Einstein’s earliest writings

In 1881, 12 year old Albert Einstein was absorbed in a book about geometry (Lehrbuch der Geometrie zum Gebrauche an höheren Lehranstalten. Von Eduard Heis und Thomas Joseph Eschweiler) - a text that no modern twelve year old would read. On p76 something struck the young Einstein and he felt compelled to annotate his geometry book with what now stands as his earliest surviving writing. 

The text (on p76 of the book) describes the concept of a cylindrical surface as one that can be “developed” or unrolled into a plane. By considering the cylinder’s side as formed by a sequence of planes, one can imagine these planes rotating around their edges until they align flat with neighboring planes. This process divides the cylinder’s surface into prismatic spaces, whose lateral surfaces can similarly flatten. As the number of planes increases, these prismatic surfaces approximate the cylinder, showing how its geometry can be represented in a plane.

Einstein, then adds in the side margin “Der Beweis hat keinen Sinn, denn so gut wir annehmen können, daß sich diese prismatischen Räume entrollen lassen dürfen wir dies vom Cylinder sagen!”

In English - “The proof has no sense, because as well as we can assume that these prismatic spaces can be unrolled, we can say this from the cylinder!”

But was he right? 

Einstein’s critique of the proof regarding cylindrical surfaces highlights a logical flaw in the reasoning. He correctly points out that if one assumes the lateral surfaces of the prismatic spaces can be unrolled into a plane, then the same must hold for the cylinder itself, as the prismatic spaces are approximations of the cylinder. His statement underscores that the reasoning assumes the conclusion it aims to prove, making the proof circular and invalid from a rigorous mathematical perspective.

So yes - at age 12 Einstein correctly disproved part of his textbook.