The length of a minute of longitude at longitude f⁰ (phi degrees) is smaller than one at the equator (l nautical mile) by the factor of cos f.

Thus if two positions (navels) A and B form the hypotenuse of a right-angled triangle whose other sides are the line of longitude through A and line of latitude f (phi) through B and if the angle at C is q0  (Fig 1) then cos q = BC/AC. If AC = N nautical miles (N minutes of latitude) then : cos q =N(cos j)/N = cos j

 So the angle of A to true east (or west) as observed from B is the same as the latitude of B.

We can identify several of these cos-lat triangles formed between ancient centres as follows.

The Cos-Lat triangle at Westminster

Westminster Abbey was built over a much earlier centre of worship - ¬probably a stone or wood circle associated with the Old Religion. Ancient tradition and other evidence points to a form of henge circle occupying the top of a low hill - ¬what is known as a "tothill". This is still remembered when we find Tothill Street running from near the Abbey down towards Buckingham Palace.

The whole area beside the River Thames was once tidal marshes and the area of Westminster was largely the delta of the River Teoburna (Tyburn). Thus there was very little wholly dry land. However a small dry island (Thorney or Thornea) occupied the area which today is Westminster. It was the top of this island that was developed as a "tothill". A tothill was a centre from which local astronomical and other scientific observation were made. The Westminster tothill was roughly the centre of Thorney island.

Probably to absorb its ancient sanctity Westminster Abbey was built over the centre of whatever artefacts had been erected by the astronomer-priests of the Old Religion,

The road from Dover - the only means of ingress or egress to the Continent for those without a boat -came to the banks of the Thames at the only practical fordable spot and it came ashore on Thorney. Thus Thorney once thronged with passing humanity and ancient tradition makes the island the most hallowed spot in southern Britain. And it has lost none of its sacredness.

The approximate centre of Westminster Abbey will be called navel A. Half a nautical mile to the southwest there is another preserved open area which is now the Westminster School Playing Fields (Fig 2 below)

The cross ( C) of the latitude through B and the longitude through A form (with AB) a cos-lat triangle in that BA to the centre of Westminster Abbey is 0.5 nautical miles (n.mi) while BC measures 0.5 cos 51⁰ 30' n.mi . Thus the angle ABC is the same as the local latitude (51⁰30')

We can assume that there once was a standing stone or post at navel A over which the Abbey was subsequently raised and so astronomer- surveyors at B (again suitably marked) had, by this means, preserved the local latitude. True east (B-C) could have been found and marked at either of the equinoxes.

Cos ABC could be measured directly - by finding the point C where the true north to A and the true west to B crossed- but also on a more limited scale.

The surveyors could take a standard measure of length - a standard rod - and measure say 10 rods due east from B to a point X, as shown in Fig 3 alongside.  An observer standing on the line BA could adjust his position to Y so that it lay due north of X and the distance BY is found to be n rods. Thus cos j is 10/n. In this case XY would be 16 rods to a very close approximation.

As Fig 2 shows the navel B lies in the southeast corner of the Playing Fields providing enough open area to the latitude triangle.

The Cos-Lat triangle at Troy

In the environs of ancient Troy and in the delta of the Scamander river there are two mounds (See fig 4 below).

The one closest to Troy is the Mound of Ilus, while that which is exactly 2 nautical miles to the northwest is the Tumulus of Achilles.

When we draw the triangle formed by the line of latitude through Ilus , the meridian through Achilles and its hypotenuse - the 2 n.mi. distance between Ilus and Achilles - then another cos-lat triangle is formed which is four-times bigger than the one at Westminster.

The base is 2 minutes of longitude i.e 2 cos j n.mi. The latitude of Troy is very close to 40 degrees N (39⁰58' N) and this is the angle between the two mounds measured from due W.

The Cos-Lat triangle between Badbury Rings and Knowlton Circles

Knowlton Circles (SU 025 100) is a complex of earthworks 5 nautical miles to the north east of the hill fort of Badbury Rings (ST 964 030) (Fig 5) below. Stones from what is thought to be an ancient temple have recently been discovered there by dowsing.

The meridians through Knowlton's main circle (in which there is a ruined church) and the centre of Badbury Rings are 5 minutes of longitude apart. Thus the sight line from Badbury through Knowlton is looking in a direction that is the same to due east as the local latitude. (50⁰ 50').

The Badbury - Knowlton connection is part of a complex which includes Pimperne the largest (but as yet unexcavated) long barrow in England (ST 917 105).

Pimperne forms with Badbury the diagonal of a small geodetic square of 4' lat, x 4' long, and it lies on the same parallel as Knowlton (Fig 3) The two are 9 minutes of longitude apart with the meridian through Badbury dividing this in the ratio of 4:5. Thus the sun at midday is due south at Pimperne 36 seconds of time before it is due south at Knowlton, but it is 16 seconds later than Pimperne and 20 seconds before Knowlton.

As the diagram shows Long Crichel Church (ST 978 103) divides the Pimperne-Knowlton distance in the ratio of 5:4 - the exact opposite of how Badbury Rings divides the meridians through Pimperne and Knowlton.