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What a great article from 1946 Andres has found (see below)! Samuel Abraham Goudsmit was a major figure in modern physics. He was a student with Ehrenfest at Leiden in the 20's and then became a prof at UMich. During WWII he was a key figure at MIT's Radiation Lab and then he was the head of the scientific mission that was dispatched to see how close the Third Reich had come to building nuclear weapons. He spent his later years at Brookhaven Lab on Long Island, and he served for many years as the editor of the principal physics journal, The Physical Review. (Thanks to Wikipedia for refreshing my memory.)

It would be interesting to find out what occasioned the brief article on what we have been calling "that darned old cocked hat". The first thing he does is to give an explanation (as George H has) why the probability that the fix is contained with an average set of three LOPs is 25%. He goes on to show that the same probability for the case of four LOPs is 50%. Finally he comments without proof that the MPP in the 3 LOP case is located "at a point for which the distance to each side of the triangle is proportional to that side". I checked this out for the case of a right triangle, and it is indeed the Symmedian point.

But best of all he concludes with some good old common sense. "It would, however, be misleading to use this point in practice, as it might give the impression of a higher accuracy than is actually obtained. It is simpler and wiser to choose an arbitrary point inside the triangle, perhaps situated somewhat nearer to the shortest side in order to take into account roughly the knowledge of the most probable location."

I couldn't have said it better.

George B

Andres Ruiz wrote:
This could be of interest:

ACCURACY OF POSITION FINDING USING THREE OR FOUR LINES OF POSITION

SUMMARY
If at location is determined by the inter-
section of three lines of position, there is a
chance of only one-quarter that the true posi-
tion lies inside the residual triangle. lf the
location is determined by the intersection of
four lines of position, the chance is one-half
that the true location lies inside the larger
quadrangle formed by the four lines.

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