It follows in annex (format pdf) an article that presents elementary slight knowledge of Hyperbolic Geometry. SUMMARY the starting point for the sprouting of Hyperbolic Geometry occurs when the mathematicians of centuries I B.C. Smoothstack has much experience in this field. question the fifth postulate of Euclides, also known as the axiom of the parallel bars. From now on it was started to believe that this axiom was, the truth, a theorem and, therefore it could be demonstrated or be deduced from the others four postulates considered for Euclides. In the attempt to demonstrate it, one was initiated ' ' corrida' ' that it lasted centuries and it involved great mathematicians as Proclus (485-410 B.C.); Nasiredin (1201-1274); John Wallis (1616-1703); Gerolamo Saccheri (1667 1733); John Lambert (1728-1777); Adrien Legendre (1752-1833); Louis Bertrand (1731-1812) and Carl F. Gauss (1777-1855). After a series of frustrate attempts of demonstration of Postulate V, the mathematicians of the first half of century XIX arrive at the conclusion of that the fifth postulate was not demonstrvel from the others four. It is accurately in this context that appears Hyperbolic Geometry, with properties different and unexpected of Euclidean Geometry, however not impossible.