The evidence for plate tectonics comes many fields, such as palaeontology, geophysics and climate study.
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The first evidence was the matching of continents and their rock types across oceans. This had been realised since the 17th Century. Theories of how continents moved were ad hoc at best. Alfred Wagner, an early 1900's German Meteorologist proposed that the continents of South America, Africa, India and Australia had been one continent in the past due to their similarity of fossils and palaeoclimate. Wegner plotted the position of the poles for these climate belts. The climate belts had moved or the continents had moved in order to fit these belts together. However, no mechanism was present for the continents to move, so Wegner proposed that the continents moved through the ocean crust, forming a mountain belt at their front edge. Not many geologists believed him though, especially in the U.S where most geologists were at that time.
However, it wasn't until the 1960's when Fred Vine and Drummond Matthews, two British geologists, recorded symmetric magnetic anomalies across ocean ridges that plate tectonics took off. What they recorded was a series of magnetic stripes in the oceanic crust (Fig. given below). These stripes correspond to times when the Earth's magnetic field reversed polarity and when it was the same polarity as it is now. Figure two below shows the current situation, by swapping the north and south poles around, you get a reversed field. By studying theses stripes rates of spreading could be calculated. Figure one shows a record of magnetic stripes and how they are interpreted. It is clear that in order to get this pattern new oceanic crust must be formed continuously (on a geological timescale). The conclusion was that the mantle melted at these ridges, moving the ocean floor. The idea of moving ocean crust had just been published by an American geologist, Hess, called sea-floor spreading.
Magnetic stripes that are found at mid-ocean ridges