Category Archives: Mathematical Topics

Points of an Ellipse

February 3, 2023 – 2:08 pm

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

Continue reading →

By David Pierce | Also posted in Conic Sections, Mathematics, Visual Art | Tagged | Comments (0)

More of What It Is

September 21, 2020 – 7:08 pm

I say that mathematics is the deductive science; and yet there would seem to be mathematicians who disagree. I take up two cases here.

From Archimedes, De Planorum Aequilibriis,
in Heiberg’s edition (Leipzig: Teubner, 1881)

Continue reading →

By David Pierce | Also posted in absolute presuppositions, Archimedes, Calculus, Collingwood, Euclid, Mathematics, Philosophy, Philosophy of Mathematics | Tagged , , | Comments (2)

An Exercise in Analytic Geometry

August 5, 2020 – 6:26 pm

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.

Continue reading →

By David Pierce | Also posted in Conic Sections, Descartes, Education, Exposition, Mathematics | Tagged | Comments (4)

Salvation

February 24, 2020 – 7:06 am

Because Herman Wouk was going to put physicists in a novel, Richard Feynman advised him to learn calculus: “It’s the language God talks.” I think I know what Feynman meant. Calculus is the means by which we express the laws of the physical universe. This is the universe that, according to the mythology, God brought into existence with such commands as, “Let there be light.” Calculus has allowed us to refine those words of creation from the Biblical account. Credited as a discover of calculus, as well as of physical laws, Isaac Newton was given an epitaph (ultimately not used) by Alexander Pope:

Nature and Nature’s laws lay hid in night:
God said, Let Newton be! and all was light.

I don’t know, but maybe Steven Strogatz quotes Pope’s words in his 2019 book, Infinite Powers: How Calculus Reveals the Secrets of the Universe. This is where I found out about Wouk’s visit with Feynman. I saw the book recently (Saturday, February 22, 2020) in Pandora Kitabevi here in Istanbul. I looked in the book for a certain topic that was of interest to me, but did not find it; then I found a serious misunderstanding.

book cover: Steven Strogatz, Infinite Powers Continue reading →

By David Pierce | Also posted in Archimedes, Calculus, Euclid, Mathematics, Philosophy of Mathematics | Tagged , , , , , , , , | Comments (4)

Elliptical Affinity

April 17, 2019 – 2:21 pm

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 →

By David Pierce | Also posted in Conic Sections, Descartes, Exposition, Mathematics | Tagged , , , , , , , , , | Comments (7)

The Hyperbola

October 10, 2014 – 3:50 pm

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


Continue reading →

By David Pierce | Also posted in Conic Sections, Education, Exposition, History, Mathematics | Tagged , , | Comments (4)

The Parabola

October 8, 2014 – 11:19 am

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


Continue reading →

By David Pierce | Also posted in Conic Sections, Exposition, History, Mathematics, Nesin Mathematics Village | Tagged , , | Comments (1)

NL III: “Body As Mind”

January 25, 2014 – 7:17 am

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 →

By David Pierce | Also posted in Collingwood, Conic Sections, Descartes, Education, Euclid, History, Mathematics, New Leviathan, overlap of classes, Philosophy, Philosophy of History, Plato, Principles of Art, Psychology, Science | Tagged , , , , , , | Comments (6)

Learning mathematics

May 14, 2013 – 11:23 am

This is mostly reminiscences about high school. I also give some opinions about how mathematics ought to be learned. The post originally formed one piece with my last article, “Limits.”

I learned calculus, and the epsilon-delta definition of limit, in Washington D.C., in my last two years at St Albans School, in a course taught by a peculiar fellow named Donald J. Brown. The first of these two years was officially called Precalculus Honors, but some time in that year, we started in on calculus proper.

Continue reading →

By David Pierce | Also posted in Archimedes, Calculus, Education, Euclid, G. H. Hardy, Logic, Mathematics, Philosophy of Mathematics, Strunk and White | Tagged , , , , , , , , , , , | Comments (5)

Limits

May 4, 2013 – 12:54 pm

This is about limits in mathematics: both the technical notion that arises in calculus, and the barriers to comprehension that one might reach in one’s own studies. I am going to say a few technical things about the technical notion, but there is no reason why this should be a barrier to your reading: you can just skip the paragraphs that have special symbols in them.

Looking up something else in the online magazine called Slate, I noted a reprint of an article called “What It Feels Like to Be Bad at Math” from a blog called Math With Bad Drawings by Ben Orlin. Now teaching high-school mathematics, Mr Orlin recalls his difficulties in an undergraduate topology course. His memories help him understand the difficulties of his own students. When students do not study, why is this? It is because studying makes them conscious of how much they do not understand. They feel stupid, and they do not like this feeling. Continue reading →

By David Pierce | Also posted in Archimedes, Calculus, Education, Mathematics, Philosophy of Mathematics, Plato | Tagged , , , , | Comments (6)