Hereinafter we are going to show that Bolyai Janos was preoccupied by the problem of the Duplication of the Cube, which was unknown until now by the rich Bolyai-literature.
This problem was solved using the Parabola, the Hyperbola and the Cissoide already in the ancient times. The Cissoide was created by Diocles especially for the constuction of the Duplication of the Cube without Compass and Straightedge. The hereinafter "deciphered" document of Bolyai was written during his university studies. In his study he presents the solutions discovered by then and tries to give a new one. We transcribed his notations to the present-day use and complemented it where it was necessary.
The mathematics historically background and the explication is very detailed described by Van derWaerden in Erwachende Wissenschaft [7], which is to find on English, German and Hungarian, too. That's because we dispense with this [8].