# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: A simple three-body fix puzzle
From: UNK
Date: 2010 Dec 10, 21:39 +0000
Peter,

I was aware of these calculus free proofs. But none was given by Antoine.

By the way, calculus is not only unnecessary in this case, it is also unnecessary in the general case. Completion of the square always works. This is how Lemoine did it in his paper. (I don't have the reference handy.). In Mystic It took me more than half an hour on the blackboard to go through it and we all needed a beer afterward. The point of the exercise was to proof that even Bowditch's cook could have understood why the MPP is where it is.

Herbert Prinz

On 2010-12-10 19:37, P H wrote:
Calculus indeed provides one way of proving this.  It is not necessary in this case, though.  I attach two non-calculus proofs; the first is geometric, and the second is algebraic ("completion of a square").

Peter Hakel

From: Herbert Prinz <666@poorherbert.org>
To: NavList@fer3.com
Sent: Fri, December 10, 2010 8:00:24 AM
Subject: [NavList] Re: A simple three-body fix puzzle

[parts deleted by PH]

To those who will find my "proof" neither easier nor more immediate than Antoine's I will respond that what you find easy depends on what you already know. For example, Antoine concludes his proof with the bold assertion

"Since the sum " up + us " is constant (and equal to d), the quantity " up**2 + us**2 " is minimum when up = us."

Well, who says? Most of us know that, but strictly, it should not be taken for granted. This is exactly the place where the differential calculus is hidden that Antoine wanted to avoid in the first place.

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