In Book I of his Éléments, Euclid defines the objects of geometry and adds five "requests", which Proclus called postulates, to lay the foundations of his geometry, together with a number of "common notions". The first four postulates concern the construction of lines and circles and the universality of the right angle. Over the centuries, they have caused no difficulty. The same cannot be said of the most famous of his postulates, or axioms—the fifth in modern editions—known as the parallel axiom and now stated as follows: "Through a point not on a given line, exactly one line can be drawn parallel to that line." After twenty-two centuries of fruitless research, hyperbolic geometry was born. On November 3, 1823, János Bolyai (1802–1860) wrote to his father Farkas that he had "created a new world, a different world, out of nothing". At the same time, independently, Nikolai Lobachevsky (1792–1856) obtained a similar result.
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Projective geometry
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From then on, mathematicians no longer spoke of Geometry, but of geometries. Following Felix Klein's lead, geometry came to focus on finding the elements invariant under a characteristic group of transformations. In ordinary Euclidean geometry, for example, a figure "does not change" in shape or size under isometries such as rotations and translations; nor does its shape change under a change of scale, or dilation. Together, these transformations form the similarity group, which, in the complex plane, is simply an affine map of the form
Growing out of the study of perspective, or central projection, projective geometry has no notion of distance or orthogonality and does not distinguish among conic sections.
