Early European arithmetics was largely influenced by the Romans as their number system remained in use for over 500 years. As arithmetic was virtually impossible with Roman numerals, merchants and accountants used the abacus until the Roman number system was replaced by the Hindu-Arabic notation we still use today. The battle between the algorists, who used algorithms (calculating methods) with Hindu-Arabic numerals and the abacists, who used an abacus and Roman numerals, raged for centuries.

Italian Mathematicians

by Catherine Marien

by Catherine Marien

The Roman abacus was the first portable calculator used by merchants and clerks that greatly reduced the time needed to perform the basic operations of arithmetic using Roman numerals. Predating the Chinese Suan Pan, the Roman abacus is based on a biquinary system with the longer grooves denoting ten units (1, 10, 100, 1000, etc.) and the shorter grooves denoting five units, or fives (5, 50, 500, 5000, etc.), as in the Roman numeral system. Image source: Opera historica et philologica by Marcus Welser,1682.

While it were the Indians who invented zero and the place-value system, it were the Arab scholars who brought the Indian discoveries to the Western world. Through their conquests the Arabs reached as far as Southern Italy and Sicily. Saying that mathematical science has a long tradition in Italy is an understatement. It was the enormous prestige of Italian scholarship acquired in the sixteenth century that spread some of the concepts brought by the Arabs to the rest of pre-cartesian Europe. Both Leonardo of Pisa (Fibonacci) and Luca Pacioli were instrumental in popularizing the Hindu-Arabic system, especially in mathematical functions (merchants, accountants) It was Fibonacci's term zephirum which gave rise to the modern term of zero, by the way of the Venetian zefiro.

The earliest printed arithmetic was published anonymously in Treviso in 1478.

Luca Pacioli's Summa, published in 1494, is considered to be the first book on algebra. It is also known as the first work on accountancy practices.

Renaissance mathematicians such as Gerolamo Cardano were responsible for the most important developments in algebra since the Babylonians.

The length of a foot, divided into twelve inches, as a measure of length originated with the Romans, although it was probably slightly shorter. The Romans also had a palm, which was a quarter of a foot. These measures along with the Roman measures for weight based on pounds and ounces, spread through Europe and from there to the rest of the world.

(In alphabetical order:)

The earliest printed arithmetic was published anonymously in Treviso in 1478.

Luca Pacioli's Summa, published in 1494, is considered to be the first book on algebra. It is also known as the first work on accountancy practices.

Renaissance mathematicians such as Gerolamo Cardano were responsible for the most important developments in algebra since the Babylonians.

The length of a foot, divided into twelve inches, as a measure of length originated with the Romans, although it was probably slightly shorter. The Romans also had a palm, which was a quarter of a foot. These measures along with the Roman measures for weight based on pounds and ounces, spread through Europe and from there to the rest of the world.

(In alphabetical order:)

Maria Gaetana Agnesi

(1718 - 1799)

Maria Gaetana Agnesi was a mathematician and philosopher from Milan famous for writing the first book discussing both differential and integral calculus, Instituzioni analitiche ad uso della gioventù italiana, published in 1748. Among other things the book discussed the Witch of Agnesi, a curve earlier studied and constructed by Pierre de Fermat and Guido Grandi.

(1718 - 1799)

Maria Gaetana Agnesi was a mathematician and philosopher from Milan famous for writing the first book discussing both differential and integral calculus, Instituzioni analitiche ad uso della gioventù italiana, published in 1748. Among other things the book discussed the Witch of Agnesi, a curve earlier studied and constructed by Pierre de Fermat and Guido Grandi.

Archimedes (c. 287 BC- c. 212 BC)

Archimedes was an ancient mathematician, physicist, astronomer and engineer born in Syracyse, Sicily, then a colony of Magna Graecia.

Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity.

One of his mathematical contributions was the development of a sophisticated method to approximate the value of π (pi). His method involved drawing drawing a smaller polygon inside a circle and a larger polygon outside the circle, giving upper and lower limits for a value of π.

Archimedes was an ancient mathematician, physicist, astronomer and engineer born in Syracyse, Sicily, then a colony of Magna Graecia.

Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity.

One of his mathematical contributions was the development of a sophisticated method to approximate the value of π (pi). His method involved drawing drawing a smaller polygon inside a circle and a larger polygon outside the circle, giving upper and lower limits for a value of π.

The more sides are added to the polygon, the more accurate the limiting values. Archimedes settled on 96 sides and calculated each side, obtaining a value for π between 223/71 (3.14085) and 22/7 (3.14286), or an average value of about 3.1418. It was not until the 17th century that better methods of calculations of π were developed. Before that time more accurate values were found, but they were all based on the Archimedean method.

He also discovered that π could be used to calculate the area of a circle, as the area was equal to π multiplied by the square of the radius of the circle.

He is also credited with discovering, while he was in his bath, that the volume of an irregular shape can be found by measuring the volume of water it displaces, a discovery that supposedl led him to run naked down the street shouting 'Eureka!'.

In On the Sphere and Cylinder, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. This is the Archimedean propertyof real numbers.

Not all of his written work has survived, and seven of his treatises are known to exist only through references made to them by other authors.

He also discovered that π could be used to calculate the area of a circle, as the area was equal to π multiplied by the square of the radius of the circle.

He is also credited with discovering, while he was in his bath, that the volume of an irregular shape can be found by measuring the volume of water it displaces, a discovery that supposedl led him to run naked down the street shouting 'Eureka!'.

In On the Sphere and Cylinder, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. This is the Archimedean propertyof real numbers.

Not all of his written work has survived, and seven of his treatises are known to exist only through references made to them by other authors.

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Eugenio Beltrami (1835-1899)

Eugenio Beltrami was a cremonese mathematician famous for his work in non-Euclidian geometry and for his theories of surfaces of constant curvature.

He made a physical representation of the pseudosphere, showing that the surface of the pseudosphere satisfied Lobachevsky's geometry. These considerations of non-Euclidean geometry influenced the dadaist and surrealist painters.

He also pioneered the singular value decomposition for matrices with an article that appeared in the Journal of Mathematics for the Use of the Students of the Italian Universities.

Eugenio Beltrami was a cremonese mathematician famous for his work in non-Euclidian geometry and for his theories of surfaces of constant curvature.

He made a physical representation of the pseudosphere, showing that the surface of the pseudosphere satisfied Lobachevsky's geometry. These considerations of non-Euclidean geometry influenced the dadaist and surrealist painters.

He also pioneered the singular value decomposition for matrices with an article that appeared in the Journal of Mathematics for the Use of the Students of the Italian Universities.

Beltrami's Opere Matematiche (1902-20), a four-volume work published posthumously, contains his comments on a variety of physical and mathematical subjects, including electricity, thermodynamics, magnetism, optics and elasticity.

Rafael Bombelli (1526 - 1572)

In his Algebra published in 1572, this mathematician from Bologna extended the field of numbers to square roots and cube roots and first investigated and described imaginary numbers and complex numbers. An important part of his work was not published until the twentieth century.

Gerolamo Cardano

(Latin Hieronymus Cardanus,1501 - 1576)

Rafael Bombelli (1526 - 1572)

In his Algebra published in 1572, this mathematician from Bologna extended the field of numbers to square roots and cube roots and first investigated and described imaginary numbers and complex numbers. An important part of his work was not published until the twentieth century.

Gerolamo Cardano

(Latin Hieronymus Cardanus,1501 - 1576)

Gerolamo was a Pavian Renaissance mathematician and physician who was also found of astrology and games of chance. He was emprisoned for several months for calculating the horoscope of Jesus Christ. His interest in dice and card games and his Liber de ludo alea, published a century after his death, were early expressions of combinatorial mathematics. However, he remains famous as the most important mathematicians of his time for his contribution in algebra, considered the greatest advance, since the Babylonians. In his 1545 book Ars Magna, he explained how to solve cubic (third-order) and quartic (fourth-order) equations .

The solution to one particular case of the cubic, x3 + ax = b (in modern notation), was communicated to him by Niccolò Fontana Tartaglia, although it had probably been discovered earlier by Scipione del Ferro (c.1465-1526) from the university of Bologna who passed to information to his student Antonio Maria Fior. Tartaglia had independently discovered the solution and shared it with Cardano on condition that he would not reveal it. When Cardano discovered that the solution was not original to Tartaglia he decided to publish it, which led to a decade-long fight. The quartic was solved by Cardano's amanuensis Ludovico Ferrari. Both were acknowledged in his book. By opening up the possibility of algebra and algebraic geometry extending into more than three dimensions, Cardano laid the basis of Riemann geometrics and the four-dimensional space-time continuum with which Einstein would remodel the universe.

He acknowledged the existence of what are now called imaginary and complex numbers, although he did not understand their properties and discarded them as invalid or unuseful. In Opus novum de proportionibus he introduced the binomial coefficients and the binomial theorem. He also published the first systematic work on probability a century before Pascal and Fermat.

Leonardo Fibonacci (c. 1175-1250)

He acknowledged the existence of what are now called imaginary and complex numbers, although he did not understand their properties and discarded them as invalid or unuseful. In Opus novum de proportionibus he introduced the binomial coefficients and the binomial theorem. He also published the first systematic work on probability a century before Pascal and Fermat.

Leonardo Fibonacci (c. 1175-1250)

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Niccolo Fontana (c.1500 - 1557), better known as Tartaglia

Niccolo Fontana was a mathematician and engineer now best known for his decade-long conflict with Cardano.

Guido Fubini (1879-1943)

A Venetian mathematician known for Fubini's theorem.

Galileo Galilei (1564-1642)

Developed what is known as Galileo's paradox. Galileo concluded that the attributes of less, equal, and greater are not applicable to infinite, but only to finite sets. This conclusion was based on two apparently contradictory statements about the positive whole numbers. First, with finite sets, a part is always smaller than the whole. So, as integers include both squares and non-squares the set of integers must be larger than the set of squares. Second, we can say that as there is and infinite number of integers, there is an infinite number of squares. But can the infinite number of squares be larger than the infinite numbers of integers ? For each square we can find exactly one number that is its square root, and there are as many squares as there are numbers as for every number we can calculate its square, so, apparently, there cannot be more of one than of the other. So, with infinite sets one part of the set can be just as large as the whole, which is known as Galileo's paradox.

Luca Pacioli (1445-1514 or 1517)

Luca Pacioli compiled a treatise on number puzzles and magic that lay undiscovered in the archives of the university of Bologna until it was published in 2008. He collected and studied magic squares, arrangements of numbers in a square grid so that each horizontal, vertical and diagonal line of numbers adds up to the same total, called the magic constant. With his 1509 book De Divina Proportione Pacioli also gave new impetus to the theory of the golden ratio by illustraring the golden ratio as applied to human faces by artists, architects, scientists, and mystics. Pacioli also produced the first European text to use zero properly.

Giovanni Girolamo Saccheri (1667-1733)

His work on alternative geometries had little impact during his lifetime, but was rediscovered by Eugenio Beltrami in the mid-19th century.

Leonida Tonelli (1885 - 1946)

Author of the Tonelli's theorem, a successor of Fubini's theorem. The conclusion of Tonelli's theorem is identical to that of Fubini's theorem, but the assumptions are different. Tonelli's theorem states that on the product of two s-finite measure spaces, a product measure integral can be evaluated by way of an iterated integral for nonnegative measurable functions, regardless of whether they have finite integral.

Evangelista Torricelli (1608-1647)

A physicist and mathematician from Faenza famous for the discovery of the Torricelli's trumpet or horn whose surface area is infinite, but whose volume is finite. The discovery prompted a fierce controversy about the nature of infinity, and is supposed by some to have led to the idea of a "completed infinity".

Niccolo Fontana was a mathematician and engineer now best known for his decade-long conflict with Cardano.

Guido Fubini (1879-1943)

A Venetian mathematician known for Fubini's theorem.

Galileo Galilei (1564-1642)

Developed what is known as Galileo's paradox. Galileo concluded that the attributes of less, equal, and greater are not applicable to infinite, but only to finite sets. This conclusion was based on two apparently contradictory statements about the positive whole numbers. First, with finite sets, a part is always smaller than the whole. So, as integers include both squares and non-squares the set of integers must be larger than the set of squares. Second, we can say that as there is and infinite number of integers, there is an infinite number of squares. But can the infinite number of squares be larger than the infinite numbers of integers ? For each square we can find exactly one number that is its square root, and there are as many squares as there are numbers as for every number we can calculate its square, so, apparently, there cannot be more of one than of the other. So, with infinite sets one part of the set can be just as large as the whole, which is known as Galileo's paradox.

Luca Pacioli (1445-1514 or 1517)

Luca Pacioli compiled a treatise on number puzzles and magic that lay undiscovered in the archives of the university of Bologna until it was published in 2008. He collected and studied magic squares, arrangements of numbers in a square grid so that each horizontal, vertical and diagonal line of numbers adds up to the same total, called the magic constant. With his 1509 book De Divina Proportione Pacioli also gave new impetus to the theory of the golden ratio by illustraring the golden ratio as applied to human faces by artists, architects, scientists, and mystics. Pacioli also produced the first European text to use zero properly.

Giovanni Girolamo Saccheri (1667-1733)

His work on alternative geometries had little impact during his lifetime, but was rediscovered by Eugenio Beltrami in the mid-19th century.

Leonida Tonelli (1885 - 1946)

Author of the Tonelli's theorem, a successor of Fubini's theorem. The conclusion of Tonelli's theorem is identical to that of Fubini's theorem, but the assumptions are different. Tonelli's theorem states that on the product of two s-finite measure spaces, a product measure integral can be evaluated by way of an iterated integral for nonnegative measurable functions, regardless of whether they have finite integral.

Evangelista Torricelli (1608-1647)

A physicist and mathematician from Faenza famous for the discovery of the Torricelli's trumpet or horn whose surface area is infinite, but whose volume is finite. The discovery prompted a fierce controversy about the nature of infinity, and is supposed by some to have led to the idea of a "completed infinity".

Leonard of Pisa, known as Fibonacci, learned about Hindu-Arabic numerals as a boy while traveling in the Near East and Northern Africa with his father. He later wrote a tract on arithmetic entitled Liber Abaci (a tract about the Abacus) and was instrumental in popularizing Hindu-Arabic numerals in the Western world, the system we still use today (before that Roman numerals were used).

talian Mathematics Between the Two World Wars

(Science Networks. Historical Studies)

Angelo Guerraggio, Pietro Nastasi

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(Science Networks. Historical Studies)

Angelo Guerraggio, Pietro Nastasi

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Further Reading

The World of Maria Gaetana Agnesi, Mathematician of God

by Massimo Mazzotti

More information:

by Massimo Mazzotti

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Famous Puzzles of Great Mathematicians

Miodrag S. Petkovic

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Miodrag S. Petkovic

More information:

The Man of Numbers: Fibonacci's Arithmetic Revolution

Keith Devlin

More information:

Keith Devlin

More information:

The Book of My Life

(New York Review Books Classics)

Girolamo Cardano

More information:

(New York Review Books Classics)

Girolamo Cardano

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