Before everyone knew about the solid inner core, they thought that the inner core was liquid or molten rock. In 1880 the seismograph, which detects and records the movement of seismic waves, was invented.In 1929 a large earthquake occurred near New Zealand. Danish seismologist Inge Lehmann "the only Danish seismologist," as she once referred to herself—studied the shock waves and was puzzled by what she saw. A few P-waves, which should have been deflected by the core, were in fact recorded at seismic stations.
Lehmann theorized that these waves had traveled some distance into the core and then bounced off some kind of boundary. Her interpretation of this data was the foundation of a 1936 paper in which she theorized that Earth's center consisted of two parts: a solid inner core surrounded by a liquid outer core, separated by what has come to be called the Lehmann Discontinuity. Lehmann's hypothesis was confirmed in 1970 when more sensitive seismographs detected waves deflecting off this solid core.
Inge Lehmann, Danish Seismologist, who found out about the core.
Born in Denmark in 1888, Lehmann was a pioneer among women and scientists. Her early education was at a progressive school where boys and girls were treated exactly alike. This was a sharp contrast to the mathematical and scientific community she later encountered, about which she once protested to her nephew, Niles Groes, "You should know how many incompetent men I had to compete with—in vain." Groes recalls, "I remember Inge one Sunday in her beloved garden...with a big table filled with cardboard oatmeal boxes. In the boxes were cardboard cards with information on earthquakes...all over the world. This was before computer processing was available, but the system was the same. With her cardboard cards and her oatmeal boxes, Inge registered the velocity of propagation of the earthquakes to all parts of the globe. By means of this information, she deduced new theories of the inner parts of the Earth."
This video explains how we can know whats inside the earth:-
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Unsolved Mysteries
Non-FictionThis is an informational book on some of the world's unsolved mysteries, there are some videos linked so you have a broader understanding of the concept.