GeometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
(
GreekGreek is an independent branch of the IndoEuropean family of languages. Native to the southern Balkans, it has the longest documented history of any IndoEuropean language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the other being the study of numbers (
arithmeticArithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daytoday counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
).
Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
, who introduced mathematical rigor and the axiomatic method still in use today. His book,
The ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.
In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See
areas of mathematicsMathematics has become a vastly diverse subject over history, and there is a corresponding need to categorize the different areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different...
and
algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
.)
Early geometry
The earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the
ancient Indus ValleyThe Indus Valley Civilization was a Bronze Age civilization that was located in the northwestern region of the Indian subcontinent, consisting of what is now mainly modernday Pakistan and northwest India...
(see Harappan Mathematics), and ancient
BabyloniaBabylonia was an ancient cultural region in centralsouthern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in centralsouthern Mesopotamia (presentday Iraq), with Babylon as its capital. Babylonia emerged as...
(see
Babylonian mathematicsBabylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...
) from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in
surveyingSee Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or threedimensional position of points and the distances and angles between them...
,
constructionIn the fields of architecture and civil engineering, construction is a process that consists of the building or assembling of infrastructure. Far from being a single activity, large scale construction is a feat of human multitasking...
,
astronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of
calculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
. For example, both the
EgyptiansEgyptians are nation an ethnic group made up of Mediterranean North Africans, the indigenous people of Egypt.Egyptian identity is closely tied to geography. The population of Egypt is concentrated in the lower Nile Valley, the small strip of cultivable land stretching from the First Cataract to...
and the
BabylonBabylon was an Akkadian citystate of ancient Mesopotamia, the remains of which are found in presentday Al Hillah, Babil Province, Iraq, about 85 kilometers south of Baghdad...
ians were aware of versions of the
Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
about 1500 years before
PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
; the Egyptians had a correct formula for the volume of a
frustumIn geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....
of a square pyramid; the Babylonians had a trigonometry table.
Egyptian geometry
The ancient Egyptians knew that they could approximate the area of a circle as follows:



 Area of Circle ≈ [ (Diameter) x 8/9 ]^{2}.
Problem 50 of the
AhmesAhmes was an ancient Egyptian scribe who lived during the Second Intermediate Period and the beginning of the Eighteenth Dynasty . He wrote the Rhind Mathematical Papyrus, a work of Ancient Egyptian mathematics that dates to approximately 1650 BC; he is the earliest contributor to mathematics...
papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter. This assumes that
π' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
is 4×(8/9)² (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the
BabyloniaBabylonia was an ancient cultural region in centralsouthern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in centralsouthern Mesopotamia (presentday Iraq), with Babylon as its capital. Babylonia emerged as...
ns (25/8 = 3.125, within 0.53 percent), but was not otherwise surpassed until
ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
' approximation of 211875/67441 = 3.14163, which had an error of just over 1 in 10,000.
Interestingly, Ahmes knew of the modern 22/7 as an approximation for pi, and used it to split a hekat, hekat x 22/x x 7/22 = hekat; however, Ahmes continued to use the traditional 256/81 value for pi for computing his hekat volume found in a cylinder.
Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for π of 3.111...
The two problems together indicate a range of values for Pi between 3.11 and 3.16.
Problem 14 in the
Moscow Mathematical PapyrusThe Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
gives the only ancient example finding the volume of a
frustumIn geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....
of a pyramid, describing the correct formula:
Babylonian geometry
The Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as onetwelfth the square of the circumference, which would be correct if
π is estimated as 3. The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The
Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
was also known to the Babylonians. Also, there was a recent discovery in which a tablet used
π as 3 and 1/8. The Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a timemile used for measuring the travel of the Sun, therefore, representing time.
Classical Greek geometry
For the ancient
GreekGreece , officially the Hellenic Republic , and historically Hellas or the Republic of Greece in English, is a country in southeastern Europe....
mathematiciansGreek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trialanderror to logical deduction; they recognized that geometry studies "eternal forms", or abstractions, of which physical objects are only approximations; and they developed the idea of the
"axiomatic method"In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
, still in use today.
Thales and Pythagoras
ThalesThales of Miletus was a preSocratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition...
(635543 BC) of
MiletusMiletus was an ancient Greek city on the western coast of Anatolia , near the mouth of the Maeander River in ancient Caria...
(now in southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived.
PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
(582496 BC) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and traveled to
BabylonBabylon was an Akkadian citystate of ancient Mesopotamia, the remains of which are found in presentday Al Hillah, Babil Province, Iraq, about 85 kilometers south of Baghdad...
and
EgyptEgypt , officially the Arab Republic of Egypt, Arabic: , is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Southwest Asia. Egypt is thus a transcontinental country, and a major power in Africa, the Mediterranean Basin, the Middle East and the Muslim world...
. The theorem that bears his name may not have been his discovery, but he was probably one of the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of
incommensurable lengthsIn mathematics, two nonzero real numbers a and b are said to be commensurable if a/b is a rational number.History of the concept:...
and
irrational numberIn mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b nonzero, and is therefore not a rational number....
s. (There is no evidence that Thales provided any deductive proofs, and in fact, deductive mathematical proofs did not appear until after Parmenides. At best, all that we can say about Thales is that he introduced various geometric theorems to the Greeks. The idea that mathematics was from its inception deductive is false. At the time of Thales, mathematics was inductive. This means that Thales would have "provided" empirical and direct proofs, but not deductive proofs.)
Plato
PlatoPlato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
(427347 BC), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, "Let none ignorant of geometry enter here." Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but compass and straightedge – never measuring instruments such as a marked
rulerA ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines...
or a
protractorIn geometry, a protractor is a circular or semicircular tool for measuring an angle or a circle. The units of measurement utilized are usually degrees.Some protractors are simple halfdiscs; these have existed since ancient times...
, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of possible
compass and straightedgeCompassandstraightedge or rulerandcompass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....
constructions, and three classic construction problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system.
AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
(384322 BC), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see
LogicIn philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
) which was not substantially improved upon until the 19th century.
Euclid
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
(c. 325265 BC), of
AlexandriaAlexandria is the secondlargest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
, probably a student of one of Plato’s students, wrote a treatise in 13 books (chapters), titled
The Elements of GeometryEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, in which he presented geometry in an ideal
axiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be selfevident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
atic form, which came to be known as
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. The treatise is not a compendium of all that the Hellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by
Ptolemy IPtolemy I Soter I , also known as Ptolemy Lagides, c. 367 BC – c. 283 BC, was a Macedonian general under Alexander the Great, who became ruler of Egypt and founder of both the Ptolemaic Kingdom and the Ptolemaic Dynasty...
, King of Egypt.
The Elements began with definitions of terms, fundamental geometric principles (called
axioms or
postulates), and general quantitative principles (called
common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read.
 Any two points can be joined by a straight line.
 Any finite straight line can be extended in a straight line.
 A circle can be drawn with any center and any radius.
 All right angles are equal to each other.
 If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
).
Archimedes
ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
(287212 BC), of
SyracuseSyracuse is a historic city in Sicily, the capital of the province of Syracuse. The city is notable for its rich Greek history, culture, amphitheatres, architecture, and as the birthplace of the preeminent mathematician and engineer Archimedes. This 2,700yearold city played a key role in...
,
SicilySicily is a region of Italy, and is the largest island in the Mediterranean Sea. Along with the surrounding minor islands, it constitutes an autonomous region of Italy, the Regione Autonoma Siciliana Sicily has a rich and unique culture, especially with regard to the arts, music, literature,...
, when it was a Greek citystate, is often considered to be the greatest of the Greek mathematicians, and occasionally even named as one of the three greatest of all time (along with
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
and
Carl Friedrich GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.
After Archimedes
After Archimedes, Hellenistic mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over.
ProclusProclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
(410485), author of
Commentary on the First Book of Euclid, was one of the last important players in Hellenistic geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek citystates produced excellent engineers, but no mathematicians of note.
The great
Library of AlexandriaThe Royal Library of Alexandria, or Ancient Library of Alexandria, in Alexandria, Egypt, was the largest and most significant great library of the ancient world. It flourished under the patronage of the Ptolemaic dynasty and functioned as a major center of scholarship from its construction in the...
was later burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,00070,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later.
Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in nonreligious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the fourth century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign.
Vedic period
The
Satapatha Brahmana (9th century BCE) contains rules for ritual geometric constructions that are similar to the
Sulba Sutras.
The
Śulba Sūtras (literally, "Aphorisms of the Chords" in
Vedic SanskritVedic Sanskrit is an old IndoAryan language. It is an archaic form of Sanskrit, an early descendant of ProtoIndoIranian. It is closely related to Avestan, the oldest preserved Iranian language...
) (c. 700400 BCE) list rules for the construction of sacrificial fire altars. Most mathematical problems considered in the
Śulba Sūtras spring from "a single theological requirement," that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks.
According to , the
Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians."
The diagonal rope () of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal () produce separately."
Since the statement is a
sūtra, it is necessarily compressed and what the ropes
produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student.
They contain lists of Pythagorean triples, which are particular cases of Diophantine equations.
They also contain statements (that with hindsight we know to be approximate) about
squaring the circleSquaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...
and "circling the square."
BaudhayanaBaudhāyana, was an Indian mathematician, whowas most likely also a priest. He is noted as the author of the earliest Sulba Sūtra—appendices to the Vedas giving rules for the construction of altars—called the , which contained several important mathematical results. He is older than the other...
(c. 8th century BCE) composed the
Baudhayana Sulba Sutra, the bestknown
Sulba Sutra, which contains examples of simple Pythagorean triples, such as:
,
,
,
, and
as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square." It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together."
According to mathematician S. G. Dani, the Babylonian cuneiform tablet
Plimpton 322Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University...
written ca. 1850 BCE "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple, indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India." Dani goes on to say:
"As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras. The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and
would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily."
In all, three
Sulba Sutras were composed. The remaining two, the
Manava Sulba Sutra composed by
ManavaManava is an author of the Indian geometric text of Sulba Sutras.The Manava Sulbasutra is not the oldest , nor is it one of the most important, there being at least three Sulbasutras which are considered more important...
(
fl.Floruit , abbreviated fl. , is a Latin verb meaning "flourished", denoting the period of time during which something was active...
750650 BCE) and the
Apastamba Sulba Sutra, composed by
ApastambaThe Dharmasutra of Āpastamba forms a part of the larger Kalpasūtra of Āpastamba. It contains thirty praśnas, which literally means ‘questions’ or books. The subjects of this Dharmasūtra are well organized and preserved in good condition...
(c. 600 BCE), contained results similar to the
Baudhayana Sulba Sutra.
Classical period
In the
Bakhshali manuscriptThe Bakhshali Manuscript is an Ancient Indian mathematical manuscript written on birch bark which was found near the village of Bakhshali in 1881 in what was then the NorthWest Frontier Province of British India...
, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."
AryabhataAryabhata was the first in the line of great mathematicianastronomers from the classical age of Indian mathematics and Indian astronomy...
's
AryabhatiyaĀryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opus and only extant work of the 5th century Indian mathematician, Āryabhaṭa. Structure and style:...
(499 CE) includes the computation of areas and volumes.
BrahmaguptaBrahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
wrote his astronomical work
{{histOfScience}}
GeometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
(GreekGreek is an independent branch of the IndoEuropean family of languages. Native to the southern Balkans, it has the longest documented history of any IndoEuropean language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the other being the study of numbers (arithmeticArithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daytoday counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
).
Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
, who introduced mathematical rigor and the axiomatic method still in use today. His book, The ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.
In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathematicsMathematics has become a vastly diverse subject over history, and there is a corresponding need to categorize the different areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different...
and algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
.)
Early geometry
The earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus ValleyThe Indus Valley Civilization was a Bronze Age civilization that was located in the northwestern region of the Indian subcontinent, consisting of what is now mainly modernday Pakistan and northwest India...
(see Harappan Mathematics), and ancient BabyloniaBabylonia was an ancient cultural region in centralsouthern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in centralsouthern Mesopotamia (presentday Iraq), with Babylon as its capital. Babylonia emerged as...
(see Babylonian mathematicsBabylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...
) from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveyingSee Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or threedimensional position of points and the distances and angles between them...
, constructionIn the fields of architecture and civil engineering, construction is a process that consists of the building or assembling of infrastructure. Far from being a single activity, large scale construction is a feat of human multitasking...
, astronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
. For example, both the EgyptiansEgyptians are nation an ethnic group made up of Mediterranean North Africans, the indigenous people of Egypt.Egyptian identity is closely tied to geography. The population of Egypt is concentrated in the lower Nile Valley, the small strip of cultivable land stretching from the First Cataract to...
and the BabylonBabylon was an Akkadian citystate of ancient Mesopotamia, the remains of which are found in presentday Al Hillah, Babil Province, Iraq, about 85 kilometers south of Baghdad...
ians were aware of versions of the Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
about 1500 years before PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
; the Egyptians had a correct formula for the volume of a frustumIn geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....
of a square pyramid; the Babylonians had a trigonometry table.
Egyptian geometry
{{mainEgyptian mathematics}}
The ancient Egyptians knew that they could approximate the area of a circle as follows:



 Area of Circle ≈ [ (Diameter) x 8/9 ]^{2}.
Problem 50 of the AhmesAhmes was an ancient Egyptian scribe who lived during the Second Intermediate Period and the beginning of the Eighteenth Dynasty . He wrote the Rhind Mathematical Papyrus, a work of Ancient Egyptian mathematics that dates to approximately 1650 BC; he is the earliest contributor to mathematics...
papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter. This assumes that π' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
is 4×(8/9)² (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the BabyloniaBabylonia was an ancient cultural region in centralsouthern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in centralsouthern Mesopotamia (presentday Iraq), with Babylon as its capital. Babylonia emerged as...
ns (25/8 = 3.125, within 0.53 percent), but was not otherwise surpassed until ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
' approximation of 211875/67441 = 3.14163, which had an error of just over 1 in 10,000.
Interestingly, Ahmes knew of the modern 22/7 as an approximation for pi, and used it to split a hekat, hekat x 22/x x 7/22 = hekat; however, Ahmes continued to use the traditional 256/81 value for pi for computing his hekat volume found in a cylinder.
Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for π of 3.111...
The two problems together indicate a range of values for Pi between 3.11 and 3.16.
Problem 14 in the Moscow Mathematical PapyrusThe Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
gives the only ancient example finding the volume of a frustumIn geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....
of a pyramid, describing the correct formula:
Babylonian geometry
{{mainBabylonian mathematics}}
The Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as onetwelfth the square of the circumference, which would be correct if π is estimated as 3. The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 3 and 1/8. The Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a timemile used for measuring the travel of the Sun, therefore, representing time.
Classical Greek geometry
For the ancient GreekGreece , officially the Hellenic Republic , and historically Hellas or the Republic of Greece in English, is a country in southeastern Europe....
mathematiciansGreek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trialanderror to logical deduction; they recognized that geometry studies "eternal forms", or abstractions, of which physical objects are only approximations; and they developed the idea of the "axiomatic method"In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
, still in use today.
Thales and Pythagoras
ThalesThales of Miletus was a preSocratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition...
(635543 BC) of MiletusMiletus was an ancient Greek city on the western coast of Anatolia , near the mouth of the Maeander River in ancient Caria...
(now in southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
(582496 BC) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and traveled to BabylonBabylon was an Akkadian citystate of ancient Mesopotamia, the remains of which are found in presentday Al Hillah, Babil Province, Iraq, about 85 kilometers south of Baghdad...
and EgyptEgypt , officially the Arab Republic of Egypt, Arabic: , is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Southwest Asia. Egypt is thus a transcontinental country, and a major power in Africa, the Mediterranean Basin, the Middle East and the Muslim world...
. The theorem that bears his name may not have been his discovery, but he was probably one of the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengthsIn mathematics, two nonzero real numbers a and b are said to be commensurable if a/b is a rational number.History of the concept:...
and irrational numberIn mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b nonzero, and is therefore not a rational number....
s. (There is no evidence that Thales provided any deductive proofs, and in fact, deductive mathematical proofs did not appear until after Parmenides. At best, all that we can say about Thales is that he introduced various geometric theorems to the Greeks. The idea that mathematics was from its inception deductive is false. At the time of Thales, mathematics was inductive. This means that Thales would have "provided" empirical and direct proofs, but not deductive proofs.)
Plato
PlatoPlato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
(427347 BC), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, "Let none ignorant of geometry enter here." Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but compass and straightedge – never measuring instruments such as a marked rulerA ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines...
or a protractorIn geometry, a protractor is a circular or semicircular tool for measuring an angle or a circle. The units of measurement utilized are usually degrees.Some protractors are simple halfdiscs; these have existed since ancient times...
, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of possible compass and straightedgeCompassandstraightedge or rulerandcompass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....
constructions, and three classic construction problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
(384322 BC), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see LogicIn philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
) which was not substantially improved upon until the 19th century.
Euclid
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
(c. 325265 BC), of AlexandriaAlexandria is the secondlargest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
, probably a student of one of Plato’s students, wrote a treatise in 13 books (chapters), titled The Elements of GeometryEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, in which he presented geometry in an ideal axiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be selfevident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
atic form, which came to be known as Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. The treatise is not a compendium of all that the Hellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy IPtolemy I Soter I , also known as Ptolemy Lagides, c. 367 BC – c. 283 BC, was a Macedonian general under Alexander the Great, who became ruler of Egypt and founder of both the Ptolemaic Kingdom and the Ptolemaic Dynasty...
, King of Egypt.
The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read.
 Any two points can be joined by a straight line.
 Any finite straight line can be extended in a straight line.
 A circle can be drawn with any center and any radius.
 All right angles are equal to each other.
 If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
).
Archimedes
ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
(287212 BC), of SyracuseSyracuse is a historic city in Sicily, the capital of the province of Syracuse. The city is notable for its rich Greek history, culture, amphitheatres, architecture, and as the birthplace of the preeminent mathematician and engineer Archimedes. This 2,700yearold city played a key role in...
, SicilySicily is a region of Italy, and is the largest island in the Mediterranean Sea. Along with the surrounding minor islands, it constitutes an autonomous region of Italy, the Regione Autonoma Siciliana Sicily has a rich and unique culture, especially with regard to the arts, music, literature,...
, when it was a Greek citystate, is often considered to be the greatest of the Greek mathematicians, and occasionally even named as one of the three greatest of all time (along with Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
and Carl Friedrich GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.
After Archimedes
After Archimedes, Hellenistic mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. ProclusProclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
(410485), author of Commentary on the First Book of Euclid, was one of the last important players in Hellenistic geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek citystates produced excellent engineers, but no mathematicians of note.
The great Library of AlexandriaThe Royal Library of Alexandria, or Ancient Library of Alexandria, in Alexandria, Egypt, was the largest and most significant great library of the ancient world. It flourished under the patronage of the Ptolemaic dynasty and functioned as a major center of scholarship from its construction in the...
was later burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,00070,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later.
Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in nonreligious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the fourth century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign.
Vedic period
The Satapatha Brahmana (9th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.
The Śulba Sūtras (literally, "Aphorisms of the Chords" in Vedic SanskritVedic Sanskrit is an old IndoAryan language. It is an archaic form of Sanskrit, an early descendant of ProtoIndoIranian. It is closely related to Avestan, the oldest preserved Iranian language...
) (c. 700400 BCE) list rules for the construction of sacrificial fire altars. Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement," that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks.
According to {{HarvHayashi2005p=363}}, the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians." The diagonal rope ({{IASTakṣṇayārajju}}) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal ({{IASTtiryaṇmānī}}) produce separately."
Since the statement is a sūtra, it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student.
They contain lists of Pythagorean triples, which are particular cases of Diophantine equations.
They also contain statements (that with hindsight we know to be approximate) about squaring the circleSquaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...
and "circling the square."
BaudhayanaBaudhāyana, was an Indian mathematician, whowas most likely also a priest. He is noted as the author of the earliest Sulba Sūtra—appendices to the Vedas giving rules for the construction of altars—called the , which contained several important mathematical results. He is older than the other...
(c. 8th century BCE) composed the Baudhayana Sulba Sutra, the bestknown Sulba Sutra, which contains examples of simple Pythagorean triples, such as: , , , , and as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square." It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together."
According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University...
written ca. 1850 BCE "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple, indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India." Dani goes on to say:
"As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras. The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and
would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily."
In all, three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by ManavaManava is an author of the Indian geometric text of Sulba Sutras.The Manava Sulbasutra is not the oldest , nor is it one of the most important, there being at least three Sulbasutras which are considered more important...
(fl.Floruit , abbreviated fl. , is a Latin verb meaning "flourished", denoting the period of time during which something was active...
750650 BCE) and the Apastamba Sulba Sutra, composed by ApastambaThe Dharmasutra of Āpastamba forms a part of the larger Kalpasūtra of Āpastamba. It contains thirty praśnas, which literally means ‘questions’ or books. The subjects of this Dharmasūtra are well organized and preserved in good condition...
(c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra.
Classical period
In the Bakhshali manuscriptThe Bakhshali Manuscript is an Ancient Indian mathematical manuscript written on birch bark which was found near the village of Bakhshali in 1881 in what was then the NorthWest Frontier Province of British India...
, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero." AryabhataAryabhata was the first in the line of great mathematicianastronomers from the classical age of Indian mathematics and Indian astronomy...
's AryabhatiyaĀryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opus and only extant work of the 5th century Indian mathematician, Āryabhaṭa. Structure and style:...
(499 CE) includes the computation of areas and volumes.
BrahmaguptaBrahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
wrote his astronomical work
{{histOfScience}}
GeometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
(GreekGreek is an independent branch of the IndoEuropean family of languages. Native to the southern Balkans, it has the longest documented history of any IndoEuropean language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the other being the study of numbers (arithmeticArithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daytoday counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
).
Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
, who introduced mathematical rigor and the axiomatic method still in use today. His book, The ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.
In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathematicsMathematics has become a vastly diverse subject over history, and there is a corresponding need to categorize the different areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different...
and algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
.)
Early geometry
The earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus ValleyThe Indus Valley Civilization was a Bronze Age civilization that was located in the northwestern region of the Indian subcontinent, consisting of what is now mainly modernday Pakistan and northwest India...
(see Harappan Mathematics), and ancient BabyloniaBabylonia was an ancient cultural region in centralsouthern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in centralsouthern Mesopotamia (presentday Iraq), with Babylon as its capital. Babylonia emerged as...
(see Babylonian mathematicsBabylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...
) from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveyingSee Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or threedimensional position of points and the distances and angles between them...
, constructionIn the fields of architecture and civil engineering, construction is a process that consists of the building or assembling of infrastructure. Far from being a single activity, large scale construction is a feat of human multitasking...
, astronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
. For example, both the EgyptiansEgyptians are nation an ethnic group made up of Mediterranean North Africans, the indigenous people of Egypt.Egyptian identity is closely tied to geography. The population of Egypt is concentrated in the lower Nile Valley, the small strip of cultivable land stretching from the First Cataract to...
and the BabylonBabylon was an Akkadian citystate of ancient Mesopotamia, the remains of which are found in presentday Al Hillah, Babil Province, Iraq, about 85 kilometers south of Baghdad...
ians were aware of versions of the Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
about 1500 years before PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
; the Egyptians had a correct formula for the volume of a frustumIn geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....
of a square pyramid; the Babylonians had a trigonometry table.
Egyptian geometry
{{mainEgyptian mathematics}}
The ancient Egyptians knew that they could approximate the area of a circle as follows:



 Area of Circle ≈ [ (Diameter) x 8/9 ]^{2}.
Problem 50 of the AhmesAhmes was an ancient Egyptian scribe who lived during the Second Intermediate Period and the beginning of the Eighteenth Dynasty . He wrote the Rhind Mathematical Papyrus, a work of Ancient Egyptian mathematics that dates to approximately 1650 BC; he is the earliest contributor to mathematics...
papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter. This assumes that π' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
is 4×(8/9)² (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the BabyloniaBabylonia was an ancient cultural region in centralsouthern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in centralsouthern Mesopotamia (presentday Iraq), with Babylon as its capital. Babylonia emerged as...
ns (25/8 = 3.125, within 0.53 percent), but was not otherwise surpassed until ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
' approximation of 211875/67441 = 3.14163, which had an error of just over 1 in 10,000.
Interestingly, Ahmes knew of the modern 22/7 as an approximation for pi, and used it to split a hekat, hekat x 22/x x 7/22 = hekat; however, Ahmes continued to use the traditional 256/81 value for pi for computing his hekat volume found in a cylinder.
Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for π of 3.111...
The two problems together indicate a range of values for Pi between 3.11 and 3.16.
Problem 14 in the Moscow Mathematical PapyrusThe Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
gives the only ancient example finding the volume of a frustumIn geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....
of a pyramid, describing the correct formula:
Babylonian geometry
{{mainBabylonian mathematics}}
The Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as onetwelfth the square of the circumference, which would be correct if π is estimated as 3. The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 3 and 1/8. The Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a timemile used for measuring the travel of the Sun, therefore, representing time.
Classical Greek geometry
For the ancient GreekGreece , officially the Hellenic Republic , and historically Hellas or the Republic of Greece in English, is a country in southeastern Europe....
mathematiciansGreek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trialanderror to logical deduction; they recognized that geometry studies "eternal forms", or abstractions, of which physical objects are only approximations; and they developed the idea of the "axiomatic method"In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
, still in use today.
Thales and Pythagoras
ThalesThales of Miletus was a preSocratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition...
(635543 BC) of MiletusMiletus was an ancient Greek city on the western coast of Anatolia , near the mouth of the Maeander River in ancient Caria...
(now in southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
(582496 BC) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and traveled to BabylonBabylon was an Akkadian citystate of ancient Mesopotamia, the remains of which are found in presentday Al Hillah, Babil Province, Iraq, about 85 kilometers south of Baghdad...
and EgyptEgypt , officially the Arab Republic of Egypt, Arabic: , is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Southwest Asia. Egypt is thus a transcontinental country, and a major power in Africa, the Mediterranean Basin, the Middle East and the Muslim world...
. The theorem that bears his name may not have been his discovery, but he was probably one of the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengthsIn mathematics, two nonzero real numbers a and b are said to be commensurable if a/b is a rational number.History of the concept:...
and irrational numberIn mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b nonzero, and is therefore not a rational number....
s. (There is no evidence that Thales provided any deductive proofs, and in fact, deductive mathematical proofs did not appear until after Parmenides. At best, all that we can say about Thales is that he introduced various geometric theorems to the Greeks. The idea that mathematics was from its inception deductive is false. At the time of Thales, mathematics was inductive. This means that Thales would have "provided" empirical and direct proofs, but not deductive proofs.)
Plato
PlatoPlato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
(427347 BC), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, "Let none ignorant of geometry enter here." Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but compass and straightedge – never measuring instruments such as a marked rulerA ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines...
or a protractorIn geometry, a protractor is a circular or semicircular tool for measuring an angle or a circle. The units of measurement utilized are usually degrees.Some protractors are simple halfdiscs; these have existed since ancient times...
, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of possible compass and straightedgeCompassandstraightedge or rulerandcompass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....
constructions, and three classic construction problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
(384322 BC), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see LogicIn philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
) which was not substantially improved upon until the 19th century.
Euclid
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
(c. 325265 BC), of AlexandriaAlexandria is the secondlargest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
, probably a student of one of Plato’s students, wrote a treatise in 13 books (chapters), titled The Elements of GeometryEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, in which he presented geometry in an ideal axiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be selfevident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
atic form, which came to be known as Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. The treatise is not a compendium of all that the Hellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy IPtolemy I Soter I , also known as Ptolemy Lagides, c. 367 BC – c. 283 BC, was a Macedonian general under Alexander the Great, who became ruler of Egypt and founder of both the Ptolemaic Kingdom and the Ptolemaic Dynasty...
, King of Egypt.
The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read.
 Any two points can be joined by a straight line.
 Any finite straight line can be extended in a straight line.
 A circle can be drawn with any center and any radius.
 All right angles are equal to each other.
 If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
).
Archimedes
ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
(287212 BC), of SyracuseSyracuse is a historic city in Sicily, the capital of the province of Syracuse. The city is notable for its rich Greek history, culture, amphitheatres, architecture, and as the birthplace of the preeminent mathematician and engineer Archimedes. This 2,700yearold city played a key role in...
, SicilySicily is a region of Italy, and is the largest island in the Mediterranean Sea. Along with the surrounding minor islands, it constitutes an autonomous region of Italy, the Regione Autonoma Siciliana Sicily has a rich and unique culture, especially with regard to the arts, music, literature,...
, when it was a Greek citystate, is often considered to be the greatest of the Greek mathematicians, and occasionally even named as one of the three greatest of all time (along with Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
and Carl Friedrich GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.
After Archimedes
After Archimedes, Hellenistic mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. ProclusProclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
(410485), author of Commentary on the First Book of Euclid, was one of the last important players in Hellenistic geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek citystates produced excellent engineers, but no mathematicians of note.
The great Library of AlexandriaThe Royal Library of Alexandria, or Ancient Library of Alexandria, in Alexandria, Egypt, was the largest and most significant great library of the ancient world. It flourished under the patronage of the Ptolemaic dynasty and functioned as a major center of scholarship from its construction in the...
was later burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,00070,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later.
Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in nonreligious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the fourth century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign.
Vedic period
The Satapatha Brahmana (9th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.
The Śulba Sūtras (literally, "Aphorisms of the Chords" in Vedic SanskritVedic Sanskrit is an old IndoAryan language. It is an archaic form of Sanskrit, an early descendant of ProtoIndoIranian. It is closely related to Avestan, the oldest preserved Iranian language...
) (c. 700400 BCE) list rules for the construction of sacrificial fire altars. Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement," that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks.
According to {{HarvHayashi2005p=363}}, the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians." The diagonal rope ({{IASTakṣṇayārajju}}) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal ({{IASTtiryaṇmānī}}) produce separately."
Since the statement is a sūtra, it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student.
They contain lists of Pythagorean triples, which are particular cases of Diophantine equations.
They also contain statements (that with hindsight we know to be approximate) about squaring the circleSquaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...
and "circling the square."
BaudhayanaBaudhāyana, was an Indian mathematician, whowas most likely also a priest. He is noted as the author of the earliest Sulba Sūtra—appendices to the Vedas giving rules for the construction of altars—called the , which contained several important mathematical results. He is older than the other...
(c. 8th century BCE) composed the Baudhayana Sulba Sutra, the bestknown Sulba Sutra, which contains examples of simple Pythagorean triples, such as: , , , , and as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square." It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together."
According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University...
written ca. 1850 BCE "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple, indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India." Dani goes on to say:
"As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras. The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and
would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily."
In all, three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by ManavaManava is an author of the Indian geometric text of Sulba Sutras.The Manava Sulbasutra is not the oldest , nor is it one of the most important, there being at least three Sulbasutras which are considered more important...
(fl.Floruit , abbreviated fl. , is a Latin verb meaning "flourished", denoting the period of time during which something was active...
750650 BCE) and the Apastamba Sulba Sutra, composed by ApastambaThe Dharmasutra of Āpastamba forms a part of the larger Kalpasūtra of Āpastamba. It contains thirty praśnas, which literally means ‘questions’ or books. The subjects of this Dharmasūtra are well organized and preserved in good condition...
(c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra.
Classical period
In the Bakhshali manuscriptThe Bakhshali Manuscript is an Ancient Indian mathematical manuscript written on birch bark which was found near the village of Bakhshali in 1881 in what was then the NorthWest Frontier Province of British India...
, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero." AryabhataAryabhata was the first in the line of great mathematicianastronomers from the classical age of Indian mathematics and Indian astronomy...
's AryabhatiyaĀryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opus and only extant work of the 5th century Indian mathematician, Āryabhaṭa. Structure and style:...
(499 CE) includes the computation of areas and volumes.
BrahmaguptaBrahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
wrote his astronomical work {{IASTThe main work of Brahmagupta, Brāhmasphuṭasiddhānta , written c.628, contains ideas including a good understanding of the mathematical role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and some quadratic...
in 628 CE. Chapter 12, containing 66 SanskritSanskrit , is a historical IndoAryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateralIn Euclidean geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. Other names for these quadrilaterals are chordal quadrilateral and inscribed...
:
Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicularIn geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side.
Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas).
Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by

where s, the semiperimeterIn geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name...
, given by:
Brahmagupta's Theorem on rational triangles: A triangle with rational sides and rational area is of the form:
for some rational numbers and .
Chinese geometry
{{see alsoChinese mathematics}}
The first definitive work (or at least oldest existent) on geometry in China was the Mo Jing, the Mohist canon of the early utilitarian philosopher MoziMozi Lat.]] as Micius, ca. 470 BC – ca. 391 BC), original name Mo Di , was a Chinese philosopher during the Hundred Schools of Thought period . Born in Tengzhou, Shandong Province, China, he founded the school of Mohism, and argued strongly against Confucianism and Daoism...
(470 BCYear 470 BC was a year of the preJulian Roman calendar. At the time, it was known as the Year of the Consulship of Potitus and Mamercus...
390 BCYear 390 BC was a year of the preJulian Roman calendar. At the time, it was known as the Year of the Tribunate of Ambustus, Longus, Ambustus, Fidenas, Ambustus and Cornelius...
). It was compiled years after his death by his later followers around the year 330 BC. Although the Mo Jing is the oldest existent book on geometry in China, there is the possibility that even older written material exists. However, due to the infamous Burning of the BooksBurning of the books and burying of the scholars is a phrase that refers to a policy and a sequence of events in the Qin Dynasty of Ancient China, between the period of 213 and 206 BC. During these events, the Hundred Schools of Thought were pruned; legalism survived...
in the political maneauver by the Qin DynastyThe Qin Dynasty was the first imperial dynasty of China, lasting from 221 to 207 BC. The Qin state derived its name from its heartland of Qin, in modernday Shaanxi. The strength of the Qin state was greatly increased by the legalist reforms of Shang Yang in the 4th century BC, during the Warring...
ruler Qin Shihuang (r. 221 BCYear 221 BC was a year of the preJulian Roman calendar. At the time it was known as the Year of the Consulship of Asina and Rufus/Lepidus...
210 BCYear 210 BC was a year of the preJulian Roman calendar. At the time it was known as the Year of the Consulship of Marcellus and Laevinus...
), multitudes of written literature created before his time was purged. In addition, the Mo Jing presents geometrical concepts in mathematics that are perhaps too advanced not to have had a previous geometrical base or mathematic background to work upon.
The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point. Much like EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's first and third definitions and PlatoPlato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
's 'beginning of a line', the Mo Jing stated that "a point may stand at the end (of a line) or at its beginning like a headpresentation in childbirth. (As to its invisibility) there is nothing similar to it." Similar to the atomists of DemocritusDemocritus was an Ancient Greek philosopher born in Abdera, Thrace, Greece. He was an influential preSocratic philosopher and pupil of Leucippus, who formulated an atomic theory for the cosmos....
, the Mo Jing stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved. It stated that two lines of equal length will always finish at the same place, while providing definitions for the comparison of lengths and for parallels, along with principles of space and bounded space. It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch. The book provided definitions for circumference, diameter, and radius, along with the definition of volume.
The Han DynastyThe Han Dynasty was the second imperial dynasty of China, preceded by the Qin Dynasty and succeeded by the Three Kingdoms . It was founded by the rebel leader Liu Bang, known posthumously as Emperor Gaozu of Han. It was briefly interrupted by the Xin Dynasty of the former regent Wang Mang...
(202 BCYear 202 BC was a year of the preJulian Roman calendar. At the time it was known as the Year of the Consulship of Geminus and Nero...
220 AD) period of China witnessed a new flourishing of mathematics. One of the oldest Chinese mathematical texts to present geometric progressionIn mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed nonzero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...
s was the Suàn shù shūThe Suàn shù shū , or the Writings on Reckoning , is one of the earliest known Chinese mathematical treatises...
of 186 BC, during the Western Han era. The mathematician, inventor, and astronomer Zhang HengZhang Heng was a Chinese astronomer, mathematician, inventor, geographer, cartographer, artist, poet, statesman, and literary scholar from Nanyang, Henan. He lived during the Eastern Han Dynasty of China. He was educated in the capital cities of Luoyang and Chang'an, and began his career as a...
(78Year 78 was a common year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Novius and Commodus...
139Year 139 was a common year starting on Wednesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Hadrianus and Praesens...
AD) used geometrical formulas to solve mathematical problems. Although rough estimates for pi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
(π) were given in the Zhou Li (compiled in the 2nd century BC), it was Zhang Heng who was the first to make a concerted effort at creating a more accurate formula for pi. This in turn would be made more accurate by later Chinese such as Zu ChongzhiZu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.Life and works:...
(429Year 429 was a common year starting on Tuesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Florentius and Dionysius...
500Year 500 was a leap year starting on Saturday of the Julian calendar. At the time, it was known as the Year of the Consulship of Patricius and Hypatius...
AD). Zhang Heng approximated pi as 730/232 (or approx 3.1466), although he used another formula of pi in finding a spherical volume, using the square root of 10 (or approx 3.162) instead. Zu Chongzhi's best approximation was between 3.1415926 and 3.1415927, with ^{355}⁄_{113} (密率, Milü, detailed approximation) and ^{22}⁄_{7} (约率, Yuelü, rough approximation) being the other notable approximation. In comparison to later works, the formula for pi given by the French mathematician Franciscus Vieta (15401603) fell halfway between Zu's approximations.
The Nine Chapters on the Mathematical Art
The Nine Chapters on the Mathematical ArtThe Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 1st century CE...
, the title of which first appeared by 179 AD on a bronze inscription, was edited and commented on by the 3rd century mathematician Liu HuiLiu Hui was a mathematician of the state of Cao Wei during the Three Kingdoms period of Chinese history. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the Mathematical Art .He was a...
from the Kingdom of Cao WeiCao Wei was one of the states that competed for control of China during the Three Kingdoms period. With the capital at Luoyang, the state was established by Cao Pi in 220, based upon the foundations that his father Cao Cao laid...
. This book included many problems where geometry was applied, such as finding surface areas for squares and circles, the volumes of solids in various three dimensional shapes, and included the use of the Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
. The book provided illustrated proof for the Pythagorean theorem, contained a written dialogue between of the earlier Duke of ZhouThe Duke of Zhou played a major role in consolidating the newlyfounded Zhou Dynasty . He was the brother of King Wu of Zhou, the first king of the ancient Chinese Zhou Dynasty...
and Shang Gao on the properties of the right angle triangle and the Pythagorean theorem, while also referring to the astronomical gnomonThe gnomon is the part of a sundial that casts the shadow. Gnomon is an ancient Greek word meaning "indicator", "one who discerns," or "that which reveals."It has come to be used for a variety of purposes in mathematics and other fields....
, the circle and square, as well as measurements of heights and distances. The editor Liu Hui listed pi as 3.141014 by using a 192 sided polygonIn geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
, and then calculated pi as 3.14159 using a 3072 sided polygon. This was more accurate than Liu Hui's contemporary Wang FanWang Fan , style name Yongyuan , was an official and astronomer of the state of Eastern Wu during the Three Kingdoms period of Chinese history. He was from Lujiang . He was proficient in mathematics and astronomy. He calculated the distance from the Sun to the Earth, but his geometric model was not...
, a mathematician and astronomer from Eastern WuEastern Wu, also known as Sun Wu, was one the three states competing for control of China during the Three Kingdoms period after the fall of the Han Dynasty. It was based in the Jiangnan region of China...
, would render pi as 3.1555 by using ^{142}⁄_{45}. Liu Hui also wrote of mathematical surveyingSee Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or threedimensional position of points and the distances and angles between them...
to calculate distance measurements of depth, height, width, and surface area. In terms of solid geometry, he figured out that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge. He also figured out that a wedge with trapezoidIn Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...
base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. Furthermore, Liu Hui described Cavalieri's principleIn geometry, Cavalieri's principle, sometimes called the method of indivisibles, named after Bonaventura Cavalieri, is as follows:* 2dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane...
on volume, as well as Gaussian eliminationIn linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
. From the Nine Chapters, it listed the following geometrical formulas that were known by the time of the Former Han Dynasty (202 BCE–9 CE).
Areas for the
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 Square
 Rectangle
 Circle
 Isosceles triangle
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 Rhomboid
Traditionally, in twodimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are oblique.A parallelogram with sides of equal length is a rhombus but not a rhomboid....
 Trapezoid
In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...
 Double trapezium
 Segment of a circle
 Annulus ('ring' between two concentric circles)
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Volumes for the
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 Parallelepiped with two square surfaces
 Parallelepiped with no square surfaces
 Pyramid
 Frustum
In geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....
of pyramid with square base
 Frustum of pyramid with rectangular base of unequal sides
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 Cube
 Prism
In geometry, a prism is a polyhedron with an nsided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All crosssections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
 Wedge with rectangular base and both sides sloping
 Wedge with trapezoid base and both sides sloping
 Tetrahedral wedge
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 Frustum of a wedge of the second type (used for applications in engineering)
 Cylinder
 Cone with circular base
 Frustum of a cone
 Sphere
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Continuing the geometrical legacy of ancient China, there were many later figures to come, including the famed astronomer and mathematician Shen KuoShen Kuo or Shen Gua , style name Cunzhong and pseudonym Mengqi Weng , was a polymathic Chinese scientist and statesman of the Song Dynasty...
(10311095 AD), Yang HuiYang Hui , courtesy name Qianguang , was a Chinese mathematician from Qiantang , Zhejiang province during the late Song Dynasty . Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting 'Yang Hui's Triangle'...
(12381298 AD) who discovered Pascal's TriangleIn mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
, Xu GuangqiXu Guangqi , was a Chinese scholarbureaucrat, agricultural scientist, astronomer, and mathematician in the Ming Dynasty. Xu was a colleague and collaborator of the Italian Jesuits Matteo Ricci and Sabatino de Ursis and they translated several classic Western texts into Chinese, including part of...
(15621633 AD), and many others.
Islamic geometry
{{see alsoIslamic mathematics}}
The IslamIslam . The most common are and . : Arabic pronunciation varies regionally. The first vowel ranges from ~~. The second vowel ranges from ~~~...
ic CaliphateThe term caliphate, "dominion of a caliph " , refers to the first system of government established in Islam and represented the political unity of the Muslim Ummah...
established across the Middle EastThe Middle East is a region that encompasses Western Asia and Northern Africa. It is often used as a synonym for Near East, in opposition to Far East...
, North AfricaNorth Africa or Northern Africa is the northernmost region of the African continent, linked by the Sahara to SubSaharan Africa. Geopolitically, the United Nations definition of Northern Africa includes eight countries or territories; Algeria, Egypt, Libya, Morocco, South Sudan, Sudan, Tunisia, and...
, SpainSpain , officially the Kingdom of Spain languages]] under the European Charter for Regional or Minority Languages. In each of these, Spain's official name is as follows:;;;;;;), is a country and member state of the European Union located in southwestern Europe on the Iberian Peninsula...
, PortugalPortugal , officially the Portuguese Republic is a country situated in southwestern Europe on the Iberian Peninsula. Portugal is the westernmost country of Europe, and is bordered by the Atlantic Ocean to the West and South and by Spain to the North and East. The Atlantic archipelagos of the...
, Persia and parts of Persia, began around 640 CEYear 640 was a leap year starting on Saturday of the Julian calendar. The denomination 640 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Europe :* Tulga succeeds his father Suinthila as king...
. Islamic mathematicsIn the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600...
during this period was primarily algebraic rather than geometric, though there were important works on geometry. Scholarship in Europe declined and eventually the Hellenistic works of antiquityClassical antiquity is a broad term for a long period of cultural history centered on the Mediterranean Sea, comprising the interlocking civilizations of ancient Greece and ancient Rome, collectively known as the GrecoRoman world...
were lost to them, and survived only in the Islamic centers of learning.
Although the Muslim mathematicians are most famed for their work on algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and number systemIn mathematics, a 'number system' is a set of numbers, , together with one or more operations, such as addition or multiplication....
s, they also made considerable contributions to geometry, trigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
and mathematical astronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, and were responsible for the development of algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. Geometrical magnitudes were treated as "algebraic objects" by most Muslim mathematicians however.
The successors of Muḥammad ibn Mūsā alḴwārizmī who was Persian Scholar, mathematician and Astronomer who invented the AlgorithmIn mathematics and computer science, an algorithm is an effective method expressed as a finite list of welldefined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
in MathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
which is the base for Computer ScienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
(born 780Year 780 was a leap year starting on Saturday of the Julian calendar. The denomination 780 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Byzantine Empire :* Constantine VI becomes Byzantine...
) undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.
AlMahaniAbuAbdullah Muhammad ibn Īsa Māhānī was a Persian mathematician and astronomer from Mahan, Kermān, Persia.A series of observations of lunar and solar eclipses and planetary conjunctions, made by him from 853 to 866, was in fact used by Ibn Yunus....
(born 820Year 820 was a leap year starting on Sunday of the Julian calendar. Asia :* Tahir, the son of a slave, is rewarded with the governorship of Khurasan for supporting the caliphate...
) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. AlKaraji' was a 10th century Persian Muslim mathematician and engineer. His three major works are AlBadi' fi'lhisab , AlFakhri fi'ljabr wa'lmuqabala , and AlKafi fi'lhisab .Because alKaraji's original works in Arabic are lost, it is not...
(born 953Year 953 was a common year starting on Saturday of the Julian calendar. Europe :* Liudolf, Duke of Swabia and Conrad the Red rebel against German King Otto I....
) completely freed algebra from geometrical operations and replaced them with the arithmeticArithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daytoday counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
al type of operations which are at the core of algebra today.
Thabit family and other early geometers
Thabit ibn Qurra' was a mathematician, physician, astronomer and translator of the Islamic Golden Age.Ibn Qurra made important discoveries in algebra, geometry and astronomy...
(known as Thebit in LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient ProtoIndoEuropean language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
) (born 836Year 836 was a leap year starting on Saturday of the Julian calendar. Asia :* Abbasid caliph alMutasim establishes a new capital at Samarra, Iraq. Europe :...
) contributed to a number of areas in mathematics, where he played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s, integral calculus, theorems in spherical trigonometrySpherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...
, analytic geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
, and nonEuclidean geometryNonEuclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of staticsStatics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...
. An important geometrical aspect of Thabit's work was his book on the composition of ratios. In this book, Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. By introducing arithmetical operations on quantities previously regarded as geometric and nonnumerical, Thabit started a trend which led eventually to the generalisation of the number concept.
In some respects, Thabit is critical of the ideas of Plato and Aristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments. Another important contribution Thabit made to geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
was his generalization of the Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
, which he extended from special right trianglesA special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 454590. This is called an "anglebased" right triangle...
to all triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
s in general, along with a general proofIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
.
Ibrahim ibn SinanIbrahim ibn Sinan ibn Thabit ibn Qurra was a Syriac speaking Mandean from Harran in northern Mesopotamia/Assyria. He was mathematician and astronomer who studied geometry and in particular tangents to circles. He also made advances in the theory of integration...
ibn Thabit (born 908Year 908 was a leap year starting on Friday of the Julian calendar. Asia :* The Battle of Belach Mugna is fought.* Zhu Wen kills the last Tang Dynasty emperor. Deaths :* AlMuktafi, Abbasid caliph...
), who introduced a method of integrationIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
more general than that of ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
, and alQuhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular Ibn alHaytham, studied opticsOptics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
and investigated the optical properties of mirrors made from conic sectionIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s.
Astronomy, timekeeping and geographyGeography is the science that studies the lands, features, inhabitants, and phenomena of Earth. A literal translation would be "to describe or write about the Earth". The first person to use the word "geography" was Eratosthenes...
provided other motivations for geometrical and trigonometrical research. For example Ibrahim ibn Sinan and his grandfather Thabit ibn Qurra' was a mathematician, physician, astronomer and translator of the Islamic Golden Age.Ibn Qurra made important discoveries in algebra, geometry and astronomy...
both studied curves required in the construction of sundials. Abu'lWafa and Abu Nasr MansurAbu Nasr Mansur ibn Ali ibn Iraq was a Persian Muslim mathematician. He is well known for his work with the spherical sine law....
both applied spherical geometrySpherical geometry is the geometry of the twodimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....
to astronomy.
Geometric architecture
Recent discoveries have shown that geometrical quasicrystalA quasiperiodic crystal, or, in short, quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry...
patterns were first employed in the girih tilesGirih tiles are a set of five tiles that were used in the creation of tiling patterns for decoration of buildings in Islamic architecture...
found in medieval Islamic architectureIslamic architecture encompasses a wide range of both secular and religious styles from the foundation of Islam to the present day, influencing the design and construction of buildings and structures in Islamic culture....
dating back over five centuries ago. In 2007, Professor Peter LuPeter James Lu, PhD is a postdoctoral research fellow in the Department of Physics and the School of Engineering and Applied Sciences at Harvard University in Cambridge, Massachusetts...
of Harvard UniversityHarvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
and Professor Paul SteinhardtPaul J. Steinhardt is the Albert Einstein Professor of Science at Princeton University and a professor of theoretical physics. He received his B.S. at the California Institute of Technology and his Ph.D. in Physics at Harvard University...
of Princeton UniversityPrinceton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
published a paper in the journal Science suggesting that girih tilings possessed properties consistent with selfsimilar fractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reducedsize copy of the whole," a property called selfsimilarity...
quasicrystalline tilings such as the Penrose tilingA Penrose tiling is a nonperiodic tiling generated by an aperiodic set of prototiles named after Sir Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original...
s, predating them by five centuries.
The 17th century
When Europe began to emerge from its Dark AgesThe Middle Ages is a periodization of European history from the 5th century to the 15th century. The Middle Ages follows the fall of the Western Roman Empire in 476 and precedes the Early Modern Era. It is the middle period of a threeperiod division of Western history: Classic, Medieval and Modern...
, the Hellenistic and IslamIslam . The most common are and . : Arabic pronunciation varies regionally. The first vowel ranges from ~~. The second vowel ranges from ~~~...
ic texts on geometry found in Islamic libraries were translated from Arabic into LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient ProtoIndoEuropean language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
. The rigorous deductive methods of geometry found in Euclid’s Elements of Geometry were relearned, and further development of geometry in the styles of both Euclid (Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) and Khayyam (algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
) continued, resulting in an abundance of new theorems and concepts, many of them very profound and elegant.
In the early 17th century, there were two important developments in geometry. The first and most important was the creation of analytic geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
, or geometry with coordinatesIn geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
and equationAn equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
s, by René Descartes (1596–1650) and Pierre de FermatPierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
(1601–1665). This was a necessary precursor to the development of calculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
and a precise quantitative science of physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. The second geometric development of this period was the systematic study of projective geometryIn mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
by Girard Desargues (1591–1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other. There had been some early work in this area by Hellenistic geometers, notably PappusPappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
(c. 340). The greatest flowering of the field occurred with JeanVictor PonceletJeanVictor Poncelet was a French engineer and mathematician who served most notably as the commandant general of the École Polytechnique...
(1788–1867).
In the late 17th century, calculus was developed independently and almost simultaneously by Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
(1642–1727) and Gottfried Wilhelm von Leibniz (1646–1716). This was the beginning of a new field of mathematics now called analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
. Though not itself a branch of geometry, it is applicable to geometry, and it solved two families of problems that had long been almost intractable: finding tangent lines to odd curves, and finding areas enclosed by those curves. The methods of calculus reduced these problems mostly to straightforward matters of computation.
NonEuclidean geometry
The old problem of proving Euclid’s Fifth Postulate, the "Parallel PostulateIn geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. Though Omar Khayyám was also unsuccessful in proving the parallel postulate, his criticisms of Euclid's theories of parallels and his proof of properties of figures in nonEuclidean geometries contributed to the eventual development of nonEuclidean geometryNonEuclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
. By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. Saccheri, LambertJohann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.Biography:...
, and LegendreAdrienMarie Legendre was a French mathematician.The Moon crater Legendre is named after him. Life :...
each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
, Johann BolyaiJános Bolyai was a Hungarian mathematician, known for his work in nonEuclidean geometry.Bolyai was born in the Transylvanian town of Kolozsvár , then part of the Habsburg Empire , the son of Zsuzsanna Benkő and the wellknown mathematician Farkas Bolyai.Life:By the age of 13, he had mastered...
, and Lobatchewsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a selfconsistent geometry in which that postulate was false. In this they were successful, thus creating the first nonEuclidean geometry. By 1854, Bernhard RiemannGeorg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
, a student of Gauss, had applied methods of calculus in a groundbreaking study of the intrinsic (selfcontained) geometry of all smooth surfaces, and thereby found a different nonEuclidean geometry. This work of Riemann later became fundamental for EinsteinAlbert Einstein was a Germanborn theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
's theory of relativityThe theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
.
It remained to be proved mathematically that the nonEuclidean geometry was just as selfconsistent as Euclidean geometry, and this was first accomplished by BeltramiEugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics...
in 1868. With this, nonEuclidean geometry was established on an equal mathematical footing with Euclidean geometry.
While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense (interstellar, not earthbound) distances. With the development of relativity theory in physics, this question became vastly more complicated.
Introduction of mathematical rigor
All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms were given by David HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
in 1894 in his dissertation Grundlagen der Geometrie (Foundations of Geometry). Some other complete sets of axioms had been given a few years earlier, but did not match Hilbert's in economy, elegance, and similarity to Euclid's axioms.
Analysis situs, or topology
In the mid18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a metric space was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus and analysisrelated concepts came to be known as analysis situs, and later as topologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and nonEuclidean geometry. Topology soon became a separate field of major importance, rather than a subfield of geometry or analysis.
The 20th century
Developments in algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
included the study of curves and surfaces over finite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s as demonstrated by the works of among others André WeilAndré Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
, Alexander GrothendieckAlexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
, and JeanPierre SerreJeanPierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.Early years:...
as well as over the real or complex numbers. Finite geometryA finite geometry is any geometric system that has only a finite number of points.Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact as many points as there are real numbers...
itself, the study of spaces with only finitely many points, found applications in coding theoryCoding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, errorcorrection and more recently also for network coding...
and cryptographyCryptography is the practice and study of techniques for secure communication in the presence of third parties...
. With the advent of the computer, new disciplines such as computational geometryComputational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...
or digital geometryDigital geometry deals with discrete sets considered to be digitized models or images of objects of the 2D or 3D Euclidean space.Simply put, digitizing is replacing an object by a discrete set of its points...
deal with geometric algorithms, discrete representations of geometric data, and so forth.
See also
{{wikisourceFlatland}}
 Flatland
Flatland: A Romance of Many Dimensions is an 1884 satirical novella by the English schoolmaster Edwin Abbott Abbott. Writing pseudonymously as "A Square", Abbott used the fictional twodimensional world of Flatland to offer pointed observations on the social hierarchy of Victorian culture...
, Book written by " A^{2} " about two and threedimensional spaceThreedimensional space is a geometric 3parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
, to understand the concept of four dimensions
 History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....
 Important publications in geometry.
 Interactive geometry software
Interactive geometry software are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry. In most IGS, one starts construction by putting a few points and using them to define new objects such as lines, circles or other points...
 List of geometry topics
External links
{{DEFAULTSORT:History Of Geometry}}