In another thread (The 47th Problem), molder wrote:
> People said I was dreaming if I thought the ancients used Trigonometry.
I’d go one step further and venture that some people might say he was mad to do so. However, I have some sympathy with his view, but feel that the matter might be more simple than Jim suggests.
In my analysis of stone circles, I have noticed that, in addition to geometry, the builders may well have been using common approximations to trigonometric ratios. This is particularly striking if using the perimetric and diametric units I suggest as being derived from Thom’s megalithic yard (the perimetric unit is one-sixteenth of a megalithic yard multiplied by pi).
There are many sites at which such ratios appear to have been used, but perhaps the most striking in potential might be the Recumbent Stone Circle (RSC) Esslie the Greater in Kinkardineshire, Scotland.
Image may be NSFW.
Clik here to view.
The design would be what Thom refers to as egg-shaped Type I based on a Pythagorean triangle of [3, 4, 5], but I prefer to see such designs based on a defining inner circle that is a simple fraction of the diameter of the originating circle (as with Thom’s flattened circles) and an angle that is a simple rational (common) fraction of a revolution (either 90 or 60 degrees at the apex).
There are 3 rings. The diameter of the outer circle is about 23m, that of the inner is about 6.2m and the intermediate circle is about 17.5m. It is interesting to note a potential geometrical progression from the inner circle through the intermediate to the originating circle. Could this be:
1. A circle (defining the inner kerb)
2. within an equilateral triangle
3. within a circle
4. within a square
5. within a circle (defining the intermediate outer kerb)
6. within a pentagon
7 within a circle (defining the flankers of the recumbent stone)
8 within a decagon (upon which the recumbent sits)
9 within an originating circle
10. which is the base of an oval defined geometrically from the inner circle at 1?
Thom estimates the diameter of the outer circle at 28MY, the intermediate circle at 22MY and gives no estimate for the inner circle. The 14:11 ratio no doubt appealed to him.
However, assigning a value of 120 diametric units (7.5MY) to the inner circle, the diameters of the other circles would be 340 units (21.25MY) and 440 units (27.5MY) being 17.6m and 22.8m if based on Thom’s yard and on the suggested geometry.
Ardent supporters of Thom will declare that this cannot possibly be correct as Thom pronounced otherwise, but consider that the geometrical progression would be supported by approximate trigonometric ratios of:
sin45 = 12/17
cos36 = 17/21
cos18 = 21/22
In the figure below, the gaps are expressed in numbers of perimetric units.
Image may be NSFW.
Clik here to view.
Of course, this is not to say that this was done, and archaeologists will doubtless declare it to be a pure accident of number, but, then, archaeologists would not dream of testing to determine whether a similar thing might happen at other sites. It does (and, notably, in addition, tan30 = 15/26, or as in this case 26/45).
I suggest that all stone circles appear to be pure geometrical constructions with their axes and gaps common fractions of a revolution, based on number and incorporating simple trigonometric ratios. Now people will really know I must be mad!
> People said I was dreaming if I thought the ancients used Trigonometry.
I’d go one step further and venture that some people might say he was mad to do so. However, I have some sympathy with his view, but feel that the matter might be more simple than Jim suggests.
In my analysis of stone circles, I have noticed that, in addition to geometry, the builders may well have been using common approximations to trigonometric ratios. This is particularly striking if using the perimetric and diametric units I suggest as being derived from Thom’s megalithic yard (the perimetric unit is one-sixteenth of a megalithic yard multiplied by pi).
There are many sites at which such ratios appear to have been used, but perhaps the most striking in potential might be the Recumbent Stone Circle (RSC) Esslie the Greater in Kinkardineshire, Scotland.
Image may be NSFW.
Clik here to view.
The design would be what Thom refers to as egg-shaped Type I based on a Pythagorean triangle of [3, 4, 5], but I prefer to see such designs based on a defining inner circle that is a simple fraction of the diameter of the originating circle (as with Thom’s flattened circles) and an angle that is a simple rational (common) fraction of a revolution (either 90 or 60 degrees at the apex).
There are 3 rings. The diameter of the outer circle is about 23m, that of the inner is about 6.2m and the intermediate circle is about 17.5m. It is interesting to note a potential geometrical progression from the inner circle through the intermediate to the originating circle. Could this be:
1. A circle (defining the inner kerb)
2. within an equilateral triangle
3. within a circle
4. within a square
5. within a circle (defining the intermediate outer kerb)
6. within a pentagon
7 within a circle (defining the flankers of the recumbent stone)
8 within a decagon (upon which the recumbent sits)
9 within an originating circle
10. which is the base of an oval defined geometrically from the inner circle at 1?
Thom estimates the diameter of the outer circle at 28MY, the intermediate circle at 22MY and gives no estimate for the inner circle. The 14:11 ratio no doubt appealed to him.
However, assigning a value of 120 diametric units (7.5MY) to the inner circle, the diameters of the other circles would be 340 units (21.25MY) and 440 units (27.5MY) being 17.6m and 22.8m if based on Thom’s yard and on the suggested geometry.
Ardent supporters of Thom will declare that this cannot possibly be correct as Thom pronounced otherwise, but consider that the geometrical progression would be supported by approximate trigonometric ratios of:
sin45 = 12/17
cos36 = 17/21
cos18 = 21/22
In the figure below, the gaps are expressed in numbers of perimetric units.
Image may be NSFW.
Clik here to view.
Of course, this is not to say that this was done, and archaeologists will doubtless declare it to be a pure accident of number, but, then, archaeologists would not dream of testing to determine whether a similar thing might happen at other sites. It does (and, notably, in addition, tan30 = 15/26, or as in this case 26/45).
I suggest that all stone circles appear to be pure geometrical constructions with their axes and gaps common fractions of a revolution, based on number and incorporating simple trigonometric ratios. Now people will really know I must be mad!