This past spring, 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.
From Weeks & Adkins, Second Course in Algebra, p. 395
The exercise uses no more mathematical tools than may be found in an algebra course in high school; yet it serves the purposes of university mathematics, as I understand them.
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.
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
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
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
