The title says it all. The irrationality of $\pi$ was proved by Lambert in the 18th century, but the Greeks at the time of Pythagoras already knew that $\sqrt2$ and the golden ratio were irrational. Did the classical Greek mathematicians already suspect that $\pi$ is also irrational or did nobody think about this before Lambert? Did he settle an old question or did his result come more or less out of the blue?

  • $\begingroup$ Just want to point out that even if the circle could be squared, $\pi$ could still be irrational, because square roots can be constructed with ruler and compass. $\endgroup$
    – Spencer
    Commented Oct 10, 2020 at 22:53

1 Answer 1


"suspect" is hard to track...

François Viète in Variorum de rebus responsorum mathematics liber VIII (1593) discovered the first infinite product in the history of mathematics by giving an expression of $\pi$ with what is now called Viète's formula.

John Wallis, like Viète, expressed $\pi$ in the form of an infinite formula, but involving only rational operations.

William Brouncker transformed Wallis' formula into a continued fraction.

Leonhard Euler produced a lot of formulas involving $\pi$ without founding any "periodicty".

Thus, at least from Euler's time, the suspect that it is irrational must be quite common, and a new issue emerged: what type of number $\pi$ is, algebraic or transcendental ?

But see James Gregory (1638 – 1675)'s Vera circuli et hyperbolae quadratura(1667) that attempts

"to prove that π and e are transcendental but contains a subtle error."

See also: Lennart Berggren & Peter Borwein, Pi: A Source Book (Springer, 2004):

Gregory attempted to show that the area of a general sector of an ellipse, circle or hyperbola could not be expressed in terms of the areas of the inscribed and circumscribed triangle and quadrilateral using arithmetical operations and root extractions.

  • $\begingroup$ And all transcendental numbers are irrational. OTOH, they could suspect it was irrational before guessing it was transcendental. $\endgroup$
    – Mary
    Commented Sep 23, 2020 at 12:48
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    $\begingroup$ @Mary - ??? transcendental number: "The name "transcendental" comes from the Latin transcendĕre 'to climb over or beyond, surmount', and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that sin(x) is not an algebraic function of x. Euler, in the 18th century, was probably the first person to define transcendental numbers in the modern sense." $\endgroup$ Commented Sep 23, 2020 at 14:48
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    $\begingroup$ They started to suspect this in the ancient Greece, since the problem of "quadrature of a circle" apparently had no solution (as they found from experience). This problem is equivalent to showing that $\pi=a+b\sqrt{n}$ where $a,b$ are rational and $n$ is an integer. $\endgroup$ Commented Sep 23, 2020 at 15:04
  • $\begingroup$ @MauroALLEGRANZA the question was not about transcendental numbers, but irrational ones, and therefore without that information, your answer does not, in fact, answer it. $\endgroup$
    – Mary
    Commented Sep 23, 2020 at 21:14

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