Projective geometry

Projective geometry originated with the French mathematician Girard Desargues (1591–
1661) to deal with those properties of geometric figures that are not altered by projecting
their image, or “shadow,” onto another surface.
Differential geometry
The German mathematician Carl Friedrich Gauss (1777–1855), in connection with
practical problems of surveying and geodesy, initiated the field of differential geometry.
Using differential calculus, he characterized the intrinsic properties of curves and
surfaces. For instance, he showed that the intrinsic curvature of a cylinder is the same as
that of a plane, as can be seen by cutting a cylinder along its axis and flattening, but not
the same as that of a sphere, which cannot be flattened without distortion.
Non-Euclidean geometries
Beginning in the 19th century, various mathematicians substituted alternatives to
Euclid’s parallel postulate, which, in its modern form, reads, “given a line and a point
not on the line, it is possible to draw exactly one line through the given point parallel to
the line.” They hoped to show that the alternatives were logically impossible. Instead,
they discovered that consistent non-Euclidean geometries exist.
Topology
Topology, the youngest and most sophisticated branch of geometry, focuses on the
properties of geometric objects that remain unchanged upon continuous deformation—
shrinking, stretching, and folding, but not tearing. The continuous development of
topology dates from 1911, when the Dutch mathematician L.E.J. Brouwer (1881–1966)
introduced methods generally applicable to the topic.
History of geometry
The earliest known unambiguous examples of written records—dating from Egypt and
Mesopotamia about 3100 BCE—demonstrate that ancient peoples had already begun to
devise mathematical rules and techniques useful for surveying land areas, constructing
buildings, and measuring storage containers. Beginning about the 6th century BCE, the
Greeks gathered and extended this practical knowledge and from it generalized the
abstract subject now known as geometry, from the combination of the Greek
words geo (“Earth”) and metron (“measure”) for the measurement of the Earth.

In addition to describing some of the achievements of the ancient Greeks, notably

Euclid’s logical development of geometry in the Elements, this article examines some
applications of geometry to astronomy, cartography, and painting from classical Greece
through medieval Islam and Renaissance Europe. It concludes with a brief discussion of
extensions to non-Euclidean and multidimensional geometries in the modern age.
Ancient geometry: practical and empirical
The origin of geometry lies in the concerns of everyday life. The traditional account,
preserved in Herodotus’s History (5th century BCE), credits the Egyptians with
inventing surveying in order to reestablish property values after the annual flood of the
Nile. Similarly, eagerness to know the volumes of solid figures derived from the need to
evaluate tribute, store oil and grain, and build dams and pyramids. Even the
three abstruse geometrical problems of ancient times—to double a cube, trisect an angle,
and square a circle, all of which will be discussed later—probably arose from practical
matters, from religious ritual, timekeeping, and construction, respectively, in pre-Greek
societies of the Mediterranean. And the main subject of later Greek geometry, the theory
of conic sections, owed its general importance, and perhaps also its origin, to its
application to optics and astronomy.

While many ancient individuals, known and unknown, contributed to the subject, none
equaled the impact of Euclid and his Elements of geometry, a book now 2,300 years old
and the object of as much painful and painstaking study as the Bible. Much less is
known about Euclid, however, than about Moses. In fact, the only thing known with a
fair degree of confidence is that Euclid taught at the Library of Alexandria during the
reign of Ptolemy I (323–285/283 BCE). Euclid wrote not only on geometry but also on
astronomy and optics and perhaps also on mechanics and music. Only the Elements,
which was extensively copied and translated, has survived intact.

Euclid’s Elements was so complete and clearly written that it literally obliterated the
work of his predecessors. What is known about Greek geometry before him comes
primarily from bits quoted by Plato and Aristotle and by later mathematicians and
commentators. Among other precious items they preserved are some results and the
general approach of Pythagoras (c. 580–c. 500 BCE) and his followers.
The Pythagoreans convinced themselves that all things are, or owe their relationships to,
numbers. The doctrine gave mathematics supreme importance in the investigation and
understanding of the world. Plato developed a similar view, and philosophers influenced
by Pythagoras or Plato often wrote ecstatically about geometry as the key to the
interpretation of the universe. Thus ancient geometry gained an association with
the sublime to complement its earthy origins and its reputation as the exemplar of
precise reasoning.
Finding the right angle
Ancient builders and surveyors needed to be able to construct right angles in the field on
demand. The method employed by the Egyptians earned them the name “rope pullers”
in Greece, apparently because they employed a rope for laying out their construction
guidelines. One way that they could have employed a rope to construct right triangles
was to mark a looped rope with knots so that, when held at the knots and pulled tight,
the rope must form a right triangle. The simplest way to perform the trick is to take a
rope that is 12 units long, make a knot 3 units from one end and another 5 units from
the other end, and then knot the ends together to form a loop. However, the
Egyptian scribes have not left us instructions about these procedures, much less any
hint that they knew how to generalize them to obtain the Pythagorean theorem: the
square on the line opposite the right angle equals the sum of the squares on the other
two sides. Similarly, the Vedic scriptures of ancient India contain sections
called sulvasutras, or “rules of the rope,” for the exact positioning of sacrificial altars.
The required right angles were made by ropes marked to give the triads (3, 4, 5) and (5,
12, 13).

In Babylonian clay tablets (c. 1700–1500 BCE) modern historians have discovered
problems whose solutions indicate that the Pythagorean theorem and some
special triads were known more than a thousand years before Euclid. A right triangle
made at random, however, is very unlikely to have all its sides measurable by the same
unit—that is, every side a whole-number multiple of some common unit of
measurement. This fact, which came as a shock when discovered by the Pythagoreans,
gave rise to the concept and theory of incommensurability.
Locating the inaccessible

Chinese and Greek geometric theorems

A comparison of a Chinese and a Greek geometric theoremThe figure illustrates the equivalence of the
Chinese complementary rectangles theorem and the Greek similar triangles theorem.(more)
By ancient tradition, Thales of Miletus, who lived before Pythagoras in the 6th
century BCE, invented a way to measure inaccessible heights, such as the Egyptian
pyramids. Although none of his writings survives, Thales may well have known about a
Babylonian observation that for similar triangles (triangles having the same shape but
not necessarily the same size) the length of each corresponding side is increased (or
decreased) by the same multiple. The ancient Chinese arrived at measures of
inaccessible heights and distances by another route, using “complementary” rectangles,
as seen in the next figure, which can be shown to give results equivalent to those of the
Greek method involving triangles.
Estimating the wealth
A Babylonian cuneiform tablet written some 3,500 years ago treats problems
about dams, wells, water clocks, and excavations. It also has an exercise on circular
enclosures with an implied value of π = 3. The contractor for King Solomon’s swimming
pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23), used the
same value. However, the Hebrews should have taken their π from the Egyptians before
crossing the Red Sea, for the Rhind papyrus (c. 2000 BCE; our principal source for
ancient Egyptian mathematics) implies π = 3.1605.