Above you can hopefully see a rare photo of none other than Ersjdamoo himself. He for the moment is stumped by Proposition 2-6 (chapter 2, proposition 6) of John A. Parker’s book, The Quadrature of the Circle (1874 edition).
The trouble began when Parker, in his explanation of Proposition 2-6, came out of the blue with, “…the perpendicular, a, b, mathematically determined after Playfair and Legendre’s method…”
What on earth was the Playfair/Legendre method? After some digging, it was found that a John Playfair and an Adrien-Marie Legendre had once existed. (Background: “Tyrannical Decree Against the Quadrature”, Ersjdamoo’s Blog entry of August 19, 2013.)
In my copy of J. Ralston Skinner’s book, Key to the Hebrew-Egyptian Mystery in the Source of Measures (1894), there is found, way in the back, in a “Supplement To Source of Measures”, valuable material on “The Legendre and Playfair Method.”
A critique of the Playfair/Legendre method begins the discourse. There can be erroneous deductions connected with the subject-matter of approximate values. Case in point: Isaac Newton’s Lemma 1 from his Philosophiæ Naturalis Principia Mathematica. Lemma 1 is “manifestly untrue,” it is claimed. Lemma 1 had claimed that, “Quantities and the ratio of quantities which, in any finite time, converge continually to equality, and, before that time, approach nearer, the one to the other, than by any given difference, ultimately become equal.”
Legendre and Playfair had used such an idea of an ultimate equality, over time, and applied it to curved and straight lines. A polygon was bisected to 6144 sides. Given enough bisections, the polygon would become a circle, in an infinite time. Could this be a way to “square the circle”?
Which all brings us to Parker’s 6th Proposition in his chapter 2: “The circumference of a circle, such that its half being multiplied by radius, to which all other radii are equal, shall express the whole area of the circle, by the properties of straight lines, is greater in value in the sixth decimal place of figures than the same circumference in any polygon of 6144 sides, and greater also than the approximation of geometers at the same decimal place in any line of figures.”
And there I shall leave it for the moment, with the preliminary admission, Ersjdamoo is stumped.