Tuesday, 28 July 2009

Rediscovered, or Did He Cheat?

In a recent post I commented on the "discovery" by a British High School student,Anthony Bayes, of a graphic relationship that allowed every point on the coordinate plane to represent a quadratic equation of the form x2+bx+c=0.

I was impressed that a high school age student would come up with the idea of a transformation from a quadratic to a point on the plane. THEN.... a few days ago I came across a post from the Mathematical Gazette, January of 1913, entitled "A Graphic Solution of the Equation xn-px+q=0, much of which dealt with the idea that the curve 4y=x2 (which the author called the Discriminant Curve) would serve as a graphic solution approach for any quadratic of the form x2-px+q=0.
Note the use of -p, a step up from what young Anthony had used, making it a slightly more useful approach, and perhaps testifying to the independence of his discovery. The author of the piece, A. Lodge, (with no other information given), did not claim it as original, nor site another use.

So the solutions to the equation x2-px+q=0 are found by the points where the tangents to 4y=x2 intersect the x-axis.

The article also pointed out, and I admit I had never thought of this, that the solutions to x2-px+q=0 are the same as the solutions to $x^2-\frac{p}{k}x+\frac{q}{k^2}=0$. It is easy enough to prove, but for the use of a nomograph like the approximations using the graph of 4y=x2, it allows you to work on a much narrower range of x and y values.

The article also referred to solutions of higher powers of trinomials that could be solved in the same way, but my little mind has not grasped it enough to explain it yet, so I save that for another day...
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