Archimedes
Famous as: Mathematician, Engineer, Inventor, Physicist
Nationality: Greek
Born on: 287 BC
Born in: Syracuse
father: Phidias
discoveries / inventions: Antikythera, Mechanism, Archimedes' Screw, Hydrostatics, Levers, Infinitesimals
Works & Achievements: Credited with Archimedes' principle of density, siege machines, screw pump, formulae for the volumes of surfaces, approximate value of ð
Archimedes of Syracuse was an outstanding Greek mathematician, inventor, physicist, engineer and also an astronomer. Although not much is known about his life, he is considered as one of the most eminent scientists in classical antiquity. He established strong foundations in the field of physics, particularly in statics, hydrostatics and explained the principle of the lever. In his lifetime, he made many incredible inventions such as designing innovative machines, including screw pumps and siege machines, After intensive experiments, it is concluded that the machines designed by Archimedes are capable of lifting attacking ships out of the water and even setting ships on fire using an array of mirrors. Undoubtedly, Archimedes is considered the greatest scientist and mathematician of ancient times. He applied the 'method of exhaustion' in calculating the area under the arc of a parabola with the summation of an endless series and gave a marvelously precise approximation of pi, the symbol. He also identified the spiral that bears his name, designed formulae for the volumes of surfaces of revolution and also invented a technique for expressing extremely large numbers.
Famous Discoveries And Inventions Archimedes' Principle The most popular tale about Archimedes is regarding how he discovered a method for calculating the volume of objects with irregular shape. According to Vitruvius, a crown for a temple had been made for King Hiero II, who had supplied the pure gold to be used and Archimedes was asked to verify whether any silver had been used by the deceitful goldsmith. Archimedes was expected to solve the problem without damaging the crown and thus the option of melting it down into a regular shape was ruled out. One day, while taking a bath, he discerned that the level of the water in the tub increased as he got in, and comprehended that this effect could be used to determine the volume of the crown. As water is incompressible practically, so the crown after submerging would displace an amount of water equal to its own density and it would be possible to calculate the density of the crown if mass of the crown was divided by the volume of water displaced. Archimedes was so excited that he ran on the streets naked (he forgot to dress up), crying out ‘Eureka!’ meaning ’I have found it!’ The test was conducted successfully, concluding that silver had certainly been mixed with the gold.
(http://www.thefamouspeople.com/profiles/archimedes-422.php)
Famous as: Mathematician, Engineer, Inventor, Physicist
Nationality: Greek
Born on: 287 BC
Born in: Syracuse
father: Phidias
discoveries / inventions: Antikythera, Mechanism, Archimedes' Screw, Hydrostatics, Levers, Infinitesimals
Works & Achievements: Credited with Archimedes' principle of density, siege machines, screw pump, formulae for the volumes of surfaces, approximate value of ð
Archimedes of Syracuse was an outstanding Greek mathematician, inventor, physicist, engineer and also an astronomer. Although not much is known about his life, he is considered as one of the most eminent scientists in classical antiquity. He established strong foundations in the field of physics, particularly in statics, hydrostatics and explained the principle of the lever. In his lifetime, he made many incredible inventions such as designing innovative machines, including screw pumps and siege machines, After intensive experiments, it is concluded that the machines designed by Archimedes are capable of lifting attacking ships out of the water and even setting ships on fire using an array of mirrors. Undoubtedly, Archimedes is considered the greatest scientist and mathematician of ancient times. He applied the 'method of exhaustion' in calculating the area under the arc of a parabola with the summation of an endless series and gave a marvelously precise approximation of pi, the symbol. He also identified the spiral that bears his name, designed formulae for the volumes of surfaces of revolution and also invented a technique for expressing extremely large numbers.
Famous Discoveries And Inventions Archimedes' Principle The most popular tale about Archimedes is regarding how he discovered a method for calculating the volume of objects with irregular shape. According to Vitruvius, a crown for a temple had been made for King Hiero II, who had supplied the pure gold to be used and Archimedes was asked to verify whether any silver had been used by the deceitful goldsmith. Archimedes was expected to solve the problem without damaging the crown and thus the option of melting it down into a regular shape was ruled out. One day, while taking a bath, he discerned that the level of the water in the tub increased as he got in, and comprehended that this effect could be used to determine the volume of the crown. As water is incompressible practically, so the crown after submerging would displace an amount of water equal to its own density and it would be possible to calculate the density of the crown if mass of the crown was divided by the volume of water displaced. Archimedes was so excited that he ran on the streets naked (he forgot to dress up), crying out ‘Eureka!’ meaning ’I have found it!’ The test was conducted successfully, concluding that silver had certainly been mixed with the gold.
(http://www.thefamouspeople.com/profiles/archimedes-422.php)
Andrew Wiles
The only currently living mathematician on this list, Andrew Wiles is most well known for his proof of Fermat’s Last Theorem: That no positive integers, a, b and c can satisfy the equation a^n+b^n=c^n For n greater then 2. (If n=2 it is the Pythagoras Formula). Although the contributions to math are not, perhaps, as grand as other on this list, he did ‘invent’ large portions of new mathematics for his proof of the theorem. Besides, his dedication is often admired by most, as he quite literally shut himself away for 7 years to formulate a solution. When it was found that the solution contained an error, he returned to solitude for a further year before the solution was accepted. To put in perspective how ground breaking and new the math was, it had been said that you could count the number of mathematicians in the world on one hand who, at the time, could understand and validate his proof. Nonetheless, the effects of such are likely to only increase as time passes (and more and more people can understand it).(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
The only currently living mathematician on this list, Andrew Wiles is most well known for his proof of Fermat’s Last Theorem: That no positive integers, a, b and c can satisfy the equation a^n+b^n=c^n For n greater then 2. (If n=2 it is the Pythagoras Formula). Although the contributions to math are not, perhaps, as grand as other on this list, he did ‘invent’ large portions of new mathematics for his proof of the theorem. Besides, his dedication is often admired by most, as he quite literally shut himself away for 7 years to formulate a solution. When it was found that the solution contained an error, he returned to solitude for a further year before the solution was accepted. To put in perspective how ground breaking and new the math was, it had been said that you could count the number of mathematicians in the world on one hand who, at the time, could understand and validate his proof. Nonetheless, the effects of such are likely to only increase as time passes (and more and more people can understand it).(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
Blaise-Pascal
Famous as: French Mathematician, Physicist, Inventor, Writer and Catholic Philosopher
Nationality: French
religion: Roman Catholic
Born on: 19 June 1623 Famous 19th June Birthdays
Zodiac Sign: Gemini Famous Geminis
Born in: Clermont-Ferrand,, Auvergne, France
Died on: 19 August 1662
place of death: Paris, France
father: Étienne Pascal
mother: Antoinette Begon
siblings: Jacqueline Pascal
Married: No
discoveries / inventions: Hydraulic Press, Syringe
Works & Achievements: Invented mechanical calculator, hydraulic press and wrote famous religious works like "Lettres provinciales and the Pensees".
Blaise Pascal was a French mathematician, physicist, inventor, writer and Catholic philosopher. He did pioneering works in calculating machines and came up with the mechanical calculator. He was an influential mathematician and made significant researches in areas like projective geometry and probability theory. He also made important contribution in arithmetic triangle and cycloid. Along with Fermat, Pascal came up with the calculus of probabilities, which later led to the foundation of mathematical theory of probabilities. He also worked in natural and applied sciences and made important contributions in concepts like fluids, pressure and vacuum. Honoring his works and contributions, his name Pascal has been made the SI unit of pressure and also a programming language. His other notable contributions include Pascal's law, Pascal's triangle and Pascal's wager. His religious works, "Lettres provinciales and the Pensees" had a religious influence all over France and created a new level of style in French prose.
Contribution In Mathematics & Science Pascal always remained an influential mathematician throughout his life. His convenient tabular presentation of binomial coefficients described in his Traité du triangle arithmétique, released in 1653, later became famous as Pascal’s triangle. In 1654, following a friend,the Chevalier de Méré’s interest in gambling problem, Pascal discussed this subject with Fermat, which later led to the foundation of mathematical theory of probabilities. One of the gambling problems was of two players who wanted to finish a game early, and given the then condition of the game, wanted to share the stakes fairly, based on the fact that each player had equal chances of winning the match from that point. In this context, Pascal used a probabilistic argument also known as Pascal's wager. The work done by Pascal and Fermat later helped Leibniz formulate infinitesimal calculus. Pascal also made important contribution to the philosophy of mathematics with his works like De l'Esprit géométrique and De l'Art de persuader. Pascal contribution to the physical sciences includes his works in fields of hydrodynamics and hydrostatics which were mostly based on hydraulic principles. He also had the credit of inventing syringe and hydraulic press. Following the views of Galileo and Torricelli, he opposed the Aristotelian notion which says that a creation is a thing of substance, whether visible or invisible. He advocated the presence of vacuum in substances. He said that it is the vacuum which keeps the mercury floating in a barometer and even fills the space above the mercury in the tube. In his work in 1647, “Experiences nouvelles touchant le vide” he gave more experiments regarding his statement on vacuum. These experiments performed by Pascal were praised throughout the Europe and established his principle and also the value of barometer. (http://www.thefamouspeople.com/profiles/blaise-pascal-131.php)
Famous as: French Mathematician, Physicist, Inventor, Writer and Catholic Philosopher
Nationality: French
religion: Roman Catholic
Born on: 19 June 1623 Famous 19th June Birthdays
Zodiac Sign: Gemini Famous Geminis
Born in: Clermont-Ferrand,, Auvergne, France
Died on: 19 August 1662
place of death: Paris, France
father: Étienne Pascal
mother: Antoinette Begon
siblings: Jacqueline Pascal
Married: No
discoveries / inventions: Hydraulic Press, Syringe
Works & Achievements: Invented mechanical calculator, hydraulic press and wrote famous religious works like "Lettres provinciales and the Pensees".
Blaise Pascal was a French mathematician, physicist, inventor, writer and Catholic philosopher. He did pioneering works in calculating machines and came up with the mechanical calculator. He was an influential mathematician and made significant researches in areas like projective geometry and probability theory. He also made important contribution in arithmetic triangle and cycloid. Along with Fermat, Pascal came up with the calculus of probabilities, which later led to the foundation of mathematical theory of probabilities. He also worked in natural and applied sciences and made important contributions in concepts like fluids, pressure and vacuum. Honoring his works and contributions, his name Pascal has been made the SI unit of pressure and also a programming language. His other notable contributions include Pascal's law, Pascal's triangle and Pascal's wager. His religious works, "Lettres provinciales and the Pensees" had a religious influence all over France and created a new level of style in French prose.
Contribution In Mathematics & Science Pascal always remained an influential mathematician throughout his life. His convenient tabular presentation of binomial coefficients described in his Traité du triangle arithmétique, released in 1653, later became famous as Pascal’s triangle. In 1654, following a friend,the Chevalier de Méré’s interest in gambling problem, Pascal discussed this subject with Fermat, which later led to the foundation of mathematical theory of probabilities. One of the gambling problems was of two players who wanted to finish a game early, and given the then condition of the game, wanted to share the stakes fairly, based on the fact that each player had equal chances of winning the match from that point. In this context, Pascal used a probabilistic argument also known as Pascal's wager. The work done by Pascal and Fermat later helped Leibniz formulate infinitesimal calculus. Pascal also made important contribution to the philosophy of mathematics with his works like De l'Esprit géométrique and De l'Art de persuader. Pascal contribution to the physical sciences includes his works in fields of hydrodynamics and hydrostatics which were mostly based on hydraulic principles. He also had the credit of inventing syringe and hydraulic press. Following the views of Galileo and Torricelli, he opposed the Aristotelian notion which says that a creation is a thing of substance, whether visible or invisible. He advocated the presence of vacuum in substances. He said that it is the vacuum which keeps the mercury floating in a barometer and even fills the space above the mercury in the tube. In his work in 1647, “Experiences nouvelles touchant le vide” he gave more experiments regarding his statement on vacuum. These experiments performed by Pascal were praised throughout the Europe and established his principle and also the value of barometer. (http://www.thefamouspeople.com/profiles/blaise-pascal-131.php)
EUCLID
Famous as: Mathematician
Nationality: Greek
Born on: 330 BC
Born in: Alexandria
place of death: NA
education: Plato's Academy, Athens, Greece
Works & Achievements: He was responsible in connoting Geometrical knowledge and also wrote the famous Euclid's Elements and framed the skeleton for geometry that would be used in mathematics by the Western World for over 2000 years.
Although little is known about Euclid's early and personal life, he was known as the forerunner of geometrical knowledge and went on to contribute greatly in the field of mathematics. Also known as the 'father of Geometry', Euclid was known to have taught the subject of mathematics in Ancient Egypt during the reign of Ptolemy I. He was well-known, having written the most permanent mathematical works of all time, known as the 'Elements' that comprised of the 13 gigantic volumes filled with geometrical theories and knowledge. This would then go on to arouse the Western World and a series of Mathematicians around the globe for over 2000 years breaking all boundaries and defining new ones in the field of Math. Euclid used the 'synthetic approach' towards producing his theorems, definitions and axioms in math. Apart from being a tutor at the Alexandria library, Euclid coined and structured the different elements of mathematics, such as Porisms, geometric systems, infinite values, factorizations, and the congruence of shapes that went on to contour Euclidian Geometry. His works were heavily influenced by Pythagoras, Aristotle, Eudoxus, and Thales to name a few.
Career
Euclid was known as the ‘father of geometry’ for a reason. He discovered the subject and gave it its value, making it one of the most complex forms of mathematics at the time. After moving to Alexandria, Euclid spent most of his time at the Alexandria library, like many other eminent scholars who spent their time there wisely. The museum was built by Ptolemy, which was central to literature, arts and sciences. It was here that Euclid began developing geometrical ideas, arithmetic’s, theories and irrational numbers into a section called “geometry”. He began developing his theorems and collated it into a colossal treatise called ‘The Elements’. During the course of his vaguely known career, he developed 13 editions to the ‘Elements’ that covered a wide spectrum of subjects ranging from axioms and statements to solid geometry and algorithm concepts. Along with stating these various theories, he began backing these ideas with methods and logical proof that would approve of the statements produced by Euclid.
His treatise consisted of over 467 propositions to plain and solid geometry, proposes and adages that suggested and agreed to his theories relating to his geometrical ideas. There was a certain case with the Pythagoras equation for the triangle that Euclid used as an example while writing the ‘Elements’. He stated that ‘the equation was always true when it was the matter of every right-angle triangle’. The ‘Elements’ sold more copies than the Bible and was used and printed countless times by mathematicians and publishers, who have used the information, even up to the 20th century. There was no end to Euclid’s geometry, and he continued to develop theorems on various aspects of math such as ‘prime numbers’ and other, basic ‘arithmetic’. With a series of logical steps developed by Euclid, he believed in making the unknown known to the world. The system that Euclid went on to describe in the ‘Elements’ was commonly known as the only form of geometry the world had witnessed and seen up until the 19th century. However, mathematicians of the modern era developed new theorems and ideas pertaining to geometry and divided the subject to ‘Euclidean Geometry’ and ‘Non-Euclidean Geometry’.
He called this the ‘synthetic approach’ that was not based on the logic of trial and error, but on presenting facts from theory. At a time when knowledge was limited, Euclid even began to take on knowledge based quests on subjects relating to a different field such as ‘arithmetic and numbers’. He deciphered that it would be humanly impossible to find out the ‘largest prime number’. He backed this with an example stating that if 1 was added to the largest known prime number, the product will lead to another prime number. This classic example was the proof of Euclid’s clarity of thought and precision at his time and age.
Axioms
Euclid stated that axioms were statements that were just believed to be true, but he realized that by blindly following statements, there would be no point in devising mathematical theories and formulae. He realized that even axioms had to be backed with solid proofs. Therefore, he started to develop logical evidences that would testify his axioms and theorems in geometry. In order to further understand these axioms, he divided them into groups of two called ‘postulates’. One group would be called the ‘common notions’ which were agreed statements of science. His second set of postulates was synonymous with geometry. The first set of notions mentioned statements such as the “whole is greater than the part” and “things which are equal to the same thing are also equal to one another”. These are only two of the five statements written by Euclid. The remaining five statements in the second set of postulates are a little more specific to the subject of Geometry and state theories such as “All right angle are equal” and “straight lines can be drawn between any two points”.
Euclid’s career flourished as a Mathematician and the ‘Elements’ was eventually translated from Greek to Arabic and then into English by John Dee in the early periods of 1570. There were more than 1000 editions of the ‘Elements’ printed ever since its inception, which eventually secured a place in early 20th century classrooms as well. There have been a myriad of Mathematicians who tried to refute and break Euclid’s theories in geometry and mathematics, but these attempts were always futile. An Italian Mathematician called Girolamo Saccheri tried to outdo the works of Euclid, but gave up when he couldn’t pinpoint a single flaw in his theories. It would take another century for a new group of Mathematicians to present new theories in the subject of geometry.
(http://www.thefamouspeople.com/profiles/euclid-436.php)
Famous as: Mathematician
Nationality: Greek
Born on: 330 BC
Born in: Alexandria
place of death: NA
education: Plato's Academy, Athens, Greece
Works & Achievements: He was responsible in connoting Geometrical knowledge and also wrote the famous Euclid's Elements and framed the skeleton for geometry that would be used in mathematics by the Western World for over 2000 years.
Although little is known about Euclid's early and personal life, he was known as the forerunner of geometrical knowledge and went on to contribute greatly in the field of mathematics. Also known as the 'father of Geometry', Euclid was known to have taught the subject of mathematics in Ancient Egypt during the reign of Ptolemy I. He was well-known, having written the most permanent mathematical works of all time, known as the 'Elements' that comprised of the 13 gigantic volumes filled with geometrical theories and knowledge. This would then go on to arouse the Western World and a series of Mathematicians around the globe for over 2000 years breaking all boundaries and defining new ones in the field of Math. Euclid used the 'synthetic approach' towards producing his theorems, definitions and axioms in math. Apart from being a tutor at the Alexandria library, Euclid coined and structured the different elements of mathematics, such as Porisms, geometric systems, infinite values, factorizations, and the congruence of shapes that went on to contour Euclidian Geometry. His works were heavily influenced by Pythagoras, Aristotle, Eudoxus, and Thales to name a few.
Career
Euclid was known as the ‘father of geometry’ for a reason. He discovered the subject and gave it its value, making it one of the most complex forms of mathematics at the time. After moving to Alexandria, Euclid spent most of his time at the Alexandria library, like many other eminent scholars who spent their time there wisely. The museum was built by Ptolemy, which was central to literature, arts and sciences. It was here that Euclid began developing geometrical ideas, arithmetic’s, theories and irrational numbers into a section called “geometry”. He began developing his theorems and collated it into a colossal treatise called ‘The Elements’. During the course of his vaguely known career, he developed 13 editions to the ‘Elements’ that covered a wide spectrum of subjects ranging from axioms and statements to solid geometry and algorithm concepts. Along with stating these various theories, he began backing these ideas with methods and logical proof that would approve of the statements produced by Euclid.
His treatise consisted of over 467 propositions to plain and solid geometry, proposes and adages that suggested and agreed to his theories relating to his geometrical ideas. There was a certain case with the Pythagoras equation for the triangle that Euclid used as an example while writing the ‘Elements’. He stated that ‘the equation was always true when it was the matter of every right-angle triangle’. The ‘Elements’ sold more copies than the Bible and was used and printed countless times by mathematicians and publishers, who have used the information, even up to the 20th century. There was no end to Euclid’s geometry, and he continued to develop theorems on various aspects of math such as ‘prime numbers’ and other, basic ‘arithmetic’. With a series of logical steps developed by Euclid, he believed in making the unknown known to the world. The system that Euclid went on to describe in the ‘Elements’ was commonly known as the only form of geometry the world had witnessed and seen up until the 19th century. However, mathematicians of the modern era developed new theorems and ideas pertaining to geometry and divided the subject to ‘Euclidean Geometry’ and ‘Non-Euclidean Geometry’.
He called this the ‘synthetic approach’ that was not based on the logic of trial and error, but on presenting facts from theory. At a time when knowledge was limited, Euclid even began to take on knowledge based quests on subjects relating to a different field such as ‘arithmetic and numbers’. He deciphered that it would be humanly impossible to find out the ‘largest prime number’. He backed this with an example stating that if 1 was added to the largest known prime number, the product will lead to another prime number. This classic example was the proof of Euclid’s clarity of thought and precision at his time and age.
Axioms
Euclid stated that axioms were statements that were just believed to be true, but he realized that by blindly following statements, there would be no point in devising mathematical theories and formulae. He realized that even axioms had to be backed with solid proofs. Therefore, he started to develop logical evidences that would testify his axioms and theorems in geometry. In order to further understand these axioms, he divided them into groups of two called ‘postulates’. One group would be called the ‘common notions’ which were agreed statements of science. His second set of postulates was synonymous with geometry. The first set of notions mentioned statements such as the “whole is greater than the part” and “things which are equal to the same thing are also equal to one another”. These are only two of the five statements written by Euclid. The remaining five statements in the second set of postulates are a little more specific to the subject of Geometry and state theories such as “All right angle are equal” and “straight lines can be drawn between any two points”.
Euclid’s career flourished as a Mathematician and the ‘Elements’ was eventually translated from Greek to Arabic and then into English by John Dee in the early periods of 1570. There were more than 1000 editions of the ‘Elements’ printed ever since its inception, which eventually secured a place in early 20th century classrooms as well. There have been a myriad of Mathematicians who tried to refute and break Euclid’s theories in geometry and mathematics, but these attempts were always futile. An Italian Mathematician called Girolamo Saccheri tried to outdo the works of Euclid, but gave up when he couldn’t pinpoint a single flaw in his theories. It would take another century for a new group of Mathematicians to present new theories in the subject of geometry.
(http://www.thefamouspeople.com/profiles/euclid-436.php)
JACOB BERNOULLI
Famous as: Mathematician
Nationality: Swiss
Born on: 27 December 1654 AD
Zodiac Sign: Capricorn Famous Capricons
Born in: Basel, Switzerland
Died on: 16 August 1705 AD
place of death: Basel, Switzerland
father: Nicolaus Bernoulli
mother: Margaretha Bernoulli
siblings: Johann Bernoulli
Spouse: Judith Stupanus
education: University of Basel
discoveries / inventions: Constant E
Works & Achievements: Bernoulli differential equation, Bernoulli numbers, Bernoulli trial, Bernoulli's inequality and Lemniscate of Bernoulli
Jacob Bernoulli can be rightly called the initiator of the Bernoulli family's mathematical dynasty. A class of his own, Jacob was bright and intelligent right from the very beginning. His well-researched concepts brought about a revolution in Swiss mathematics. Jacob Bernoulli is credited for being the first person to develop the technique for solving separable differential equations. He is also responsible for the Bernoulli numbers, by which he derived the exponential series. In addition to that, the subject of probability, which is now called the Bernoulli law of large numbers, also falls in the kitty of this prominent and renowned mathematician. With this article, know more about the life and history of Jacob Bernoulli.
Contribution to Mathematics Jacob Bernoulli’s foremost significant contributions to mathematics were noted in the pamphlets which accounted information on the parallels of logic and algebra and his work on probability. These were published in the year 1685. Two years later, Bernoulli’s work on geometry was published, which gave a construction to divide any triangle into four equal parts with two perpendicular lines. In 1689, Jacob Bernoulli published an important work on infinite series and his law of large numbers in probability theory. According to his interpretation, if an experiment is repeated a number of times, then the relative frequency with which an event occurs equals the probability of the event. The law of large numbers is a mathematical interpretation of this result. From 1682 to 1704, Jacob Bernoulli published five treatises on infinite series. He also researched on the exponential series which came out of examining compound interest. In the year 1690, Jacob Bernoulli proved that the problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation, in the paper published in Acta Eruditorum. According to it, the isochrone, or simply the curve of a constant descent, is a curve along which a particle descends under gravity from any point to the bottom in exactly the same time, regardless of what the starting point is. After finding the differential equation, Bernoulli then solved it by what we now know as separation of variables. His paper of 1690 is exceptionally essential in the history of calculus, since it was then that the term integral appeared for the first time with its integration meaning. Other than these, Jacob also ascertained a general method to determine evolutes of a curve as the envelope of its circles of curvature. In 1692, he investigated on the caustic curves, particularly, those associated with the curves of the parabola, the logarithmic spiral and epicycloids. Two years later, he first conceived the lemniscate of Bernoulli and later the next year, he carried out a research wherein he applied calculus in the building of suspension bridges. In 1696, Bernoulli solved the equation, now known as "the Bernoulli equation". However, Jacob Bernoulli’s most notable work was the Ars Conjectandi (The Art of Conjecture), which was published posthumously in 1713, by his nephew Nicholas. Herein he went on to brief about the known results in probability theory and in enumeration, often providing alternative proofs of known results. The work also included the application of probability theory to games of chance and his introduction of the theorem known as the law of large numbers. The terms “Bernoulli trial” and “Bernoulli numbers” are the products from this work.
(http://www.thefamouspeople.com/profiles/jacob-bernoulli-543.php)
Famous as: Mathematician
Nationality: Swiss
Born on: 27 December 1654 AD
Zodiac Sign: Capricorn Famous Capricons
Born in: Basel, Switzerland
Died on: 16 August 1705 AD
place of death: Basel, Switzerland
father: Nicolaus Bernoulli
mother: Margaretha Bernoulli
siblings: Johann Bernoulli
Spouse: Judith Stupanus
education: University of Basel
discoveries / inventions: Constant E
Works & Achievements: Bernoulli differential equation, Bernoulli numbers, Bernoulli trial, Bernoulli's inequality and Lemniscate of Bernoulli
Jacob Bernoulli can be rightly called the initiator of the Bernoulli family's mathematical dynasty. A class of his own, Jacob was bright and intelligent right from the very beginning. His well-researched concepts brought about a revolution in Swiss mathematics. Jacob Bernoulli is credited for being the first person to develop the technique for solving separable differential equations. He is also responsible for the Bernoulli numbers, by which he derived the exponential series. In addition to that, the subject of probability, which is now called the Bernoulli law of large numbers, also falls in the kitty of this prominent and renowned mathematician. With this article, know more about the life and history of Jacob Bernoulli.
Contribution to Mathematics Jacob Bernoulli’s foremost significant contributions to mathematics were noted in the pamphlets which accounted information on the parallels of logic and algebra and his work on probability. These were published in the year 1685. Two years later, Bernoulli’s work on geometry was published, which gave a construction to divide any triangle into four equal parts with two perpendicular lines. In 1689, Jacob Bernoulli published an important work on infinite series and his law of large numbers in probability theory. According to his interpretation, if an experiment is repeated a number of times, then the relative frequency with which an event occurs equals the probability of the event. The law of large numbers is a mathematical interpretation of this result. From 1682 to 1704, Jacob Bernoulli published five treatises on infinite series. He also researched on the exponential series which came out of examining compound interest. In the year 1690, Jacob Bernoulli proved that the problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation, in the paper published in Acta Eruditorum. According to it, the isochrone, or simply the curve of a constant descent, is a curve along which a particle descends under gravity from any point to the bottom in exactly the same time, regardless of what the starting point is. After finding the differential equation, Bernoulli then solved it by what we now know as separation of variables. His paper of 1690 is exceptionally essential in the history of calculus, since it was then that the term integral appeared for the first time with its integration meaning. Other than these, Jacob also ascertained a general method to determine evolutes of a curve as the envelope of its circles of curvature. In 1692, he investigated on the caustic curves, particularly, those associated with the curves of the parabola, the logarithmic spiral and epicycloids. Two years later, he first conceived the lemniscate of Bernoulli and later the next year, he carried out a research wherein he applied calculus in the building of suspension bridges. In 1696, Bernoulli solved the equation, now known as "the Bernoulli equation". However, Jacob Bernoulli’s most notable work was the Ars Conjectandi (The Art of Conjecture), which was published posthumously in 1713, by his nephew Nicholas. Herein he went on to brief about the known results in probability theory and in enumeration, often providing alternative proofs of known results. The work also included the application of probability theory to games of chance and his introduction of the theorem known as the law of large numbers. The terms “Bernoulli trial” and “Bernoulli numbers” are the products from this work.
(http://www.thefamouspeople.com/profiles/jacob-bernoulli-543.php)
JOHN NAPIER
Famous as: Mathematician
Nationality: Scottish
Born on: 29 March 1904 AD
Zodiac Sign: Aries Famous Arians
Born in: Merchiston Tower, Edinburgh
Died on: 04 April 1617 AD
place of death: Edinburgh
father: Sir Archibald Napier
mother: Janet Bothwell
siblings: Adam Bothwell
Spouses: Elizabeth Stirling, Agnes Chisholm
education: University of St Andrews
discoveries / inventions: Logarithms
Works & Achievements: Originated the concept of logarithm, Creator of Napier's Bones and Napier's analogies
David Hume's personification of the title "a great man" more than aptly describes the prominence and distinction of John Napier. A distinguished Scottish mathematician and theological writer, John Napier is famously credited as the man who originated the concept of logarithm. With his innovative discoveries and research, Napier created a storm in the field of mathematical calculations. While his concept of logarithms gained most limelight, Napier's other contributions in the field of spherical trigonometry, the invention of the divining rods and pressing forward the use of decimal fraction are second to none. It was due to his ground-breaking inventions that Napier earned the respect of some of the most illustrious astronomers and scientists of the age. Know more about the life and contributions of this ace mathematician through the following lines.
Contributions in Mathematics Napier’s interest in astronomy led way to his interest in mathematics. His long hours of leisure were spent exploring and devising new methods of computation that could help astronomers during their research. Logarithm, as we now know it, was a seed of thought of Napier who, from his research, brought out a newer and simpler way for performing large number calculations. He found out that through the use of exponentials, the operations which involved multiplication and division of very large numbers reduced to being just an addition of the exponents. He gradually went on to elaborate his computational system wherein roots, products and quotients could be easily found out from tables showing powers of a fixed number at the base. Napier’s finding was first made public in the year 1614, through his book, Mirifici logarithmorum canonis descriptio (Description of the Marvellous Canon of Logarithm). The book, though, only had brief details of the steps that led to the discovery, importantly contained his first set of logarithmic table. Not only did the table found an instant acceptance from the astronomers and scientists around the world, it paved way to the cooperative movement including the development of Base 10. In his second book, Mirifici Logarithmorum Canonis Constructio (Construction of the Marvelous Canon of Logarithms), which was published posthumously, Napier further advanced the concept of decimal fraction, which was first introduced by Simon Stevin, a Flemish mathematician. His research, which suggested that a simple point could separate a whole number and fractional parts of the number, was an instant hit in Great Britain. The advances in the field of computation through logarithms not only made calculations by hand quicker, but it also opened doors to further scientific advancements done in the field of astronomy, dynamics, physics and even astrology. Although his invention of logarithm outshined his other mathematical findings, Napier, nevertheless, holds the credit for his inventions in the arena of spherical trigonometry. Two formulae known as "Napier's analogies" used in solving spherical triangles and an invention called "Napier's bones" used for mechanically multiplying, dividing and taking square roots and cube roots also fall in the kitty of this ace mathematician. The latter concepts were published in his book, Rabdologiæ seu Numerationis per Virgulas libri duo (Study of Divining Rods).
(http://www.thefamouspeople.com/profiles/john-napier-546.php)
Famous as: Mathematician
Nationality: Scottish
Born on: 29 March 1904 AD
Zodiac Sign: Aries Famous Arians
Born in: Merchiston Tower, Edinburgh
Died on: 04 April 1617 AD
place of death: Edinburgh
father: Sir Archibald Napier
mother: Janet Bothwell
siblings: Adam Bothwell
Spouses: Elizabeth Stirling, Agnes Chisholm
education: University of St Andrews
discoveries / inventions: Logarithms
Works & Achievements: Originated the concept of logarithm, Creator of Napier's Bones and Napier's analogies
David Hume's personification of the title "a great man" more than aptly describes the prominence and distinction of John Napier. A distinguished Scottish mathematician and theological writer, John Napier is famously credited as the man who originated the concept of logarithm. With his innovative discoveries and research, Napier created a storm in the field of mathematical calculations. While his concept of logarithms gained most limelight, Napier's other contributions in the field of spherical trigonometry, the invention of the divining rods and pressing forward the use of decimal fraction are second to none. It was due to his ground-breaking inventions that Napier earned the respect of some of the most illustrious astronomers and scientists of the age. Know more about the life and contributions of this ace mathematician through the following lines.
Contributions in Mathematics Napier’s interest in astronomy led way to his interest in mathematics. His long hours of leisure were spent exploring and devising new methods of computation that could help astronomers during their research. Logarithm, as we now know it, was a seed of thought of Napier who, from his research, brought out a newer and simpler way for performing large number calculations. He found out that through the use of exponentials, the operations which involved multiplication and division of very large numbers reduced to being just an addition of the exponents. He gradually went on to elaborate his computational system wherein roots, products and quotients could be easily found out from tables showing powers of a fixed number at the base. Napier’s finding was first made public in the year 1614, through his book, Mirifici logarithmorum canonis descriptio (Description of the Marvellous Canon of Logarithm). The book, though, only had brief details of the steps that led to the discovery, importantly contained his first set of logarithmic table. Not only did the table found an instant acceptance from the astronomers and scientists around the world, it paved way to the cooperative movement including the development of Base 10. In his second book, Mirifici Logarithmorum Canonis Constructio (Construction of the Marvelous Canon of Logarithms), which was published posthumously, Napier further advanced the concept of decimal fraction, which was first introduced by Simon Stevin, a Flemish mathematician. His research, which suggested that a simple point could separate a whole number and fractional parts of the number, was an instant hit in Great Britain. The advances in the field of computation through logarithms not only made calculations by hand quicker, but it also opened doors to further scientific advancements done in the field of astronomy, dynamics, physics and even astrology. Although his invention of logarithm outshined his other mathematical findings, Napier, nevertheless, holds the credit for his inventions in the arena of spherical trigonometry. Two formulae known as "Napier's analogies" used in solving spherical triangles and an invention called "Napier's bones" used for mechanically multiplying, dividing and taking square roots and cube roots also fall in the kitty of this ace mathematician. The latter concepts were published in his book, Rabdologiæ seu Numerationis per Virgulas libri duo (Study of Divining Rods).
(http://www.thefamouspeople.com/profiles/john-napier-546.php)
PYTHAGORAS
Famous as: Philosopher and Mathematician
Nationality: Greek
Born on: 570 BC
Born in: Samos
Died on: 495 BC
place of death: Metapontum
father: Mnesarchus
mother: Pythais
Spouse: Theano
children: Damo, Myia, Arignote, Telauges
education: Pythagoreanism
Works & Achievements: Pythagoras of Samos is known for his mathematical works such as the 'Pythagoras Theorem' and also for his philosophical teachings.
Pythagoras of Samos was a well-known mathematician, scientist and a religious teacher. He was born in Samos and is often hailed as the first great mathematician. Pythagoras is remembered today for his famous theorem in geometry, the 'Pythagoras Theorem'. His mentors were Thales, Pherekydes and Anaximander, who inspired him to pursue mathematics and astronomy. Pythagoras also made important discoveries in music, astronomy and medicine. He accepted priesthood and performed the rites that were required in order to enter one of the temples in Egypt, known as Diospolis. He set up a brotherhood with some of his followers, who practiced his way of life and pursued his religious ideologies. He became one of the most distinguished teachers of religion in ancient Greece. Read on to know more about the childhood and career of this ancient Greek philosopher and mathematician.
Mathematical Concepts
Pythagoras studied properties of numbers and classified them as even numbers, odd numbers, triangular numbers and perfect numbers etc. The ‘Pythagoras theorem’ is one of the earliest theorems in geometry, which states that in right-angle triangles, the square of the hypotenuse is equal to the sum of square the other two sides. This theorem was already proposed during the reign of the Babylonian King Hammurabi, but Pythagoras applied it to mathematics and science and refined the concept. Pythagoras also asserted that dynamics of the structure of the universe lies on the interaction of the contraries or the opposites, such as, light and darkness, limited and unlimited, square and oblong, straight and crooked, right and left, singularity and plurality, male and female, motionless and movement and good and bad.
(http://www.thefamouspeople.com/profiles/pythagoras-504.php)
Famous as: Philosopher and Mathematician
Nationality: Greek
Born on: 570 BC
Born in: Samos
Died on: 495 BC
place of death: Metapontum
father: Mnesarchus
mother: Pythais
Spouse: Theano
children: Damo, Myia, Arignote, Telauges
education: Pythagoreanism
Works & Achievements: Pythagoras of Samos is known for his mathematical works such as the 'Pythagoras Theorem' and also for his philosophical teachings.
Pythagoras of Samos was a well-known mathematician, scientist and a religious teacher. He was born in Samos and is often hailed as the first great mathematician. Pythagoras is remembered today for his famous theorem in geometry, the 'Pythagoras Theorem'. His mentors were Thales, Pherekydes and Anaximander, who inspired him to pursue mathematics and astronomy. Pythagoras also made important discoveries in music, astronomy and medicine. He accepted priesthood and performed the rites that were required in order to enter one of the temples in Egypt, known as Diospolis. He set up a brotherhood with some of his followers, who practiced his way of life and pursued his religious ideologies. He became one of the most distinguished teachers of religion in ancient Greece. Read on to know more about the childhood and career of this ancient Greek philosopher and mathematician.
Mathematical Concepts
Pythagoras studied properties of numbers and classified them as even numbers, odd numbers, triangular numbers and perfect numbers etc. The ‘Pythagoras theorem’ is one of the earliest theorems in geometry, which states that in right-angle triangles, the square of the hypotenuse is equal to the sum of square the other two sides. This theorem was already proposed during the reign of the Babylonian King Hammurabi, but Pythagoras applied it to mathematics and science and refined the concept. Pythagoras also asserted that dynamics of the structure of the universe lies on the interaction of the contraries or the opposites, such as, light and darkness, limited and unlimited, square and oblong, straight and crooked, right and left, singularity and plurality, male and female, motionless and movement and good and bad.
(http://www.thefamouspeople.com/profiles/pythagoras-504.php)
RENE DESCARTES
Famous as: Mathematician, Philosopher and Writer
Nationality: French
religion: Roman Catholic
Born on: 31 March 1596 AD
Zodiac Sign: Aries Famous Arians
Born in: La Haye en Touraine, Touraine, France
Died on: 11 February 1650 AD
place of death: Stockholm, Sweden
father: Joachim Descartes
mother: Jeanne Brochard
Married: No
education: University of Poitiers, Prytanée National Militaire
discoveries / inventions: Law Of Conservation Of Mechanical Momentum
Works & Achievements: His most famous works include La Geometrie, Discourse on the Method, Principles of Philosophy, Passions of the Soul and Meditations on First Philosophy.
Rene Descartes was an eminent French Mathematician, philosopher and writer, who has been popularly referred to as 'Father of Modern Philosophy'. Descartes was the foremost amongst all to highlight the importance of reason for the growth of natural sciences. He regarded philosophy as a belief system that contained immense knowledge. To this day, his work on philosophy " Meditations on First Philosophy" is taught as a standard text in many universities. His philosophical statement "Cogito ergo sum" meaning "I think, therefore I am", mentioned in his book 'Discourse on the Method' took him to fame. In his natural philosophy he refuted the 'analysis of corporeal substance into matter and form' and rejected any appeal to divine or natural ends in explaining natural phenomena. His contribution in mathematics was immense that he has been called the 'father of analytical geometry'. Descartes was also proponent of continental rationalism along with Leibniz, Gottfried and Spinoza in the seventeenth century.
Career
Descartes came back to France in 1622. It was during his stay in Paris that he wrote his first essay— Regulae ad Directionem Ingenii (Rules for the Direction of the Mind). In 1628 René Déscartes moved to Dutch Republic and got himself enrolled in the University of Franeker and the Leiden University to study mathematics. He lived in Dutch Republic for over 20 years, during which he published many works on philosophy and mathematics. Descartes withheld the publication of his work “Treatise on the World” following censorship of Galileo works by Catholic Church in 1633. However, he produced part of his writings in his essays namely La Géométrie, La Dioptrique and Les Météores. He presented his work such as Meditations on First Philosophy (1641) and Principles of Philosophy (1644) on metaphysics. After Cartesian philosophy faced criticism at the University of Utrecht in 1643, Descartes established contact with Princess Elisabeth of Bohemia through correspondence, writing topics on psychology and morality, which he compiled in Passions of the Soul (1649) with dedication to the Princess. He argued that moral philosophy must include the study of the body as well. He dealt with this in his books “The Description of the Human Body” and “Passions of the Soul”, where he argues that human body is more like a machine and therefore, it has material properties. The King of France rewarded Déscartes a pension in 1647. However, his books were banned by the Pope in 1663.
Legacy
Descartes left rich legacy in mathematics by ideas on Cartesian geometry and creation of XYZ as representation for unknown equation. His works became foundation for development of calculus theory by Leibinz and Newton. Besides, he also made contribution in the field of optics.
(http://www.thefamouspeople.com/profiles/ren-dscartes-499.php)
Famous as: Mathematician, Philosopher and Writer
Nationality: French
religion: Roman Catholic
Born on: 31 March 1596 AD
Zodiac Sign: Aries Famous Arians
Born in: La Haye en Touraine, Touraine, France
Died on: 11 February 1650 AD
place of death: Stockholm, Sweden
father: Joachim Descartes
mother: Jeanne Brochard
Married: No
education: University of Poitiers, Prytanée National Militaire
discoveries / inventions: Law Of Conservation Of Mechanical Momentum
Works & Achievements: His most famous works include La Geometrie, Discourse on the Method, Principles of Philosophy, Passions of the Soul and Meditations on First Philosophy.
Rene Descartes was an eminent French Mathematician, philosopher and writer, who has been popularly referred to as 'Father of Modern Philosophy'. Descartes was the foremost amongst all to highlight the importance of reason for the growth of natural sciences. He regarded philosophy as a belief system that contained immense knowledge. To this day, his work on philosophy " Meditations on First Philosophy" is taught as a standard text in many universities. His philosophical statement "Cogito ergo sum" meaning "I think, therefore I am", mentioned in his book 'Discourse on the Method' took him to fame. In his natural philosophy he refuted the 'analysis of corporeal substance into matter and form' and rejected any appeal to divine or natural ends in explaining natural phenomena. His contribution in mathematics was immense that he has been called the 'father of analytical geometry'. Descartes was also proponent of continental rationalism along with Leibniz, Gottfried and Spinoza in the seventeenth century.
Career
Descartes came back to France in 1622. It was during his stay in Paris that he wrote his first essay— Regulae ad Directionem Ingenii (Rules for the Direction of the Mind). In 1628 René Déscartes moved to Dutch Republic and got himself enrolled in the University of Franeker and the Leiden University to study mathematics. He lived in Dutch Republic for over 20 years, during which he published many works on philosophy and mathematics. Descartes withheld the publication of his work “Treatise on the World” following censorship of Galileo works by Catholic Church in 1633. However, he produced part of his writings in his essays namely La Géométrie, La Dioptrique and Les Météores. He presented his work such as Meditations on First Philosophy (1641) and Principles of Philosophy (1644) on metaphysics. After Cartesian philosophy faced criticism at the University of Utrecht in 1643, Descartes established contact with Princess Elisabeth of Bohemia through correspondence, writing topics on psychology and morality, which he compiled in Passions of the Soul (1649) with dedication to the Princess. He argued that moral philosophy must include the study of the body as well. He dealt with this in his books “The Description of the Human Body” and “Passions of the Soul”, where he argues that human body is more like a machine and therefore, it has material properties. The King of France rewarded Déscartes a pension in 1647. However, his books were banned by the Pope in 1663.
Legacy
Descartes left rich legacy in mathematics by ideas on Cartesian geometry and creation of XYZ as representation for unknown equation. His works became foundation for development of calculus theory by Leibinz and Newton. Besides, he also made contribution in the field of optics.
(http://www.thefamouspeople.com/profiles/ren-dscartes-499.php)
Isaac Newton and Wilhelm Leibniz
I have placed these two together as they are both often given the honor of being the ‘inventor’ of modern infinitesimal calculus, and as such have both made monolithic contributions to the field. To start, Leibniz is often given the credit for introducing modern standard notation, notably the integral sign. He made large contributions to the field of Topology. Whereas all round genius Isaac Newton has, because of the grand scientific epic Principia, generally become the primary man hailed by most to be the actual inventor of calculus. Nonetheless, what can be said is that both men made considerable vast contributions in their own manner.(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
I have placed these two together as they are both often given the honor of being the ‘inventor’ of modern infinitesimal calculus, and as such have both made monolithic contributions to the field. To start, Leibniz is often given the credit for introducing modern standard notation, notably the integral sign. He made large contributions to the field of Topology. Whereas all round genius Isaac Newton has, because of the grand scientific epic Principia, generally become the primary man hailed by most to be the actual inventor of calculus. Nonetheless, what can be said is that both men made considerable vast contributions in their own manner.(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
Leonardo Pisano Blgollo
Blgollo, also known as Leonardo Fibonacci, is perhaps one of the middle ages greatest mathematicians. Living from 1170 to 1250, he is best known for introducing the infamous Fibonacci Series to the western world. Although known to Indian mathematicians since approximately 200 BC, it was, nonetheless, a truly insightful sequence, appearing in biological systems frequently. In addition, from this Fibonacci also contributed greatly to the introduction of the Arabic numbering system. Something he is often forgotten for.
Haven spent a large portion of his childhood within North Africa he learned the Arabic numbering system, and upon realizing it was far simpler and more efficient then the bulky Roman numerals, decided to travel the Arab world learning from the leading mathematicians of the day. Upon returning to Italy in 1202, he published his Liber Abaci, whereupon the Arabic numbers were introduced and applied to many world situations to further advocate their use. As a result of his work the system was gradually adopted and today he is considered a major player in the development of modern mathematics.(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
Blgollo, also known as Leonardo Fibonacci, is perhaps one of the middle ages greatest mathematicians. Living from 1170 to 1250, he is best known for introducing the infamous Fibonacci Series to the western world. Although known to Indian mathematicians since approximately 200 BC, it was, nonetheless, a truly insightful sequence, appearing in biological systems frequently. In addition, from this Fibonacci also contributed greatly to the introduction of the Arabic numbering system. Something he is often forgotten for.
Haven spent a large portion of his childhood within North Africa he learned the Arabic numbering system, and upon realizing it was far simpler and more efficient then the bulky Roman numerals, decided to travel the Arab world learning from the leading mathematicians of the day. Upon returning to Italy in 1202, he published his Liber Abaci, whereupon the Arabic numbers were introduced and applied to many world situations to further advocate their use. As a result of his work the system was gradually adopted and today he is considered a major player in the development of modern mathematics.(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
Alan Turing
Computer Scientist and Cryptanalyst Alan Turing is regarded my many, if not most, to be one of the greatest minds of the 20th Century. Having worked in the Government Code and Cypher School in Britain during the second world war, he made significant discoveries and created ground breaking methods of code breaking that would eventually aid in cracking the German Enigma Encryptions. Undoubtedly affecting the outcome of the war, or at least the time-scale.
After the end of the war he invested his time in computing. Having come up with idea of a computing style machine before the war, he is considered one of the first true computer scientists. Furthermore, he wrote a range of brilliant papers on the subject of computing that are still relevant today, notably on Artificial Intelligence, on which he developed the Turing test which is still used to evaluate a computers ‘intelligence’. Remarkably, he began in 1948 working with D. G. Champernowne, an undergraduate acquaintance on a computer chess program for a machine not yet in existence. He would play the ‘part’ of the machine in testing such programs.(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
Computer Scientist and Cryptanalyst Alan Turing is regarded my many, if not most, to be one of the greatest minds of the 20th Century. Having worked in the Government Code and Cypher School in Britain during the second world war, he made significant discoveries and created ground breaking methods of code breaking that would eventually aid in cracking the German Enigma Encryptions. Undoubtedly affecting the outcome of the war, or at least the time-scale.
After the end of the war he invested his time in computing. Having come up with idea of a computing style machine before the war, he is considered one of the first true computer scientists. Furthermore, he wrote a range of brilliant papers on the subject of computing that are still relevant today, notably on Artificial Intelligence, on which he developed the Turing test which is still used to evaluate a computers ‘intelligence’. Remarkably, he began in 1948 working with D. G. Champernowne, an undergraduate acquaintance on a computer chess program for a machine not yet in existence. He would play the ‘part’ of the machine in testing such programs.(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
G. F. Bernhard Riemann
Bernhard Riemann, born to a poor family in 1826, would rise to become one of the worlds prominent mathematicians in the 19th Century. The list of contributions to geometry are large, and he has a wide range of theorems bearing his name. To name just a few: Riemannian Geometry, Riemannian Surfaces and the Riemann Integral. However, he is perhaps most famous (or infamous) for his legendarily difficult Riemann Hypothesis; an extremely complex problem on the matter of the distributions of prime numbers. Largely ignored for the first 50 years following its appearance, due to few other mathematicians actually understanding his work at the time, it has quickly risen to become one of the greatest open questions in modern science, baffling and confounding even the greatest mathematicians. Although progress has been made, its has been incredibly slow. However, a prize of $1 million has been offered from the Clay Maths Institute for a proof, and one would almost undoubtedly receive a Fields medal if under 40 (The Nobel prize of mathematics). The fallout from such a proof is hypothesized to be large: Major encryption systems are thought to be breakable with such a proof, and all that rely on them would collapse. As well as this, a proof of the hypothesis is expected to use ‘new mathematics’. It would seem that, even in death, Riemann’s work may still pave the way for new contributions to the field, just as he did in life.(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
Bernhard Riemann, born to a poor family in 1826, would rise to become one of the worlds prominent mathematicians in the 19th Century. The list of contributions to geometry are large, and he has a wide range of theorems bearing his name. To name just a few: Riemannian Geometry, Riemannian Surfaces and the Riemann Integral. However, he is perhaps most famous (or infamous) for his legendarily difficult Riemann Hypothesis; an extremely complex problem on the matter of the distributions of prime numbers. Largely ignored for the first 50 years following its appearance, due to few other mathematicians actually understanding his work at the time, it has quickly risen to become one of the greatest open questions in modern science, baffling and confounding even the greatest mathematicians. Although progress has been made, its has been incredibly slow. However, a prize of $1 million has been offered from the Clay Maths Institute for a proof, and one would almost undoubtedly receive a Fields medal if under 40 (The Nobel prize of mathematics). The fallout from such a proof is hypothesized to be large: Major encryption systems are thought to be breakable with such a proof, and all that rely on them would collapse. As well as this, a proof of the hypothesis is expected to use ‘new mathematics’. It would seem that, even in death, Riemann’s work may still pave the way for new contributions to the field, just as he did in life.(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
Carl Friedrich Gauss
Child prodigy Gauss, the ‘Prince of Mathematics’, made his first major discovery whilst still a teenager, and wrote the incredible Disquisitiones Arithmeticae, his magnum opus, by the time he was 21. Many know Gauss for his outstanding mental ability – quoted to have added the numbers 1 to 100 within seconds whilst attending primary school (with the aid of a clever trick). The local Duke, recognizing his talent, sent him to Collegium Carolinum before he left for Gottingen (at the time it was the most prestigious mathematical university in the world, with many of the best attending). After graduating in 1798 (at the age of 22), he began to make several important contributions in major areas of mathematics, most notably number theory (especially on Prime numbers). He went on to prove the fundamental theorem of algebra, and introduced the Gaussian gravitational constant in physics, as well as much more – all this before he was 24! Needless to say, he continued his work up until his death at the age of 77, and had made major advances in the field which have echoed down through time.(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
Child prodigy Gauss, the ‘Prince of Mathematics’, made his first major discovery whilst still a teenager, and wrote the incredible Disquisitiones Arithmeticae, his magnum opus, by the time he was 21. Many know Gauss for his outstanding mental ability – quoted to have added the numbers 1 to 100 within seconds whilst attending primary school (with the aid of a clever trick). The local Duke, recognizing his talent, sent him to Collegium Carolinum before he left for Gottingen (at the time it was the most prestigious mathematical university in the world, with many of the best attending). After graduating in 1798 (at the age of 22), he began to make several important contributions in major areas of mathematics, most notably number theory (especially on Prime numbers). He went on to prove the fundamental theorem of algebra, and introduced the Gaussian gravitational constant in physics, as well as much more – all this before he was 24! Needless to say, he continued his work up until his death at the age of 77, and had made major advances in the field which have echoed down through time.(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
Leonhard Euler
Science Next Previous Random List Share Humans Top 10 Greatest Mathematicians M. R. Sexton December 7, 2010
Check out our new companion site: http://knowledgenuts.com Often called the language of the universe, mathematics is fundamental to our understanding of the world and, as such, is vitally important in a modern society such as ours. Everywhere you look it is likely mathematics has made an impact, from the faucet in your kitchen to the satellite that beams your television programs to your home. As such, great mathematicians are undoubtedly going to rise above the rest and have their name embedded within history. This list documents some such people. I have rated them based on contributions and how they effected mathematics at the time, as well as their lasting effect. I also suggest one looks deeper into the lives of these men, as they are truly fascinating people and their discoveries are astonishing – too much to include here. As always, such lists are highly subjective, and as such please include your own additions in the comments!(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
Science Next Previous Random List Share Humans Top 10 Greatest Mathematicians M. R. Sexton December 7, 2010
Check out our new companion site: http://knowledgenuts.com Often called the language of the universe, mathematics is fundamental to our understanding of the world and, as such, is vitally important in a modern society such as ours. Everywhere you look it is likely mathematics has made an impact, from the faucet in your kitchen to the satellite that beams your television programs to your home. As such, great mathematicians are undoubtedly going to rise above the rest and have their name embedded within history. This list documents some such people. I have rated them based on contributions and how they effected mathematics at the time, as well as their lasting effect. I also suggest one looks deeper into the lives of these men, as they are truly fascinating people and their discoveries are astonishing – too much to include here. As always, such lists are highly subjective, and as such please include your own additions in the comments!(http://listverse.com/2010/12/07/top-10-greatest-mathematicians/)
Pierre de Fermat (1601-1665) France
Pierre de Fermat was the most brilliant mathematician of his era and, along with Déscartes, one of the most influential. Although mathematics was just his hobby (Fermat was a government lawyer), Fermat practically founded Number Theory, and also played key roles in the discoveries of Analytic Geometry and Calculus. He was also an excellent geometer (e.g. discovering a triangle's Fermat point), and (in collaboration with Blaise Pascal) discovered probability theory. Fellow geniuses are the best judges of genius, and Blaise Pascal had this to say of Fermat: "For my part, I confess that [Fermat's researches about numbers] are far beyond me, and I am competent only to admire them." E.T. Bell wrote "it can be argued that Fermat was at least Newton's equal as a pure mathematician." Fermat's most famous discoveries in number theory include the ubiquitously-used Fermat's Little Theorem; the n = 4 case of his conjectured Fermat's Last Theorem (he may have proved the n = 3 case as well); the fact that every natural number is the sum of three triangle numbers; and Fermat's Christmas Theorem (that any prime (4n+1) can be represented as the sum of two squares in exactly one way) which may be considered the most difficult theorem of arithmetic which had been proved up to that date. Fermat proved the Christmas Theorem with difficulty using "infinite descent," but details are unrecorded, so the theorem is often named the Fermat-Euler Prime Number Theorem, with the first published proof being by Euler more than a century after Fermat's claim. (Most of Fermat's proofs were never published, but it is wrong to suppose that Fermat's work comprised mostly false or unproven conjectures. This misconception arises from his so-called "Last Theorem" which was actually just a private scribble.)
Fermat developed a system of analytic geometry which both preceded and surpassed that of Déscartes; he developed methods of differential and integral calculus which Newton acknowledged as an inspiration. Solving df(x)/dx = 0 to find extrema of f(x) is perhaps the most useful idea in applied mathematics; this technique originated with Fermat. Fermat was also the first European to find the integration formula for the general polynomial; he used his calculus to find centers of gravity, etc.
Fermat's contemporaneous rival René Déscartes is more famous than Fermat, and Déscartes' writings were more influential. Whatever one thinks of Déscartes as a philosopher, however, it seems clear that Fermat was the better mathematician. Fermat and Déscartes did work in physics and independently discovered the (trigonometric) law of refraction, but Fermat gave the correct explanation, and used it remarkably to anticipate the Principle of Least Action later enunciated by Maupertius (though Maupertius himself, like Déscartes, had an incorrect explanation of refraction). Fermat and Déscartes independently discovered analytic geometry, but it was Fermat who extended it to more than two dimensions, and followed up by developing elementary calculus. (http://fabpedigree.com/james/mathmen.htm#Fermat)
Pierre de Fermat was the most brilliant mathematician of his era and, along with Déscartes, one of the most influential. Although mathematics was just his hobby (Fermat was a government lawyer), Fermat practically founded Number Theory, and also played key roles in the discoveries of Analytic Geometry and Calculus. He was also an excellent geometer (e.g. discovering a triangle's Fermat point), and (in collaboration with Blaise Pascal) discovered probability theory. Fellow geniuses are the best judges of genius, and Blaise Pascal had this to say of Fermat: "For my part, I confess that [Fermat's researches about numbers] are far beyond me, and I am competent only to admire them." E.T. Bell wrote "it can be argued that Fermat was at least Newton's equal as a pure mathematician." Fermat's most famous discoveries in number theory include the ubiquitously-used Fermat's Little Theorem; the n = 4 case of his conjectured Fermat's Last Theorem (he may have proved the n = 3 case as well); the fact that every natural number is the sum of three triangle numbers; and Fermat's Christmas Theorem (that any prime (4n+1) can be represented as the sum of two squares in exactly one way) which may be considered the most difficult theorem of arithmetic which had been proved up to that date. Fermat proved the Christmas Theorem with difficulty using "infinite descent," but details are unrecorded, so the theorem is often named the Fermat-Euler Prime Number Theorem, with the first published proof being by Euler more than a century after Fermat's claim. (Most of Fermat's proofs were never published, but it is wrong to suppose that Fermat's work comprised mostly false or unproven conjectures. This misconception arises from his so-called "Last Theorem" which was actually just a private scribble.)
Fermat developed a system of analytic geometry which both preceded and surpassed that of Déscartes; he developed methods of differential and integral calculus which Newton acknowledged as an inspiration. Solving df(x)/dx = 0 to find extrema of f(x) is perhaps the most useful idea in applied mathematics; this technique originated with Fermat. Fermat was also the first European to find the integration formula for the general polynomial; he used his calculus to find centers of gravity, etc.
Fermat's contemporaneous rival René Déscartes is more famous than Fermat, and Déscartes' writings were more influential. Whatever one thinks of Déscartes as a philosopher, however, it seems clear that Fermat was the better mathematician. Fermat and Déscartes did work in physics and independently discovered the (trigonometric) law of refraction, but Fermat gave the correct explanation, and used it remarkably to anticipate the Principle of Least Action later enunciated by Maupertius (though Maupertius himself, like Déscartes, had an incorrect explanation of refraction). Fermat and Déscartes independently discovered analytic geometry, but it was Fermat who extended it to more than two dimensions, and followed up by developing elementary calculus. (http://fabpedigree.com/james/mathmen.htm#Fermat)
Gottfried Wilhelm von Leibniz (1646-1716) Germany
Leibniz was one of the most brilliant and prolific intellectuals ever; and his influence in mathematics (especially his co-invention of the infinitesimal calculus) was immense. His childhood IQ has been estimated as second-highest in all of history, behind only Goethe's. Descriptions which have been applied to Leibniz include "one of the two greatest universal geniuses" (da Vinci was the other); "the most important logician between Aristotle and Boole;" and the "Father of Applied Science." Leibniz described himself as "the most teachable of mortals." Mathematics was just a self-taught sideline for Leibniz, who was a philosopher, lawyer, historian, diplomat and renowned inventor. Because he "wasted his youth" before learning mathematics, he probably ranked behind the Bernoullis as well as Newton in pure mathematical talent, and thus he may be the only mathematician among the Top Fifteen who was never the greatest living algorist or theorem prover. We won't try to summarize Leibniz' contributions to philosophy and diverse other fields including biology; as just three examples: he predicted the Earth's molten core, introduced the notion of subconscious mind, and built the first calculator that could do multiplication. Leibniz also had political influence: he consulted to both the Holy Roman and Russian Emperors; another of his patrons was Sophia Wittelsbach, who was only distantly in line for the British throne, but was made Heir Presumptive. (Sophia died before Queen Anne, but her son was crowned King George I of England.)
Leibniz pioneered the common discourse of mathematics, including its continuous, discrete, and symbolic aspects. (His ideas on symbolic logic weren't pursued and it was left to Boole to reinvent this almost two centuries later.) Mathematical innovations attributed to Leibniz include the notations ∫f(x)dx, df(x)/dx, and even ∛x; the concepts of matrix determinant and Gaussian elimination; the theory of geometric envelopes; and the binary number system. He invented more mathematical terms than anyone, including "function," "analysis situ," "variable," "abscissa," "parameter," and "coordinate." His works seem to anticipate cybernetics and information theory; and Mandelbrot acknowledged Leibniz' anticipation of self-similarity. Like Newton, Leibniz discovered The Fundamental Theorem of Calculus; his contribution to calculus was much more influential than Newton's, and his superior notation is used to this day. As Leibniz himself pointed out, since the concept of mathematical analysis was already known to ancient Greeks, the revolutionary invention was notation ("calculus"), because with "symbols [which] express the exact nature of a thing briefly ... the labor of thought is wonderfully diminished."
Leibniz' thoughts on mathematical physics had some influence. He developed laws of motion that gave different insights from those of Newton. His cosmology was opposed to that of Newton but, anticipating theories of Mach and Einstein, is more in accord with modern physics. Mathematical physicists influenced by Leibniz include not only Mach, but perhaps Hamilton and Poincaré themselves.
Although others found it independently (including perhaps Madhava three centuries earlier), Leibniz discovered and proved a striking identity for π:
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
(http://fabpedigree.com/james/mathmen.htm#Leibniz)
Leibniz was one of the most brilliant and prolific intellectuals ever; and his influence in mathematics (especially his co-invention of the infinitesimal calculus) was immense. His childhood IQ has been estimated as second-highest in all of history, behind only Goethe's. Descriptions which have been applied to Leibniz include "one of the two greatest universal geniuses" (da Vinci was the other); "the most important logician between Aristotle and Boole;" and the "Father of Applied Science." Leibniz described himself as "the most teachable of mortals." Mathematics was just a self-taught sideline for Leibniz, who was a philosopher, lawyer, historian, diplomat and renowned inventor. Because he "wasted his youth" before learning mathematics, he probably ranked behind the Bernoullis as well as Newton in pure mathematical talent, and thus he may be the only mathematician among the Top Fifteen who was never the greatest living algorist or theorem prover. We won't try to summarize Leibniz' contributions to philosophy and diverse other fields including biology; as just three examples: he predicted the Earth's molten core, introduced the notion of subconscious mind, and built the first calculator that could do multiplication. Leibniz also had political influence: he consulted to both the Holy Roman and Russian Emperors; another of his patrons was Sophia Wittelsbach, who was only distantly in line for the British throne, but was made Heir Presumptive. (Sophia died before Queen Anne, but her son was crowned King George I of England.)
Leibniz pioneered the common discourse of mathematics, including its continuous, discrete, and symbolic aspects. (His ideas on symbolic logic weren't pursued and it was left to Boole to reinvent this almost two centuries later.) Mathematical innovations attributed to Leibniz include the notations ∫f(x)dx, df(x)/dx, and even ∛x; the concepts of matrix determinant and Gaussian elimination; the theory of geometric envelopes; and the binary number system. He invented more mathematical terms than anyone, including "function," "analysis situ," "variable," "abscissa," "parameter," and "coordinate." His works seem to anticipate cybernetics and information theory; and Mandelbrot acknowledged Leibniz' anticipation of self-similarity. Like Newton, Leibniz discovered The Fundamental Theorem of Calculus; his contribution to calculus was much more influential than Newton's, and his superior notation is used to this day. As Leibniz himself pointed out, since the concept of mathematical analysis was already known to ancient Greeks, the revolutionary invention was notation ("calculus"), because with "symbols [which] express the exact nature of a thing briefly ... the labor of thought is wonderfully diminished."
Leibniz' thoughts on mathematical physics had some influence. He developed laws of motion that gave different insights from those of Newton. His cosmology was opposed to that of Newton but, anticipating theories of Mach and Einstein, is more in accord with modern physics. Mathematical physicists influenced by Leibniz include not only Mach, but perhaps Hamilton and Poincaré themselves.
Although others found it independently (including perhaps Madhava three centuries earlier), Leibniz discovered and proved a striking identity for π:
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
(http://fabpedigree.com/james/mathmen.htm#Leibniz)
Adrien-Marie Legendre
Also Known As: Adrien M. Legendre
Famous as: Mathematician
Nationality: French
Born on: 18 September 1752 AD
Zodiac Sign: Virgo Famous Virgos
Born in: Paris, France
Died on: 10 January 1833 AD
place of death: Paris, France
Spouse: Marguerite-Claudine Couhin
Works & Achievements: Introduced Legendre polynomials and Legendre transformation.
French mathematician Adrien-Marie Legendre invented mathematical theories of Legendre transformation and elliptic functions. He worked on quadratic reciprocity law in number theory. He made remarkable contribution to application of analysis to number theory and division of primes. His valuable research on elliptic functions has significant place in the arena of mathematics. Several theories of Legendre acted as a source of inspiration for many mathematical theories like Galois Theory, some parts of Gauss' statistics and number theory and Abel's work on elliptic functions. He developed the least squares method, which has wide ranging application in linear regression, signal processing, statistics, and curve fitting. He gave the concept of Legendre transformation, which is used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics. Legendre transformation is also used in thermodynamics to obtain the enthalpy and the Helmholtz and Gibbs (free) energies from the internal energy.(http://www.thefamouspeople.com/profiles/adrien-marie-legendre-591.php)
Also Known As: Adrien M. Legendre
Famous as: Mathematician
Nationality: French
Born on: 18 September 1752 AD
Zodiac Sign: Virgo Famous Virgos
Born in: Paris, France
Died on: 10 January 1833 AD
place of death: Paris, France
Spouse: Marguerite-Claudine Couhin
Works & Achievements: Introduced Legendre polynomials and Legendre transformation.
French mathematician Adrien-Marie Legendre invented mathematical theories of Legendre transformation and elliptic functions. He worked on quadratic reciprocity law in number theory. He made remarkable contribution to application of analysis to number theory and division of primes. His valuable research on elliptic functions has significant place in the arena of mathematics. Several theories of Legendre acted as a source of inspiration for many mathematical theories like Galois Theory, some parts of Gauss' statistics and number theory and Abel's work on elliptic functions. He developed the least squares method, which has wide ranging application in linear regression, signal processing, statistics, and curve fitting. He gave the concept of Legendre transformation, which is used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics. Legendre transformation is also used in thermodynamics to obtain the enthalpy and the Helmholtz and Gibbs (free) energies from the internal energy.(http://www.thefamouspeople.com/profiles/adrien-marie-legendre-591.php)
Carl Ludwig Siegel
Famous as: Mathematician
Nationality: German
Born on: 31 December 1896 AD
Zodiac Sign: Capricorn Famous Capricons
Born in: Berlin, German Empire
Died on: 04 April 1981 AD
place of death: Göttingen, West Germany
Married: No
education: Georg-August University of Göttingen, Humboldt University of Berlin
Works & Achievements: Number theory and Celestial mechanics
awards: 1978 - Wolf Prize in Mathematics
- copey medal
The man behind the number theory, Carl Ludwig Siegel was a significant mathematician of the 20th century. Son of a postal worker, Siegel, in spite of his humble background, made it big in the world of science through his sheer determination and labor. A profound student of history of mathematics, Siegel put his studies to good use and thus was successful in bringing about the number theory and celestial mechanics. Additionally, he also proved important theorems in the theory of analytical functions of several complex numbers. For his immense contribution in the field of mathematics, Siegel was the recipient of many honorary doctorates, and a member of the most renowned academies in his life. Explore this biography to know more about this profile and works.
Early Life Carl Ludwig Siegel was born in Berlin in the year 1896. His father worked for the post office. At the age of nineteen, Siegel enrolled himself at the Humboldt University in the midst of World War I, wherein he studied mathematics, astronomy and physics. It was there that Siegel attended lectures by Max Planck and Ferdinand Georg Frobenius. These lectures made an impression on young Siegel’s mind so much so that it made Siegel abandon astronomy and switch to number theory, which became the main research topic of his career. Despite being an antimilitarist, Siegel, in 1917, had to forgo his studies as he was called for military service. However, army life did not suit Siegel as he could not adapt to the strict schedule, and hence, he was discharged as a failure. Post war, Siegel resumed his studies, for which he enrolled himself at the Georg-August University of Göttingen, as a teaching and research assistant. Under the instruction and training of Edmund Landau, Siegel received his PhD degree in 1920. It was during his years at Göttingen that Siegel undertook many of his groundbreaking research and published them. In 1922, Siegel was appointed as a professor at the Johann Wolfgang Goethe-Universität of Frankfurt am Main as the successor of Arthur Moritz Schönflies. A close friend of docents Ernst Hellinger and Max Dehn, Siegel used his influence to help the two as he too strongly opposed Nazism. It was due to this that Siegel was not appointed as a successor to the chair of Constantin Carathéodory in Munich.
Role in the Seminar Along with Dehn, Hellinger and Epstein, Siegel took part in a seminar in Frankfurt in 1922, the subject matter of which was history of mathematics. the seminar was one-of-a-kind as it was to last for thirteen years. One important rule that every participant needed to follow was that they had to study all the mathematical works in their original languages. the participants studied works of Euclid, Archimedes, Fibonacci, Cardan, Stevin, Viète, Kepler, Desargues, Descartes, Fermat, Huygens, Barrow, and Gregory. The main aim of the seminar was to educate the participants and increase their understanding of the results presented as well as providing teachers with the satisfaction of getting a handsome opportunity of examining the exceptional works of the mathematicians of the bygone era. upon Siegel’s appointement, the number of students enrolling for the seminar increased significantly. Such was the state that by 1928, Siegel taught almost 143 students in differential and integral calculus course.
Later Years Hitler’s return to power in 1933 brought about a phase in Siegel’s life that he did not cherish much. According to the Civil Service Law, no Jews could hold the position of teacher in the universities. Though this did not affect Siegel since he was an Aryan, he did not like what was going on. As a result, Siegel spent a year at the Institute for Advanced Study at Princeton in the United States in 1935. Upon returning, he learned that the state of the Jews had worsened and that his friends, Dehn, Hellinger and Epstein were forcefully removed from their positions. The foursome remained in Frankfurt but was unable to teach. It was then that Siegel was offered professorship in Göttingen. As such, in the year 1938, Siegel returned to Göttingen. However, this homecoming didn’t last long as Nazi regime had taken Germany to war in 1939 and Siegel felt that he could no longer remain in his native land. Thus, a year later, i.e. in 1940, Siegel moved to United States via Norway. Therein he joined the Institute for Advanced Study in Princeton, where he had already spent a sabbatical in 1935. Siegel was given a permanent professorship in the year 1946. It was only after the end of World War II that Siegel returned to Göttingen in 1951. Siegel accepted the post of a professor which was with him until his retirement in 1959.
His Career Siegel’s groundbreaking work in the field of number theory won him appreciation and accolades from around the world. In addition to this, Siegel is also accounted for the substantial contribution he made to the transdence theory, devising a new method for the algebriac independence of values of certain E-functions. His research on the analytic theory of quadratic forms in 1935 was historically significant as he broke new ground in considering quadratic forms in which the coefficients were from an algebraic number field.The Siegel modular forms are recognised as part of the moduli theory of abelian varieties. Siegel is also known to have made eight major contributions in celestial mechanics, which was next to number theory in his list of favourite topics. It was due to his immense work and contribution to mathematics that Siegel was honored with the Wolf Prize award in Mathematics in the year 1978.
Death
Carl Ludwig Siegel breathed his last on April 4, 1981. He was 84 years old at the time of his death.
(http://www.thefamouspeople.com/profiles/carl-ludwig-siegel-540.php)
Famous as: Mathematician
Nationality: German
Born on: 31 December 1896 AD
Zodiac Sign: Capricorn Famous Capricons
Born in: Berlin, German Empire
Died on: 04 April 1981 AD
place of death: Göttingen, West Germany
Married: No
education: Georg-August University of Göttingen, Humboldt University of Berlin
Works & Achievements: Number theory and Celestial mechanics
awards: 1978 - Wolf Prize in Mathematics
- copey medal
The man behind the number theory, Carl Ludwig Siegel was a significant mathematician of the 20th century. Son of a postal worker, Siegel, in spite of his humble background, made it big in the world of science through his sheer determination and labor. A profound student of history of mathematics, Siegel put his studies to good use and thus was successful in bringing about the number theory and celestial mechanics. Additionally, he also proved important theorems in the theory of analytical functions of several complex numbers. For his immense contribution in the field of mathematics, Siegel was the recipient of many honorary doctorates, and a member of the most renowned academies in his life. Explore this biography to know more about this profile and works.
Early Life Carl Ludwig Siegel was born in Berlin in the year 1896. His father worked for the post office. At the age of nineteen, Siegel enrolled himself at the Humboldt University in the midst of World War I, wherein he studied mathematics, astronomy and physics. It was there that Siegel attended lectures by Max Planck and Ferdinand Georg Frobenius. These lectures made an impression on young Siegel’s mind so much so that it made Siegel abandon astronomy and switch to number theory, which became the main research topic of his career. Despite being an antimilitarist, Siegel, in 1917, had to forgo his studies as he was called for military service. However, army life did not suit Siegel as he could not adapt to the strict schedule, and hence, he was discharged as a failure. Post war, Siegel resumed his studies, for which he enrolled himself at the Georg-August University of Göttingen, as a teaching and research assistant. Under the instruction and training of Edmund Landau, Siegel received his PhD degree in 1920. It was during his years at Göttingen that Siegel undertook many of his groundbreaking research and published them. In 1922, Siegel was appointed as a professor at the Johann Wolfgang Goethe-Universität of Frankfurt am Main as the successor of Arthur Moritz Schönflies. A close friend of docents Ernst Hellinger and Max Dehn, Siegel used his influence to help the two as he too strongly opposed Nazism. It was due to this that Siegel was not appointed as a successor to the chair of Constantin Carathéodory in Munich.
Role in the Seminar Along with Dehn, Hellinger and Epstein, Siegel took part in a seminar in Frankfurt in 1922, the subject matter of which was history of mathematics. the seminar was one-of-a-kind as it was to last for thirteen years. One important rule that every participant needed to follow was that they had to study all the mathematical works in their original languages. the participants studied works of Euclid, Archimedes, Fibonacci, Cardan, Stevin, Viète, Kepler, Desargues, Descartes, Fermat, Huygens, Barrow, and Gregory. The main aim of the seminar was to educate the participants and increase their understanding of the results presented as well as providing teachers with the satisfaction of getting a handsome opportunity of examining the exceptional works of the mathematicians of the bygone era. upon Siegel’s appointement, the number of students enrolling for the seminar increased significantly. Such was the state that by 1928, Siegel taught almost 143 students in differential and integral calculus course.
Later Years Hitler’s return to power in 1933 brought about a phase in Siegel’s life that he did not cherish much. According to the Civil Service Law, no Jews could hold the position of teacher in the universities. Though this did not affect Siegel since he was an Aryan, he did not like what was going on. As a result, Siegel spent a year at the Institute for Advanced Study at Princeton in the United States in 1935. Upon returning, he learned that the state of the Jews had worsened and that his friends, Dehn, Hellinger and Epstein were forcefully removed from their positions. The foursome remained in Frankfurt but was unable to teach. It was then that Siegel was offered professorship in Göttingen. As such, in the year 1938, Siegel returned to Göttingen. However, this homecoming didn’t last long as Nazi regime had taken Germany to war in 1939 and Siegel felt that he could no longer remain in his native land. Thus, a year later, i.e. in 1940, Siegel moved to United States via Norway. Therein he joined the Institute for Advanced Study in Princeton, where he had already spent a sabbatical in 1935. Siegel was given a permanent professorship in the year 1946. It was only after the end of World War II that Siegel returned to Göttingen in 1951. Siegel accepted the post of a professor which was with him until his retirement in 1959.
His Career Siegel’s groundbreaking work in the field of number theory won him appreciation and accolades from around the world. In addition to this, Siegel is also accounted for the substantial contribution he made to the transdence theory, devising a new method for the algebriac independence of values of certain E-functions. His research on the analytic theory of quadratic forms in 1935 was historically significant as he broke new ground in considering quadratic forms in which the coefficients were from an algebraic number field.The Siegel modular forms are recognised as part of the moduli theory of abelian varieties. Siegel is also known to have made eight major contributions in celestial mechanics, which was next to number theory in his list of favourite topics. It was due to his immense work and contribution to mathematics that Siegel was honored with the Wolf Prize award in Mathematics in the year 1978.
Death
Carl Ludwig Siegel breathed his last on April 4, 1981. He was 84 years old at the time of his death.
(http://www.thefamouspeople.com/profiles/carl-ludwig-siegel-540.php)
John von Neumann
Famous as: Mathematician
Nationality: Hungarian,American
Born on: 28 December 1903 AD
Zodiac Sign: Capricorn Famous Capricons
Born in: Budapest
Died on: 08 February 1957 AD
place of death: Washington, D.C.
father: Neumann Miksa
mother: Kann Margit
Spouse: Klara Dan
children: Marina von Neumann Whitman
education: University of Budapest, ETH Zurich
Works & Achievements: Game theory, quantum logic, linear programming, mathematical statistics etc.
awards: 1956 - Enrico Fermi Award
1938 - Bôcher Memorial Prize
John von Neumann was a famous Hungarian-American mathematician, who is still revered for his unparalleled contributions to disciplines like mathematics, science, economics and many more. Born in a wealthy family to Jewish parents, he was a child prodigy, exhibiting great analytic and computing skills. He was able to divide eight - digit numbers in mind in very early ages, which proves his great analytical and computing skills. He was a passionate learner and was interested in many subjects like physics, economics, statistics, history etc., apart from mathematics. He was a prominent member of the Manhattan Project and the Institute for Advanced Study in Princeton. His unusual abilities astonished many great personalities like mathematician Jean Dieudonne Peter Lax, Hans Bethe etc. Hans Bethe once said, "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man". Read through this biography to know more about the career and life of this genius.
Early Life John von Neumann was born in Budapest, the capital city of Hungary, on 28 December 1903. He was the elder son of Neumann Miksa and Kann Margit. The nature of the numbers and the mathematical logic induced interest in him at a very early age. However, mathematics was not the only subject which fascinated him; Neumann was greatly interested in history as well that he went through forty four volumes of universal history at an early age of eight. This indicated that Neumann found comfort dealing with both logical and social areas. He was fortunate enough to have very supportive parents, who stood by his interests. In 1914, at the age of ten, he joined the Lutheran Gymnasium, one among the three best organizations in Budapest. He published his first paper in the Journal of the German Mathematical Society in 1922, which dealt with the zeros of certain minimal polynomials.
Berlin, Zurich and Budapest Neumann had little interest towards chemistry and engineering but, his father encouraged him to pursue engineering as it was considered as an elite career during that time. He attended Pázmány Péter University in Budapest to pursue advanced doctorate in mathematics along with completing his undergraduate course, diploma in chemical engineering fromETH Zurich. In his thesis for PhD, he wanted to give an attempt to the axiomatization of the set theory which was developed by George Cantor. It was an achievement indeed that a seventeen year old boy pursued both undergraduate and PhD simultaneously. He received good scores for his exams for both the courses and thus, completed under graduation degree in chemical engineering and PhD in mathematics at the age of twenty-two.
Quantum Mechanics After receiving his degrees in 1926, Neumann attended the university in Gottingen, Germany, where he dealt with quantum mechanics. He was creative and an original thinker, and came up with complete and logical concepts. He worked on the theories of quantum mechanics in 1926 to reconcile and improve them. Neumann worked to find out the similarities between wave mechanics and matrix mechanics. Also, von Neumann worked on the rules of ‘abstract Hilbert space’ and devised a mathematical structure in terms of quantum theory.
Personal life During the time from 1927-1929, after Neumann formulated the quantum mechanics, he attended extensive conferences and colloquia. By 1929, he wrote about 32 papers in German. These papers were was well-organized so that the other mathematicians could incorporate Neumann’s works into their theories. By this time, he became famous in the academic world, owing to his creative yet new ideas. By the end of 1929, he was offered a post of lecturer at Princeton in America. It was during the same time, when he married Mariette Kovesi, his childhood friend. In 1935, Mariette gave birth to a girl child, Marina. The marital relationship did not last long; they separated in 1936. Mariette went back to Budapest and Neumann drifted around Europe for a brief period and returned to United States. He met Klari Dan on his way to Budapest and married her in 1938.
Death John von Neumann was diagnosed with cancer. Despite his ailment and weakness, he took part in the ceremonies organized to honor him, in a wheel chair, and kept contact with family and friends during this time. He passed away on 8 February 1957.
Notable Contributions Neumann contributed to the Los Alamos project devised by the American government in the development of the nuclear weapons and also in developing the concept and design of explosive lenses. The mathematical modelling which he used at Los Alamos helped him in the development of the modern computer. Along with gathering resources, he also funded the project for the development of the modern computer. He worked on the architecture of the computer as well. His efforts made other scientists realize that computer is not just a ‘bigger calculator’. Quantum logic, game theory, linear programming and mathematical statistics are some of the many contributions he had made to the field of science. (http://www.thefamouspeople.com/profiles/john-von-neumann-481.php)
Famous as: Mathematician
Nationality: Hungarian,American
Born on: 28 December 1903 AD
Zodiac Sign: Capricorn Famous Capricons
Born in: Budapest
Died on: 08 February 1957 AD
place of death: Washington, D.C.
father: Neumann Miksa
mother: Kann Margit
Spouse: Klara Dan
children: Marina von Neumann Whitman
education: University of Budapest, ETH Zurich
Works & Achievements: Game theory, quantum logic, linear programming, mathematical statistics etc.
awards: 1956 - Enrico Fermi Award
1938 - Bôcher Memorial Prize
John von Neumann was a famous Hungarian-American mathematician, who is still revered for his unparalleled contributions to disciplines like mathematics, science, economics and many more. Born in a wealthy family to Jewish parents, he was a child prodigy, exhibiting great analytic and computing skills. He was able to divide eight - digit numbers in mind in very early ages, which proves his great analytical and computing skills. He was a passionate learner and was interested in many subjects like physics, economics, statistics, history etc., apart from mathematics. He was a prominent member of the Manhattan Project and the Institute for Advanced Study in Princeton. His unusual abilities astonished many great personalities like mathematician Jean Dieudonne Peter Lax, Hans Bethe etc. Hans Bethe once said, "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man". Read through this biography to know more about the career and life of this genius.
Early Life John von Neumann was born in Budapest, the capital city of Hungary, on 28 December 1903. He was the elder son of Neumann Miksa and Kann Margit. The nature of the numbers and the mathematical logic induced interest in him at a very early age. However, mathematics was not the only subject which fascinated him; Neumann was greatly interested in history as well that he went through forty four volumes of universal history at an early age of eight. This indicated that Neumann found comfort dealing with both logical and social areas. He was fortunate enough to have very supportive parents, who stood by his interests. In 1914, at the age of ten, he joined the Lutheran Gymnasium, one among the three best organizations in Budapest. He published his first paper in the Journal of the German Mathematical Society in 1922, which dealt with the zeros of certain minimal polynomials.
Berlin, Zurich and Budapest Neumann had little interest towards chemistry and engineering but, his father encouraged him to pursue engineering as it was considered as an elite career during that time. He attended Pázmány Péter University in Budapest to pursue advanced doctorate in mathematics along with completing his undergraduate course, diploma in chemical engineering fromETH Zurich. In his thesis for PhD, he wanted to give an attempt to the axiomatization of the set theory which was developed by George Cantor. It was an achievement indeed that a seventeen year old boy pursued both undergraduate and PhD simultaneously. He received good scores for his exams for both the courses and thus, completed under graduation degree in chemical engineering and PhD in mathematics at the age of twenty-two.
Quantum Mechanics After receiving his degrees in 1926, Neumann attended the university in Gottingen, Germany, where he dealt with quantum mechanics. He was creative and an original thinker, and came up with complete and logical concepts. He worked on the theories of quantum mechanics in 1926 to reconcile and improve them. Neumann worked to find out the similarities between wave mechanics and matrix mechanics. Also, von Neumann worked on the rules of ‘abstract Hilbert space’ and devised a mathematical structure in terms of quantum theory.
Personal life During the time from 1927-1929, after Neumann formulated the quantum mechanics, he attended extensive conferences and colloquia. By 1929, he wrote about 32 papers in German. These papers were was well-organized so that the other mathematicians could incorporate Neumann’s works into their theories. By this time, he became famous in the academic world, owing to his creative yet new ideas. By the end of 1929, he was offered a post of lecturer at Princeton in America. It was during the same time, when he married Mariette Kovesi, his childhood friend. In 1935, Mariette gave birth to a girl child, Marina. The marital relationship did not last long; they separated in 1936. Mariette went back to Budapest and Neumann drifted around Europe for a brief period and returned to United States. He met Klari Dan on his way to Budapest and married her in 1938.
Death John von Neumann was diagnosed with cancer. Despite his ailment and weakness, he took part in the ceremonies organized to honor him, in a wheel chair, and kept contact with family and friends during this time. He passed away on 8 February 1957.
Notable Contributions Neumann contributed to the Los Alamos project devised by the American government in the development of the nuclear weapons and also in developing the concept and design of explosive lenses. The mathematical modelling which he used at Los Alamos helped him in the development of the modern computer. Along with gathering resources, he also funded the project for the development of the modern computer. He worked on the architecture of the computer as well. His efforts made other scientists realize that computer is not just a ‘bigger calculator’. Quantum logic, game theory, linear programming and mathematical statistics are some of the many contributions he had made to the field of science. (http://www.thefamouspeople.com/profiles/john-von-neumann-481.php)
Richard Dedekind
Famous as: Mathematician
Nationality: German
Born on: 06 October 1831 AD
Zodiac Sign: Libra Famous Libras
Died on: 12 February 1916 AD
father: Julius Levin Ulrich Dedekind
mother: Caroline Marie Hanriette Emperius
siblings: Julia
Works & Achievements: 'Lectures on Number Theory', 'Continuity and Irrational Numbers' and Modular Lattices
A well known German mathematician, Richard Dedekind made immense contribution in the field of abstract algebra, especially, algebraic number theory, the foundations of the real numbers and ring theory. His most celebrated work was 'Dedekind cuts', which he put forth in the beginning of career. He presented his findings in the book Stetigkeit und Irrationale Zahlen' (Continuity and Irrational Numbers) in 1872. In his book, 'Was sind und was sollen die Zahlen (What are numbers and what should they be), he put forward the definition of an infinite set and analysis of number theory. Along with working on mathematics, he pursued a teaching career. He spent most of his life in his hometown Braunschweig, teaching and working on mathematics. Dedekind was bestowed with honorary memberships of the Berlin Academy, the Academy of Rome, to the Göttingen Academy, the Academie des Sciences in Paris and the Leopoldino-Carolina Naturae Curiosorum Academia. Read on and learn more about this genius.
Childhood and Early Life Born on 6 October, 1831 in Braunschweig, Julius Wilhelm Richard Dedekind was the youngest of four children. His father Julius Levin Ulrich Dedekind was an administrator at Collegium Carolinum in his hometown. Dedekind had three elder siblings. He was educated at Martino-Catharineum school in Braunschweig and Collegium Carolinum. During school days, Dedekind did not show any particular interest in Mathematics. He studied chemistry and physics. He was not particularly happy with vague logic of Physics, which pulled him towards Mathematics. In 1850 he entered the University of Göttingen to study under Moritz Stern. Dedekind earned his doctorate degree in 1852 for his thesis ‘Über die Theorie der Eulerschen Integrale’ (On the Theory of Eulerian integrals).
Career Dedekind started his career teaching at a Polytechnic in Zürich in 1858. However, in 1862, he came back to his hometown Braunschweig to teach at Technische Hochschule, previously known as Collegium Carolinum. It was while giving lessons on calculus at Polytechnic that Dedekind came up with the concept of Schnitt or Dedekind cut, which is now considered as a ‘standard definition of the real numbers’. Later, he wrote on Dedekind cuts and irrational numbers and published these works on a pamphlet namely Stetigkeit und irrationale Zahlen (Continuity and irrational numbers). Dedekind held close association with Cantor after they met in 1874. He had great admiration for Cantor’s work on infinite sets. Besides his own works such as Dedekind's theorem, Dedekind edited works of Gauss, Riemann and Dirichlet. Dedekind went on to publish Dirichlet's lectures on number theory titled ‘Vorlesungen über Zahlentheorie’ (Lectures on Number Theory) in German in 1863. His investigation on Dirichlet’s work directed his later analysis on ideals and algebraic number fields. Dedekin also presented works on ideals. Dedekind, along with Heinrich Martin Weber, gave algebraic proof to the Riemann-Roch theorem in 1882. Later in 1888, he presented short essay entitled ‘Was sind und was sollen die Zahlen’ (What are numbers and what should they be?). In 1900 he published works on modular lattices in algebra. He continued working at Technische Hochschule until his retirement in 1894. He continued publishing works and teaching even after the retirement.
Personal Life and Death Dedekind remained unmarried for his lifetime and lived with his sister Julia. He took occasional holidays in the Black Forest in Germany and in Switzerland. In one of such trip, he met Cantor- a fellow mathematician where the two got to work on set theory. He breathed his last on 12 February 1916 in Braunschweig, Germany.
Major Works
Famous as: Mathematician
Nationality: German
Born on: 06 October 1831 AD
Zodiac Sign: Libra Famous Libras
Died on: 12 February 1916 AD
father: Julius Levin Ulrich Dedekind
mother: Caroline Marie Hanriette Emperius
siblings: Julia
Works & Achievements: 'Lectures on Number Theory', 'Continuity and Irrational Numbers' and Modular Lattices
A well known German mathematician, Richard Dedekind made immense contribution in the field of abstract algebra, especially, algebraic number theory, the foundations of the real numbers and ring theory. His most celebrated work was 'Dedekind cuts', which he put forth in the beginning of career. He presented his findings in the book Stetigkeit und Irrationale Zahlen' (Continuity and Irrational Numbers) in 1872. In his book, 'Was sind und was sollen die Zahlen (What are numbers and what should they be), he put forward the definition of an infinite set and analysis of number theory. Along with working on mathematics, he pursued a teaching career. He spent most of his life in his hometown Braunschweig, teaching and working on mathematics. Dedekind was bestowed with honorary memberships of the Berlin Academy, the Academy of Rome, to the Göttingen Academy, the Academie des Sciences in Paris and the Leopoldino-Carolina Naturae Curiosorum Academia. Read on and learn more about this genius.
Childhood and Early Life Born on 6 October, 1831 in Braunschweig, Julius Wilhelm Richard Dedekind was the youngest of four children. His father Julius Levin Ulrich Dedekind was an administrator at Collegium Carolinum in his hometown. Dedekind had three elder siblings. He was educated at Martino-Catharineum school in Braunschweig and Collegium Carolinum. During school days, Dedekind did not show any particular interest in Mathematics. He studied chemistry and physics. He was not particularly happy with vague logic of Physics, which pulled him towards Mathematics. In 1850 he entered the University of Göttingen to study under Moritz Stern. Dedekind earned his doctorate degree in 1852 for his thesis ‘Über die Theorie der Eulerschen Integrale’ (On the Theory of Eulerian integrals).
Career Dedekind started his career teaching at a Polytechnic in Zürich in 1858. However, in 1862, he came back to his hometown Braunschweig to teach at Technische Hochschule, previously known as Collegium Carolinum. It was while giving lessons on calculus at Polytechnic that Dedekind came up with the concept of Schnitt or Dedekind cut, which is now considered as a ‘standard definition of the real numbers’. Later, he wrote on Dedekind cuts and irrational numbers and published these works on a pamphlet namely Stetigkeit und irrationale Zahlen (Continuity and irrational numbers). Dedekind held close association with Cantor after they met in 1874. He had great admiration for Cantor’s work on infinite sets. Besides his own works such as Dedekind's theorem, Dedekind edited works of Gauss, Riemann and Dirichlet. Dedekind went on to publish Dirichlet's lectures on number theory titled ‘Vorlesungen über Zahlentheorie’ (Lectures on Number Theory) in German in 1863. His investigation on Dirichlet’s work directed his later analysis on ideals and algebraic number fields. Dedekin also presented works on ideals. Dedekind, along with Heinrich Martin Weber, gave algebraic proof to the Riemann-Roch theorem in 1882. Later in 1888, he presented short essay entitled ‘Was sind und was sollen die Zahlen’ (What are numbers and what should they be?). In 1900 he published works on modular lattices in algebra. He continued working at Technische Hochschule until his retirement in 1894. He continued publishing works and teaching even after the retirement.
Personal Life and Death Dedekind remained unmarried for his lifetime and lived with his sister Julia. He took occasional holidays in the Black Forest in Germany and in Switzerland. In one of such trip, he met Cantor- a fellow mathematician where the two got to work on set theory. He breathed his last on 12 February 1916 in Braunschweig, Germany.
Major Works
- Lectures on Number Theory
- Continuity and irrational numbers
- What are numbers and what should they be?
- Modular Lattices
Srinivasa Ramanujan
Famous as: Mathematician
Nationality: Indian
Born on: 22 December 1887 AD
Zodiac Sign: Sagittarius Famous Sagitarians
Born in: Erode
Died on: 26 April 1920 AD
place of death: Chetput
father: K. Srinivasa Iyengar
mother: Komalat Ammal
siblings: Sadagopan
Spouse: Janaki Ammal
education: Trinity College, Cambridge (1919–1920), University of Cambridge (1914–1919), University of Cambridge (1916), Government Arts College, Kumbakonam (1904–1906), Town Higher Secondary School (1904), Pachaiyappa's College, University of
Works & Achievements: Ramanujan constant, Ramanujan prime, Ramanujan theta function, Ramanujan's master theorem, Mock theta functions, Ramanujan conjecture, Ramanujan-Soldner constant, Ramanujan's sum.
Rightly regarded as 'natural genius' by the English mathematician G.H. Hardy, Srinivasa Ramanujan displayed an amazing talent in mathematics, even though he did not receive formal training in that subject. He contributed to several areas of mathematics such as the number theory, mathematical analysis, infinite series and continued fractions. This great mathematician of the 20th century added much to the field of advance mathematics with his fascinating theories and proofs, which are in use even today. Also, in 1997, 'The Ramanujan Journal' was published by an American mathematician Bruce .C. Berndt, which showed Ramanujan's areas of study. He formulated many formulas to solve problems, but his untimely death put an end to his great exploration to the unseen beauty and enormity of this subject. Within a short-life, he independently compiled about 3900 results involving identities and equations. Ramanujan used to jot down some of the proofs and theorems in his notebooks that had been studied by many mathematicians, after his death. Scroll further and read more about the profile, life, career and timeline of Srinivasa Ramanujan.
Career Ramanujan was focused to pass the First Arts examination, which would be his ticket to the University of Madras. Hence, he went to Pachaiyappa’s College in Madras in 1906 and put all his efforts in studying and attended all the lectures. Unfortunately, after three months of his dedicated study, he became ill. He appeared for the Fine Arts examination and cleared in mathematics, but failed in all the other subjects. This stopped him from pursuing his dream of getting into the University of Madras. He left college without a degree and pursued independent research in Mathematics. In 1908, he studied fractions and divergent series. His health deteriorated and this time, it became worse and he had to undergo an operation in 1909. It took considerable time for him to recover. Ramanujan spent more time and effort in developing his mathematical ability and solved problems in the Journal of the ‘Indian Mathematical Society’, developing relations between elliptic modular equations. His brilliant work on the Bernoulli numbers in 1911, in the Journal of the Indian mathematical society, grabbed the recognition for all his hard work over the years. Though he did not have a University qualification, he became quite famous in Madras as a mathematical genius. He required means of income and so, he approached the founder of the Indian Mathematical Society in 1911. Hence, he was appointed in a temporary post at the accountant’s General Office in Madras. Afterwards, he also approached Ramachandra Rao, the Collector at Nellore, for a job. In 1912, Ramanujan applied at the Madras Port Trust in the section of accounts for the clerical post. Recommendations from the university mathematicians helped him to get through the selection process. Hence, he joined the office on 1 March 1912. In the office, he was surrounded with great mathematicians who enhanced Ramanujan’s knowledge in the subject.
Ramanujan-Hardy Number (1729) When Ramanujan fell ill, Hardy came to his residence to visit Ramanujan in a cab with a number 1729. That day, he made a comment to Ramanujan saying that the number appeared to be very dull number. Ramanujan corrected him instantly saying that is an interesting number and explained that it is the smallest natural number that could be expressed as the sum of two positive cubes, in two different ways (i.e., 13 + 123 and 93 + 103.)
Personal Life and Death In July, 14, 1909, Ramanujan married a ten year old girl, S. Janaki Ammal. However, he did not live with her until she was twelve years old. He succumbed to tuberculosis (T.B) in 1920 and was eventually admitted in the hospital. All the efforts went in vain and he passed away at the age of 32, on 26 April 1920.(http://www.thefamouspeople.com/profiles/srinivasa-ramanujan-503.php)
Famous as: Mathematician
Nationality: Indian
Born on: 22 December 1887 AD
Zodiac Sign: Sagittarius Famous Sagitarians
Born in: Erode
Died on: 26 April 1920 AD
place of death: Chetput
father: K. Srinivasa Iyengar
mother: Komalat Ammal
siblings: Sadagopan
Spouse: Janaki Ammal
education: Trinity College, Cambridge (1919–1920), University of Cambridge (1914–1919), University of Cambridge (1916), Government Arts College, Kumbakonam (1904–1906), Town Higher Secondary School (1904), Pachaiyappa's College, University of
Works & Achievements: Ramanujan constant, Ramanujan prime, Ramanujan theta function, Ramanujan's master theorem, Mock theta functions, Ramanujan conjecture, Ramanujan-Soldner constant, Ramanujan's sum.
Rightly regarded as 'natural genius' by the English mathematician G.H. Hardy, Srinivasa Ramanujan displayed an amazing talent in mathematics, even though he did not receive formal training in that subject. He contributed to several areas of mathematics such as the number theory, mathematical analysis, infinite series and continued fractions. This great mathematician of the 20th century added much to the field of advance mathematics with his fascinating theories and proofs, which are in use even today. Also, in 1997, 'The Ramanujan Journal' was published by an American mathematician Bruce .C. Berndt, which showed Ramanujan's areas of study. He formulated many formulas to solve problems, but his untimely death put an end to his great exploration to the unseen beauty and enormity of this subject. Within a short-life, he independently compiled about 3900 results involving identities and equations. Ramanujan used to jot down some of the proofs and theorems in his notebooks that had been studied by many mathematicians, after his death. Scroll further and read more about the profile, life, career and timeline of Srinivasa Ramanujan.
Career Ramanujan was focused to pass the First Arts examination, which would be his ticket to the University of Madras. Hence, he went to Pachaiyappa’s College in Madras in 1906 and put all his efforts in studying and attended all the lectures. Unfortunately, after three months of his dedicated study, he became ill. He appeared for the Fine Arts examination and cleared in mathematics, but failed in all the other subjects. This stopped him from pursuing his dream of getting into the University of Madras. He left college without a degree and pursued independent research in Mathematics. In 1908, he studied fractions and divergent series. His health deteriorated and this time, it became worse and he had to undergo an operation in 1909. It took considerable time for him to recover. Ramanujan spent more time and effort in developing his mathematical ability and solved problems in the Journal of the ‘Indian Mathematical Society’, developing relations between elliptic modular equations. His brilliant work on the Bernoulli numbers in 1911, in the Journal of the Indian mathematical society, grabbed the recognition for all his hard work over the years. Though he did not have a University qualification, he became quite famous in Madras as a mathematical genius. He required means of income and so, he approached the founder of the Indian Mathematical Society in 1911. Hence, he was appointed in a temporary post at the accountant’s General Office in Madras. Afterwards, he also approached Ramachandra Rao, the Collector at Nellore, for a job. In 1912, Ramanujan applied at the Madras Port Trust in the section of accounts for the clerical post. Recommendations from the university mathematicians helped him to get through the selection process. Hence, he joined the office on 1 March 1912. In the office, he was surrounded with great mathematicians who enhanced Ramanujan’s knowledge in the subject.
Ramanujan-Hardy Number (1729) When Ramanujan fell ill, Hardy came to his residence to visit Ramanujan in a cab with a number 1729. That day, he made a comment to Ramanujan saying that the number appeared to be very dull number. Ramanujan corrected him instantly saying that is an interesting number and explained that it is the smallest natural number that could be expressed as the sum of two positive cubes, in two different ways (i.e., 13 + 123 and 93 + 103.)
Personal Life and Death In July, 14, 1909, Ramanujan married a ten year old girl, S. Janaki Ammal. However, he did not live with her until she was twelve years old. He succumbed to tuberculosis (T.B) in 1920 and was eventually admitted in the hospital. All the efforts went in vain and he passed away at the age of 32, on 26 April 1920.(http://www.thefamouspeople.com/profiles/srinivasa-ramanujan-503.php)
Kurt Gödel
Famous as: Mathematician, Philosopher
Nationality: American
Born on: 28 April 1906 AD
Zodiac Sign: Taurus Famous Taureans
Born in: Brno
Died on: 14 January 1978 AD
place of death: Princeton
father: Rudolf Gödel
mother: Marianne Gödel
siblings: Rudolf
Spouse: Adele Nimbursky
education: University of Vienna
Works & Achievements: Kurt Gödel, a math genius, was known for his 'incompleteness theorem', which is also famous as the 'Gödel's theorem'. He also gained popularity by proving the fundamental results on axioms in mathematics.
awards: Albert Einstein Award (1951); National Medal of Science (USA) in Mathematical
Statistical
and Computational Sciences (1974)
Kurt Friedrich Gödel was known for his mathematical work in the 20th century, which was the time when most mathematicians were concentrating on the logic and set theory concepts in mathematics. He is also said to have influenced his contemporaries with his scientific and philosophical thinking, which is why he is also considered as a philosopher. His work in modern mathematics left a huge impact on mathematicians even to this day. He dedicated his life to theoretical work in mathematics until 1942, when he got greatly influenced by philosophy. His commitment towards his work was so high that, he detached himself from the world around and hardly took part in any of the social activities. He preferred secluded life and kept in touch only with very few people. He never got into arguments with anyone, and appeared to be highly sensitive when it came to accepting criticism. His extraordinary and brilliant work in mathematics was the result of his dedication and hard work. Read on and learn more about this genius.
Childhood Kurt Gödel was born on April 28, 1906 in Brno, Austria as second son of Viennese textile business man Rudolf Gödel and German-origin Marianne Handschuh. Kurt Gödel’s brother, Rudolf II Gödel (named after his father) was a famous physician of his times, who helped Kurt during the later stages of his life. Little Kurt was always known as “Mr. Why” during his childhood days, owing to his curious nature. Gödel was sent to Evangelische Volksschule from 1912 to 1916 and later he went on to study in Deutsches Staats Realgymnasium from 1916 to 1924, where he excelled with honors in different languages and mathematics. Gödel’s interests in mathematics increased at the age of 14, when his brother left for Vienna to study Medicine. Gödel suffered from rheumatic fever when he was a child, but it is said that he convinced himself that he suffered from a weak heart, after reading a book on medicine. This clearly indicates that he suffered from paranoia and mental instability since he was a kid. Career By 1923, at the age of 18, Gödel gained entry into the University of Vienna, where he chose to study theoretical Physics. Besides studying just physics, Gödel also showed interest in mathematics and philosophy. He attended lectures on number theory by Prof. Phillip Furtwangler, which made him take up mathematics as a career. As a teenager, Gödel had also studied Gabelsberger shorthand, Goethe’s Theory of Colors and the writings of Immanuel Kant. Gödel actively took part in the Vienna Circle, an association of Philosophers, headed by Moritz Schlick. Later, when Gödel gained interested in mathematical logic, he studied “Introduction to Mathematical Philosophy” by Bertrand Russell. Gödel went on to pursue mathematics and logic with Hans Hahn and Karl Menger and completed his doctoral thesis in 1929 under Hans Hahn. After he was awarded doctorate in 1930, he became an unpaid lecturer (Privatdozent) at the University of Vienna. The Vienna Academy of Science published his thesis and some of his other work. Later, Gödel gained entry into the Institute of Advanced Study.
Work and Achievements Kurt Gödel wrote two papers even before he turned 25 which gained him lot of recognition worldwide, out of which it was the ‘Incompleteness Theorem’ that gained immense popularity. This theorem, which is now popularly known as Gödel’s theorem, states that “given any formal system S capable of expressing arithmetic, if S is consistent then there exists a proposition A of arithmetic which is not formally decidable within S i.e., neither A nor its negotiation is provable in S”. He published his incompleteness theorems in Uber Formal unentscheidbare Satze der Principia Mathematica in the year 1931. In 1934, Gödel travelled to Princeton, where he gave a number of lectures on “Undecidable Propositions of Formal Mathematical Systems” at the Institute of Advanced Studies (IAS). Following his lecture, Gödel paid frequent visits to the IAS in 1935 and became very close to Einstein and Morgenstern. Frequent travelling affected his health and hence, he took a break from work and resumed teaching at the University in 1937. Back in Vienna, when Hitler abolished the Privatdozent role, Gödel had to apply for another position at the University of Vienna. However, his application was turned down by the University because of his association with Jewish friends. Later, in 1939, Gödel left Vienna with his wife because of the chaos that had started with the World War II. They travelled to Princeton where he was offered a teaching position at the IAS. At the IAS, Gödel got back on his feet and continued his work in mathematics. He even went on to publish his paper on “Consistency of the Axiom of Choice and of the Generalized Continuum- Hypothesis with the Axioms of Set Theory”. Gödel first received the Albert Einstein Award from Institute of Advanced Studies in the year 1951 which consisted of a gold medal and specific prize money. Later, in 1974, Gödel received the National Medal of Science in the Math and Computer Science discipline from Gerald Ford,the then President of USA, in an award ceremony held at the White House. The award was quoted “For laying the foundation for today’s flourishing study of mathematical logic”. Personal Life Gödel met Adele Nimbursky in 1929, when he was 21. She was six years elder to him and was already divorced. Considering Adele’s history, Gödel’s parents were completely against this relationship. However, despite the disapproval, they did get married in autumn 1938 and spent the summer of 1942 in Blue Hill Inn at Maine. Gödel’s father Rudolf Gödel passed away in 1929, the same year when he submitted the thesis for his doctorate on axioms. Gödel’s mother bought a new villa in Vienna, where she lived with her two sons. This is the time when Gödel developed a love for operas. Gödel was considered Jewish as he had many intelligent Jewish friends and he always moved along with them. He was in fact, attacked by a group of youngsters who mistook him to be Jewish, while he was going for a stroll on the streets of Vienna with his wife Adele. Later Years and Death In 1933, Gödel travelled to USA for a while fearing the increased attacks of the Nazis in Germany. He went into a mental shock especially after his dear friend Moritz Schlick was murdered by a Nazi student. In USA, Gödel met Albert Einstein and they became very good friends. During his stay at USA, Gödel developed an interest in recursive functions and delivered a speech at the annual meeting of the American Mathematical Society. He also engrossed himself in philosophy and physics by reading the works of Gottfried Leibniz during his stay at IAS. Though Gödel went on to become a permanent member of the IAS in 1946, he was rejected the US citizenship under the judgment of Philip Forman. Later, he became so religious that he circulated his own detailed version of Leibniz’s Anselm of Canterbury’s Ontological Proof of God’s Existence.
Gödel suffered from mental instability as paranoia had set in during the later years of his life. He lived in the fear of being poisoned and ate food that was prepared only by his wife. Later, when Adele was hospitalized, Gödel refused to eat and began to starve. This affected his health which eventually led to his death on January 14th, 1978. It is believed that he weighed only 65 pounds when he died.
Legacy
The Kurt Gödel Society was founded in 1987, which is an international organization for the promotion of research in areas like philosophy, mathematics and logic. John Dawson Jr. was so moved by Gödel’s work that he published a biography on Gödel and called it “Logical Dilemmas: Life and Work of Kurt Gödel”, in the year 1997. Also, Douglas Hofstadter came up with a book titled “Gödel, Escher, Bach: An Eternal Golden Braid” in the year 1979. Apart from this, the University of Vienna hosts a ‘Kurt Gödel Research Centre’ for Mathematical Logic where students, today concentrate on set theory, cardinal axioms etc. Even after years of Kurt Gödel passed away, his work in mathematics has laid such a huge impact on the mathematicians today that his legacy continues to live among us.
Famous as: Mathematician, Philosopher
Nationality: American
Born on: 28 April 1906 AD
Zodiac Sign: Taurus Famous Taureans
Born in: Brno
Died on: 14 January 1978 AD
place of death: Princeton
father: Rudolf Gödel
mother: Marianne Gödel
siblings: Rudolf
Spouse: Adele Nimbursky
education: University of Vienna
Works & Achievements: Kurt Gödel, a math genius, was known for his 'incompleteness theorem', which is also famous as the 'Gödel's theorem'. He also gained popularity by proving the fundamental results on axioms in mathematics.
awards: Albert Einstein Award (1951); National Medal of Science (USA) in Mathematical
Statistical
and Computational Sciences (1974)
Kurt Friedrich Gödel was known for his mathematical work in the 20th century, which was the time when most mathematicians were concentrating on the logic and set theory concepts in mathematics. He is also said to have influenced his contemporaries with his scientific and philosophical thinking, which is why he is also considered as a philosopher. His work in modern mathematics left a huge impact on mathematicians even to this day. He dedicated his life to theoretical work in mathematics until 1942, when he got greatly influenced by philosophy. His commitment towards his work was so high that, he detached himself from the world around and hardly took part in any of the social activities. He preferred secluded life and kept in touch only with very few people. He never got into arguments with anyone, and appeared to be highly sensitive when it came to accepting criticism. His extraordinary and brilliant work in mathematics was the result of his dedication and hard work. Read on and learn more about this genius.
Childhood Kurt Gödel was born on April 28, 1906 in Brno, Austria as second son of Viennese textile business man Rudolf Gödel and German-origin Marianne Handschuh. Kurt Gödel’s brother, Rudolf II Gödel (named after his father) was a famous physician of his times, who helped Kurt during the later stages of his life. Little Kurt was always known as “Mr. Why” during his childhood days, owing to his curious nature. Gödel was sent to Evangelische Volksschule from 1912 to 1916 and later he went on to study in Deutsches Staats Realgymnasium from 1916 to 1924, where he excelled with honors in different languages and mathematics. Gödel’s interests in mathematics increased at the age of 14, when his brother left for Vienna to study Medicine. Gödel suffered from rheumatic fever when he was a child, but it is said that he convinced himself that he suffered from a weak heart, after reading a book on medicine. This clearly indicates that he suffered from paranoia and mental instability since he was a kid. Career By 1923, at the age of 18, Gödel gained entry into the University of Vienna, where he chose to study theoretical Physics. Besides studying just physics, Gödel also showed interest in mathematics and philosophy. He attended lectures on number theory by Prof. Phillip Furtwangler, which made him take up mathematics as a career. As a teenager, Gödel had also studied Gabelsberger shorthand, Goethe’s Theory of Colors and the writings of Immanuel Kant. Gödel actively took part in the Vienna Circle, an association of Philosophers, headed by Moritz Schlick. Later, when Gödel gained interested in mathematical logic, he studied “Introduction to Mathematical Philosophy” by Bertrand Russell. Gödel went on to pursue mathematics and logic with Hans Hahn and Karl Menger and completed his doctoral thesis in 1929 under Hans Hahn. After he was awarded doctorate in 1930, he became an unpaid lecturer (Privatdozent) at the University of Vienna. The Vienna Academy of Science published his thesis and some of his other work. Later, Gödel gained entry into the Institute of Advanced Study.
Work and Achievements Kurt Gödel wrote two papers even before he turned 25 which gained him lot of recognition worldwide, out of which it was the ‘Incompleteness Theorem’ that gained immense popularity. This theorem, which is now popularly known as Gödel’s theorem, states that “given any formal system S capable of expressing arithmetic, if S is consistent then there exists a proposition A of arithmetic which is not formally decidable within S i.e., neither A nor its negotiation is provable in S”. He published his incompleteness theorems in Uber Formal unentscheidbare Satze der Principia Mathematica in the year 1931. In 1934, Gödel travelled to Princeton, where he gave a number of lectures on “Undecidable Propositions of Formal Mathematical Systems” at the Institute of Advanced Studies (IAS). Following his lecture, Gödel paid frequent visits to the IAS in 1935 and became very close to Einstein and Morgenstern. Frequent travelling affected his health and hence, he took a break from work and resumed teaching at the University in 1937. Back in Vienna, when Hitler abolished the Privatdozent role, Gödel had to apply for another position at the University of Vienna. However, his application was turned down by the University because of his association with Jewish friends. Later, in 1939, Gödel left Vienna with his wife because of the chaos that had started with the World War II. They travelled to Princeton where he was offered a teaching position at the IAS. At the IAS, Gödel got back on his feet and continued his work in mathematics. He even went on to publish his paper on “Consistency of the Axiom of Choice and of the Generalized Continuum- Hypothesis with the Axioms of Set Theory”. Gödel first received the Albert Einstein Award from Institute of Advanced Studies in the year 1951 which consisted of a gold medal and specific prize money. Later, in 1974, Gödel received the National Medal of Science in the Math and Computer Science discipline from Gerald Ford,the then President of USA, in an award ceremony held at the White House. The award was quoted “For laying the foundation for today’s flourishing study of mathematical logic”. Personal Life Gödel met Adele Nimbursky in 1929, when he was 21. She was six years elder to him and was already divorced. Considering Adele’s history, Gödel’s parents were completely against this relationship. However, despite the disapproval, they did get married in autumn 1938 and spent the summer of 1942 in Blue Hill Inn at Maine. Gödel’s father Rudolf Gödel passed away in 1929, the same year when he submitted the thesis for his doctorate on axioms. Gödel’s mother bought a new villa in Vienna, where she lived with her two sons. This is the time when Gödel developed a love for operas. Gödel was considered Jewish as he had many intelligent Jewish friends and he always moved along with them. He was in fact, attacked by a group of youngsters who mistook him to be Jewish, while he was going for a stroll on the streets of Vienna with his wife Adele. Later Years and Death In 1933, Gödel travelled to USA for a while fearing the increased attacks of the Nazis in Germany. He went into a mental shock especially after his dear friend Moritz Schlick was murdered by a Nazi student. In USA, Gödel met Albert Einstein and they became very good friends. During his stay at USA, Gödel developed an interest in recursive functions and delivered a speech at the annual meeting of the American Mathematical Society. He also engrossed himself in philosophy and physics by reading the works of Gottfried Leibniz during his stay at IAS. Though Gödel went on to become a permanent member of the IAS in 1946, he was rejected the US citizenship under the judgment of Philip Forman. Later, he became so religious that he circulated his own detailed version of Leibniz’s Anselm of Canterbury’s Ontological Proof of God’s Existence.
Gödel suffered from mental instability as paranoia had set in during the later years of his life. He lived in the fear of being poisoned and ate food that was prepared only by his wife. Later, when Adele was hospitalized, Gödel refused to eat and began to starve. This affected his health which eventually led to his death on January 14th, 1978. It is believed that he weighed only 65 pounds when he died.
Legacy
The Kurt Gödel Society was founded in 1987, which is an international organization for the promotion of research in areas like philosophy, mathematics and logic. John Dawson Jr. was so moved by Gödel’s work that he published a biography on Gödel and called it “Logical Dilemmas: Life and Work of Kurt Gödel”, in the year 1997. Also, Douglas Hofstadter came up with a book titled “Gödel, Escher, Bach: An Eternal Golden Braid” in the year 1979. Apart from this, the University of Vienna hosts a ‘Kurt Gödel Research Centre’ for Mathematical Logic where students, today concentrate on set theory, cardinal axioms etc. Even after years of Kurt Gödel passed away, his work in mathematics has laid such a huge impact on the mathematicians today that his legacy continues to live among us.
Isaac (Sir) Newton (1642-1727) England
Newton was an industrious lad who built marvelous toys (e.g. a model windmill powered by a mouse on treadmill). At about age 22, on leave from University, this genius began revolutionary advances in mathematics, optics, dynamics, thermodynamics, acoustics and celestial mechanics. He is famous for his Three Laws of Motion (inertia, force, reciprocal action) but, as Newton himself acknowledged, these Laws weren't fully novel: Hipparchus, Ibn al-Haytham, Galileo and Huygens had all developed much basic mechanics already, and Newton credits the First Law itself to Aristotle. However Newton was also apparently the first person to conclude that the ordinary gravity we observe on Earth is the very same force that keeps the planets in orbit. His Law of Universal Gravitation was revolutionary and due to Newton alone. (Christiaan Huygens, the other great mechanist of the era, had independently deduced that Kepler's laws imply inverse-square gravitation, but he considered the action at a distance in Newton's theory to be "absurd.") Newton's other intellectual interests included chemistry, theology, astrology and alchemy. Although this list is concerned only with mathematics, Newton's greatness is indicated by the wide range of his physics: even without his revolutionary Laws of Motion and his Cooling Law of thermodynamics, he'd be famous just for his work in optics, where he explained diffraction and observed that white light is a mixture of all the rainbow's colors. (Although his corpuscular theory competed with Huygen's wave theory, Newton understood that his theory was incomplete without waves.) Newton also designed the first reflecting telescope, first reflecting microscope, and the sextant.
Although others also developed the techniques independently, Newton is regarded as the Father of Calculus (which he called "fluxions"); he shares credit with Leibniz for the Fundamental Theorem of Calculus (that integration and differentiation are each other's inverse operation). He applied calculus for several purposes: finding areas, tangents, the lengths of curves and the maxima and minima of functions. In addition to several other important advances in analytic geometry, his mathematical works include the Binomial Theorem, his eponymous numeric method, the idea of polar coordinates, and power series for exponential and trigonometric functions. (His equation ex = ∑ xk / k! has been called the "most important series in mathematics.") He contributed to algebra and the theory of equations; he was first to state Bézout's Theorem; he generalized Déscartes' rule of signs. (The generalized rule of signs was incomplete and finally resolved two centuries later by Sturm and Sylvester.) He developed a series for the arcsin function. He developed facts about cubic equations (just as the "shadows of a cone" yield all quadratic curves, Newton found a curve whose "shadows" yield all cubic curves). He proved that same-mass spheres of any radius have equal gravitational attraction: this fact is key to celestial motions. He discovered Puiseux series almost two centuries before they were re-invented by Puiseux. (Like some of the greatest ancient mathematicians, Newton took the time to compute an approximation to π; his was better than Vieta's, though still not as accurate as al-Kashi's.)
Newton is so famous for his calculus, optics and laws of motion, it is easy to overlook that he was also one of the greatest geometers. He solved the Delian cube-doubling problem; he solved the Problem of Pappus. Even before the invention of the calculus of variations, Newton was doing difficult work in that field, e.g. his calculation of the "optimal bullet shape." Among many marvelous theorems, he proved several about quadrilaterals and their in- or circum-scribing ellipses, and constructed the parabola defined by four given points. He anticipated Poncelet's Principle of Continuity. An anecdote often cited to demonstrate his brilliance is the problem of the brachistochrone, which had baffled the best mathematicians in Europe, and came to Newton's attention late in life. He solved it in a few hours and published the answer anonymously. But on seeing the solution Jacob Bernoulli immediately exclaimed "I recognize the lion by his footprint."
In 1687 Newton published Philosophiae Naturalis Principia Mathematica, surely the greatest scientific book ever written. The motion of the planets was not understood before Newton, although the heliocentric system allowed Kepler to describe the orbits. In Principia Newton analyzed the consequences of his Laws of Motion and introduced the Law of Universal Gravitation. With the key mystery of celestial motions finally resolved, the Great Scientific Revolution began. (In his work Newton also proved important theorems about inverse-cube forces, work largely unappreciated until Chandrasekhar's modern-day work.) Newton once wrote "Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things." Sir Isaac Newton was buried at Westminster Abbey in a tomb inscribed "Let mortals rejoice that so great an ornament to the human race has existed."
Newton ranks #2 on Michael Hart's famous list of the Most Influential Persons in History. (Muhammed the Prophet of Allah is #1.) Whatever the criteria, Newton would certainly rank first or second on any list of physicists, or scientists in general, but some listmakers would demote him slightly on a list of pure mathematicians: his emphasis was physics not mathematics, and the contribution of Leibniz (Newton's rival for the title Inventor of Calculus) lessens the historical importance of Newton's calculus. One reason I've ranked him at #1 is a comment by Gottfried Leibniz himself: "Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part."
(http://fabpedigree.com/james/mathmen.htm#Newton)
Newton was an industrious lad who built marvelous toys (e.g. a model windmill powered by a mouse on treadmill). At about age 22, on leave from University, this genius began revolutionary advances in mathematics, optics, dynamics, thermodynamics, acoustics and celestial mechanics. He is famous for his Three Laws of Motion (inertia, force, reciprocal action) but, as Newton himself acknowledged, these Laws weren't fully novel: Hipparchus, Ibn al-Haytham, Galileo and Huygens had all developed much basic mechanics already, and Newton credits the First Law itself to Aristotle. However Newton was also apparently the first person to conclude that the ordinary gravity we observe on Earth is the very same force that keeps the planets in orbit. His Law of Universal Gravitation was revolutionary and due to Newton alone. (Christiaan Huygens, the other great mechanist of the era, had independently deduced that Kepler's laws imply inverse-square gravitation, but he considered the action at a distance in Newton's theory to be "absurd.") Newton's other intellectual interests included chemistry, theology, astrology and alchemy. Although this list is concerned only with mathematics, Newton's greatness is indicated by the wide range of his physics: even without his revolutionary Laws of Motion and his Cooling Law of thermodynamics, he'd be famous just for his work in optics, where he explained diffraction and observed that white light is a mixture of all the rainbow's colors. (Although his corpuscular theory competed with Huygen's wave theory, Newton understood that his theory was incomplete without waves.) Newton also designed the first reflecting telescope, first reflecting microscope, and the sextant.
Although others also developed the techniques independently, Newton is regarded as the Father of Calculus (which he called "fluxions"); he shares credit with Leibniz for the Fundamental Theorem of Calculus (that integration and differentiation are each other's inverse operation). He applied calculus for several purposes: finding areas, tangents, the lengths of curves and the maxima and minima of functions. In addition to several other important advances in analytic geometry, his mathematical works include the Binomial Theorem, his eponymous numeric method, the idea of polar coordinates, and power series for exponential and trigonometric functions. (His equation ex = ∑ xk / k! has been called the "most important series in mathematics.") He contributed to algebra and the theory of equations; he was first to state Bézout's Theorem; he generalized Déscartes' rule of signs. (The generalized rule of signs was incomplete and finally resolved two centuries later by Sturm and Sylvester.) He developed a series for the arcsin function. He developed facts about cubic equations (just as the "shadows of a cone" yield all quadratic curves, Newton found a curve whose "shadows" yield all cubic curves). He proved that same-mass spheres of any radius have equal gravitational attraction: this fact is key to celestial motions. He discovered Puiseux series almost two centuries before they were re-invented by Puiseux. (Like some of the greatest ancient mathematicians, Newton took the time to compute an approximation to π; his was better than Vieta's, though still not as accurate as al-Kashi's.)
Newton is so famous for his calculus, optics and laws of motion, it is easy to overlook that he was also one of the greatest geometers. He solved the Delian cube-doubling problem; he solved the Problem of Pappus. Even before the invention of the calculus of variations, Newton was doing difficult work in that field, e.g. his calculation of the "optimal bullet shape." Among many marvelous theorems, he proved several about quadrilaterals and their in- or circum-scribing ellipses, and constructed the parabola defined by four given points. He anticipated Poncelet's Principle of Continuity. An anecdote often cited to demonstrate his brilliance is the problem of the brachistochrone, which had baffled the best mathematicians in Europe, and came to Newton's attention late in life. He solved it in a few hours and published the answer anonymously. But on seeing the solution Jacob Bernoulli immediately exclaimed "I recognize the lion by his footprint."
In 1687 Newton published Philosophiae Naturalis Principia Mathematica, surely the greatest scientific book ever written. The motion of the planets was not understood before Newton, although the heliocentric system allowed Kepler to describe the orbits. In Principia Newton analyzed the consequences of his Laws of Motion and introduced the Law of Universal Gravitation. With the key mystery of celestial motions finally resolved, the Great Scientific Revolution began. (In his work Newton also proved important theorems about inverse-cube forces, work largely unappreciated until Chandrasekhar's modern-day work.) Newton once wrote "Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things." Sir Isaac Newton was buried at Westminster Abbey in a tomb inscribed "Let mortals rejoice that so great an ornament to the human race has existed."
Newton ranks #2 on Michael Hart's famous list of the Most Influential Persons in History. (Muhammed the Prophet of Allah is #1.) Whatever the criteria, Newton would certainly rank first or second on any list of physicists, or scientists in general, but some listmakers would demote him slightly on a list of pure mathematicians: his emphasis was physics not mathematics, and the contribution of Leibniz (Newton's rival for the title Inventor of Calculus) lessens the historical importance of Newton's calculus. One reason I've ranked him at #1 is a comment by Gottfried Leibniz himself: "Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part."
(http://fabpedigree.com/james/mathmen.htm#Newton)
Georg Friedrich Bernhard Riemann (1826-1866) Germany
Riemann was a phenomenal genius whose work was exceptionally deep, creative and rigorous; he made revolutionary contributions in many areas of pure mathematics, and also inspired the development of physics. He had poor physical health and died at an early age, yet is still considered to be among the most productive mathematicians ever. He was the master of complex analysis, which he connected to both topology and number theory. He applied topology to analysis, and analysis to number theory, making revolutionary contributions to all three fields. He took non-Euclidean geometry far beyond his predecessors. He introduced the Riemann integral which clarified analysis. Riemann's other masterpieces include differential geometry, tensor analysis, the theory of functions, and, especially, the theory of manifolds. He generalized the notions of distance and curvature and, therefore, described new possibilities for the geometry of space itself. Several important theorems and concepts are named after Riemann, e.g. the Riemann-Roch Theorem, a key connection among topology, complex analysis and algebraic geometry. He was so prolific and original that some of his work went unnoticed (for example, Weierstrass became famous for showing a nowhere-differentiable continuous function; later it was found that Riemann had casually mentioned one in a lecture years earlier). Like his mathematical peers (Gauss, Archimedes, Newton), Riemann was intensely interested in physics. His theory unifying electricity, magnetism and light was supplanted by Maxwell's theory; however modern physics, beginning with Einstein's relativity, relies on Riemann's notions of the geometry of space.Riemann's teacher was Carl Gauss, who helped steer the young genius towards pure mathematics. Gauss selected "On the hypotheses that Lie at the Foundations of Geometry" as Riemann's first lecture; with this famous lecture Riemann advanced Gauss' initial effort in differential geometry, extended it to multiple dimensions, and introduced the new and important theory of differential manifolds. Five years later, to celebrate his election to the Berlin Academy, Riemann presented a lecture "On the Number of Prime Numbers Less Than a Given Quantity," for which "Number" he presented and proved an
exact formula, albeit weirdly complicated and seemingly intractable. Numerous papers have been written on the distribution of primes, but Riemann's contribution is incomparable, despite that his Berlin Academy lecture was his only paper ever on the topic, and number theory was far from his specialty. In the lecture he posed the "Hypothesis of Riemann's zeta function" which is now considered the most important and famous unsolved problem in mathematics. (Asked what he would first do, if he were magically awakened after centuries, David Hilbert replied "I would ask whether anyone had proved the Riemann Hypothesis.") ζ() was defined for convergent cases in Euler's mini-bio, which Riemann extended via analytic continuation for all cases. The Riemann Hypothesis "simply" states that in all solutions of ζ(s = a+bi) = 0, either s has real part a=1/2 or imaginary partb=0.
Despite his great creativity (Gauss praised Riemann's "gloriously fertile originality"), Riemann once said: "If only I had the theorems! Then I should find the proofs easily enough."
(http://fabpedigree.com/james/mathmen.htm#Riemann)
Riemann was a phenomenal genius whose work was exceptionally deep, creative and rigorous; he made revolutionary contributions in many areas of pure mathematics, and also inspired the development of physics. He had poor physical health and died at an early age, yet is still considered to be among the most productive mathematicians ever. He was the master of complex analysis, which he connected to both topology and number theory. He applied topology to analysis, and analysis to number theory, making revolutionary contributions to all three fields. He took non-Euclidean geometry far beyond his predecessors. He introduced the Riemann integral which clarified analysis. Riemann's other masterpieces include differential geometry, tensor analysis, the theory of functions, and, especially, the theory of manifolds. He generalized the notions of distance and curvature and, therefore, described new possibilities for the geometry of space itself. Several important theorems and concepts are named after Riemann, e.g. the Riemann-Roch Theorem, a key connection among topology, complex analysis and algebraic geometry. He was so prolific and original that some of his work went unnoticed (for example, Weierstrass became famous for showing a nowhere-differentiable continuous function; later it was found that Riemann had casually mentioned one in a lecture years earlier). Like his mathematical peers (Gauss, Archimedes, Newton), Riemann was intensely interested in physics. His theory unifying electricity, magnetism and light was supplanted by Maxwell's theory; however modern physics, beginning with Einstein's relativity, relies on Riemann's notions of the geometry of space.Riemann's teacher was Carl Gauss, who helped steer the young genius towards pure mathematics. Gauss selected "On the hypotheses that Lie at the Foundations of Geometry" as Riemann's first lecture; with this famous lecture Riemann advanced Gauss' initial effort in differential geometry, extended it to multiple dimensions, and introduced the new and important theory of differential manifolds. Five years later, to celebrate his election to the Berlin Academy, Riemann presented a lecture "On the Number of Prime Numbers Less Than a Given Quantity," for which "Number" he presented and proved an
exact formula, albeit weirdly complicated and seemingly intractable. Numerous papers have been written on the distribution of primes, but Riemann's contribution is incomparable, despite that his Berlin Academy lecture was his only paper ever on the topic, and number theory was far from his specialty. In the lecture he posed the "Hypothesis of Riemann's zeta function" which is now considered the most important and famous unsolved problem in mathematics. (Asked what he would first do, if he were magically awakened after centuries, David Hilbert replied "I would ask whether anyone had proved the Riemann Hypothesis.") ζ() was defined for convergent cases in Euler's mini-bio, which Riemann extended via analytic continuation for all cases. The Riemann Hypothesis "simply" states that in all solutions of ζ(s = a+bi) = 0, either s has real part a=1/2 or imaginary partb=0.
Despite his great creativity (Gauss praised Riemann's "gloriously fertile originality"), Riemann once said: "If only I had the theorems! Then I should find the proofs easily enough."
(http://fabpedigree.com/james/mathmen.htm#Riemann)
Bonaventura Cavalieri (1598 - 1647)From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.
Almost contemporaneously with the publication in 1637 of Descartes' geometry, the principles of the integral calculus, so far as they are concerned with summation, were being worked out in Italy. This was effected by what was called the principle of indivisibles, and was the invention of Cavalieri. It was applied by him and his contemporaries to numerous problems connected with the quadrature of curves and surfaces, the determination on volumes, and the positions of centres of mass. It served the same purpose as the tedious method of exhaustions used by the Greeks; in principle the methods are the same, but the notation of indivisibles is more concise and convenient. It was, in its turn, superceded at the beginning of the eighteenth century by the integral calculus.
Bonaventura Cavalieri was born at Milan in 1598, and died at Bologna on November 27, 1647. He became a Jesuit at an early age; on the recommendation of the Order he was in 1629 made professor of mathematics at Bologna; and he continued to occupy the chair there until his death. I have already mentioned Cavalieri's name in connection with the introduction of the use of logarithms into Italy, and have alluded to his discovery of the expression for the area of a spherical triangle in terms of the spherical excess. He was one of the most influential mathematicians of his time, but his subsequent reputation rests mainly on his invention of the principle of indivisibles.
The principle of indivisibles had been used by Kepler in 1604 and 1615 in a somewhat crude form. It was first stated by Cavalieri in 1629, but he did not publish his results till 1635. In his early enunciation of the principle in 1635 Cavalieri asserted that a line was made up of an infinite number of points (each without magnitude), a surface of infinite number of lines (each without breadth), and a volume of an infinite number of surfaces (each without thickness). To meet the objections of Guldinus and others, the statement was recast, and in its final form as used by the mathematicians of the seventeenth century it was published in Cavalieri's Exercitationes Geometricae in 1647; the third exercise is devoted to a defence of the theory. This book contains the earliest demonstration of the properties of Pappus. Cavalieri's works on indivisibles were reissued with his later corrections in 1653.
The method of indivisibles rests, in effect, on the assumption that any magnitude may be divided into an infinite number of small quantities which can be made to bear any required ratios (ex. gr. equality) one to the other. The analysis given by Cavalieri is hardly worth quoting except as being one of the first steps taken towards the formation of an infinitesimal calculus. One example will suffice. Suppose it be required to find the area of a right-angled triangle. Let the base be made up of, or contain n points (or indivisibles), and similarly let the other side contain na points, then the ordinates at the successive points of the base will contain a, 2a ..., napoints. Therefore the number of points in the area is a + 2a + ... + na; the sum of which is ½ n²a + ½na. Since n is very large, we may neglect ½na for it is inconsiderable compared with ½n²a. Hence the area is equal to ½(na)n, that is, ½ × altitude × base. There is no difficulty in criticizing such a proof, but, although the form in which it is presented is indefensible, the substance of it is correct.
It would be misleading to give the above as the only specimen of the method of indivisibles, and I therefore quote another example, taken from a later writer, which will fairly illustrate the use of the method when modified and corrected by the method of limits.
Let it be required to find the area outside a parabola APC and bounded by the curve, the tangent at A, and a line DC parallel to AB the diameter at A. Complete the parallelogram ABCD. Divide AD into n equal parts, let AM contain r of them, and let MN be the (r + 1)th part. Draw MP and NQ parallel to AB, and draw PRparallel to AD. Then when n becomes indefinitely large, the curvilinear area APCD will be the the limit of the sum of all parallelograms like PN. Nowarea PN : area BD = MP.MN : DC.AD.But by the properties of the parabolaMP : DC = AM² : AD² = r² : n²,and MN : AD = 1 : n. Hence MP.MN : DC.AD = r² : n³. Therefore area PN : area BD = r² : n³. Therefore, ultimately,area APCD : area BD= 1² + 2² + ... + (n-1)² : n³= n (n-1)(2n-1) : n³which, in the limit, = 1 : 3.
It is perhaps worth noticing that Cavalieri and his successors always used the method to find the ratio of two areas, volumes, or magnitudes of the same kind and dimensions, that is, they never thought of an area as containing so many units of area. The idea of comparing a magnitude with a unit of the same kind seems to have been due to Wallis.
It is evident that in its direct form the method is applicable to only a few curves. Cavalieri proved that, if m be a positive integer, then the limit, when n is infinite, of is 1/(m+1), which is equivalent to saying that he found the integral of x to from x = 0 to x = 1; he also discussed the quadrature of the hyperbola.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).
Transcribed by
D.R. Wilkins
([email protected])
School of Mathematics (
http://www.maths.tcd.ie/pub/HistMath/People/Cavalieri/RouseBall/RB_Cavalieri.html
William, Viscount Brouncker (c.1620 - 1684)
From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.
William, Viscount Brouncker, one of the founders of the Royal Society of London, born about 1620, and died on April 5, 1684, was among the most brilliant mathematicians of this time, and was in intimate relations with Wallis, Fermat, and other leading mathematicians. I mentioned above his curious reproduction of Brahmagupta's solution of a certain indeterminate equation. Brouncker proved that the area enclosed between the equilateral hyperbola xy = 1, the axis of x, and the ordinates x = 1 and x = 2, is equal either to
or toHe also worked out other similar expressions for different areas bounded by the hyperbola and straight lines. He wrote on the rectification of the parabola and of the cycloid. It is noticeable that he used infinite series to express quantities whose values he could not otherwise determine. In answer to a request of Wallis to attempt the quadrature of the circle he shewed that the ratio of the area of a circle to the area of the circumscribed square, that is, the ratio of to 4, is equal to the ratio ofto 1. Continued fractions had been employed by Bombelli in 1572, and had been systematically used by Cataldi in his treatise on finding the square roots of numbers, published at Bologna in 1613. Their properties and theory were given by Huygens, 1703 and Euler, 1744.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).
Transcribed by
D.R. Wilkins
([email protected])
School of Mathematics
Trinity College, Dublin
(http://www.maths.tcd.ie/pub/HistMath/People/Brouncker/RouseBall/RB_Brouncker.html)
Christian Huygens (1629 - 1695)
From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.
Christian Huygens was born at the Hague on April 14, 1629, and died in the same town on July 8, 1695. He generally wrote his name as Hugens, but I follow the usual custom in spelling it as above: it is also sometimes written as Huyghens. His life was uneventful, and there is little more to record in it than a statement of his various memoirs and researches.
In 1651 he published an essay in which he shewed the fallacy in a system of quadratures proposed by Grégoire de Saint-Vincent, who was well versed in the geometry of the Greeks, but had not grasped the essential points in the more modern methods. This essay was followed by tracts on the quadrature of the conics and the approximate rectification of the circle.
In 1654 his attention was directed to the improvement of the telescope. In conjunction with his brother he devised a new and better way of grinding and polishing lenses. As a result of these improvements he was able during the following two years, 1655 and 1656, to resolve numerous astronomical questions; as, for example, the nature of Saturn's appendage. His astronomical observations required some exact means of measuring time, and he was thus led in 1656 to invent the pendulum clock, as described in his tract Horologium, 1658. The time-pieces previously in use had been balance-clocks.
In the year 1657 Huygens wrote a small work on the calculus of probabilities founded on the correspondence of Pascal and Fermat. He spent a couple of years in England about this time. His reputation was now so great that in 1665 Louis XIV. offered him a pension if he would live in Paris, which accordingly then became his place of residence.
In 1668 he sent to the Royal Society of London, in answer to a problem they had proposed, a memoir in which (simultaneously with Wallis and Wren) he proved by experiment that the momentum in a certain direction before the collision of two bodies is equal to the momentum in that direction after the collision. This was one of the points in mechanics on which Descartes had been mistaken.
The most important of Huygens's work was his Horologium Oscillatorium published at Paris in 1673. The first chapter is devoted to pendulum clocks. The second chapter contains a complete account of the descent of heavy bodies under their own weights in a vacuum, either vertically down or on smooth curves. Amongst other propositions he shews that the cycloid is tautochronous. In the third chapter he defines evolutes and involutes, proves some of their more elementary properties, and illustrates his methods by finding the evolutes of the cycloid and the parabola. These are the earliest instances in which the envelope of a moving line was determined. In the fourth chapter he solves the problem of the compound pendulum, and shews that the centres of oscillation and suspension are interchangeable. In the fifth and last chapter he discusses again the theory of clocks, points out that if the bob of the pendulum were, by means of cycloidal clocks, made to oscillate in a cycloid the oscillations would be isochronous; and finishes by shewing that the centrifugal force on a body which moves around a circle of radius r with a uniform velocity vvaries directly as v² and inversely as r. This work contains the first attempt to apply dynamics to bodies of finite size, and not merely to particles.
In 1675 Huygens proposed to regulate the motion of watches by the use of the balance spring, in the theory of which he had been perhaps anticipated in a somewhat ambiguous and incomplete statement made by Hooke in 1658. Watches or portable clocks had been invented early in the sixteenth century, and by the end of that century were not very uncommon, but they were clumsy and unreliable, being driven by a main spring and regulated by a conical pulley and verge escapement; moreover, until 1687 they had only one hand. The first watch whose motion was regulated by a balance spring was made at Paris under Huygens's directions, and presented by him to Louis XIV.
The increasing intolerance of the Catholics led to his return to Holland in 1681, and after the revocation of the edict of Nantes he refused to hold any further communication with France. He now devoted himself to the construction of lenses of enormous focal length: of these three of focal lengths 123 feet, 180 feet, and 210 feet, were subsequently given by him to the Royal Society of London, in whose possession they still remain. It was about this time that he discovered the achromatic eye-piece (for a telescope) which is known by his name. In 1689 he came from Holland to England in order to make the acquaintance of Newton, whosePrincipia had been published in 1687. Huygens fully recognized the intellectual merits of the work, but seems to have deemed any theory incomplete which did not explain gravitation by mechanical means.
On his return in 1690 Huygens published his treatise on light in which the undulatory theory was expounded and explained. Most of this had been written as early as 1678. The general idea of the theory had been suggested by Robert Hooke in 1664, but he had not investigated its consequences in any detail. Only three ways have been suggested in which light can be produced mechanically. Either the eye may be supposed to send out something which, so to speak, feels the object (as the Greeks believed); or the object perceived may send out something which hits or affects the eye (as assumed in the emission theory); or there may be some medium between the eye and the object, and the object may cause some change in the form or condition of this intervening medium and thus affect the eye (as Hooke and Huygens supposed in the wave or undulatory theory). According to this last theory space is filled with an extremely rare ether, and light is caused by a series of waves or vibrations in this ether which are set in motion by the pulsations of the luminous body. From this hypothesis Huygens deduced the laws of reflexion and refraction, explained the phenomenon of double refraction, and gave a construction for the extraordinary ray in biaxal crystals; while he found by experiment the chief phenomena of polarization.
The immense reputation and unrivalled powers of Newton led to disbelief in a theory which he rejected, and to the general adoption of Newton's emission theory. Within the present century crucial experiments have been devised which give different results according as one or the other theory is adopted; all these experiments agree with the results of the undulatory theory and differ from the results of the Newtonian theory; the latter is therefore untenable. Until, however, the theory of interference, suggested by Young, was worked out by Fresnel, the hypothesis of Huygens failed to account for all the facts, and even now the properties which, under it, have to be attributed to the intervening medium or ether involve difficulties of which we still seek a solution. Hence the problem as to how the effects of light are really produced cannot be said to be finally solved.
Besides these works Huygens took part in most of the controversies and challenges which then played so large a part in the mathematical world, and wrote several minor tracts. In one of these he investigated the form and properties of the catenary. In another he stated in general terms the rule for finding maxima and minima of which Fermat had made use, and shewed that the subtangent of an algebraical curve f(x,y) = 0 was equal to , where is the derived function of f(x,y) regarded as a function of y. In some posthumous works, issued at Leyden in 1703, he further shewed how from the focal lengths of the component lenses the magnifying power of a telescope could be determined; and explained some of the phenomena connected with haloes and parhelia.
I should add that almost all his demonstrations, like those of Newton, are rigidly geometrical, and he would seem to have made no use of the differential or fluxional calculus, though he admitted the validity of the methods used therein. Thus, even when first written, his works were expressed in an archaic language, and perhaps received less attention than their intrinsic merits deserved.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).
Transcribed by
D.R. Wilkins
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School of Mathematics
Trinity College, Dublin
(http://www.maths.tcd.ie/pub/HistMath/People/Huygens/RouseBall/RB_Huygens.html)
Isaac Barrow (1630 - 1677)
From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.
Isaac Barrow was born in London in 1630, and died at Cambridge in 1677. He went to school first at Charterhouse (where he was so troublesome that his father was heard to pray that if it pleased God to take any of his children he could best spare Isaac), and subsequently to Felstead. He completed his education at Trinity College, Cambridge; after taking his degree in 1648, he was elected to a fellowship in 1649; he then resided for a few years in college, but in 1655 he was driven out by the persecution of the Independents. He spent the next four years in the East of Europe, and after many adventures returned to England in 1659. He was ordained the next year, and appointed to the professorship of Greek at Cambridge. In 1662 he was made professor of geometry at Gresham College, and in 1663 was selected as the first occupier of the Lucasian chair at Cambridge. He resigned the latter to his pupil Newton in 1669, whose superior abilities he recognized and frankly acknowledged. For the remainder of his life he devoted himself to the study of divinity. He was appointed master of Trinity College in 1672, and held the post until his death.
He is described as ``low in stature, lean, and of a pale complexion,'' slovenly in his dress, and an inveterate smoker. He was noted for his strength and courage, and once when travelling in the East he saved the ship by his own prowess from capture by pirates. A ready and caustic wit made him a favourite of Charles II., and induced the courtiers to respect even if they did not appreciate him. He wrote with a sustained and somewhat stately eloquence, and with his blameless life and scrupulous conscientiousness was an impressive personage of the time.
His earliest work was a complete edition of the Elements of Euclid, which he issued in Latin in 1655, and in English in 1660; in 1657 he published an edition of theData. His lectures, delivered in 1664, 1665, and 1666, were published in 1683 under the title Lectiones Mathematicae; these are mostly on the metaphysical basis for mathematical truths. His lectures for 1667 were published in the same year, and suggest the analysis by which Archimedes was led to his chief results. In 1669 he issued his Lectiones Opticae et Geometricae. It is said in the preface that Newton revised and corrected these lectures, adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. This, which is his most important work in mathematics, was republished with a few minor alterations in 1674. In 1675 he published an edition with numerous comments of the first four books of theConics of Apollonius, and of the extant works of Archimedes and Theodosius.
In the optical lectures many problems connected with the reflexion and refraction of light are treated with ingenuity. The geometrical focus of a point seen by reflexion or refraction is defined; and it is explained that the image of an object is the locus of the geometrical foci of every point on it. Barrow also worked out a few of the easier properties of thin lenses, and considerably simplified the Cartesian explanation of the rainbow.
The geometrical lectures contain some new ways of determining the areas and tangents of curves. The most celebrated of these is the method given for the determination of tangents to curves, and this is sufficiently important to require a detailed notice, because it illustrates the way in which Barrow, Hudde and Sluze were working on the lines suggested by Fermat towards the methods of the differential calculus.
Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it were known; hence, if the length of the subtangentMT could be found (thus determining the point T), then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a pointQ adjacent to P were drawn, he got a small triangle PQR (which he called the differential triangle, because its sides PR and PQ were the differences of the abscissae and ordinates of P and Q), so thatTM : MP = QR : RP.To find QR : RP he supposed that x, y were the co-ordinates of P, and x - e, y - a those of Q (Barrow actually used p for x and m for y, but I alter these to agree with modern practice). Substituting the co-ordinates of Q in the equation of the curve, and neglecting the squares and higher powers of e and a as compared with their first powers, he obtained e : a. The ratio a/e was subsequently (in accordance with a suggestion made by Sluze) termed the angular coefficient of the tangent at the point.
Barrow applied this method to the curves (i) x² (x² + y²) = r²y²; (ii) x³ + y³ = r³; (iii) x³ + y³ = rxy, called la galande; (iv) y = (r - x) tan x/2r, the quadratrix; and (v) y = r tan x/2r. It will be sufficient here if I take as an illustration the simpler case of the parabola y² = px. Using the notation given above, we have for the pointP, y² = px; and for the point Q, (y - a)² = p(x - e). Subtracting we get 2ay - a² = pe. But, if a be an infinitesimal quantity, a² must be infinitely smaller and therefore may be neglected when compared with the quantities 2ay and pe. Hence 2ay = pe, that is, e : a = 2y : p. Therefore TP : y = e : a = 2y : p. Hence TM = 2y²/p = 2x.This is exactly the procedure of the differential calculus, except that there we have a rule by which we can get the ratio a/e or dy/dx directly without the labour of going through a calculation similar to the above for every separate case.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).
Transcribed by
D.R. Wilkins
([email protected])
School of Mathematics
Trinity College, Dublin
James Gregory (1638 - 1675)
From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.
James Gregory, born at Drumoak near Aberdeen in 1638, and died at Edinburgh in October 1675, was successively professor at St. Andrews and Edinburgh. In 1660 he published his Optica Promota, in which the reflecting telescope known by his name is described. In 1667 he issued his Vera Circuli et Hyperbolae Quadratura, in which he shewed how the areas of the circle and hyperbola could be obtained in the form of infinite convergent series, and here (I believe for the first time) we find a distinction drawn between convergent and divergent series. This work contains a remarkable geometrical proposition to the effect that the ratio of the area of any arbitrary sector of a circle to that of the inscribed or circumscribed regular polygons is not expressible by a finite number of terms. Hence he inferred that the quadrature of the circle was impossible; this was accepted by Montucla, but it is not conclusive, for it is conceivable that some particular sector might be squared, and this particular sector might be the whole circle. This book contains also the earliest enunciation of the expansions in series of sin x, cos x, x or arc sin x, and x or arc cos x. It was reprinted in 1668 with an appendix, Geometriae Pars, in which Gregory explained how the volumes of solids of revolution could be determined. In 1671, or perhaps earlier, he established the theorem that
the result being true only if lie between - and . This is the theorem on which many of the subsequent calculations of approximations to the numerical value of have been based.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).
Transcribed by
D.R. Wilkins
([email protected])
School of Mathematics
Trinity College, Dublin
(http://www.maths.tcd.ie/pub/HistMath/People/Gregory/RouseBall/RB_JGregory.html)
Some Contemporaries of Descartes, Fermat, Pascal and Huygens
Bachet
Claude Gaspard Bachet de Méziriac was born at Bourg in 1581, and died in 1638. He wrote the Problèmes plaisants, of which the first edition was issued in 1612, a second and enlarged edition was brought out in 1624; this contains an interesting collection of arithmetical tricks and questions, many of which are quoted in my Mathematical Recreations and Essays. He also wrote Les éléments arithmétiques, which exists in manuscript; and a translation of the Arithmetic of Diophantus. Bachet was the earliest writer who discussed the solution of indeterminate equations by means of continued fractions.
Mersenne
Marin Mersenne, born in 1588 and died at Paris in 1648, was a Franciscan friar, who made it his business to be acquainted and correspond with the French mathematicians of that date and many of their foreign contemporaries. In 1634 he published a translation of Galileo's mechanics; in 1644 he issued his Cogita Physico-Mathematica, by which he is best known, containing an account of some experiments in physics; he also wrote a synopsis of mathematics, which was printed in 1664.
The preface to the Cogitata contains a statement (possibly due to Fermat) that, in order that may be prime, the only values of p, not greater than 257, which are possible are 1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257; the number 67 is probably a misprint for 61. With this correction the statement appears to be true, and it has been verified for all except twenty-one values of p, namely 71, 89, 101, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 181, 193, 199, 227, 229, 241, and 257. Of these values, Mersenne asserted that p = 127 and p = 257 make prime, and that the other nineteen values make a composite number. It has been asserted that the statement has been verified when p = 89 and 127, but these verifications rest on long numerical calculations made by single computators and not published; until these demonstrations have been confirmed we may say that twenty-one cases still await verification or require further investigation. The factors of when p = 89 are not known, the calculation merely showing that the number could not be prime. It is most likely that these results are particular cases of some general theorem on the subject which remains to be discovered.
The theory of perfect numbers depends directly on that of Mersenne's numbers. It is probable that all perfect numbers are included in the formula , where is a prime. Euclid proved that any number of this form is perfect. Euler shewed that the formula includes all even perfect numbers; and there is reason to believe - though a rigid demonstration is wanting - that an odd number cannot be perfect. If we assume that the last of these statements is true, then every perfect number is of the above form. Thus if p = 2, 3, 5, 7, 13, 17, 19, 31, 61, then, by Mersenne's rule, the corresponding values of are prime; they are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951; and the corresponding perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, and 2658455991569831744654692615953842176.
Roberval
Gilles Personier (de) Roberval, born at Roberval in 1602 and died at Paris in 1675, described himself from the place of his birth as de Roberval, a seigniorial title to which he had no right. He discussed the nature of the tangents to curves, solved some of the easier questions connected with the cycloid, generalized Archimedes's theorems on the spiral, wrote on mechanics, and on the method of indivisibles, which he rendered more precise and logical. He was a professor in the university of Paris, and in correspondence with nearly all the leading mathematicians of his time.
Van Schooten
Frans van Schooten, to whom we owe an edition of Vieta's works, succeeded his father (who had taught mathematics to Huygens, Hudde, and Sluze) as professor at Leyden in 1646. He brought out in 1659 a Latin translation of Descartes's Géométrie, and in 1657 a collection of mathematical exercises in which he recommended the use of co-ordinates in space of three dimensions. He died in 1661.
Saint-Vincent
Grégoire de Saint-Vincent, a Jesuit, born at Bruges in 1584 and died at Ghent in 1667, discovered the expansion of log(1 + x) in ascending powers of x. Although a circle-squarer he is worthy of mention for the numerous theorems of interest which he discovered in his search after the impossible, and Montucla ingeniously remarks that ``no one ever squared the circle with so much ability or (except for his principal object) with so much success.'' He wrote two books on the subject, one published in 1647 and the other in 1668, which cover some two or three thousand closely printed pages; the fallacy in the quadrature was pointed out by Huygens. In the former work he used indivisibles. An earlier work entitled Theoremata Mathematica, published in 1624, contains a clear account of the method of exhaustions, which is applied to several quadratures, notably that of the hyperbola.
Torricelli
Evangelista Torricelli, born at Faenza on Oct. 15, 1608, and died at Florence in 1647, wrote on the quadrature of the cycloid and conics; the rectifications of the logarithmic spiral; the theory of the barometer; the value of gravity found by observing the motion of two weights connected by a string passing over a fixed pulley; the theory of projectiles; and the motion of fluids.
Hudde
Johann Hudde, burgomaster of Amsterdam, was born there in 1633, and died in the same town in 1704. He wrote two tracts in 1659: one was on the reduction of equations which have equal roots; in the other he stated what is equivalent to the proposition that if f(x,y) = 0 be the algebraical equation of a curve, then the subtangent is
but being ignorant of the notation of the calculus his enunciation is involved.
Frénicle
Bernard Frénicle de Bessy, born in Paris circ. 1605 and died in 1670, wrote numerous papers on combinations and on the theory of numbers, also on magic squares. It may be interesting to add that he challenged Huygens to solve the following system of equations in integers, x² + y² = z², x² = u² + v², x - y = u - v. A solution was given by M. Pépin in 1880.
De Laloubère
Antoine de Laloubère, a Jesuit, born in Languedoc in 1600 and died at Toulouse in 1664, is chiefly celebrated for an incorrect solution of Pascal's problems on the cycloid, which he gave in 1660, but he has a better claim to distinction in having been the first mathematician to study the properties of the helix.
N. Mercator
Nicholas Mercator (sometimes known as Kauffmann) was born in Holstein about 1620, but resided most of his life in England. He went to France in 1683, where he designed and constructed the fountains at Versailles, but the payment agreed on was refused unless he would turn Catholic; he died of vexation and poverty in Paris in 1687. He wrote a treatise on logarithms entitled Logarithmo-technica, published in 1668, and discovered the series
he proved this by writing the equation of a hyperbola in the formto which Wallis's method of quadrature could be applied. The same series had been independently discovered by Saint-Vincent.
WrenSir Christopher Wren was born at Knoyle, Wiltshire, on October 20, 1632, and died in London on February 25, 1723. Wren's reputation as a mathematician has been overshadowed by his fame as an architect, but he was Savilian professor of astronomy at Oxford from 1661 to 1673, and for some time president of the Royal Society. Together with Wallis and Huygens he investigated the laws of collision of bodies; he also discovered the two systems of generating lines on the hyperboloid of one sheet, though it is probable that he confined his attention to a hyperboloid of revolution. Besides these he wrote papers on the resistance of fluids, and the motion of the pendulum. He was a friend of Newton and (like Huygens, Hooke, Halley, and others) had made attempts to shew that the force under which the planets move varies inversely as the square of the distance from the sun.
Wallis, Brouncker, Wren, and Boyle
(the last-named being a chemist and physicist rather than a mathematician) were the leading philosophers who founded the Royal Society of London. The society arose from the self-styled ``indivisible college'' in London in 1645; most of its members moved to Oxford during the civil war, where Hooke, who was then an assistant in Boyle's laboratory, joined in their meetings; the society was formally constituted in London in 1660, and was incorporated on July 15, 1662. The French Academy was founded in 1666, and the Berlin Academy in 1700. The Accademia dei Lincei was founded in 1603, but was dissolved in 1630.
Hooke
Robert Hooke, born at Freshwater on July 18, 1635, and died in London on March 3, 1703, was educated at Westminster, and Christ Church, Oxford, and in 1665 became professor of geometry at Gresham College, a post which he occupied till his death. He is still known by the law which he discovered, that the tension exerted by a stretched string is (within certain limits) proportional to the extension, or, in other words, that the stress is proportional to the strain. He invented and discussed the conical pendulum, and was the first to state explicitly that the motions of the heavenly bodies were merely dynamical problems. He was as jealous as he was vain and irritable, and accused both Newton and Huygens of unfairly appropriating his results. Like Huygens, Wren, and Halley, he made efforts to find the law of force under which the planets move about the sun, and he believed the law to be that of the inverse square of the distance. He, like Huygens, discovered that the small oscillations of a coiled spiral spring were practically isochronous, and was thus led to recommend (possibly in 1658) the use of the balance spring in watches. He had a watch of this kind made in London in 1675; it was finished just three months later than a similar one made in Paris under the directions of Huygens.
Collins
John Collins, born near Oxford on March 5, 1625, and died in London on November 10, 1683, was a man of great natural ability, but of slight education. Being devoted to mathematics, he spent his spare time in correspondence with the leading mathematicians of the time, for whom he was always ready to do anything in his power, and he has been described - not inaptly - as the English Mersenne. To him we are indebted for much information on the details of the discoveries of the period.
Pell
Another mathematician who devoted a considerable part of his time to making known the discoveries of others, and to correspondence with leading mathematicians, was John Pell. Pell was born in Sussex on March 1, 1610, and died in London on December 10, 1685. He was educated at Trinity College, Cambridge; he occupied in succession the mathematical chairs at Amsterdam and Breda; he then entered the English diplomatic service; but finally settled in 1661 in London, where he spent the last twenty years of his life. His chief works were an edition, with considerable new matter, of the Algebra by Branker and Rhonius, London, 1668; and a table of square numbers, London, 1672.
Sluze
René François Walther de Sluze (Slusius), canon of Liége, born on July 7, 1622, and died on March 19, 1685, found for the subtangent of a curve f(x,y) = 0 an expression which is equivalent to
he wrote numerous tracts, and in particular discussed at some length spirals and points of inflexion.
Viviani
Vincenzo Viviani, a pupil of Galileo and Torricelli, born at Florence on April 5, 1622, and died there on September 22, 1703, brought out in 1659 a restoration of the lost book of Apollonius on conic sections, and in 1701 a restoration of the work of Aristaeus. He explained in 1677 how an angle could be trisected by the aid of the equilateral hyperbola or the conchoid. In 1692 he proposed the problem to construct four windows in a hemispherical vault so that the remainder of the surface can be accurately determined; a celebrated problem, of which analytical solutions were given by Wallis, Leibnitz, David Gregory, and James Bernoulli.
Tchirnhausen
Ehrenfried Walther von Tschirnhausen was born at Kislingswalde on April 10, 1631, and died at Dresden on October 11, 1708. In 1682 he worked out the theory of caustics by reflexion, or, as they were usually called, catacaustics, and shewed that they were rectifiable. This was the second case in which the envelope of a moving line was determined. He constructed burning mirrors of great power. The transformation by which he removed certain intermediate terms from a given algebraical equation is well known; it was published in the Acta Eruditorum for 1683.
De la Hire
Philippe De la Hire (or Lahire), born in Paris on March 18, 1640, and died there on April 21, 1719, wrote on graphical methods, 1673; on the conic sections, 1685; a treatise on epicycloids, 1694; one on roulettes, 1702; and, lastly, another on conchoids, 1708. His works on conic sections and epicycloids were founded on the teaching of Desargues, whose favourite pupil he was. He also translated the essay of Moschopulus on magic squares, and collected many of the theorems on them which were previously known; this was published in 1705.
Ole Roemer
, born at Aarhuus on September 25, 1644, and died at Copenhagen on September 19, 1710, was the first to measure the velocity of light; this was done in 1675 by means of the eclipses of Jupiter's satellites. He brought the transit and mural circle into common use, the altazimuth having been previously generally employed, and it was on his recommendation that astronomical observations of stars were subsequently made in general on the meridian. He was also the first to introduce micrometers and reading microscopes into an observatory. He also deduced from the properties of epicycloids the form of the teeth in toothed-wheels best fitted to secure a uniform motion.
Rolle
Michel Rolle, born at Ambert on April 21, 1652, and died in Paris on November 8, 1719, wrote an algebra in 1689, which contains the theorem on the position of the roots of an equation which is known by his name. He published in 1696 a treatise on the solutions of equations, whether determinate or indeterminate, and he produced several other minor works. He taught that the differential calculus, which, as we shall see later, had been introduced towards the close of the seventeenth century, was nothing but a collection of ingenious fallacies.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).
Transcribed by
D.R. Wilkins
([email protected])
School of Mathematics
Trinity College, Dublin
(http://www.maths.tcd.ie/pub/HistMath/People/17thCentury/RouseBall/RB_Math17C.html)
Almost contemporaneously with the publication in 1637 of Descartes' geometry, the principles of the integral calculus, so far as they are concerned with summation, were being worked out in Italy. This was effected by what was called the principle of indivisibles, and was the invention of Cavalieri. It was applied by him and his contemporaries to numerous problems connected with the quadrature of curves and surfaces, the determination on volumes, and the positions of centres of mass. It served the same purpose as the tedious method of exhaustions used by the Greeks; in principle the methods are the same, but the notation of indivisibles is more concise and convenient. It was, in its turn, superceded at the beginning of the eighteenth century by the integral calculus.
Bonaventura Cavalieri was born at Milan in 1598, and died at Bologna on November 27, 1647. He became a Jesuit at an early age; on the recommendation of the Order he was in 1629 made professor of mathematics at Bologna; and he continued to occupy the chair there until his death. I have already mentioned Cavalieri's name in connection with the introduction of the use of logarithms into Italy, and have alluded to his discovery of the expression for the area of a spherical triangle in terms of the spherical excess. He was one of the most influential mathematicians of his time, but his subsequent reputation rests mainly on his invention of the principle of indivisibles.
The principle of indivisibles had been used by Kepler in 1604 and 1615 in a somewhat crude form. It was first stated by Cavalieri in 1629, but he did not publish his results till 1635. In his early enunciation of the principle in 1635 Cavalieri asserted that a line was made up of an infinite number of points (each without magnitude), a surface of infinite number of lines (each without breadth), and a volume of an infinite number of surfaces (each without thickness). To meet the objections of Guldinus and others, the statement was recast, and in its final form as used by the mathematicians of the seventeenth century it was published in Cavalieri's Exercitationes Geometricae in 1647; the third exercise is devoted to a defence of the theory. This book contains the earliest demonstration of the properties of Pappus. Cavalieri's works on indivisibles were reissued with his later corrections in 1653.
The method of indivisibles rests, in effect, on the assumption that any magnitude may be divided into an infinite number of small quantities which can be made to bear any required ratios (ex. gr. equality) one to the other. The analysis given by Cavalieri is hardly worth quoting except as being one of the first steps taken towards the formation of an infinitesimal calculus. One example will suffice. Suppose it be required to find the area of a right-angled triangle. Let the base be made up of, or contain n points (or indivisibles), and similarly let the other side contain na points, then the ordinates at the successive points of the base will contain a, 2a ..., napoints. Therefore the number of points in the area is a + 2a + ... + na; the sum of which is ½ n²a + ½na. Since n is very large, we may neglect ½na for it is inconsiderable compared with ½n²a. Hence the area is equal to ½(na)n, that is, ½ × altitude × base. There is no difficulty in criticizing such a proof, but, although the form in which it is presented is indefensible, the substance of it is correct.
It would be misleading to give the above as the only specimen of the method of indivisibles, and I therefore quote another example, taken from a later writer, which will fairly illustrate the use of the method when modified and corrected by the method of limits.
Let it be required to find the area outside a parabola APC and bounded by the curve, the tangent at A, and a line DC parallel to AB the diameter at A. Complete the parallelogram ABCD. Divide AD into n equal parts, let AM contain r of them, and let MN be the (r + 1)th part. Draw MP and NQ parallel to AB, and draw PRparallel to AD. Then when n becomes indefinitely large, the curvilinear area APCD will be the the limit of the sum of all parallelograms like PN. Nowarea PN : area BD = MP.MN : DC.AD.But by the properties of the parabolaMP : DC = AM² : AD² = r² : n²,and MN : AD = 1 : n. Hence MP.MN : DC.AD = r² : n³. Therefore area PN : area BD = r² : n³. Therefore, ultimately,area APCD : area BD= 1² + 2² + ... + (n-1)² : n³= n (n-1)(2n-1) : n³which, in the limit, = 1 : 3.
It is perhaps worth noticing that Cavalieri and his successors always used the method to find the ratio of two areas, volumes, or magnitudes of the same kind and dimensions, that is, they never thought of an area as containing so many units of area. The idea of comparing a magnitude with a unit of the same kind seems to have been due to Wallis.
It is evident that in its direct form the method is applicable to only a few curves. Cavalieri proved that, if m be a positive integer, then the limit, when n is infinite, of is 1/(m+1), which is equivalent to saying that he found the integral of x to from x = 0 to x = 1; he also discussed the quadrature of the hyperbola.
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William, Viscount Brouncker (c.1620 - 1684)
From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.
William, Viscount Brouncker, one of the founders of the Royal Society of London, born about 1620, and died on April 5, 1684, was among the most brilliant mathematicians of this time, and was in intimate relations with Wallis, Fermat, and other leading mathematicians. I mentioned above his curious reproduction of Brahmagupta's solution of a certain indeterminate equation. Brouncker proved that the area enclosed between the equilateral hyperbola xy = 1, the axis of x, and the ordinates x = 1 and x = 2, is equal either to
or toHe also worked out other similar expressions for different areas bounded by the hyperbola and straight lines. He wrote on the rectification of the parabola and of the cycloid. It is noticeable that he used infinite series to express quantities whose values he could not otherwise determine. In answer to a request of Wallis to attempt the quadrature of the circle he shewed that the ratio of the area of a circle to the area of the circumscribed square, that is, the ratio of to 4, is equal to the ratio ofto 1. Continued fractions had been employed by Bombelli in 1572, and had been systematically used by Cataldi in his treatise on finding the square roots of numbers, published at Bologna in 1613. Their properties and theory were given by Huygens, 1703 and Euler, 1744.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).
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Christian Huygens (1629 - 1695)
From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.
Christian Huygens was born at the Hague on April 14, 1629, and died in the same town on July 8, 1695. He generally wrote his name as Hugens, but I follow the usual custom in spelling it as above: it is also sometimes written as Huyghens. His life was uneventful, and there is little more to record in it than a statement of his various memoirs and researches.
In 1651 he published an essay in which he shewed the fallacy in a system of quadratures proposed by Grégoire de Saint-Vincent, who was well versed in the geometry of the Greeks, but had not grasped the essential points in the more modern methods. This essay was followed by tracts on the quadrature of the conics and the approximate rectification of the circle.
In 1654 his attention was directed to the improvement of the telescope. In conjunction with his brother he devised a new and better way of grinding and polishing lenses. As a result of these improvements he was able during the following two years, 1655 and 1656, to resolve numerous astronomical questions; as, for example, the nature of Saturn's appendage. His astronomical observations required some exact means of measuring time, and he was thus led in 1656 to invent the pendulum clock, as described in his tract Horologium, 1658. The time-pieces previously in use had been balance-clocks.
In the year 1657 Huygens wrote a small work on the calculus of probabilities founded on the correspondence of Pascal and Fermat. He spent a couple of years in England about this time. His reputation was now so great that in 1665 Louis XIV. offered him a pension if he would live in Paris, which accordingly then became his place of residence.
In 1668 he sent to the Royal Society of London, in answer to a problem they had proposed, a memoir in which (simultaneously with Wallis and Wren) he proved by experiment that the momentum in a certain direction before the collision of two bodies is equal to the momentum in that direction after the collision. This was one of the points in mechanics on which Descartes had been mistaken.
The most important of Huygens's work was his Horologium Oscillatorium published at Paris in 1673. The first chapter is devoted to pendulum clocks. The second chapter contains a complete account of the descent of heavy bodies under their own weights in a vacuum, either vertically down or on smooth curves. Amongst other propositions he shews that the cycloid is tautochronous. In the third chapter he defines evolutes and involutes, proves some of their more elementary properties, and illustrates his methods by finding the evolutes of the cycloid and the parabola. These are the earliest instances in which the envelope of a moving line was determined. In the fourth chapter he solves the problem of the compound pendulum, and shews that the centres of oscillation and suspension are interchangeable. In the fifth and last chapter he discusses again the theory of clocks, points out that if the bob of the pendulum were, by means of cycloidal clocks, made to oscillate in a cycloid the oscillations would be isochronous; and finishes by shewing that the centrifugal force on a body which moves around a circle of radius r with a uniform velocity vvaries directly as v² and inversely as r. This work contains the first attempt to apply dynamics to bodies of finite size, and not merely to particles.
In 1675 Huygens proposed to regulate the motion of watches by the use of the balance spring, in the theory of which he had been perhaps anticipated in a somewhat ambiguous and incomplete statement made by Hooke in 1658. Watches or portable clocks had been invented early in the sixteenth century, and by the end of that century were not very uncommon, but they were clumsy and unreliable, being driven by a main spring and regulated by a conical pulley and verge escapement; moreover, until 1687 they had only one hand. The first watch whose motion was regulated by a balance spring was made at Paris under Huygens's directions, and presented by him to Louis XIV.
The increasing intolerance of the Catholics led to his return to Holland in 1681, and after the revocation of the edict of Nantes he refused to hold any further communication with France. He now devoted himself to the construction of lenses of enormous focal length: of these three of focal lengths 123 feet, 180 feet, and 210 feet, were subsequently given by him to the Royal Society of London, in whose possession they still remain. It was about this time that he discovered the achromatic eye-piece (for a telescope) which is known by his name. In 1689 he came from Holland to England in order to make the acquaintance of Newton, whosePrincipia had been published in 1687. Huygens fully recognized the intellectual merits of the work, but seems to have deemed any theory incomplete which did not explain gravitation by mechanical means.
On his return in 1690 Huygens published his treatise on light in which the undulatory theory was expounded and explained. Most of this had been written as early as 1678. The general idea of the theory had been suggested by Robert Hooke in 1664, but he had not investigated its consequences in any detail. Only three ways have been suggested in which light can be produced mechanically. Either the eye may be supposed to send out something which, so to speak, feels the object (as the Greeks believed); or the object perceived may send out something which hits or affects the eye (as assumed in the emission theory); or there may be some medium between the eye and the object, and the object may cause some change in the form or condition of this intervening medium and thus affect the eye (as Hooke and Huygens supposed in the wave or undulatory theory). According to this last theory space is filled with an extremely rare ether, and light is caused by a series of waves or vibrations in this ether which are set in motion by the pulsations of the luminous body. From this hypothesis Huygens deduced the laws of reflexion and refraction, explained the phenomenon of double refraction, and gave a construction for the extraordinary ray in biaxal crystals; while he found by experiment the chief phenomena of polarization.
The immense reputation and unrivalled powers of Newton led to disbelief in a theory which he rejected, and to the general adoption of Newton's emission theory. Within the present century crucial experiments have been devised which give different results according as one or the other theory is adopted; all these experiments agree with the results of the undulatory theory and differ from the results of the Newtonian theory; the latter is therefore untenable. Until, however, the theory of interference, suggested by Young, was worked out by Fresnel, the hypothesis of Huygens failed to account for all the facts, and even now the properties which, under it, have to be attributed to the intervening medium or ether involve difficulties of which we still seek a solution. Hence the problem as to how the effects of light are really produced cannot be said to be finally solved.
Besides these works Huygens took part in most of the controversies and challenges which then played so large a part in the mathematical world, and wrote several minor tracts. In one of these he investigated the form and properties of the catenary. In another he stated in general terms the rule for finding maxima and minima of which Fermat had made use, and shewed that the subtangent of an algebraical curve f(x,y) = 0 was equal to , where is the derived function of f(x,y) regarded as a function of y. In some posthumous works, issued at Leyden in 1703, he further shewed how from the focal lengths of the component lenses the magnifying power of a telescope could be determined; and explained some of the phenomena connected with haloes and parhelia.
I should add that almost all his demonstrations, like those of Newton, are rigidly geometrical, and he would seem to have made no use of the differential or fluxional calculus, though he admitted the validity of the methods used therein. Thus, even when first written, his works were expressed in an archaic language, and perhaps received less attention than their intrinsic merits deserved.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).
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Isaac Barrow (1630 - 1677)
From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.
Isaac Barrow was born in London in 1630, and died at Cambridge in 1677. He went to school first at Charterhouse (where he was so troublesome that his father was heard to pray that if it pleased God to take any of his children he could best spare Isaac), and subsequently to Felstead. He completed his education at Trinity College, Cambridge; after taking his degree in 1648, he was elected to a fellowship in 1649; he then resided for a few years in college, but in 1655 he was driven out by the persecution of the Independents. He spent the next four years in the East of Europe, and after many adventures returned to England in 1659. He was ordained the next year, and appointed to the professorship of Greek at Cambridge. In 1662 he was made professor of geometry at Gresham College, and in 1663 was selected as the first occupier of the Lucasian chair at Cambridge. He resigned the latter to his pupil Newton in 1669, whose superior abilities he recognized and frankly acknowledged. For the remainder of his life he devoted himself to the study of divinity. He was appointed master of Trinity College in 1672, and held the post until his death.
He is described as ``low in stature, lean, and of a pale complexion,'' slovenly in his dress, and an inveterate smoker. He was noted for his strength and courage, and once when travelling in the East he saved the ship by his own prowess from capture by pirates. A ready and caustic wit made him a favourite of Charles II., and induced the courtiers to respect even if they did not appreciate him. He wrote with a sustained and somewhat stately eloquence, and with his blameless life and scrupulous conscientiousness was an impressive personage of the time.
His earliest work was a complete edition of the Elements of Euclid, which he issued in Latin in 1655, and in English in 1660; in 1657 he published an edition of theData. His lectures, delivered in 1664, 1665, and 1666, were published in 1683 under the title Lectiones Mathematicae; these are mostly on the metaphysical basis for mathematical truths. His lectures for 1667 were published in the same year, and suggest the analysis by which Archimedes was led to his chief results. In 1669 he issued his Lectiones Opticae et Geometricae. It is said in the preface that Newton revised and corrected these lectures, adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. This, which is his most important work in mathematics, was republished with a few minor alterations in 1674. In 1675 he published an edition with numerous comments of the first four books of theConics of Apollonius, and of the extant works of Archimedes and Theodosius.
In the optical lectures many problems connected with the reflexion and refraction of light are treated with ingenuity. The geometrical focus of a point seen by reflexion or refraction is defined; and it is explained that the image of an object is the locus of the geometrical foci of every point on it. Barrow also worked out a few of the easier properties of thin lenses, and considerably simplified the Cartesian explanation of the rainbow.
The geometrical lectures contain some new ways of determining the areas and tangents of curves. The most celebrated of these is the method given for the determination of tangents to curves, and this is sufficiently important to require a detailed notice, because it illustrates the way in which Barrow, Hudde and Sluze were working on the lines suggested by Fermat towards the methods of the differential calculus.
Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it were known; hence, if the length of the subtangentMT could be found (thus determining the point T), then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a pointQ adjacent to P were drawn, he got a small triangle PQR (which he called the differential triangle, because its sides PR and PQ were the differences of the abscissae and ordinates of P and Q), so thatTM : MP = QR : RP.To find QR : RP he supposed that x, y were the co-ordinates of P, and x - e, y - a those of Q (Barrow actually used p for x and m for y, but I alter these to agree with modern practice). Substituting the co-ordinates of Q in the equation of the curve, and neglecting the squares and higher powers of e and a as compared with their first powers, he obtained e : a. The ratio a/e was subsequently (in accordance with a suggestion made by Sluze) termed the angular coefficient of the tangent at the point.
Barrow applied this method to the curves (i) x² (x² + y²) = r²y²; (ii) x³ + y³ = r³; (iii) x³ + y³ = rxy, called la galande; (iv) y = (r - x) tan x/2r, the quadratrix; and (v) y = r tan x/2r. It will be sufficient here if I take as an illustration the simpler case of the parabola y² = px. Using the notation given above, we have for the pointP, y² = px; and for the point Q, (y - a)² = p(x - e). Subtracting we get 2ay - a² = pe. But, if a be an infinitesimal quantity, a² must be infinitely smaller and therefore may be neglected when compared with the quantities 2ay and pe. Hence 2ay = pe, that is, e : a = 2y : p. Therefore TP : y = e : a = 2y : p. Hence TM = 2y²/p = 2x.This is exactly the procedure of the differential calculus, except that there we have a rule by which we can get the ratio a/e or dy/dx directly without the labour of going through a calculation similar to the above for every separate case.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).
Transcribed by
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School of Mathematics
Trinity College, Dublin
James Gregory (1638 - 1675)
From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.
James Gregory, born at Drumoak near Aberdeen in 1638, and died at Edinburgh in October 1675, was successively professor at St. Andrews and Edinburgh. In 1660 he published his Optica Promota, in which the reflecting telescope known by his name is described. In 1667 he issued his Vera Circuli et Hyperbolae Quadratura, in which he shewed how the areas of the circle and hyperbola could be obtained in the form of infinite convergent series, and here (I believe for the first time) we find a distinction drawn between convergent and divergent series. This work contains a remarkable geometrical proposition to the effect that the ratio of the area of any arbitrary sector of a circle to that of the inscribed or circumscribed regular polygons is not expressible by a finite number of terms. Hence he inferred that the quadrature of the circle was impossible; this was accepted by Montucla, but it is not conclusive, for it is conceivable that some particular sector might be squared, and this particular sector might be the whole circle. This book contains also the earliest enunciation of the expansions in series of sin x, cos x, x or arc sin x, and x or arc cos x. It was reprinted in 1668 with an appendix, Geometriae Pars, in which Gregory explained how the volumes of solids of revolution could be determined. In 1671, or perhaps earlier, he established the theorem that
the result being true only if lie between - and . This is the theorem on which many of the subsequent calculations of approximations to the numerical value of have been based.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).
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Some Contemporaries of Descartes, Fermat, Pascal and Huygens
Bachet
Claude Gaspard Bachet de Méziriac was born at Bourg in 1581, and died in 1638. He wrote the Problèmes plaisants, of which the first edition was issued in 1612, a second and enlarged edition was brought out in 1624; this contains an interesting collection of arithmetical tricks and questions, many of which are quoted in my Mathematical Recreations and Essays. He also wrote Les éléments arithmétiques, which exists in manuscript; and a translation of the Arithmetic of Diophantus. Bachet was the earliest writer who discussed the solution of indeterminate equations by means of continued fractions.
Mersenne
Marin Mersenne, born in 1588 and died at Paris in 1648, was a Franciscan friar, who made it his business to be acquainted and correspond with the French mathematicians of that date and many of their foreign contemporaries. In 1634 he published a translation of Galileo's mechanics; in 1644 he issued his Cogita Physico-Mathematica, by which he is best known, containing an account of some experiments in physics; he also wrote a synopsis of mathematics, which was printed in 1664.
The preface to the Cogitata contains a statement (possibly due to Fermat) that, in order that may be prime, the only values of p, not greater than 257, which are possible are 1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257; the number 67 is probably a misprint for 61. With this correction the statement appears to be true, and it has been verified for all except twenty-one values of p, namely 71, 89, 101, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 181, 193, 199, 227, 229, 241, and 257. Of these values, Mersenne asserted that p = 127 and p = 257 make prime, and that the other nineteen values make a composite number. It has been asserted that the statement has been verified when p = 89 and 127, but these verifications rest on long numerical calculations made by single computators and not published; until these demonstrations have been confirmed we may say that twenty-one cases still await verification or require further investigation. The factors of when p = 89 are not known, the calculation merely showing that the number could not be prime. It is most likely that these results are particular cases of some general theorem on the subject which remains to be discovered.
The theory of perfect numbers depends directly on that of Mersenne's numbers. It is probable that all perfect numbers are included in the formula , where is a prime. Euclid proved that any number of this form is perfect. Euler shewed that the formula includes all even perfect numbers; and there is reason to believe - though a rigid demonstration is wanting - that an odd number cannot be perfect. If we assume that the last of these statements is true, then every perfect number is of the above form. Thus if p = 2, 3, 5, 7, 13, 17, 19, 31, 61, then, by Mersenne's rule, the corresponding values of are prime; they are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951; and the corresponding perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, and 2658455991569831744654692615953842176.
Roberval
Gilles Personier (de) Roberval, born at Roberval in 1602 and died at Paris in 1675, described himself from the place of his birth as de Roberval, a seigniorial title to which he had no right. He discussed the nature of the tangents to curves, solved some of the easier questions connected with the cycloid, generalized Archimedes's theorems on the spiral, wrote on mechanics, and on the method of indivisibles, which he rendered more precise and logical. He was a professor in the university of Paris, and in correspondence with nearly all the leading mathematicians of his time.
Van Schooten
Frans van Schooten, to whom we owe an edition of Vieta's works, succeeded his father (who had taught mathematics to Huygens, Hudde, and Sluze) as professor at Leyden in 1646. He brought out in 1659 a Latin translation of Descartes's Géométrie, and in 1657 a collection of mathematical exercises in which he recommended the use of co-ordinates in space of three dimensions. He died in 1661.
Saint-Vincent
Grégoire de Saint-Vincent, a Jesuit, born at Bruges in 1584 and died at Ghent in 1667, discovered the expansion of log(1 + x) in ascending powers of x. Although a circle-squarer he is worthy of mention for the numerous theorems of interest which he discovered in his search after the impossible, and Montucla ingeniously remarks that ``no one ever squared the circle with so much ability or (except for his principal object) with so much success.'' He wrote two books on the subject, one published in 1647 and the other in 1668, which cover some two or three thousand closely printed pages; the fallacy in the quadrature was pointed out by Huygens. In the former work he used indivisibles. An earlier work entitled Theoremata Mathematica, published in 1624, contains a clear account of the method of exhaustions, which is applied to several quadratures, notably that of the hyperbola.
Torricelli
Evangelista Torricelli, born at Faenza on Oct. 15, 1608, and died at Florence in 1647, wrote on the quadrature of the cycloid and conics; the rectifications of the logarithmic spiral; the theory of the barometer; the value of gravity found by observing the motion of two weights connected by a string passing over a fixed pulley; the theory of projectiles; and the motion of fluids.
Hudde
Johann Hudde, burgomaster of Amsterdam, was born there in 1633, and died in the same town in 1704. He wrote two tracts in 1659: one was on the reduction of equations which have equal roots; in the other he stated what is equivalent to the proposition that if f(x,y) = 0 be the algebraical equation of a curve, then the subtangent is
but being ignorant of the notation of the calculus his enunciation is involved.
Frénicle
Bernard Frénicle de Bessy, born in Paris circ. 1605 and died in 1670, wrote numerous papers on combinations and on the theory of numbers, also on magic squares. It may be interesting to add that he challenged Huygens to solve the following system of equations in integers, x² + y² = z², x² = u² + v², x - y = u - v. A solution was given by M. Pépin in 1880.
De Laloubère
Antoine de Laloubère, a Jesuit, born in Languedoc in 1600 and died at Toulouse in 1664, is chiefly celebrated for an incorrect solution of Pascal's problems on the cycloid, which he gave in 1660, but he has a better claim to distinction in having been the first mathematician to study the properties of the helix.
N. Mercator
Nicholas Mercator (sometimes known as Kauffmann) was born in Holstein about 1620, but resided most of his life in England. He went to France in 1683, where he designed and constructed the fountains at Versailles, but the payment agreed on was refused unless he would turn Catholic; he died of vexation and poverty in Paris in 1687. He wrote a treatise on logarithms entitled Logarithmo-technica, published in 1668, and discovered the series
he proved this by writing the equation of a hyperbola in the formto which Wallis's method of quadrature could be applied. The same series had been independently discovered by Saint-Vincent.
WrenSir Christopher Wren was born at Knoyle, Wiltshire, on October 20, 1632, and died in London on February 25, 1723. Wren's reputation as a mathematician has been overshadowed by his fame as an architect, but he was Savilian professor of astronomy at Oxford from 1661 to 1673, and for some time president of the Royal Society. Together with Wallis and Huygens he investigated the laws of collision of bodies; he also discovered the two systems of generating lines on the hyperboloid of one sheet, though it is probable that he confined his attention to a hyperboloid of revolution. Besides these he wrote papers on the resistance of fluids, and the motion of the pendulum. He was a friend of Newton and (like Huygens, Hooke, Halley, and others) had made attempts to shew that the force under which the planets move varies inversely as the square of the distance from the sun.
Wallis, Brouncker, Wren, and Boyle
(the last-named being a chemist and physicist rather than a mathematician) were the leading philosophers who founded the Royal Society of London. The society arose from the self-styled ``indivisible college'' in London in 1645; most of its members moved to Oxford during the civil war, where Hooke, who was then an assistant in Boyle's laboratory, joined in their meetings; the society was formally constituted in London in 1660, and was incorporated on July 15, 1662. The French Academy was founded in 1666, and the Berlin Academy in 1700. The Accademia dei Lincei was founded in 1603, but was dissolved in 1630.
Hooke
Robert Hooke, born at Freshwater on July 18, 1635, and died in London on March 3, 1703, was educated at Westminster, and Christ Church, Oxford, and in 1665 became professor of geometry at Gresham College, a post which he occupied till his death. He is still known by the law which he discovered, that the tension exerted by a stretched string is (within certain limits) proportional to the extension, or, in other words, that the stress is proportional to the strain. He invented and discussed the conical pendulum, and was the first to state explicitly that the motions of the heavenly bodies were merely dynamical problems. He was as jealous as he was vain and irritable, and accused both Newton and Huygens of unfairly appropriating his results. Like Huygens, Wren, and Halley, he made efforts to find the law of force under which the planets move about the sun, and he believed the law to be that of the inverse square of the distance. He, like Huygens, discovered that the small oscillations of a coiled spiral spring were practically isochronous, and was thus led to recommend (possibly in 1658) the use of the balance spring in watches. He had a watch of this kind made in London in 1675; it was finished just three months later than a similar one made in Paris under the directions of Huygens.
Collins
John Collins, born near Oxford on March 5, 1625, and died in London on November 10, 1683, was a man of great natural ability, but of slight education. Being devoted to mathematics, he spent his spare time in correspondence with the leading mathematicians of the time, for whom he was always ready to do anything in his power, and he has been described - not inaptly - as the English Mersenne. To him we are indebted for much information on the details of the discoveries of the period.
Pell
Another mathematician who devoted a considerable part of his time to making known the discoveries of others, and to correspondence with leading mathematicians, was John Pell. Pell was born in Sussex on March 1, 1610, and died in London on December 10, 1685. He was educated at Trinity College, Cambridge; he occupied in succession the mathematical chairs at Amsterdam and Breda; he then entered the English diplomatic service; but finally settled in 1661 in London, where he spent the last twenty years of his life. His chief works were an edition, with considerable new matter, of the Algebra by Branker and Rhonius, London, 1668; and a table of square numbers, London, 1672.
Sluze
René François Walther de Sluze (Slusius), canon of Liége, born on July 7, 1622, and died on March 19, 1685, found for the subtangent of a curve f(x,y) = 0 an expression which is equivalent to
he wrote numerous tracts, and in particular discussed at some length spirals and points of inflexion.
Viviani
Vincenzo Viviani, a pupil of Galileo and Torricelli, born at Florence on April 5, 1622, and died there on September 22, 1703, brought out in 1659 a restoration of the lost book of Apollonius on conic sections, and in 1701 a restoration of the work of Aristaeus. He explained in 1677 how an angle could be trisected by the aid of the equilateral hyperbola or the conchoid. In 1692 he proposed the problem to construct four windows in a hemispherical vault so that the remainder of the surface can be accurately determined; a celebrated problem, of which analytical solutions were given by Wallis, Leibnitz, David Gregory, and James Bernoulli.
Tchirnhausen
Ehrenfried Walther von Tschirnhausen was born at Kislingswalde on April 10, 1631, and died at Dresden on October 11, 1708. In 1682 he worked out the theory of caustics by reflexion, or, as they were usually called, catacaustics, and shewed that they were rectifiable. This was the second case in which the envelope of a moving line was determined. He constructed burning mirrors of great power. The transformation by which he removed certain intermediate terms from a given algebraical equation is well known; it was published in the Acta Eruditorum for 1683.
De la Hire
Philippe De la Hire (or Lahire), born in Paris on March 18, 1640, and died there on April 21, 1719, wrote on graphical methods, 1673; on the conic sections, 1685; a treatise on epicycloids, 1694; one on roulettes, 1702; and, lastly, another on conchoids, 1708. His works on conic sections and epicycloids were founded on the teaching of Desargues, whose favourite pupil he was. He also translated the essay of Moschopulus on magic squares, and collected many of the theorems on them which were previously known; this was published in 1705.
Ole Roemer
, born at Aarhuus on September 25, 1644, and died at Copenhagen on September 19, 1710, was the first to measure the velocity of light; this was done in 1675 by means of the eclipses of Jupiter's satellites. He brought the transit and mural circle into common use, the altazimuth having been previously generally employed, and it was on his recommendation that astronomical observations of stars were subsequently made in general on the meridian. He was also the first to introduce micrometers and reading microscopes into an observatory. He also deduced from the properties of epicycloids the form of the teeth in toothed-wheels best fitted to secure a uniform motion.
Rolle
Michel Rolle, born at Ambert on April 21, 1652, and died in Paris on November 8, 1719, wrote an algebra in 1689, which contains the theorem on the position of the roots of an equation which is known by his name. He published in 1696 a treatise on the solutions of equations, whether determinate or indeterminate, and he produced several other minor works. He taught that the differential calculus, which, as we shall see later, had been introduced towards the close of the seventeenth century, was nothing but a collection of ingenious fallacies.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).
Transcribed by
D.R. Wilkins
([email protected])
School of Mathematics
Trinity College, Dublin
(http://www.maths.tcd.ie/pub/HistMath/People/17thCentury/RouseBall/RB_Math17C.html)