The cache is NOT at the listed coordinates: they refer to a
possible parking place.
Somewhere in Scotland there's a cache equidistant from three
points whose National Grid refs are:
- NT 10687 81154 - St. Margaret's Hope, Firth of Forth
- NO 15630 01736 - Near St. Serf's Island, Loch Leven
- NN 80605 27536 - E of Carn Chois, Loch Turret
(Don't bother trying to visit those places without your water
wings. There's no point, anyway, because they are literally miles
from GZ.)
The word "Geometry" comes from Greek words meaning "Earth" and
"measure" and the ancient Greeks applied it to what they saw
Egyptians doing; measuring and mapping fields so that after the
Nile's floods had washed away the boundaries, they could restore
farmers' properties, accurately and in full, once the waters had
receded.
(No doubt it came in useful when assessing the landowners for
tax, too.)
So in the image, that's not Messrs Mason and Dixon drawing their
famous line: It's a team of ancient Egyptians who have arrived in
Scotland, hired their kilts, and have begun training for this
problem using methods that Euclid would have recognised. Provided
their ropes are the same length, any point on their (dashed) line
is equidistant from the fixed pegs representing two given
points.
The question now is "How to use the third point they have been
given?".
The Greeks knew of course: Euclid's Elements: Book IV:
Proposition 5 is
"To Circumscribe a circle about a given triangle."
And to really labour the point, here's how it's
done.
However, this puzzle uses points that are widely spaced so they
don't fall on one OS map sheet nor, indeed, in one major grid
square (such as NS). Besides, ruler-and-compass geometrical
constructions are less precise than a GPS device.
So next up is Rene Descartes who not only convinced himself of
his own existence, but also invented graphs, charts and the
application of algebra to geometry. Analytic
Geometry. In his Cartesian coordinate system, the algebraic
solution to the puzzle is here:
Algebraic solution. The word "Algebra", by the way, comes from
Arabic al-jebr meaning "reunion of broken parts".
If those formulae turn you off, you could always use trial and
error: Guess a pair of coordinates for GZ then use Pythagoras
(another Greek) to find the distance to each given point and adjust
the guess until all three distances are equal. When they are,
there'll be no need for Geochecker.
A spreadsheet would be very handy for this and it's not such a
bad idea. Isaac Newton made the method of successive approximation
respectable.
By the way, this puzzle treats the National Grid as a plane
surface. A degree of latitude and a degree of longitude are
different in length around here so lat/long coordinates give the
wrong answer in the algebraic formulae mentioned above. (You WERE
worrying about that, weren't you ?)
When you log your find, please say which method you used
to find the coords.
Getting to GZ:
Swap your toga for normal hill-walking gear: weather-proof
clothes and sturdy boots. If you start in Menstrie, for example
, parking at the posted co-ordinates, you'll have to use some
gates but you WON'T have to climb over any walls. Nor will you have
to negotiate any crags, tricky slopes or burns.
The cache is NOT in any wall: it's behind some stones in a small
dip, perhaps where the builders of a nearby drystane dyke got their
rocks.
It's sheep country so keep your dog under close control.
The landowner's permission to place the cache is gratefully
acknowledged.
Congratulations Broccoli Soup on first FTF!