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How does one proceed to trisect an angle using the cardioid? It is known that Etienne Pascal, father of Blaise Pascal, has devised a way to do it. I haven´t found the method in my search on the literature.

Best Regards.

Petruchio de São Zeno
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  • Do you have reason to believe that a cardioid is suited to trisection? Is your motivation to find a novel way to use the cardioid? or a novel way to trisect an angle? – Blue Nov 27 '18 at 06:20
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    A trisectrix is a curve that can be used to trisect an arbitrary angle. Cardioid isn't listed on above wiki entry. The Limaçon trisectrix seems to be the trisectrix closest to a cardioid. – achille hui Nov 27 '18 at 06:35
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    The Limaçon trisectrix is listed in deed. There is an observation in the book The Budget of Trisections (Underwood Dudley) at pages 9-10 that goes like this: "Although the trisection is no longer the obsession it was to Greek mathematicians, new methods have been devised in modern times. Etienne Pascalthe father of Blaise (1623-1662) the mathematician, philosopher, and writer, trisected with a cardioid." – Petruchio de São Zeno Nov 27 '18 at 07:17
  • @PetruchiodeSãoZeno: Ah. Well, then ... You already have it on good authority that it is indeed possible to trisect an angle with a cardioid. So, your question really is: What method did Pascal, Sr, (and/or others) devise? Interestingly, a 1907 American Mathematical Monthly article "The Trisection Problem" (JSTOR link) opens with a casual mention that "The solution of this problem by means of the quadratrix, conchoid, and the cardioid are well known [...]". – Blue Nov 27 '18 at 09:48
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    Thank you, Blue. I have edited the question now so that the initial answers keep sounding useful. I realized that I was in fact interested in the method itself. I am not putting in doubt what these authors have said. – Petruchio de São Zeno Nov 27 '18 at 16:25
  • Curiously, this description of the "Limaçon of Pascal", $r = b + 2 a\cos\theta$, states that "When $b=2a$, then the limacon becomes a cardioid while if $b=a$ then it becomes a trisectrix." Likewise for "limacon" entries on Wikipedia and Xah Code, the cardioid and trisectrix are mentioned as separate special cases of the curve. I wonder if "cardioid" is being (mis)used to mean "limaçon" by Dudley and others. – Blue Nov 28 '18 at 06:01
  • It seems to be a possibility although I think it's not likely, for the cardioid is a singular curve. – Petruchio de São Zeno Nov 30 '18 at 03:51

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