Inge Lehmann analysed seismographs from large earthquakes (in particular the 1929 Murchison earthquake in New Zealand) and noticed something odd about the shock waves.
The way that the shock waves were moving through the planet was inconsistent with the prevailing wisdom, that the core was entirely liquid. So, in a 1936 paper, she concluded that the Earth must have a solid inner core within the liquid outer core. She was right.
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