Archimedes and Apollonius Archimedes (), a colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians. Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original. He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the
Archimedean property of finite numbers.
Apollonius of Perga () is mainly known for his investigation of conic sections. , 1648.
17th century: Descartes René Descartes (1596–1650) developed
analytic geometry, an alternative method for formalizing geometry which focused on turning geometry into algebra. In this approach, a point on a plane is represented by its
Cartesian (
x,
y) coordinates, a line is represented by its equation, and so on. In Euclid's original approach, the
Pythagorean theorem follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems. The equation :|PQ|=\sqrt{(p_x-q_x)^2+(p_y-q_y)^2} \, defining the distance between two points
P = (
px,
py) and
Q = (
qx,
qy) is then known as the
Euclidean metric, and other metrics define
non-Euclidean geometries. In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g.,
y = 2
x + 1 (a line), or
x2 +
y2 = 7 (a circle). Also in the 17th century,
Girard Desargues, motivated by the theory of
perspective, introduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry,
projective geometry, but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced. .
18th century Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763, at least 28 different proofs had been published, but all were found incorrect. Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem of
trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until
Pierre Wantzel published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include
doubling the cube and
squaring the circle. In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two, while doubling a cube requires the solution of a third-order equation.
Euler discussed a generalization of Euclidean geometry called
affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an
equivalence relation between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint).
19th century In the early 19th century,
Carnot and
Möbius systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.
Higher dimensions In the 1840s
William Rowan Hamilton developed the
quaternions, and
John T. Graves and
Arthur Cayley the
octonions. These are
normed algebras which extend the
complex numbers. Later it was understood that the quaternions are also a Euclidean geometric system with four real Cartesian coordinates. Cayley used quaternions to study
rotations in 4-dimensional Euclidean space. At mid-century
Ludwig Schläfli developed the general concept of
Euclidean space, extending Euclidean geometry to
higher dimensions. He defined
polyschemes, later called
polytopes, which are the
higher-dimensional analogues of
polygons and
polyhedra. He developed their theory and discovered all the regular polytopes, i.e. the n-dimensional analogues of regular polygons and
Platonic solids. He found there are six
regular convex polytopes in dimension four, and three in all higher dimensions. Schläfli performed this work in relative obscurity and it was published in full only posthumously in 1901. It had little influence until it was rediscovered and
fully documented in 1948 by
H.S.M. Coxeter. In 1878
William Kingdon Clifford introduced what is now termed
geometric algebra, unifying Hamilton's quaternions with
Hermann Grassmann's algebra and revealing the geometric nature of these systems, especially in four dimensions. The operations of the
Clifford algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modeled to new positions. The
Clifford torus on the surface of the
3-sphere is the simplest and most symmetric flat embedding of the Cartesian product of two circles (in the same sense that the surface of a cylinder is "flat").
Non-Euclidean geometry The century's most influential development in geometry occurred when, around 1830,
János Bolyai and
Nikolai Ivanovich Lobachevsky separately published work on
non-Euclidean geometry, in which the parallel postulate is not valid. Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates. In the 19th century, it was also realized that Euclid's ten axioms and common notions do not suffice to prove all of the theorems stated in the
Elements. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself. The very first geometric proof in the
Elements, shown in the figure above, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third
vertex. His axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the
completeness property of the real numbers. Starting with
Moritz Pasch in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of
Hilbert,
George Birkhoff, and
Tarski.
20th century and relativity . The rays of starlight were bent by the Sun's gravity on their way to Earth. This is interpreted as evidence in favor of Einstein's prediction that gravity would cause deviations from Euclidean geometry.
Einstein's theory of
special relativity involves a four-dimensional
space-time, the
Minkowski space, which is
non-Euclidean. This shows that non-Euclidean geometries, which had been introduced a few years earlier for showing that the
parallel postulate cannot be proved, are also useful for describing the physical world. However, the three-dimensional "space part" of the Minkowski space remains the space of Euclidean geometry. This is not the case with
general relativity, for which the geometry of the space part of space-time is not Euclidean geometry. For example, if a triangle is constructed out of three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity. A relatively weak gravitational field, such as the Earth's or the Sun's, is represented by a metric that is approximately, but not exactly, Euclidean. Until the 20th century, there was no technology capable of detecting these deviations in rays of light from Euclidean geometry, but Einstein predicted that such deviations would exist. They were later verified by observations such as the slight bending of starlight by the Sun during a solar eclipse in 1919, and such considerations are now an integral part of the software that runs the
GPS system. ==As a description of the structure of space==