronomical invariants within the diagonals that can be formed within a 4 square by 4 square, greater square of 16 squares in which each corner applies to each of the four quadrants of observation. One could use 8 by 8 squares to usefully define practical alignments and one can develop hypotenuse- diagonals for the 4 by 1 lunation triangle and with a triple division, a 3 by 1 eclipse triangle.

All of the above is as an introduction to a way of showing why the mathematics of multiple squares is uniquely able to achieve an interrelated but exact set of angular relationships, causing it to leave evidence of its use certainly as far away as Denmark and Scandinavia. It is due to a property of tangent angles, the tangents in question being formed by whole number ratios when multiple squares are involved. here are some of the relationships.

Interaction of 3-4-5 Triangle, Single- and Seven-Square Angles

A square diagonal is tan (1) and tan(1) minus tan(3/4) [a 4 by 3 squares diagonal angle] equals tan(1/7) [ a seven square's diagonal angle]. This can be shown to relate the summer solstice at Carnac with the lunar maximum standstill, in the north east, in 4000 BCE. It can be graphically shown below and the angles are exact to however many decimal places.

These relationships were latent in Thom's data but it required the megalithic approach of not having sophisticated mathematical tools, to find a simple way of looking at the astronomical alignments to the horizon. Here is Thom's Quiberon Bay diagram from his Megalithic Remains in Britain and Brittany with extra squares and as per my figure 4.14 in Sacred Number:

Below is an example of a monument in Denmark that can be interpreted as a seven square. Its angle is 36 degrees, probably the 36.8 of a 3-4-5 triangle or 4 by 3 rectangle as at Carnac (but with no possibility of it being the summer solstice sunrise that far north). Crowhurst was somewhat stimulated when I sent this to him but it has taken til now to comment on his work here. The miracle of his multiple square interactions is down to the properties of whole number tangents that will be further explored below and are currently miraculous.

Interaction of One-, Two-, and Three-Square Angles

If the diagonal of a single square, angle 45 degrees, then subracting the angle of a double square diagonal's angle will arrive perfectly at the angle of the diagonal of a triple square, relative to the base of the original single square. The tangents are tan(1) - tan(1/2) = tan(1/3). This can be presented geometrically as:

Interaction of Two-, Three- and Seven-Square Diagonals

The formula for this is tan(1/2) - tan(1/3) = tan(1/7), that is an arithmetic of angles. The realisation below is not the only possible manifestation of this relationship.

Now look at how the Tumulus St Michel in Carnac, overlooking the alignments, repeats the triple square angle of the alignments in Howard Crowhurst's book. The geometry aligns to the solar and lunar extreme, giving the midsummer sunrise 3-4-5 angle via two different constructions. One can use the septimal square angle, to reduce from the 45 degree angle of the single square, or the method of two triple squares (see next below) can also achieve that angle. Both are perfect transformations in angle.

Interaction of two Three-Squares to form a 3-4-5 triangle

Since tan(1) = tan(1/2) plus tan(1/3) and
because tan(1/3) plus tan(1/7) equals tan(1/2)
then by substitution,

tan(1/3) + tan(1/3) = tan(3/4),
which is to say that the smallest angle of a 3-4-5 triangle is two times the diagonal of a three square.

There are other relations with which to conclude this article soon