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?
"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 ?
"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.