Category Archives: Conic Sections

Points of an Ellipse

This is about an image that is intended

  • to be decorative,
  • to establish the mathematical construction, with ruler and compass, of points of an ellipse.

Diagram with colored regions whose borders are discussed in the text

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An Exercise in Analytic Geometry

This past spring (of 2020), when my university in Istanbul was closed (like all others in Turkey) against the spread of the novel coronavirus, I created for my students an exercise, to serve at least as a distraction for those who could find distraction in learning.

Diagram from textbook page shows, centered at the origin of coordinates, a circle and an ellipse whose four points of intersection are traversed by two lines in red through the origin
From Weeks & Adkins, Second Course in Algebra, p. 395

Note added, April 17, 2023: An account of the mathematics involved in the exercise would ultimately be published as: Pierce, D. (2021). “Conics in Place.” Annales Universitatis Paedagogicae Cracoviensis | Studia Ad Didacticam Mathematicae Pertinentia, 13, 127–150.

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Elliptical Affinity

After Descartes gave geometry the power of algebra in 1637, a purely geometrical theorem of Apollonius that is both useful and beautiful was forgotten. This is what I conclude from looking at texts from the seventeenth century on.

In ellipse, colored triangles move to illustrate theorem Continue reading

The Hyperbola

Here is the model that I made of the hyperbola, or rather the conjugate hyperbolae, as Apollonius calls them.

Conjugate hyperbolae and their common diameter

Conjugate hyperbolae and their common diameter

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The Parabola

I do not now recall my specific inspiration; but in January of 2012, sitting at home in Istanbul, I cut up a cardboard box in order to make a model of a parabola quâ conic section.

January 14, 2012

January 14, 2012

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NL III: “Body As Mind”

Index to this series

In Chapter I of The New Leviathan, we stipulated that natural science, the “science of body,” must be free to pursue its own aims. But we ourselves are doing science of mind, and:

1. 85. The sciences of mind, unless they preach error or confuse the issue by dishonest or involuntary obscurity, can tell us nothing but what each can verify for himself by reflecting on his own mind.

All of us can be scientists of mind, if only we are capable of reflection: Continue reading