In 1823, Janos Bolyai wrote to his father: "Out of nothing I have created a new universe." By which he meant that starting from the first 4 of Euclid's postulates and a modified fifth, he developed an expansive theory that, although quite unusual, did not seem to lead to any logical contradiction. Gauss expressed his conviction in consistency of the theory he had in mind in a letter in 1824. However, hesitant of the public reaction to the idea that, by the side of Euclidean, there is another geometry, he never published anything on the subject.En dan nog veel meer over dit model. Bij de links kan je dan meer vinden over deze wiskundigen en dan ben je toch weer een eind op weg... zou ik denken.
From our perspective, the situation was exactly the same as with Euclidean geometry. Euclid built an axiomatic theory by deriving a lot of theorems from his five postulates. No one had ever proved that continuing in Euclid's footstep would not lead to a contradiction. However, a 2000 year history elevated Elements on a pedestal of infallibility. Kant even stipulated that the universe had been built according to Euclid. Lobachevsky was quite aware of the problem and in later publications tried without success to redefine the notions of line and plane.
A breakthrough came in 1868 with the publication of Saggio di interpretazoine della geometria non euclidea by the Italian mathematician Eugenio Beltrami (1835-1900). Beltrami discovered that Lobachevsky's geometry admits an interpretation in terms of Euclidean geometry. From here it follows that if Lobachevsky's geometry leads to a contradiction, Euclidean geometry is as well contradictory. In other words, consistency of Euclidean geometry implies consistency of the geometry of Lobachevsky. The construction is now known as Beltrami-Klein model of Lobachevsky geometry. Sometimes it's called the projective model because it may be extended to conic sections and builds on an unusual usage of usual straight lines.
WvR
12-3-2004