History of Math

Mathematics, study of relationships among quantities, magnitudes, and properties and of

logical operations by which unknown quantities, magnitudes, and properties may be deduced.

In the past, mathematics was regarded as the science of quantity, whether of magnitudes,

as in geometry, or of numbers, as in arithmetic, or of the generalization of these two

fields, as in algebra. Toward the middle of the 19th century, however, mathematics came to

be regarded increasingly as the science of relations, or as the science that draws

necessary conclusions. This latter view encompasses mathematical or symbolic logic, the

science of using symbols to provide an exact theory of logical deduction and inference

based on definitions, axioms, postulates, and rules for combining and transforming

primitive elements into more complex relations and theorems.

This brief survey of the history of mathematics traces the evolution of mathematical ideas

and concepts, beginning in prehistory. Indeed, mathematics is nearly as old as humanity

itself; evidence of a sense of geometry and interest in geometric pattern has been found

in the designs of prehistoric pottery and textiles and in cave paintings. Primitive

counting systems were almost certainly based on using the fingers of one or both hands, as

evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems

today.

Mathematics, study of relationships among quantities, magnitudes, and properties and of

logical operations by which unknown quantities, magnitudes, and properties may be deduced.

In the past, mathematics was regarded as the science of quantity, whether of magnitudes,

as in geometry, or of numbers, as in arithmetic, or of the generalization of these two

fields, as in algebra. Toward the middle of the 19th century, however, mathematics came to

be regarded increasingly as the science of relations, or as the science that draws

necessary conclusions. This latter view encompasses mathematical or symbolic logic, the

science of using symbols to provide an exact theory of logical deduction and inference

based on definitions, axioms, postulates, and rules for combining and transforming

primitive elements into more complex relations and theorems.

This brief survey of the history of mathematics traces the evolution of mathematical ideas

and concepts, beginning in prehistory. Indeed, mathematics is nearly as old as humanity

itself; evidence of a sense of geometry and interest in geometric pattern has been found

in the designs of prehistoric pottery and textiles and in cave paintings. Primitive

counting systems were almost certainly based on using the fingers of one or both hands, as

evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems

today.

Ancient Mathematics

The earliest records of advanced, organized mathematics date back to the ancient

Mesopotamian country of Babylonia and to Egypt of the 3rd millennium BC. There mathematics

was dominated by arithmetic, with an emphasis on measurement and calculation in geometry

and with no trace of later mathematical concepts such as axioms or proofs.

The earliest Egyptian texts, composed about 1800 BC, reveal a decimal numeration system

with separate symbols for the successive powers of 10 (1, 10, 100, and so forth), just as

in the system used by the Romans. Numbers were represented by writing down the symbol for

1, 10, 100, and so on as many times as the unit was in a given number. For example, the

symbol for 1 was written five times to represent the number 5, the symbol for 10 was

written six times to represent the number 60, and the symbol for 100 was written three

times to represent the number 300. Together, these symbols represented the number 365.

Addition was done by totaling separately the units-10s, 100s, and so forth-in the numbers

to be added. Multiplication was based on successive doublings, and division was based on

the inverse of this process.

The Egyptians used sums of unit fractions (a), supplemented by the fraction B, to express

all other fractions. For example, the fraction E was the sum of the fractions 3 and *.

Using this system, the Egyptians were able to solve all problems of arithmetic that

involved fractions, as well as some elementary problems in algebra. In geometry, the

Egyptians calculated the correct areas of triangles, rectangles, and trapezoids and the

The earliest records of advanced, organized mathematics date back to the ancient

Mesopotamian country of Babylonia and to Egypt of the 3rd millennium BC. There mathematics

was dominated by arithmetic, with an emphasis on measurement and calculation in geometry

and with no trace of later mathematical concepts such as axioms or proofs.

The earliest Egyptian texts, composed about 1800 BC, reveal a decimal numeration system

with separate symbols for the successive powers of 10 (1, 10, 100, and so forth), just as

in the system used by the Romans. Numbers were represented by writing down the symbol for

1, 10, 100, and so on as many times as the unit was in a given number. For example, the

symbol for 1 was written five times to represent the number 5, the symbol for 10 was

written six times to represent the number 60, and the symbol for 100 was written three

times to represent the number 300. Together, these symbols represented the number 365.

Addition was done by totaling separately the units-10s, 100s, and so forth-in the numbers

to be added. Multiplication was based on successive doublings, and division was based on

the inverse of this process.

The Egyptians used sums of unit fractions (a), supplemented by the fraction B, to express

all other fractions. For example, the fraction E was the sum of the fractions 3 and *.

Using this system, the Egyptians were able to solve all problems of arithmetic that

involved fractions, as well as some elementary problems in algebra. In geometry, the

Egyptians calculated the correct areas of triangles, rectangles, and trapezoids and the

volumes of figures such as bricks, cylinders, and pyramids. To find the area of a circle,

the Egyptians used the square on U of the diameter of the circle, a value of about

3.16-close to the value of the ratio known as pi, which is about 3.14.

The Babylonian system of numeration was quite different from the Egyptian system. In the

Babylonian system-which, when using clay tablets, consisted of various wedge-shaped

marks-a single wedge indicated 1 and an arrowlike wedge stood for 10 (see table). Numbers

up through 59 were formed from these symbols through an additive process, as in Egyptian

mathematics. The number 60, however, was represented by the same symbol as 1, and from

this point on a positional symbol was used. That is, the value of one of the first 59

numerals depended henceforth on its position in the total numeral. For example, a numeral

consisting of a symbol for 2 followed by one for 27 and ending in one for 10 stood for 2 ×

602 27 × 60 10. This principle was extended to the representation of fractions as

well, so that the above sequence of numbers could equally well represent 2 × 60 27 10

× (†), or 2 27 × (†) 10 × (†-2). With this sexagesimal system (base

60), as it is called, the Babylonians had as convenient a numerical system as the 10-based

system.

The Babylonians in time developed a sophisticated mathematics by which they could find the

positive roots of any quadratic equation (Equation). They could even find the roots of

certain cubic equations. The Babylonians had a variety of tables, including tables for

multiplication and division, tables of squares, and tables of compound interest. They

the Egyptians used the square on U of the diameter of the circle, a value of about

3.16-close to the value of the ratio known as pi, which is about 3.14.

The Babylonian system of numeration was quite different from the Egyptian system. In the

Babylonian system-which, when using clay tablets, consisted of various wedge-shaped

marks-a single wedge indicated 1 and an arrowlike wedge stood for 10 (see table). Numbers

up through 59 were formed from these symbols through an additive process, as in Egyptian

mathematics. The number 60, however, was represented by the same symbol as 1, and from

this point on a positional symbol was used. That is, the value of one of the first 59

numerals depended henceforth on its position in the total numeral. For example, a numeral

consisting of a symbol for 2 followed by one for 27 and ending in one for 10 stood for 2 ×

602 27 × 60 10. This principle was extended to the representation of fractions as

well, so that the above sequence of numbers could equally well represent 2 × 60 27 10

× (†), or 2 27 × (†) 10 × (†-2). With this sexagesimal system (base

60), as it is called, the Babylonians had as convenient a numerical system as the 10-based

system.

The Babylonians in time developed a sophisticated mathematics by which they could find the

positive roots of any quadratic equation (Equation). They could even find the roots of

certain cubic equations. The Babylonians had a variety of tables, including tables for

multiplication and division, tables of squares, and tables of compound interest. They

could solve complicated problems using the Pythagorean theorem; one of their tables

contains integer solutions to the Pythagorean equation, a2 b2 = c2, arranged so that

c2/a2 decreases steadily from 2 to about J. The Babylonians were able to sum arithmetic

and some geometric progressions, as well as sequences of squares. They also arrived at a

good approximation for ¸. In geometry, they calculated the areas of rectangles, triangles,

and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders.

However, the Babylonians did not arrive at the correct formula for the volume of a

pyramid.

Greek Mathematics

The Greeks adopted elements of mathematics from both the Babylonians and the Egyptians.

The new element in Greek mathematics, however, was the invention of an abstract

mathematics founded on a logical structure of definitions, axioms, and proofs. According

to later Greek accounts, this development began in the 6th century BC with Thales of

Miletus and Pythagoras of Samos, the latter a religious leader who taught the importance

of studying numbers in order to understand the world. Some of his disciples made important

discoveries about the theory of numbers and geometry, all of which were attributed to

Pythagoras.

In the 5th century BC, some of the great geometers were the atomist philosopher Democritus

of Abdera, who discovered the correct formula for the volume of a pyramid, and Hippocrates

of Chios, who discovered that the areas of crescent-shaped figures bounded by arcs of

contains integer solutions to the Pythagorean equation, a2 b2 = c2, arranged so that

c2/a2 decreases steadily from 2 to about J. The Babylonians were able to sum arithmetic

and some geometric progressions, as well as sequences of squares. They also arrived at a

good approximation for ¸. In geometry, they calculated the areas of rectangles, triangles,

and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders.

However, the Babylonians did not arrive at the correct formula for the volume of a

pyramid.

Greek Mathematics

The Greeks adopted elements of mathematics from both the Babylonians and the Egyptians.

The new element in Greek mathematics, however, was the invention of an abstract

mathematics founded on a logical structure of definitions, axioms, and proofs. According

to later Greek accounts, this development began in the 6th century BC with Thales of

Miletus and Pythagoras of Samos, the latter a religious leader who taught the importance

of studying numbers in order to understand the world. Some of his disciples made important

discoveries about the theory of numbers and geometry, all of which were attributed to

Pythagoras.

In the 5th century BC, some of the great geometers were the atomist philosopher Democritus

of Abdera, who discovered the correct formula for the volume of a pyramid, and Hippocrates

of Chios, who discovered that the areas of crescent-shaped figures bounded by arcs of

circles are equal to areas of certain triangles. This discovery is related to the famous

problem of squaring the circle-that is, constructing a square equal in area to a given

circle. Two other famous mathematical problems that originated during the century were

those of trisecting an angle and doubling a cube-that is, constructing a cube the volume

of which is double that of a given cube. All of these problems were solved, and in a

variety of ways, all involving the use of instruments more complicated than a straightedge

and a geometrical compass. Not until the 19th century, however, was it shown that the

three problems mentioned above could never have been solved using those instruments alone.

In the latter part of the 5th century BC, an unknown mathematician discovered that no unit

of length would measure both the side and diagonal of a square. That is, the two lengths

are incommensurable. This means that no counting numbers n and m exist whose ratio

expresses the relationship of the side to the diagonal. Since the Greeks considered only

the counting numbers (1, 2, 3, and so on) as numbers, they had no numerical way to express

this ratio of diagonal to side. (This ratio, ¸, would today be called irrational.) As a

consequence the Pythagorean theory of ratio, based on numbers, had to be abandoned and a

new, nonnumerical theory introduced. This was done by the 4th-century BC mathematician

Eudoxus of Cnidus, whose solution may be found in the Elements of Euclid. Eudoxus also

discovered a method for rigorously proving statements about areas and volumes by

successive approximations.

Euclid was a mathematician and teacher who worked at the famed Museum of Alexandria and

problem of squaring the circle-that is, constructing a square equal in area to a given

circle. Two other famous mathematical problems that originated during the century were

those of trisecting an angle and doubling a cube-that is, constructing a cube the volume

of which is double that of a given cube. All of these problems were solved, and in a

variety of ways, all involving the use of instruments more complicated than a straightedge

and a geometrical compass. Not until the 19th century, however, was it shown that the

three problems mentioned above could never have been solved using those instruments alone.

In the latter part of the 5th century BC, an unknown mathematician discovered that no unit

of length would measure both the side and diagonal of a square. That is, the two lengths

are incommensurable. This means that no counting numbers n and m exist whose ratio

expresses the relationship of the side to the diagonal. Since the Greeks considered only

the counting numbers (1, 2, 3, and so on) as numbers, they had no numerical way to express

this ratio of diagonal to side. (This ratio, ¸, would today be called irrational.) As a

consequence the Pythagorean theory of ratio, based on numbers, had to be abandoned and a

new, nonnumerical theory introduced. This was done by the 4th-century BC mathematician

Eudoxus of Cnidus, whose solution may be found in the Elements of Euclid. Eudoxus also

discovered a method for rigorously proving statements about areas and volumes by

successive approximations.

Euclid was a mathematician and teacher who worked at the famed Museum of Alexandria and

who also wrote on optics, astronomy, and music. The 13 books that make up his Elements

contain much of the basic mathematical knowledge discovered up to the end of the 4th

century BC on the geometry of polygons and the circle, the theory of numbers, the theory

of incommensurables, solid geometry, and the elementary theory of areas and volumes.

The century that followed Euclid was marked by mathematical brilliance, as displayed in

the works of Archimedes of Syracuse and a younger contemporary, Apollonius of Perga.

Archimedes used a method of discovery, based on theoretically weighing infinitely thin

slices of figures, to find the areas and volumes of figures arising from the conic

sections. These conic sections had been discovered by a pupil of Eudoxus named Menaechmus,

and they were the subject of a treatise by Euclid, but Archimedes' writings on them are

the earliest to survive. Archimedes also investigated centers of gravity and the stability

of various solids floating in water. Much of his work is part of the tradition that led,

in the 17th century, to the discovery of the calculus. Archimedes was killed by a Roman

soldier during the sack of Syracuse. His younger contemporary, Apollonius, produced an

eight-book treatise on the conic sections that established the names of the sections:

ellipse, parabola, and hyperbola. It also provided the basic treatment of their geometry

until the time of the French philosopher and scientist René Descartes in the 17th century.

After Euclid, Archimedes, and Apollonius, Greece produced no geometers of comparable

stature. The writings of Hero of Alexandria in the 1st century AD show how elements of

both the Babylonian and Egyptian mensurational, arithmetic traditions survived alongside

contain much of the basic mathematical knowledge discovered up to the end of the 4th

century BC on the geometry of polygons and the circle, the theory of numbers, the theory

of incommensurables, solid geometry, and the elementary theory of areas and volumes.

The century that followed Euclid was marked by mathematical brilliance, as displayed in

the works of Archimedes of Syracuse and a younger contemporary, Apollonius of Perga.

Archimedes used a method of discovery, based on theoretically weighing infinitely thin

slices of figures, to find the areas and volumes of figures arising from the conic

sections. These conic sections had been discovered by a pupil of Eudoxus named Menaechmus,

and they were the subject of a treatise by Euclid, but Archimedes' writings on them are

the earliest to survive. Archimedes also investigated centers of gravity and the stability

of various solids floating in water. Much of his work is part of the tradition that led,

in the 17th century, to the discovery of the calculus. Archimedes was killed by a Roman

soldier during the sack of Syracuse. His younger contemporary, Apollonius, produced an

eight-book treatise on the conic sections that established the names of the sections:

ellipse, parabola, and hyperbola. It also provided the basic treatment of their geometry

until the time of the French philosopher and scientist René Descartes in the 17th century.

After Euclid, Archimedes, and Apollonius, Greece produced no geometers of comparable

stature. The writings of Hero of Alexandria in the 1st century AD show how elements of

both the Babylonian and Egyptian mensurational, arithmetic traditions survived alongside

the logical edifices of the great geometers. Very much in the same tradition, but

concerned with much more difficult problems, are the books of Diophantus of Alexandria in

the 3rd century AD. They deal with finding rational solutions to kinds of problems that

lead immediately to equations in several unknowns. Such equations are now called

Diophantine equations (see Diophantine Analysis).

Applied Mathematics in Greece

Paralleling the studies described in pure mathematics were studies made in optics,

mechanics, and astronomy. Many of the greatest mathematical writers, such as Euclid and

Archimedes, also wrote on astronomical topics. Shortly after the time of Apollonius, Greek

astronomers adopted the Babylonian system for recording fractions and, at about the same

time, composed tables of chords in a circle. For a circle of some fixed radius, such

tables give the length of the chords subtending a sequence of arcs increasing by some

fixed amount. They are equivalent to a modern sine table, and their composition marks the

beginnings of trigonometry. In the earliest such tables-those of Hipparchus in about 150

BC-the arcs increased by steps of 71°, from 0° to 180°. By the time of the astronomer

Ptolemy in the 2nd century AD, however, Greek mastery of numerical procedures had

progressed to the point where Ptolemy was able to include in his Almagest a table of

chords in a circle for steps of 3°, which, although expressed sexagesimally, is accurate

to about five decimal places.

In the meantime, methods were developed for solving problems involving plane triangles,

concerned with much more difficult problems, are the books of Diophantus of Alexandria in

the 3rd century AD. They deal with finding rational solutions to kinds of problems that

lead immediately to equations in several unknowns. Such equations are now called

Diophantine equations (see Diophantine Analysis).

Applied Mathematics in Greece

Paralleling the studies described in pure mathematics were studies made in optics,

mechanics, and astronomy. Many of the greatest mathematical writers, such as Euclid and

Archimedes, also wrote on astronomical topics. Shortly after the time of Apollonius, Greek

astronomers adopted the Babylonian system for recording fractions and, at about the same

time, composed tables of chords in a circle. For a circle of some fixed radius, such

tables give the length of the chords subtending a sequence of arcs increasing by some

fixed amount. They are equivalent to a modern sine table, and their composition marks the

beginnings of trigonometry. In the earliest such tables-those of Hipparchus in about 150

BC-the arcs increased by steps of 71°, from 0° to 180°. By the time of the astronomer

Ptolemy in the 2nd century AD, however, Greek mastery of numerical procedures had

progressed to the point where Ptolemy was able to include in his Almagest a table of

chords in a circle for steps of 3°, which, although expressed sexagesimally, is accurate

to about five decimal places.

In the meantime, methods were developed for solving problems involving plane triangles,

and a theorem-named after the astronomer Menelaus of Alexandria-was established for

finding the lengths of certain arcs on a sphere when other arcs are known. These advances

gave Greek astronomers what they needed to solve the problems of spherical astronomy and

to develop an astronomical system that held sway until the time of the German astronomer

Johannes Kepler.

Medieval and Renaissance Mathematics

Following the time of Ptolemy, a tradition of study of the mathematical masterpieces of

the preceding centuries was established in various centers of Greek learning. The

preservation of such works as have survived to modern times began with this tradition. It

was continued in the Islamic world, where original developments based on these

masterpieces first appeared.

Islamic and Indian Mathematics

After a century of expansion in which the religion of Islam spread from its beginnings in

the Arabian Peninsula to dominate an area extending from Spain to the borders of China,

Muslims began to acquire the results of the “foreign sciences.” At centers such as the

House of Wisdom in Baghdad, supported by the ruling caliphs and wealthy individuals,

translators produced Arabic versions of Greek and Indian mathematical works.

By the year 900 AD the acquisition was complete, and Muslim scholars began to build on

what they had acquired. Thus mathematicians extended the Hindu decimal positional system

finding the lengths of certain arcs on a sphere when other arcs are known. These advances

gave Greek astronomers what they needed to solve the problems of spherical astronomy and

to develop an astronomical system that held sway until the time of the German astronomer

Johannes Kepler.

Medieval and Renaissance Mathematics

Following the time of Ptolemy, a tradition of study of the mathematical masterpieces of

the preceding centuries was established in various centers of Greek learning. The

preservation of such works as have survived to modern times began with this tradition. It

was continued in the Islamic world, where original developments based on these

masterpieces first appeared.

Islamic and Indian Mathematics

After a century of expansion in which the religion of Islam spread from its beginnings in

the Arabian Peninsula to dominate an area extending from Spain to the borders of China,

Muslims began to acquire the results of the “foreign sciences.” At centers such as the

House of Wisdom in Baghdad, supported by the ruling caliphs and wealthy individuals,

translators produced Arabic versions of Greek and Indian mathematical works.

By the year 900 AD the acquisition was complete, and Muslim scholars began to build on

what they had acquired. Thus mathematicians extended the Hindu decimal positional system

of arithmetic from whole numbers to include decimal fractions, and the 12th-century

Persian mathematician Omar Khayyam generalized Hindu methods for extracting square and

cube roots to include fourth, fifth, and higher roots. In algebra, al-Karaji completed the

algebra of polynomials of Muhammad al-Khwarizmi. Al-Karaji included polynomials with an

infinite number of terms. (Al-Khwarizmi's name, incidentally, is the source of the word

algorithm, and the title of one of his books is the source of the word algebra.) Geometers

such as Ibrahim ibn Sinan continued Archimedes' investigations of areas and volumes, and

Kamal al-Din and others applied the theory of conic sections to solve optical problems.

Using the Hindu sine function and Menelaus's theorem, mathematicians from Habas al-Hasib

to Nasir ad-Din at-Tusi created the mathematical disciplines of plane and spherical

trigonometry. These did not become mathematical disciplines in the West, however, until

the publication of De Triangulis Omnimodibus by the German astronomer Regiomontanus.

Finally, a number of Muslim mathematicians made important discoveries in the theory of

numbers, while others explained a variety of numerical methods for solving equations. The

Latin West acquired much of this learning during the 12th century, the great century of

translation. Together with translations of the Greek classics, these Muslim works were

responsible for the growth of mathematics in the West during the late Middle Ages. Italian

mathematicians such as Leonardo Fibonacci and Luca Pacioli, one of the many 15th-century

writers on algebra and arithmetic for merchants, depended heavily on Arabic sources for

their knowledge.

Persian mathematician Omar Khayyam generalized Hindu methods for extracting square and

cube roots to include fourth, fifth, and higher roots. In algebra, al-Karaji completed the

algebra of polynomials of Muhammad al-Khwarizmi. Al-Karaji included polynomials with an

infinite number of terms. (Al-Khwarizmi's name, incidentally, is the source of the word

algorithm, and the title of one of his books is the source of the word algebra.) Geometers

such as Ibrahim ibn Sinan continued Archimedes' investigations of areas and volumes, and

Kamal al-Din and others applied the theory of conic sections to solve optical problems.

Using the Hindu sine function and Menelaus's theorem, mathematicians from Habas al-Hasib

to Nasir ad-Din at-Tusi created the mathematical disciplines of plane and spherical

trigonometry. These did not become mathematical disciplines in the West, however, until

the publication of De Triangulis Omnimodibus by the German astronomer Regiomontanus.

Finally, a number of Muslim mathematicians made important discoveries in the theory of

numbers, while others explained a variety of numerical methods for solving equations. The

Latin West acquired much of this learning during the 12th century, the great century of

translation. Together with translations of the Greek classics, these Muslim works were

responsible for the growth of mathematics in the West during the late Middle Ages. Italian

mathematicians such as Leonardo Fibonacci and Luca Pacioli, one of the many 15th-century

writers on algebra and arithmetic for merchants, depended heavily on Arabic sources for

their knowledge.

Western Renaissance Mathematics

Although the late medieval period saw some fruitful mathematical considerations of

problems of infinity by writers such as Nicole Oresme, it was not until the early 16th

century that a truly important mathematical discovery was made in the West. The discovery,

an algebraic formula for the solution of both the cubic and quartic equations, was

published in 1545 by the Italian mathematician Gerolamo Cardano in his Ars Magna. The

discovery drew the attention of mathematicians to complex numbers and stimulated a search

for solutions to equations of degree higher than 4. It was this search, in turn, that led

to the first work on group theory (Group) at the end of the 18th century, and to the

theory of equations developed by the French mathematician Évariste Galois in the early

19th century.

The 16th century also saw the beginnings of modern algebraic symbolism (Mathematical

Symbols), as well as the remarkable work on the solution of equations by the French

mathematician François Viète. His writings influenced many mathematicians of the following

century, including Pierre de Fermat in France and Isaac Newton in England.

Mathematics Since the 16th Century

Europeans dominated in the development of mathematics after the Renaissance.

17th Century

During the 17th century, the greatest advances were made in mathematics since the time of

Archimedes and Apollonius. The century opened with the discovery of logarithms by the

Although the late medieval period saw some fruitful mathematical considerations of

problems of infinity by writers such as Nicole Oresme, it was not until the early 16th

century that a truly important mathematical discovery was made in the West. The discovery,

an algebraic formula for the solution of both the cubic and quartic equations, was

published in 1545 by the Italian mathematician Gerolamo Cardano in his Ars Magna. The

discovery drew the attention of mathematicians to complex numbers and stimulated a search

for solutions to equations of degree higher than 4. It was this search, in turn, that led

to the first work on group theory (Group) at the end of the 18th century, and to the

theory of equations developed by the French mathematician Évariste Galois in the early

19th century.

The 16th century also saw the beginnings of modern algebraic symbolism (Mathematical

Symbols), as well as the remarkable work on the solution of equations by the French

mathematician François Viète. His writings influenced many mathematicians of the following

century, including Pierre de Fermat in France and Isaac Newton in England.

Mathematics Since the 16th Century

Europeans dominated in the development of mathematics after the Renaissance.

17th Century

During the 17th century, the greatest advances were made in mathematics since the time of

Archimedes and Apollonius. The century opened with the discovery of logarithms by the