Goofs

The Earth's mean radius is 6,371 kilometres, so even counting "crust" as being the outer surface, it is not "over six thousand five hundred kilometres" from crust to the centre, as is said during the lecture.

The professor says that the difference between dates in the Julian and Gregorian calendars is ten days. Ten days was the difference when the Gregorian calendar was introduced in 1582 and for the first 117 years of it. After that the Gregorian calendar dropped leap days in 1700 and 1800, so the difference between the two calendars had increased to twelve days by the film's period setting. So June 20th Gregorian would be June 8th in the Julian, not June 10th as Glaüben said.

(at about four minutes and 56 seconds into the film) In the film's English dubbed version, Professor Kristoff uses different measurement units in three consecutive sentences: "Because the distance between the Earth's crust and its core is over 6,500 KILOMETERS. And no man has ever descended to a depth of more than THREE MILES. So it's obvious we'll never have a glimmer of true knowledge until we are able to reach a depth of at least a HUNDRED LEAGUES."