Conic sections with and without algebra

A contributed talk for Antalya Algebra Days in the Nesin Mathematics Village, Wednesday, May 15, 2019

Abstract:

This is about how history may reveal a forgotten vision of mathematics. Mathematics is universal, but ways of understanding it are not. As has been argued in general terms [3], the Ancients did not secretly use algebra. This should not be understood pejoratively. People without keyboards or even ready supplies of paper may develop ways of understanding that we never feel the need for.
The square on the ordinate varies as the abscissa, in a parabola; in the ellipse and hyperbola, as the rectangle bounded by the two abscissas [1]. In Cartesian terms [2], the proportions are expressed by algebraic equations:
y² = ℓx,

y² = ℓx(2d − x)/2d.

With laborious algebraic manipulations, John Wallis [5] re-established that the curves defined above, if they contain (a, b), and if c = a + d, are fixed under the respective affine transformations
x′ = x − 2by/ℓ + a,

y′ = − y + b,

x′ = cx/d − 2by/ℓ + a,

y′ = −bx/d − cy/d + b,

It has been asserted [4] that Apollonius somehow understood this; but all modern proofs that I have found, as by de Witt, Euler, and Hugh Hamilton, lack the clarity of Apollonius's non-algebraic proof, which uses areas in a way that does not reduce to manipulations of lengths in the Cartesian fashion. The key is that the equations for the conics can be written as equations of a parallelogram with a triangle and a trapezoid respectively, all lying on one plane (which need only be an affine plane).
Apollonius of Perga. Conics. Books I–III. Green Lion Press, Santa Fe, NM, revised edition, 1998. Translated and with a note and an appendix by R. Catesby Taliaferro, with a preface by Dana Densmore and William H. Donahue, an introduction by Harvey Flaumenhaft, and diagrams by Donahue, edited by Densmore.
René Descartes. The Geometry of René Descartes. Dover Publi- cations, New York, 1954. Translated from the French and Latin by David Eugene Smith and Marcia L. Latham, with a facsimile of the first edition of 1637.
Michael N. Fried and Sabetai Unguru. Apollonius of Perga’s Conica: Text, Context, Subtext. Brill, Leiden Boston Köln, 2001.
Boris Rosenfeld. The up-to-date complete scientific English translation of the great treatise of Apollonius of Perga with the most comprehensive commentaries. Hosted by Svetlana Katok, accessed April 8, 2019.
John Wallis. De Sectionibus Conicis, Nova Methodo Expositis, Tractatus. Oxford, 1655. Accessed April 1, 2019.

Notes prepared for the talk, and revised afterwards (28 pages, size A5, 12 point type):