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Inspired by the question Discovery of Earth's magnetic field, the article from New Zealand's webpage Discovery of the Earth’s magnetic field, also states that

The earth's dynamo is unstable, as is shown by magnetic reversals, when the polarity of the whole magnetic field changes over.

When was the fact that Earth's magnetic field reversed discovered?

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Cox et al. 1964 gives a good account on the different steps of how this discovery was made.
The most important first step was to acknowledge the fact that some rocks (namely magmatic rocks) acquired a remanent magnetism based on the magnetic field in which they were "cooked" (Brunhes 1906).
Followed observations of different directions in remanent magnetism of some lavas by Chevallier 1925, thus implying that the surrounding magnetic field changed direction.
Matuyama 1929 then managed to place the latest reversal of Earth's magnetic field in a stratigraphic context: it is now known to have occured at 0.781 Ma. The event later took his name, "Matuyama reversed polarity interval", now known more formally as chron C1r-1r (the current chron, C1n, is by the way nicknamed "Brunhes normal polarity interval").

So, interestingly, the discovery of reversed interval of magnetic polarity actually predates the discovery of the mechanism explaining earth magnetic field.

References:
Brunhes, 1906. Recherches sur la direction d'aimantation des roches volcaniques. Journal de Physique Théorique et Appliquée, 5: 705-724.
Chevallier, 1925. L'aimantation des laves de l'Etna et l'orientation du champ terrestre en Sicile du XIIe au XVIIe siècle. Bulletin Volcanique, 2(2): 234-244.
Cox, Doell & Dalrymple, 1964. Reversals of the Earth's magnetic field. Science, 144 (3626): 1537-1543.
Matuyama, 1929. On the Direction of Magnetisation of Basalt in Japan, Tyôsen and Manchuria. Proceedings of the Imperial Academy, 5: 203-205.

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