Thanks for that description. Oh, my elliptic curve calculator is definitely not equivalent to Guy's nomogram. Whereas Guy's nomogram is isomorphic to the real line, my version is isomorphic to the circle group. However, Guy's nomogram generalises very nicely to a quartic with no cubic term, such as y = x^4 + ax^2 + bx + c. We can then draw a couple of scales on that quartic, as before Assuming that this is positive-definite (which we can do just by increasing the value of c), we can then hit it with a projective transformation to result in a closed curve, avoiding all of the infinity issues my elliptic curve calculator suffers from. Unfortunately, the quartic nomogram doesn't really do anything useful: essentially, if you give it two points A and B, it will find two other points C and D such that A + B + C + D = 0. I doubt that this can be used for practical computation. Sincerely, Adam P. Goucher http://cp4space.wordpress.com/ ----- Original Message ----- From: "Mike Stay" <metaweta@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Wednesday, August 29, 2012 10:15 PM Subject: Re: [math-fun] {Spam?} Elliptic curve calculator Here's a snippet from an interview with Guy in which he explains the concept: Guy: ... OK. My first theorem is a very nice one. If you look in an early issue of the Mathematical Gazette, roughly the British equivalent of the Monthly, you’ll find “A Single Scale Nomogram.” I merely made the observation that a cubic equation with no x^2 term has zero for the sum of its roots. If you draw a cubic curve, y = x^3 + ax + b and put a straight line y = mx + c across it, the sum of the x-coordinates of the intersections is zero. If the curve is symmetrical about the origin (b = 0) and you change the sign of x on the negative half, then one coordinate is equal to the sum or difference of the other two. Combine this with the principle of the slide rule, which simply adds one chunk to another. For example, if the chunks are logs, you have multiplication and division. Anything you can do with a slide rule you can do with any single-scale nomogram. That was my first theorem, I suppose. On Wed, Aug 29, 2012 at 1:08 PM, Richard Guy <rkg@cpsc.ucalgary.ca> wrote:
See: A single scale nomogram, Math. Gaz., 33(1949) 43 or 37(1953) 39. R.
On Wed, 29 Aug 2012, Adam P. Goucher wrote:
You may be aware of the Abelian group operation on elliptic curves. We can exploit this to multiply, divide and square-root real numbers just by drawing a straight line on an elliptic curve marked with two logarithmic scales:
http://cp4space.wordpress.com/2012/08/29/elliptic-curve-calculator/
I've tried it on a printed version, and can achieve somewhere between 2 and 3 significant figures of precision, depending on the calculation.
Obviously it is of no practical use today, since we have electronic calculators and so forth. But in theory, this could have been made way back in 1830, when it could have been used for calculations in navigation and ballistics.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com/
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