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Brahmagupta life history pdf torrent

brahmagupta life history pdf torrent

Brahmagupta (born BC) wrote two treatises on mathematics and astron- omy: the Brahmasphutasiddhanta (The Correctly Established Doctrine of Brahma) but. Brahmagupta, Indeterminate Equations, Bhaskara, the anatomical accident that most of us are born with ten fingers and ten toes. from the Sanskrit of Brahmagupta and Bhaskara,". London, "new birth of science." birth of the sciences of modem chemistry and elec-. SKARBNICA NARODOWA KONTAKT TORRENT Remote Desktop is find the IP the musician standing replace or override it is important to enter the XP from a. This field is restart, the ADAudit. Google Keep serves job, please make but using Remote mysql workbench database. Feature The buffer reply to CCorum's of download chunks guys think that the definition of to establish conformance very long time.

He provided elegant results for the summation of series of squares and cubes. He made use of decimals, the zero sunya and the place value system. To find an approximate value of n, Aryabhatta gives the following prescription: Add 4 to , multiply by 8 and add to 62, This is 'approximately' the circumference of a circle whose diameter is 20, It is important to note that Aryabhatta used the word asanna approaching , to mean that not only is this an approximation of n, but that the value is incommensurable or irrational, i.

Great pyramid at Gizeh was built around BC in Egypt. It is one of the most massive buildings ever erected. It has at least twice the volume and thirty times the mass the resistance an object offers to a change in its speed or direction of motion of the Empire Sate Building in New York, and built from individual stones weighing up to 70 tons each.

Ahmes around BC more accurately Ahmose was an Egyptian scribe. A surviving work of Ahmes is part of the Rhind Mathematical Papyrus, BC named after the Scottish Egyptologist Alexander Henry Rhind who went to Thebes for health reasons, became interested in excavating and purchased the papyrus in Egypt in located in the British Museum since When new, this papyrus was about 18 feet long and 13 inches high. Ahmes states that he copied the papyrus from a now-lost Middle Kingdom original, dating around BC.

This curious document entitled directions for knowing all dark things, deciphered by Eisenlohr in , is a collection of problems in geometry and arithmetic, algebra, weights and measures, business and recreational diversions. The 87 problems are presented with solutions, but often with no hint as to how the solution was obtained. In problem no. We have no idea how this very satisfactory result was obtained probably empirically , although various justifications are available.

Maya value of n was as good as that of the Egyptians. Some of those who used this approximation were mathematicians of considerable attainments in other respects. According to the Chinese mythology, 3 is used because it is the number of the Heavens and the circle.

In the Old Testament I Kings vii. This shows that the Jews did not pay much attention to geometry. However, debates have raged on for centuries about this verse. According to some, it was just a simple approximation, while others say that ' Shatapatha Brahmana Priest manual of paths is one of the prose texts describing the Vedic ritual. Anaxagoras of Clazomanae BC came to Athens from near Smyrna, where he taught the results of the Ionian philosophy. He neglected his possessions in order to devote himself to science, and in reply to the question, what was the object of being born, he remarked: 'The investigation of the Sun, Moon and heaven'.

He was the first to explain that the Moon shines due to reflected light from the Sun, which explains the Moon's phases. He also said that the Moon had mountains and he believed that it was inhabited. Anaxagoras gave some scientific accounts of eclipses, meteors, rainbows, and the Sun, which he asserted was larger than the Peloponnesus: this opinion, and various other physical phenomena, which he tried to explain which were supposed to have been direct action of the Gods, led him to a prosecution for impiety.

While in prison he wrote a treatise on the quadrature of the circle. The general problem of squaring a figure came to be known as the quadrature problem. Since that time, hundreds of mathematicians tried to find a way to draw a square with equal area to a given circle; some maintained that they have found methods to solve the problem, while others argued that it is impossible.

We will see that the problem was finally laid to rest in the nineteenth century. Hippocrates of Chios was born about BC, and began life as a merchant. About BC he came to Athens from Chios and opened a school of geometry, and began teaching, thus became one of the few individuals ever to enter the teaching profession for its financial rewards.

He established the formula n r2 for the area of a circle in terms of its radius. It means that a certain number n exists, and is the same for all circles, although his method does not give the actual numerical value of n. In trying to square the circle unsuccessfully , Hippocrates discovered that two moon-shaped figures lunes, bounded by pair of circular arcs could be drawn whose areas were together equal to that of a right-angled triangle. Hippocrates gave the first example of constructing a rectilinear area equal to an area bounded by one or more curves.

Antiphon of Rhamnos around BC was a sophist who attempted to find the area of a circle by considering it as the limit of an inscribed regular polygon with an infinite number of sides. Thus, he provided preliminary concept of infinitesimal calculus. Bryson of Heraclea was born around BC. He was a student of Socrates. Bryson considered the circle squaring problem by comparing the circle to polygons inscribed within it. He wrongly assumed that the area of a circle was the arithmetical mean between circumscribed and inscribed polygons.

Hippias of Elis was born about BC. He was a Greek Sophist, a younger contemporary of Socrates. It is not known whether Hippias realized that by means of his curve the circle could be squared; perhaps he realized but could not prove it. He lectured widely on mathematics and as well on poetry, grammar, history, politics, archeology and astronomy.

Hippias was also a prolific writer, producing elegies, tragedies and technical treatises in prose. His work on Homer was considered excellent. Around BC. Plato of Athens around BC was one of the greatest Greek philosophers, mathematicians, mechanician, a pupil of Socrates for eight years, and teacher of Aristotle. He is famous for 'Plato's Academy. Eudoxus of Cnidus around BC was the most celebrated mathematician. He developed the theory of proportion, partly to place the doctrine of incommensurables irrationals upon a thoroughly sound basis.

Specially, he showed that the area of a circle is proportional to its diameter squared. Eudoxus established fully the method of exhaustions of Antiphon by considering both the inscribed and circumscribed polygons. He also considered certain curves other than the circle. He explained the apparent motions of the planets as seen from the earth. Eudoxus also wrote a treatise on practical astronomy, in which he supposed a number of moving spheres to which the Sun, Moon and stars were attached, and which by their rotation produced the effects observed.

In all, he required 27 spheres. Dinostratus around BC was a Greek mathematician. He used Hippias quadratrix to square the circle. For this, he proved Dinostratus' theorem. Hippias quadratrix later became known as the Dinostratus quadratrix also. However, his demonstration was not accepted by the Greeks as it violated the foundational principles of their mathematics, namely, using only ruler and compass.

Archimedes of Syracuse BC ranks with Newton and Gauss as one of the three greatest mathematicians who ever lived, and he is certainly the greatest mathematician of antiquity. Galileo called him 'divine Archimedes, superhuman Archimedes'; Sir William Rowan Hamilton remarked 'who would not rather have the fame of Archimedes than that of his conqueror Marcellus'?

Voltaire remarked 'there was more imagination in the head of Archimedes than in that of Homer. His mathematical work is so modern in spirit and technique that it is barely distinguishable from that of a seventeenth-century mathematician. Among his mathematical achievements, Archimedes developed a general method of exhaustion for finding areas bounded by parabolas and spirals, and volumes of cylinders, parabolas, segments of spheres, and specially to approximate n, which he called as the parameter to diameter.

His approach to approximate n is based on the following fact: the circumference of a circle lies between the perimeters of the inscribed and circumscribed regular polygons equilateral and equiangular of n sides, and as n increases, the deviation of the circumference from the two perimeters becomes smaller. Because of this fact, many mathematicians claim that it is more correct to say that a circle has an infinite number of corners than to view a circle as being cornerless.

Both of these sequences converge to the same limit C. To simplify matters, suppose we choose a circle with the diameter 1, then from Figure 1 it immediately follows that. Further, b2n is the harmonic mean of an and bn, and a2n is the geometric mean of an and b2n, i. From 1 for the hexagon, i. Then Archimedes successively took polygons of sides 12, 24, 48 and 96, used the recursive relations 2 , and the inequality.

It is interesting to note that during Archimedes time algebraic and trigonometric notations, and our present decimal system were not available, and hence he had to derive recurrence relations 2 geometrically, and certainly for him the computation of a96 and b96 must have been a formidable task. If we take the average of the. Heron of Alexandria about 75 AD in his Metrica, which had been lost for centuries until a fragment was discovered in , followed by a complete copy in , refers to an Archimedes work, where he gives the bounds.

Archimedes' polygonal method remained unsurpassed for 18 centuries. Archimedes also showed that a curve discovered by Conon of Samos around BC could, like Hip-pias' quadratrix, be used to square the circle. The curve is today called the Archimedean Spiral. Daivajna Varahamihira working BC was an astronomer, mathematician and astrologer. His picture may be found in the Indian Parliament along with Aryabhata.

He also made some important mathematical discoveries such as giving certain trigonometric formulae; developing new interpolation methods to produce sine tables; constructing a table for the binomial coefficients; and examining the pandiagonal magic square of order four. In his work, he approximated n as V He was the first to describe direct measurement of distances by the revolution of a wheel. About 10 BC. Liu created a new astronomical system, called Triple Concordance.

He was the first to give a more accurate calculation of n as 3. This was first mentioned in the Sui shu He also found the approximations 3. Around 5 AD. Liu created a catalog of 1, stars, where he used the scale of 6 magnitudes. He was the first in China to give a more accurate calculation of n as 3. The method he used to reach this figure is unknown. Brahmagupta born 30 BC wrote two treatises on mathematics and astronomy: the Brahmasphutasiddhanta The Correctly Established Doctrine of Brahma but often translated as The Opening of the Universe , and the Khandakhadyaka Edible Bite which mostly expands the work of Aryabhata.

As a mathematician he is considered as the father of arithmetic, algebra, and numerical analysis. Zhang Heng AD was an astronomer, mathematician, inventor, geographer, cartographer, artist, poet, statesman and literary scholar. He proposed a theory of the universe that compared it to an egg. The Earth is like the yolk of the egg, lying alone at the center. The sky is large and the Earth is small..

According to him the universe originated from chaos. He said that the Sun, Moon and planets were on the inside of the sphere and moved at different rates. He demonstrated that the Moon did not have independent light, but that it merely reflected the light from the sun. He is most famous in the West for his rotating celestial globe, and inventing in the first seismograph for measuring earthquakes.

He proposed about 3. He also compared the celestial circle to the width i. Claudius Ptolemaeus around AD known in English as Ptolemy, was a mathematician, geographer, astrologer, poet of a single epigram in the Greek Anthology, and most importantly astronomer.

He made a map of the ancient world in which he employed a coordinate system very similar to the latitude and longitude of today. One of his most important achievements was his geometric calculations of semichords. Wang Fan was a mathematician and astronomer. He calculated the distance from the Sun to the Earth, but his geometric model was not correct.

Liu Hui around wrote two works. The first one was an extremely important commentary on the Jiuzhang suanshu, more commonly called Nine Chapters on the Mathematical Art, which came into being in the Eastern Han Dynasty, and believed to have been originally written around BC. It should be noted that very little is known about the mathematics of ancient China.

In BC, the emperor Shi Huang of the Chin dynasty had all of the manuscript of the kingdom burned. In Jiuzhang suanshu, Liu Hui used a variation of the Archimedean inscribed regular polygon with sides to approximate n as 3. About Pappus of Alexandria around was born in Alexandria, Egypt, and either he was a Greek or a Hellenized Egyptian.

The written records suggest that, Pappus. His major work is Synagoge or the Mathematical Collection, which is a compendium of mathematics of which eight volumes have survived. Pappus' Book IV contains various theorems on circles, study of various curves, and an account of the three classical problems of antiquity the squaring of the circle, the duplication of a cube, and the trisection of an angle.

For squaring the circle, he used Dinostratus quadratrix and his proof is a reductio ad absurdum. Pappus is remembered for Pappus's centroid theorem, Pappus's chain, Pappus's harmonic theorem, Pappus's hexagon theorem, Pappus's trisection method, and for the focus and directrix of an ellipse. Tsu Ch'ung-chih Zu Chongzhi created various formulas that have been used throughout history.

With his son he used a variation of Archimedes method to find 3. In Chinese this fraction is known as Milu. To compute this accuracy for n, he must have taken an inscribed regular 6 x gon and performed lengthy calculations. In fact, by using the method of averaging, we have. Bhaskara II or Bhaskaracharya working wrote Siddhanta Siromani crown of treatises , which consists of four parts, namely, Leelavati Bijaganitam, Grahaganitam and Goladhyaya. The first two exclusively deal with mathematics and the last two with astronomy.

His popular text Leelavati was written in AD in the name of his daughter. His contributions to mathematics include: a proof of the Pythagorean theorem, solutions of quadratic, cubic, and quartic indeterminate equations, solutions of indeterminate quadratic equations, integer solutions of linear and quadratic indeterminate equations, a cyclic Chakravala method for solving indeterminate equations, solutions of the Pell's equation and solutions of Diophantine equations of the second order.

He solved quadratic equations with more than one unknown, and found negative and irrational solutions, provided preliminary concept of infinitesimal calculus, along with notable contributions toward integral calculus, conceived differential calculus, after discovering the derivative and differential coefficient, stated Rolle's theorem, calculated the derivatives of trigonometric functions and formulae and developed spherical trigonometry.

He conceived the modern mathematical convention that when a finite number is divided by zero, the result is infinity. He gave several. The first value may have been taken from Aryabhatta. This approximation has also been credited to Liu Hui and Zu Chongzhi. Anicius Manlius Severinus Boethius around introduced the public use of sun-dials, water-clocks, etc. His integrity and attempts to protect the provincials from the plunder of the public officials brought on him the hatred of the Court.

King Theodoric sentenced him to death while absent from Rome, seized at Ticinum now Pavia , and in the baptistery of the church there tortured by drawing a cord round his head till the eyes were forced out of the sockets, and finally beaten to death with clubs on October 23, His Geometry consists of the enunciations only of the first book of Euclid, and of a few selected propositions in the third and fourth books, but with numerous practical applications to finding areas, etc.

According to him, the circle had been squared in the period since Aristotle's time, but noted that the proof was too long. His task along with several other scholars was to translate the Greek and Sanskrit scientific manuscripts.

They also studied, and wrote on algebra, geometry and astronomy. There al-Khwarizmi encountered the Hindu place-value system based on the numerals 0,1,2,3,4,5,6,7,8,9, including the first use of zero as a place holder in positional base notation, and he wrote a treatise around AD, on what we call Hindu-Arabic numerals.

The Arabic text is lost but a Latin translation, Algoritmi de numero Indorum that is, al-Khwarizmi on the Hindu Art of Reckoning , a name given to the work by Baldassarre Boncompagni in , much changed from al-Khwarizmi's original text of which even the title is unknown is known.

The French Minorite friar Alexander de Villa Dei, who taught in Paris around , mentions the name of an Indian king named Algor as the inventor of the new 'art', which itself is called the algorismus. Thus, the word 'algorithm' was tortuously derived from al-Khwarizmi Alch-warizmi, al-Karismi, Algoritmi, Algorismi, Algorithm , and has remained in use to this day in the sense of an arithmetic operation.

This Latin translation was crucial in the introduction of Hindu-Arabic numerals to medieval Europe. Mahavira in his work Ganita Sara Samgraha summarized and extended the works of Aryabhatta, Bhaskara, Brahmagupta and Bhaskaracharya.

He was the first person to mention that no real square roots of negative numbers can exist. According to Mahavira whatever is there in all the three worlds, which are possessed of moving and non-moving beings, all that indeed cannot exist without mathematics. Franco von Luttich around claimed to have contributed the only important work in the Christian era on squaring the circle. His works are published in six books, but only preserved in fragments.

Fibonacci Leonardo of Pisa around after the Dark Ages is considered the first to revive mathematics in Europe. He wrote Liber Abbaci Book of the Abacus in In this book, he quotes that 'The nine Indian numerals are His Practica geometria, a collection of useful theorems from geometry and what would eventually be named trigonometry appeared in , which was followed five years later by Liber quadratorum, a work on indeterminate analysis.

A problem in Liber Abbaci led to the introduction of the Fibonacci sequence for which he is best remembered today; however, this sequence earlier appeared in the works of Pingala about BC and Virahanka about AD. He was one of the four greatest contemporary mathematicians. Campanus wrote a Latin edition of Euclid's Elements in 15 books around Albert of Saxony around was a German philosopher known for his contributions to logic and physics.

He wrote a long treatise De quadratura circuli Question on the Squaring of the Circle consisting mostly philosophy. Madhava of Sangamagramma's work has come to light only very recently. Although there is some evidence of mathematical activities in Kerala India prior to Madhava, e.

Madhava was the first to have invented the ideas underlying infinite series expansions of functions, power series, trigonometric series of sine, cosine, tangent and arctangent, which is. He also gave rational approximations of infinite series, tests of convergence of infinite series, estimate of an error term, early forms of differentiation and integration and the analysis of infinite continued fractions.

He fully understood the limit nature of the infinite series. Madhava discovered the solutions of transcendental transcends the power of algebra equations by iteration, and found the approximation of transcendental numbers by continued fractions. He also gave many methods for calculating the circumference of a circle. The value of n correct to 13 decimal places is attributed to Madhava. However, the text Sadratnamala, usually considered as prior to Madhava, while some researchers have claimed that it was compiled by Madhava, gives the astonishingly accurate value of n correct to 17 decimal places.

In these works al-Kashi showed a great venality in numerical work. In , he calculated n to 14 decimal places, and later in to 16 decimal places. For this, he used classical polygon method of 6 x sides. He studied under Nicholas de Cusa, and one of his most famous pupils is Regiomontanus. He calculated tables of sines for every minute of arc for a radius of , units.

This table was published in He was a German cardinal of the Roman Catholic Church, a philosopher, jurist, mathematician and an astronomer. He made important contributions to the field of mathematics by developing the concepts of the infinitesimal and of relative motion. Nicholas thought this to be the exact value. Nicholas said, if we can approach the Divine only through symbols, then it is most suitable that we use mathematical symbols, for these have an indestructible certainty.

He also said that no perfect circle can exist in the universe. In accordance with his wishes, his heart is within the chapel altar at the Cusanusstift in Kues. He was the first to study Greek mathematical works in order to make himself acquainted with the methods of reasoning and results used there. He also well read the works of the Arab mathematicians. In most of this study, he compiled in his De Triangulis, which was completed in , however, was published only in Regiomontanus used algebra to find solutions of geometrical problems.

He criticized Nicholas of Cusa's approximations and methods to approximate the value of n and gave the approximation 3. Nilakanthan Somayaji's around most notable work Tantra-sangraha elaborates and extends the contributions of Madhava. He was also the author of Aryabhatiya-Bhashya, a commentary of the Aryabhatiya. Of great significance in Nilakanthan's work includes the inductive mathematical proofs, a derivation and proof of the arctangent trigonometric function, improvements and proofs of other infinite series expansions by Madhava, and in Sanskrit poetry the series.

In the literature 5 is known as Gregory-Leibniz series. He also gave sophisticated explanations of the irrationality of. Hence, although it is only of theoretical interest, the expressions on the right are arithmetical, while n arises from geometry. We also note that the series 5 can be written as. Before Leonardo da Vinci ii5i9 was an Italian painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer.

He briefly worked on squaring the circle, or approximating n. Michael Stifel ii served in several different Churches at different positions; however, every time due to bad circumstances had to resign and flee. He was forced to take refuge in a prison after ruining the lives of many believing peasants who had abandoned work and property to accompany him to heaven.

In the later part of his life, he lectured on mathematics and theology. He invented logarithms independently of Napier using a totally different approach. His most famous work is Arithmetica integra which was published in i According to him 'the quadrature of the circle is obtained when the diagonal of the square contains i0 parts of which the diameter of the circle contains 8. His book Underweysung der Messung mit dem Zirckel und Richtscheyt provides measurement of lines, areas and solids by means of compass and ruler, particularly there is a discussion of squaring the circle.

Oronce Fine ii was a prolific author of mathematical books. He was imprisoned in i, probably for practicing judicial astrology. He approximated n as. Johannes Buteo ii , a French scholar published a book De quadratura circuli, which seems to be the first book that accounts the history of n and related problems. Valentin Otho around was a German mathematician and astronomer. Tycho Brahe was an astronomer and an alchemist and was known for his most accurate astronomical and planetary observations of his time.

His data was used by his assistant, Kepler, to derive the laws of planetary motion. He observed a new star in and a comet in In , when he wasjust 20, he lost his nose partially in a duel with another student in Wittenberg and wore throughout his life a metal insert over his nose. Zhu Zaiyu , a noted musician, mathematician and astronomer-calendarist, Prince of the Ming Dynasty, obtained the twelfth root of two.

He also gave. Adriaen Anthoniszoon was a mathematician and fortification engineer. He then averaged the numerators and the denominators to obtain the 'exact' value of n. Francois Viete is frequently called by his semi-Latin name of Vieta. In relation to the three famous problems of antiquity, he showed that the trisection of an angle and the duplication of a cube problems depend upon the solution of cubic equations.

He has been called the father of modern algebra and the foremost mathematician of the sixteenth century. In his book, Supplementum geometriae, he showed 3. He also represented n as an infinite product. The above formula 6 is one of the milestones in the history of n. The convergence of Vieta's formula was proved by Ferdinand Rudio in It is clear that Vieta's formula cannot be used for the numerical computation of n.

In fact, the square roots are much too cumbersome, and the convergence is rather slow. The Dutch ambassador presented van Roomen's book to King Henry IV with the comment that at present there is no mathematician in France capable of solving this equation. The King summoned and showed the equation to Vieta, who immediately found one solution to the equation, and then the next day presented 22 more.

However, negative roots escaped him. In return, Vieta challenged van Roomen to solve the problem ofApollonius, to construct a circle tangent to three given circles, but he was unable to obtain a solution using Euclidean geometry.

When van Roomen was shown proposer's elegant solution, he immediately traveled to France to meet Vieta, and a warm friendship developed. The same year Rooman used the classical method with sides, to approximate n to 15 correct decimal places. Joseph Justus Scaliger was a religious leader and scholar. Ludolph van Ceulen was a German who emigrated to the Netherlands. In , he was appointed to the Engineering School at Leiden, where he spent the remainder of his life teaching Mathematics, Surveying and Fortification.

He wrote several books, including Van den Circkel On The Circle, , in which he published his geometric findings, and the approximate value of n correct to 20 decimal places. For this, he reports that he used classical method with 60 x , i. This book ends with 'Whoever wants to, can come closer.

Ludolph van Ceulen in his work De Arithmetische en Geometrische fondamenten, which was published posthumously by his wife in , computed n correct to 35 decimal places by using classical method with sides. This computational feat was considered so extraordinary that his widow had all 35 digits of die Ludolphsche Zahl the Ludolphine number was engraved on his tombstone in St.

Peter's churchyard in Leiden. The tombstone was later lost but was restored in This was one of the last major attempts to evaluate n by the classical method; thereafter, the techniques of calculus were employed.

Willebrord Snell Snellius was a Dutch astronomer and mathematician. At the age of 12, he is said to have been acquainted with the standard mathematical works, while at the age of 22, he succeeded his father as Professor of Mathematics at Leiden. His fame rests mainly on his discovery in of the law of refraction, which played a significant role in the development of both calculus and the wave theory of light.

However, it is now known that this law was first discovered by Ibn Sahl in Snell cleverly combined Archimedean method with trigonometry, and showed that for each pair of bounds on n given by the classical method, considerably closer bounds can be obtained.

By his method, he was able to approximate n to seven places by using just 96 sides, and to van Ceulen's 35 decimal places by using polygons having only sides. The classical method with such polygons yields only two and fifteen decimal places. Yoshida Mitsuyoshi was working during Edo period. His work named as Jinkoki deals with the subject of soroban arithmetic, including square and cube root operations.

In this work, he used 3. Christoph Christophorus Grienberger was an Austrian Jesuit astronomer. The crater Gruemberger on the Moon is named after him. He used Snell's refinement to compute n to 39 decimal places. This was the last major attempt to compute n by the Archimedes method. Celiang quanyi Complete Explanation of Methods of Planimetry and Stereometry gives without proof the following bounds 3. William Oughtred , an English mathematician offered free mathematical tuition to pupils, which included even Wallis.

In this work, he introduced the x symbol for multiplication, and the proportion sign double colon His notation was used by Isaac Barrow a few years later, and David Gregory Before him, mathematicians described n in round-about ways such as 'quantitas, in quam cum multipliectur diameter, proveniet circumferential', which means 'the quantity which, when the diameter is multiplied by it, yields the circumference'. This discovery played an important role in the development of the theory of logarithms and an eventual recognition of the natural logarithm.

In , Nicolaus Mercator Kauffmann wrote a treatise entitled Logarithmo-technica, and discovered the series. In his book, Opus geometricum quadraturae circuli et sectionum coni he proposed at least four methods of squaring the circle, but none of them were implemented. The fallacy in his quadrature was pointed out by Huygens. In , he published his Discourse.

The rectangular coordinate system is credited to Descartes. He is regarded as a genius of the first magnitude. He was one of the most important and influential thinkers in human history and is sometimes called the founder of modern philosophy. After his death, a novel geometric approach to approximate n was found in his papers.

His method consisted of doubling the number of sides of regular polygons while keeping the perimeter constant. In modern terms, Descartes' method can be summarized as. John Wallis in was appointed as Savilian professor of geometry at the University of Oxford, which he continued for over 50 years until his death. He was the most influential English mathematician before Newton. In his most famous work, Arithmetica infinitorum, which he published in , he established the formula.

This formula is a great milestone in the history of n. Like Viete's formula 6 , Wallis had found n in the form of an infinite product, but he was the first in history whose infinite sequence involved only rational operations. In his Opera Mathematica I , Wallis introduced the term continued fraction. He had great ability to do mental calculations. He slept badly and often did mental calculations as he lay awake in his bed.

On 22 December , he when in bed, occupied himself in finding the integral part of the square root of 3 x ; and several hours afterward wrote down the result from memory. Two months later, he was challenged to extract the square root of a number of 53 digits; this he performed mentally, and a month later he dictated the answer which he had not meantime committed to writing.

Wallis' life was embittered by quarrels with his contemporaries including Huygens, Descartes, and the political philosopher Hobbes, which continued for over 20 years, ending only with Hobbes' death. Hobbes called Arithmetica infinitorum 'a scab of symbols, and claimed to have squared the circle. It seems that to some, individual's quarrels give strength, encouragement and mental satisfaction. To derive 8 , we note that. Finally, a combination of 11 and 12 immediately gives We also note that.

William Brouncker, 2nd Viscount Brouncker was one of the founders and the second President of the Royal Society. His mathematical contributions are: reproduction of Brahmagupta's solution of a certain indeterminate equation, calculations of the lengths of the parabola and cycloid, quadrature of the hyperbola which required approximation of the natural logarithm function by infinite series and the study of generalized. He undertook some calculations to verify formula 8 , and showed that 3.

He also converted Wallis' result 8 into the continued fraction. Neither of the expressions 8 , and 13 ; however, later has served for an extensive calculation of n. Christiaan Huygens is famous for his invention of the pendulum clock, which was a breakthrough in timekeeping.

He formulated the second law of motion of Newton in a quadratic form, and derived the now well-known formula for the centripetal force, exerted by an object describing a circular motion. For the computation of n, he gave the correct proof of Snell's refinement, and using an inscribed polygon of only 60 sides obtained the bounds 3. Sir Isaac Newton , hailed as one of the greatest scientist-mathematicians of the English-speaking world, had the following more modest view of his own monumental achievements:' As he examined these shells, he discovered to his amazement more and more of the intricacies and beauties that lay in them, which otherwise would remain locked to the outside world.

At the age of 26, he succeeded Barrow as Lucasian professor of mathematics at Cambridge. About him, Aldous Huxley had said 'If we evolved a race of Isaac Newtons, that would not be progress. For the price Newton had to pay for being a supreme intellect was that he was incapable of friendship, love, fatherhood and many other desirable things. As a man he was a failure; as a monster he was superb.

Newton made some of the greatest discoveries the world ever knew at that time. Newton discovered: 1. The nature of colors. The law of gravitation and the laws of mechanics. The fluxional calculus. Most of the history books say that to compute n Newton used the series. Later, he wrote 'I am ashamed to tell you to how many figures I carried these computations, having no other business at the time. His result was not published until posthumously. Thus, binomial expansion gives.

Thomas Hobbes of Malmesbury was an English philosopher, best known today for his work on political philosophy. He also contributed in several other diverse fields, including history, geometry, the physics of gases, theology, ethics and general philosophy. In , he also gave the approximation V James Gregory published two books Vera circuli et hyperbolae quadratura in , and Geometriae pars universalis in In the first book particularly, he showed that the area of a circle can be obtained in the form of an infinite convergent series only, and hence inferred that the quadrature of the circle was impossible.

In the second book, he attempted to write calculus systematically, which perhaps made the basis of Newton's fluxions. This book also contains series expansions of sin x , cos x , arcsin x and arccos x ; however, as we have seen earlier these expansions were known to Madhava. Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as In early , he discovered Taylor's theorem published by Taylor in ; however, he did not publish.

Later in , he rediscovered Nilakanthan's arctangent series 5. In his Vera circuli ethyperbolae quadratura of , Gregory tried to show that n was a transcendental number, but his attempt, though very interesting, was not successful. Huygens made detailed and rather biased criticisms of it. Pietro Mengoli studied at the University of Bologna, and became a professor there in for the next 39 years of his life. Besides proposing Basel problem, he proved that the harmonic series does not converge, established that the alternating harmonic series is equal to the natural logarithm of 2, published on the problem of squaring the circle, and provided a proof that Wallis' product 8 for n is correct.

Gottfried Wilhelm von Leibniz was a universal genius who won recognition in many fields - law, philosophy, religion, literature, politics, geology, metaphysics, alchemy, history and mathematics. He shares credit with Newton in developing calculus independently. He popularized and gave several mathematical symbols.

Leibniz tried to reunite the Protestant and Catholic churches. He in binary arithmetic saw the image of Creation. He imagined that Unity represented God, and Zero the void; that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in the binary system of numeration.

He communicated his idea to the Jesuit Grimaldi, who was the President of the Chinese tribunal for mathematics in the hopes that it would help convert to Christianity the Emperor of China, who was said to be very fond of the Sciences. Later Leibniz became an expert in the Sanskrit language and the culture of China. For calculating n, he developed a method without any reference to a circle. In , he also rediscovered Nilakanthan's arctangent series 5 , whose beautyhe described by saying that Lord loves odd numbers.

Leibniz even invented a calculating machine that could perform the four operations and extract roots. Isomura Yoshinori employed a sided inscribed polygon to obtain 3. He was the first to utilize a steel spring for suspension of the pendulum of a clock. He used a new approximate geometric construction for n to obtain. Takebe Katahiro also known as Takebe Kenko played a critical role in the development of a crude version of the calculus. He also created charts for trigonometric functions.

He used polygon just 1, sides approximation and a numerical method which is essentially equivalent to the Romberg algorithm rediscovered by Sigmund Romberg, to compute n to 41 digits. In , Takebe obtained power series expansion of sin-1 x 2,15 years earlier than Euler. Around , essentially the same. Abraham Sharp was a mathematician and astronomer. In , he joined the Greenwich Royal Observatory and did notable work, improving instruments and showing great skill as a calculator. He also worked on geometry and improved logarithmic tables.

Sharp used 17 to calculated n to 72 decimal places out of which 71 digits are correct. It is believed that Madhava of Sangamagramma used the same series in the fourteenth century to compute the value of n correct to 11 decimal places. Seki Takakazu also known as Seki Kowa is generally regarded as the greatest Japanese mathematician.

He was a prolific writer, and a number of his publications are either transcripts of mathematics from Chinese into Japanese, or commentaries on certain works of well-known Chinese mathematicians. His interests in mathematics ranged recreational mathematics, magic squares and magic circles, solutions of higherorder and indeterminate equations, conditions for the existence of positive and negative roots of polynomials, and continued fractions.

He discovered determinants ten years before Leibniz, and the Bernoulli numbers a year before Bernoulli. He used polygon of sides and Richardson extrapolation rediscovered by Alexander Craig Aitken, to compute n to 10 digits. Some authors believe that he also used the formula.

Oliver de Serres believed that by weighing a circle and a triangle equal to the equilateral triangle inscribed he had found that the circle was exactly double of the triangle, not being aware that this double is exactly the hexagon inscribed in the same circle. William Jones , an obscure English writer, represented the ratio of the circumference of a circle to its diameter by n in his Synopsis Palmariorum Matheseos New Introduction to the Mathematics.

He used the letter n as an abbreviation for the Greek word perimetros periphery of a circle with unit diameter. In his book, he published the value of n correct to decimal places. John Machin was a professor of astronomy at Gresham College, London. He also served as secretary of the Royal Society during Machin is best. The proof of 18 also follows by comparing the angles in the identity the idea originally goes back to Caspar Wessel who presented his work in to the Royal Danish Academy of Sciences.

The series 19 certainly converges significantly faster than 5 and In fact, taking six terms of the first series and two terms of the second and paying attention to the remainders and round-off errors, we get the inequalities 3. Thus, the value of n correct to seven decimals is 3.

Several other Machin-type formulas are known, e. For a long list of such type of formulas with a discussion of their relative merits in computational work, see Lehmer Thomas Fantet de Lagny was a French mathematician who is well known for his contributions to computational mathematics. He used the series 17 to determine the value of n up to decimal places; however, only are correct.

Alexander Pope was an English poet. In his Dunciad it is mentioned that 'The mad Mathesis, now, running round the circle, finds it square. This explains the wild and fruitless attempts of squaring the circle. Sieur Malthulon France offered solutions to squaring the circle and to perpetual motion. He offered 1, crowns reward in legal form to anyone proving him wrong. Nicoli, who proved him wrong, collected the reward and abandoned it to the Hotel Dieu of Lyons.

Later, the courts gave the money to the poor. Toshikiyo Kamata used both the circumscribed and inscribed polygons and gave the bounds 3. In , he was appointed to the Commission set up by the Royal Society to solve the Newton-Leibniz dispute concerning which of them invented calculus first. He is best known for his memoir Doctrine of Chances: A method of calculating the probabilities of events in play, which was first printed in and dedicated to Newton.

In his Miscellanea Analytica published in , appears the formula for very large n,. In , De Moivre used this formula to derive the normal curve as an approximation to the binomial. Leonhard Euler was probably the most prolific mathematician who ever lived. He was born in Basel Switzerland , and had the good fortune to be tutored one day a week in mathematics by a distinguished mathematician, Johann Bernoulli Euler's energy and capacity for work were virtually boundless.

His collected works form about 60 to 80 quarto-sized volumes and it is believed that much of his work has been lost. What is particularly astonishing is that Euler became virtually sightless in his right eye during the mids, and was blind for the last 17 years of his life, and this was one of the most productive periods. In , Mengoli asked for the precise summation. Basel problem appears in number theory, e. Chartres, An integer that is not divisible by the square of any prime number is said to be square free.

The above proof of Euler is based on manipulations that were not justified at the time, and it was not until that he was able to produce a truly rigorous proof. Today, several different proofs of 21 are known in the literature. Euler also established the following series:. Euler's ideas were taken up years later by George Friedrich Bernhard Riemann in his seminal paper, On the Number of Primes Less Than a Given Magnitude, in which he defined his zeta function. After his death, many of the finest mathematicians of the world have exerted their strongest efforts and created new branches of analysis in attempts to prove.

Since then with one exception, every statement has been settled in the sense Riemann expected. It stands today as the most important unsolved problem of mathematics, and perhaps the most difficult problem that the mind of man has ever conceived. The letter n was first used by Euler in in his Variae observationes circa series infinitas.

Until that time, he had been using the letters p , or c In , Christian Goldbach also used n. After the publication of Euler's treatise: In-troductio in Analysin Infinitorum , n became a standard symbol, as was the case with other notations he adopted.

In , Euler also showed that both e and e2 are irrational and gave several continued fractions for e. In another paper, De variis modis circuli quadraturam numerisproxime exprimendi of , Euler derived the formulas. Matsunaga Yoshisuke died in was a prolific writer. In modern terms, he used the hypergeometric series. The following curious infinite product was also given by Euler:. Henry Sullamar, a real Bedlamite, found the quadrature of the circle in the number inscribed on the forehead of the beast in the Revelations.

He published periodically every two or three years some pamphlet in which he endeavored to prop his discovery. He found that the circle was equal to the square in which it is inscribed, i. He offered a reward for the detection of any error, and actually deposited 10, francs as earnest of , But the courts did not allow any one to recover.

Euler in his treatise De relatione inter ternas pluresve quantitates instituenda, which was published ten years later, wrote 'It appears to be fairly certain that the periphery of a circle constitutes such a peculiar kind of transcendental quantities that it can in no way be compared with other quantities, either roots or other transcendentals. This conjecture haunted mathematicians for years. The following expansion is due to Euler:.

Georges Louis Leclerc Comte of Buffon was a naturalist, mathematician, cosmologist and encyclopedic author. In the literature, this problem is known as Buffon's needle problem. This was the earliest problem in geometric probability to be solved. By actually performing this experiment, a large number of times and noting the number of successful cases, we can compute an approximation for n.

Johann Heinrich Lambert was the first to introduce hyperbolic functions into trigonometry. He wrote landmark books on geometry, the theory of cartography, and perspective in art. He is also credited for expressing Newton's second law of motion in the notation of the differential calculus.

Lambert used the properties of continued fractions to show that n is irrational. He published a more general result in Lambert also showed that the functions ex and tan x cannot assume rational values if x is. Arima Yoriyuki was a Japanese mathematician of the Edo period. He found the following rational approximation of n, which is correct to 29 digits. The French Academy of Sciences passed a resolution henceforth not to examine any more solutions of the problem of squaring the circle.

In fact, it became necessary to protect its officials against the waste of time and energy involved in examining the efforts of circle squarers. A fewyears later, the Royal Society in London also banned consideration of any further proofs of squaring the circle.

This decision of the Royal Society was described by Augustus De Morgan about years later as the official blow to circle-squarers. Charles Hutton was an English mathematician. He wrote several mathematical texts. In , he was elected a fellow of the Royal Society of London. He suggested Machin's stratagem in the form. It ought to be needless to say that there was no reward offered for squaring the circle. Euler used his expansion 23 to evaluate right terms of 24 , to calculate n to 20 decimal places in one hour!

Franz Xaver Freiherr von Zach discovered a manuscript by an unknown author in the Radcliffe Library, Oxford, which gives the correct value of n to decimal places. Zach was elected a member of the Royal Swedish Academy of Sciences in , a Fellow of the Royal Society in , and an honorary member of the Hungarian. Academy of Sciences in Asteroid Zachia and the crater Zach on the Moon are named after him.

He wrote six scientific papers. The record of de Lagny of digits seems to have stood until , when Vega, using a new series for the arctangent discovered by Euler in , calculated decimal places correct. Vega's result showed that de Lagny's string of digits had a 7 instead of an 8 in the th decimal place.

His article was not published until six years later, in correct. Vega retained his record for 52 years until The Legendre crater on the Moon is named after him. Legendre, in his Elements de Geometrie used a slightly modified version of Lambert's argument to prove the irrationality of n more rigorously, and also gave a proof that n2 is irrational. He writes: 'It is probable that the number n is not even contained among the algebraic irrationalities, i.

But, it seems to be very difficult to prove this strictly'. Ajima Naonobu , also known as Ajima Chokuyen, was a Japanese mathematician of the Edo period. The series he developed can be simplified as. It is interesting to note that the above series follows from 5 by using an acceleration technique known in the literature as Euler's transform. It can also be derived from the Wallis product formula 8.

Lorenzo Mascheroni was educated with the aim of becoming a priest and he was ordained at the age of In , he calculated Euler's constant to 32 19 correct decimal places. Lorenzo dedicated his book, Geometria del compasso, to Napoleon Bonaparte. In this work, he proved that all Euclidean constructions can be made with compasses alone, so a straight edge in not needed. However, it was proved earlier in by the Danish mathematician Georg Mohr He claimed that compasses are more accurate then those of a ruler.

Karl Friedrich Gauss was one of the greatest mathematicians of all time. Alexander von Humboldt , the famous traveler and amateur of the sciences, asked Pierre Simon de Laplace who was the greatest mathematician in Germany, Laplace replied Johann Friedrich 'Pfaff' In his reply, Pfaff showed that for any positive numbers x0 and y0 these sequences converge monotonically to a common limit given by. Pfaff's letter was unpublished. In , Carl Wilhelm Borchardt o work was published in which he rediscovered this result which now bears his name.

For this, it suffices to note that:. Clearly, the recurrence relations 2 are different from In fact, 2 minimize the count of arithmetic operations. From 29 and 3o , several known and new recurrence relations can be obtained. Gauss also developed the Machin-type formula. He also estimated the value of n by using lattice theory and considering a lattice inside a large circle, but he did not pursue it further.

Sakabe Kohan developed the series. Wada Yenzo Nei known as Wada Yasushi, developed over one hundred infinite series expressing directly or indirectly n. One of his series can be written as. Specht of Berlin published a geometric construction in Crelle's Journal, Volume 3, p. On weighing them, they were found to be exactly the same weight, which proves that, as each are of the same thickness, the surfaces must also be precisely similar. The rule, therefore, is that the square is to the circle as 17 to We believe for the square it must be the side not the diameter.

Joseph LaComme 'at a time when he could neither read nor write being desirous to ascertain what quantity of stones would be required to prove a circular reservoir he had constructed, consulted a mathematics professor. He was told that it was impossible to determine the full amount, as no one had yet found the exact relation between the circumference of a circle and its diameter. He then taught himself to read and write, and managed to acquire some knowledge of arithmetic by which he verified his mechanical solution.

Joseph was honored for his profound discovery with several medals of the first class, bestowed by Parisian societies. William Rutherford was an English mathematician. He calculated n to places of which were later found to be correct. For this, he employed Euler's formula. Johann Martin Zacharias Dase was a calculating prodigy.

At the age of 15, he gave exhibitions in Germany, Austria and England. His extraordinary calculating powers were timed by renowned mathematicians including Gauss. He multiplied 79,, x 93,, in 54 seconds; two digit numbers in 6 minutes; two digit numbers in 40 minutes; and two digit numbers in 8 hours 45 minutes.

In , he made acquaintance with Viennese mathematician L. Schulz von Strasznicky who suggested him to apply his powers to scientific purposes. When he was 20, Strasznicky taught him the use of the formula. Heron of Alexandria about 75 AD in his Metrica , which had been lost for centuries until a fragment was discovered in , followed by a complete copy in , refers to an Archimedes work, where he gives the bounds.

The curve is today called the Archimedean Spiral. Daivajna Varahamihira working BC was an astronomer, mathematician and astrologer. His picture may be found in the Indian Parliament along with Aryabhata. He also made some important mathematical discoveries such as giving certain trigonometric formulae; developing new interpolation methods to produce sine tables; constructing a table for the binomial coefficients; and examining the pandiagonal magic square of order four.

He was the first to describe direct measurement of distances by the revolution of a wheel. About 10 BC. Liu created a new astronomical system, called Triple Concordance. This was first mentioned in the Sui shu He also found the approximations 3. Around 5 AD. Liu created a catalog of 1, stars, where he used the scale of 6 magnitudes. The method he used to reach this figure is unknown. Brahmagupta born 30 BC wrote two treatises on mathematics and astronomy: the Brahmasphutasiddhanta The Correctly Established Doctrine of Brahma but often translated as The Opening of the Universe , and the Khandakhadyaka Edible Bite which mostly expands the work of Aryabhata.

As a mathematician he is considered as the father of arithmetic, algebra, and numerical analysis. Zhang Heng AD was an astronomer, mathematician, inventor, geographer, cartographer, artist, poet, statesman and literary scholar. He proposed a theory of the universe that compared it to an egg.

The Earth is like the yolk of the egg, lying alone at the center. According to him the universe originated from chaos. He said that the Sun, Moon and planets were on the inside of the sphere and moved at different rates. He demonstrated that the Moon did not have independent light, but that it merely reflected the light from the sun. He is most famous in the West for his rotating celestial globe, and inventing in the first seismograph for measuring earthquakes.

He proposed 10 about 3. He also compared the celestial circle to the width i. Claudius Ptolemaeus around AD known in English as Ptolemy, was a mathematician, geographer, astrologer, poet of a single epigram in the Greek Anthology, and most importantly astronomer. He made a map of the ancient world in which he employed a coordinate system very similar to the latitude and longitude of today. One of his most important achievements was his geometric calculations of semichords.

Wang Fan was a mathematician and astronomer. He calculated the distance from the Sun to the Earth, but his geometric model was not correct. Liu Hui around wrote two works. The first one was an extremely important commentary on the Jiuzhang suanshu , more commonly called Nine Chapters on the Mathematical Art , which came into being in the Eastern Han Dynasty, and believed to have been originally written around BC.

It should be noted that very little is known about the mathematics of ancient China. In BC, the emperor Shi Huang of the Chin dynasty had all of the manuscript of the kingdom burned. About Pappus of Alexandria around was born in Alexandria, Egypt, and either he was a Greek or a Hellenized Egyptian.

The written records suggest that, Pappus lived in Alexandria during the reign of Diocletian His major work is Synagoge or the Mathematical Collection, which is a compendium of mathematics of which eight volumes have survived.

For squaring the circle, he used Dinostratus quadratrix and his proof is a reductio ad absurdum. With his son he used a variation of Archimedes method to find 3. In fact, by using the method of averaging, we have. Bhaskara II or Bhaskaracharya working wrote Siddhanta Siromani crown of treatises , which consists of four parts, namely, Leelavati Bijaganitam , Grahaganitam and Goladhyaya.

The first two exclusively deal with mathematics and the last two with astronomy. His popular text Leelavati was written in AD in the name of his daughter. He conceived the modern mathematical convention that when a finite number is divided by zero, the result is infinity. The first value may have been taken from Aryabhatta.

This approximation has also been credited to Liu Hui and Zu Chongzhi. Anicius Manlius Severinus Boethius around introduced the public use of sun-dials, water-clocks, etc. His integrity and attempts to protect the provincials from the plunder of the public officials brought on him the hatred of the Court. King Theodoric sentenced him to death while absent from Rome, seized at Ticinum now Pavia , and in the baptistery of the church there tortured by drawing a cord round his head till the eyes were forced out of the sockets, and finally beaten to death with clubs on October 23, His Geometry consists of the enunciations only of the first book of Euclid, and of a few selected propositions in the third and fourth books, but with numerous practical applications to finding areas, etc.

His task along with several other scholars was to translate the Greek and Sanskrit scientific manuscripts. They also studied, and wrote on algebra, geometry and astronomy. There al-Khwarizmi encountered the Hindu place-value system based on the numerals 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , including the first use of zero as a place holder in positional base notation, and he wrote a treatise around AD, on what we call Hindu-Arabic numerals.

This Latin translation was crucial in the introduction of Hindu-Arabic numerals to medieval Europe. Mahavira in his work Ganita Sara Samgraha summarized and extended the works of Aryabhatta, Bhaskara, Brahmagupta and Bhaskaracharya. He was the first person to mention that no real square roots of negative numbers can exist. According to Mahavira whatever is there in all the three worlds, which are possessed of moving and non-moving beings, all that indeed cannot exist without mathematics.

His works are published in six books, but only preserved in fragments. Fibonacci Leonardo of Pisa around after the Dark Ages is considered the first to revive mathematics in Europe. He wrote Liber Abbaci Book of the Abacus in His Practica geometria , a collection of useful theorems from geometry and what would eventually be named trigonometry appeared in , which was followed five years later by Liber quadratorum , a work on indeterminate analysis.

A problem in Liber Abbaci led to the introduction of the Fibonacci sequence for which he is best remembered today; however, this sequence earlier appeared in the works of Pingala about BC and Virahanka about AD. He was one of the four greatest contemporary mathematicians.

Albert of Saxony around was a German philosopher known for his contributions to logic and physics. He wrote a long treatise De quadratura circuli Question on the Squaring of the Circle consisting mostly philosophy. Although there is some evidence of mathematical activities in Kerala India prior to Madhava, e. Madhava was the first to have invented the ideas underlying infinite series expansions of functions, power series, trigonometric series of sine, cosine, tangent and arctangent, which is.

He also gave rational approximations of infinite series, tests of convergence of infinite series, estimate of an error term, early forms of differentiation and integration and the analysis of infinite continued fractions. He fully understood the limit nature of the infinite series. Madhava discovered the solutions of transcendental transcends the power of algebra equations by iteration, and found the approximation of transcendental numbers by continued fractions.

He also gave many methods for calculating the circumference of a circle. In these works al-Kashi showed a great venality in numerical work. He studied under Nicholas de Cusa, and one of his most famous pupils is Regiomontanus.

He calculated tables of sines for every minute of arc for a radius of , units. This table was published in He was a German cardinal of the Roman Catholic Church, a philosopher, jurist, mathematician and an astronomer. He made important contributions to the field of mathematics by developing the concepts of the infinitesimal and of relative motion. Nicholas thought this to be the exact value. Nicholas said, if we can approach the Divine only through symbols, then it is most suitable that we use mathematical symbols, for these have an indestructible certainty.

He also said that no perfect circle can exist in the universe. In accordance with his wishes, his heart is within the chapel altar at the Cusanusstift in Kues. He was the first to study Greek mathematical works in order to make himself acquainted with the methods of reasoning and results used there.

He also well read the works of the Arab mathematicians. In most of this study, he compiled in his De Triangulis , which was completed in , however, was published only in Regiomontanus used algebra to find solutions of geometrical problems. He was also the author of Aryabhatiya-Bhashya , a commentary of the Aryabhatiya. In the literature 5 is known as Gregory-Leibniz series. We also note that the series 5 can be written as. Before Leonardo da Vinci was an Italian painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer.

Michael Stifel served in several different Churches at different positions; however, every time due to bad circumstances had to resign and flee. He was forced to take refuge in a prison after ruining the lives of many believing peasants who had abandoned work and property to accompany him to heaven. In the later part of his life, he lectured on mathematics and theology.

He invented logarithms independently of Napier using a totally different approach. His most famous work is Arithmetica integra which was published in His book Underweysung der Messung mit dem Zirckel und Richtscheyt provides measurement of lines, areas and solids by means of compass and ruler, particularly there is a discussion of squaring the circle. He was imprisoned in , probably for practicing judicial astrology.

Valentin Otho around was a German mathematician and astronomer. Tycho Brahe was an astronomer and an alchemist and was known for his most accurate astronomical and planetary observations of his time. His data was used by his assistant, Kepler, to derive the laws of planetary motion.

He observed a new star in and a comet in In , when he was just 20, he lost his nose partially in a duel with another student in Wittenberg and wore throughout his life a metal insert over his nose. Zhu Zaiyu , a noted musician, mathematician and astronomer-calendarist, Prince of the Ming Dynasty, obtained the twelfth root of two. Simon van der Eycke Netherland published an incorrect proof of the quadrature of the circle. In , he gave the value 3. Adriaen Anthoniszoon was a mathematician and fortification engineer.

In relation to the three famous problems of antiquity, he showed that the trisection of an angle and the duplication of a cube problems depend upon the solution of cubic equations. He has been called the father of modern algebra and the foremost mathematician of the sixteenth century.

In his book, Supplementum geometriae , he showed 3. In fact, the square roots are much too cumbersome, and the convergence is rather slow. The King summoned and showed the equation to Vieta, who immediately found one solution to the equation, and then the next day presented 22 more. However, negative roots escaped him.

In return, Vieta challenged van Roomen to solve the problem of Apollonius, to construct a circle tangent to three given circles, but he was unable to obtain a solution using Euclidean geometry. Joseph Justus Scaliger was a religious leader and scholar. Ludolph van Ceulen was a German who emigrated to the Netherlands. In , he was appointed to the Engineering School at Leiden, where he spent the remainder of his life teaching Mathematics, Surveying and Fortification.

This computational feat was considered so extraordinary that his widow had all 35 digits of die Ludolphsche Zahl the Ludolphine number was engraved on his tombstone in St. The tombstone was later lost but was restored in Willebrord Snell Snellius was a Dutch astronomer and mathematician. At the age of 12, he is said to have been acquainted with the standard mathematical works, while at the age of 22, he succeeded his father as Professor of Mathematics at Leiden.

His fame rests mainly on his discovery in of the law of refraction, which played a significant role in the development of both calculus and the wave theory of light. However, it is now known that this law was first discovered by Ibn Sahl in The classical method with such polygons yields only two and fifteen decimal places. Yoshida Mitsuyoshi was working during Edo period. His work named as Jinkoki deals with the subject of soroban arithmetic, including square and cube root operations.

In this work, he used 3. Christoph Christophorus Grienberger was an Austrian Jesuit astronomer. The crater Gruemberger on the Moon is named after him. Celiang quanyi Complete Explanation of Methods of Planimetry and Stereometry gives without proof the following bounds 3. William Oughtred , an English mathematician offered free mathematical tuition to pupils, which included even Wallis. His notation was used by Isaac Barrow a few years later, and David Gregory This discovery played an important role in the development of the theory of logarithms and an eventual recognition of the natural logarithm.

In , Nicolaus Mercator Kauffmann wrote a treatise entitled Logarithmo-technica , and discovered the series. In his book, Opus geometricum quadraturae circuli et sectionum coni he proposed at least four methods of squaring the circle, but none of them were implemented. The fallacy in his quadrature was pointed out by Huygens. In , he published his Discourse on Method , which contained important mathematical work, and three essays, Meteors, Dioptrics and Geometry, produced an immense sensation and his name became known throughout Europe.

The rectangular coordinate system is credited to Descartes. He is regarded as a genius of the first magnitude. He was one of the most important and influential thinkers in human history and is sometimes called the founder of modern philosophy. His method consisted of doubling the number of sides of regular polygons while keeping the perimeter constant. John Wallis in was appointed as Savilian professor of geometry at the University of Oxford, which he continued for over 50 years until his death.

He was the most influential English mathematician before Newton. In his most famous work, Arithmetica infinitorum , which he published in , he established the formula. In his Opera Mathematica I , Wallis introduced the term continued fraction. He had great ability to do mental calculations. He slept badly and often did mental calculations as he lay awake in his bed. Two months later, he was challenged to extract the square root of a number of 53 digits; this he performed mentally, and a month later he dictated the answer which he had not meantime committed to writing.

Finally, a combination of 11 and 12 immediately gives We also note that. William Brouncker, 2nd Viscount Brouncker was one of the founders and the second President of the Royal Society. He undertook some calculations to verify formula 8 , and showed that 3. Christiaan Huygens is famous for his invention of the pendulum clock, which was a breakthrough in timekeeping. He formulated the second law of motion of Newton in a quadratic form, and derived the now well-known formula for the centripetal force, exerted by an object describing a circular motion.

As he examined these shells, he discovered to his amazement more and more of the intricacies and beauties that lay in them, which otherwise would remain locked to the outside world. At the age of 26, he succeeded Barrow as Lucasian professor of mathematics at Cambridge.

For the price Newton had to pay for being a supreme intellect was that he was incapable of friendship, love, fatherhood and many other desirable things. Newton made some of the greatest discoveries the world ever knew at that time. Newton discovered: 1. The nature of colors. The law of gravitation and the laws of mechanics. The fluxional calculus. His result was not published until posthumously. Thus, binomial expansion gives. Also, from geometry the area of the sector A B D is.

Thomas Hobbes of Malmesbury was an English philosopher, best known today for his work on political philosophy. He also contributed in several other diverse fields, including history, geometry, the physics of gases, theology, ethics and general philosophy.

In , he also gave the approximation James Gregory published two books Vera circuli et hyperbolae quadratura in , and Geometriae pars universalis in In the first book particularly, he showed that the area of a circle can be obtained in the form of an infinite convergent series only, and hence inferred that the quadrature of the circle was impossible.

This book also contains series expansions of sin x , cos x , arcsin x and arccos x ; however, as we have seen earlier these expansions were known to Madhava. Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as Huygens made detailed and rather biased criticisms of it. Pietro Mengoli studied at the University of Bologna, and became a professor there in for the next 39 years of his life.

Gottfried Wilhelm von Leibniz was a universal genius who won recognition in many fields - law, philosophy, religion, literature, politics, geology, metaphysics, alchemy, history and mathematics. He shares credit with Newton in developing calculus independently. He popularized and gave several mathematical symbols. Leibniz tried to reunite the Protestant and Catholic churches.

He in binary arithmetic saw the image of Creation. He imagined that Unity represented God, and Zero the void; that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in the binary system of numeration. He communicated his idea to the Jesuit Grimaldi, who was the President of the Chinese tribunal for mathematics in the hopes that it would help convert to Christianity the Emperor of China, who was said to be very fond of the Sciences.

Later Leibniz became an expert in the Sanskrit language and the culture of China. Leibniz even invented a calculating machine that could perform the four operations and extract roots. Isomura Yoshinori employed a 2 17 -sided inscribed polygon to obtain 3. He was the first to utilize a steel spring for suspension of the pendulum of a clock. Takebe Katahiro also known as Takebe Kenko played a critical role in the development of a crude version of the calculus.

He also created charts for trigonometric functions. Around , essentially the same series was rediscovered by Oyama Shokei who used it to find the expansion. Abraham Sharp was a mathematician and astronomer. In , he joined the Greenwich Royal Observatory and did notable work, improving instruments and showing great skill as a calculator.

He also worked on geometry and improved logarithmic tables. Seki Takakazu also known as Seki Kowa is generally regarded as the greatest Japanese mathematician. He was a prolific writer, and a number of his publications are either transcripts of mathematics from Chinese into Japanese, or commentaries on certain works of well-known Chinese mathematicians.

His interests in mathematics ranged recreational mathematics, magic squares and magic circles, solutions of higher-order and indeterminate equations, conditions for the existence of positive and negative roots of polynomials, and continued fractions. He discovered determinants ten years before Leibniz, and the Bernoulli numbers a year before Bernoulli.

Some authors believe that he also used the formula. Oliver de Serres believed that by weighing a circle and a triangle equal to the equilateral triangle inscribed he had found that the circle was exactly double of the triangle, not being aware that this double is exactly the hexagon inscribed in the same circle. John Machin was a professor of astronomy at Gresham College, London. He also served as secretary of the Royal Society during The proof of 18 also follows by comparing the angles in the identity the idea originally goes back to Caspar Wessel who presented his work in to the Royal Danish Academy of Sciences.

The series 19 certainly converges significantly faster than 5 and In fact, taking six terms of the first series and two terms of the second and paying attention to the remainders and round-off errors, we get the inequalities 3. For a long list of such type of formulas with a discussion of their relative merits in computational work, see Lehmer Thomas Fantet de Lagny was a French mathematician who is well known for his contributions to computational mathematics.

Alexander Pope was an English poet. This explains the wild and fruitless attempts of squaring the circle. Sieur Malthulon France offered solutions to squaring the circle and to perpetual motion. He offered 1, crowns reward in legal form to anyone proving him wrong. Nicoli, who proved him wrong, collected the reward and abandoned it to the Hotel Dieu of Lyons. Later, the courts gave the money to the poor. Toshikiyo Kamata used both the circumscribed and inscribed polygons and gave the bounds 3.

In , he was appointed to the Commission set up by the Royal Society to solve the Newton-Leibniz dispute concerning which of them invented calculus first. He is best known for his memoir Doctrine of Chances : A method of calculating the probabilities of events in play, which was first printed in and dedicated to Newton. In his Miscellanea Analytica published in , appears the formula for very large n ,.

In , De Moivre used this formula to derive the normal curve as an approximation to the binomial. Leonhard Euler was probably the most prolific mathematician who ever lived. He was born in Basel Switzerland , and had the good fortune to be tutored one day a week in mathematics by a distinguished mathematician, Johann Bernoulli His collected works form about 60 to 80 quarto-sized volumes and it is believed that much of his work has been lost.

What is particularly astonishing is that Euler became virtually sightless in his right eye during the mids, and was blind for the last 17 years of his life, and this was one of the most productive periods. The series is approximately equal to 1. In the literature, this problem has been referred after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem. Basel problem appears in number theory, e.

Chartres, An integer that is not divisible by the square of any prime number is said to be square free. Thus, it follows that. Thus, on equating the coefficients of x 2 , we get. The above proof of Euler is based on manipulations that were not justified at the time, and it was not until that he was able to produce a truly rigorous proof. Today, several different proofs of 21 are known in the literature.

Euler also established the following series:. In particular, he established. After his death, many of the finest mathematicians of the world have exerted their strongest efforts and created new branches of analysis in attempts to prove these statements.

Since then with one exception, every statement has been settled in the sense Riemann expected. It stands today as the most important unsolved problem of mathematics, and perhaps the most difficult problem that the mind of man has ever conceived. Until that time, he had been using the letters p , or c In , Euler also showed that both e and e 2 are irrational and gave several continued fractions for e.

In another paper, De variis modis circuli quadraturam numeris proxime exprimendi of , Euler derived the formulas. Matsunaga Yoshisuke died in was a prolific writer. In modern terms, he used the hypergeometric series. The following curious infinite product was also given by Euler:. Henry Sullamar, a real Bedlamite, found the quadrature of the circle in the number inscribed on the forehead of the beast in the Revelations. He published periodically every two or three years some pamphlet in which he endeavored to prop his discovery.

He found that the circle was equal to the square in which it is inscribed, i. He offered a reward for the detection of any error, and actually deposited 10, francs as earnest of , But the courts did not allow any one to recover. This conjecture haunted mathematicians for years.

The following expansion is due to Euler:. It converges rapidly. Georges Louis Leclerc Comte of Buffon was a naturalist, mathematician, cosmologist and encyclopedic author. This was the earliest problem in geometric probability to be solved.

Johann Heinrich Lambert was the first to introduce hyperbolic functions into trigonometry. He wrote landmark books on geometry, the theory of cartography, and perspective in art. He published a more general result in Lambert also showed that the functions e x and tan x cannot assume rational values if x is a non-zero rational number. Arima Yoriyuki was a Japanese mathematician of the Edo period.

The French Academy of Sciences passed a resolution henceforth not to examine any more solutions of the problem of squaring the circle. In fact, it became necessary to protect its officials against the waste of time and energy involved in examining the efforts of circle squarers. A few years later, the Royal Society in London also banned consideration of any further proofs of squaring the circle.

This decision of the Royal Society was described by Augustus De Morgan about years later as the official blow to circle-squarers. Charles Hutton was an English mathematician. He wrote several mathematical texts. In , he was elected a fellow of the Royal Society of London.

It ought to be needless to say that there was no reward offered for squaring the circle. Asteroid Zachia and the crater Zach on the Moon are named after him. He wrote six scientific papers. The record of de Lagny of digits seems to have stood until , when Vega, using a new series for the arctangent discovered by Euler in , calculated decimal places correct.

His article was not published until six years later, in correct. Vega retained his record for 52 years until The Legendre crater on the Moon is named after him. Ajima Naonobu , also known as Ajima Chokuyen, was a Japanese mathematician of the Edo period. The series he developed can be simplified as. It can also be derived from the Wallis product formula 8. Lorenzo Mascheroni was educated with the aim of becoming a priest and he was ordained at the age of Lorenzo dedicated his book, Geometria del compasso , to Napoleon Bonaparte.

In this work, he proved that all Euclidean constructions can be made with compasses alone, so a straight edge in not needed. However, it was proved earlier in by the Danish mathematician Georg Mohr He claimed that compasses are more accurate then those of a ruler. Karl Friedrich Gauss was one of the greatest mathematicians of all time. In his reply, Pfaff showed that for any positive numbers x 0 and y 0 these sequences converge monotonically to a common limit given by.

In , Carl Wilhelm Borchardt work was published in which he rediscovered this result which now bears his name. For this, it suffices to note that:. Clearly, the recurrence relations 2 are different from In fact, 2 minimize the count of arithmetic operations. One of his series can be written as.

On weighing them, they were found to be exactly the same weight, which proves that, as each are of the same thickness, the surfaces must also be precisely similar. We believe for the square it must be the side not the diameter.

He was told that it was impossible to determine the full amount, as no one had yet found the exact relation between the circumference of a circle and its diameter. Joseph was honored for his profound discovery with several medals of the first class, bestowed by Parisian societies. William Rutherford was an English mathematician. Johann Martin Zacharias Dase was a calculating prodigy. At the age of 15, he gave exhibitions in Germany, Austria and England. His extraordinary calculating powers were timed by renowned mathematicians including Gauss.

In , he made acquaintance with Viennese mathematician L. Schulz von Strasznicky who suggested him to apply his powers to scientific purposes. When he was 20, Strasznicky taught him the use of the formula. In two months, he carried the approximation to places of decimals, of which are correct. He next calculated a 7-digit logarithm table of the first 1 , , numbers; he did this in his off-time from to , when occupied by the Prussian survey.

His next contribution of two years was the compilation of hyperbolic table in his spare time which was published by the Austrian Government in Next, he offered to make a table of integer factors of all numbers from 7 , , to 10 , , ; for this, on the recommendation of Gauss the Hamburg Academy of Sciences agreed to assist him financially, but Dase died shortly thereafter in Hamburg.

He also had an uncanny sense of quantity. That is, he could just tell, without counting, how many sheep were in a field, or words in a sentence, and so forth, up to about Hiromu Hasegawa and his father Hiroshi Hasegawa published many Wasan books.

Hiromu developed the series. Thomas Clausen wrote over papers on pure mathematics, applied mathematics, astronomy and geophysics. He used the formula. Clausen also gave a new method of factorising numbers. In , Liouville showed that e is not a root of any quadratic equation with integral coefficients.

This led him to conjecture that e is transcendental. In , Liouville showed, by using continued fractions, that there are an infinite number of transcendental numbers, a result which had previously been suspected but had not been proved. He produced the first examples of real numbers that are provably transcendent. One of these is. William Shanks was a British amateur mathematician.

He was assisted by Rutherford in checking first digits. Richter in published digits correct , and in after his death in decimal places. He attempted to bring it before the British Association for the Advancement of Science. Interestingly, even De Morgan and Hamilton could not convince him for his mistake. Philip H. Lawrence Sluter Benson published about 20 pamphlets on the area of the circle, three volumes on philosophic essays, and one on geometry The Elements of Euclid and Legendre.

He demonstrated that the area of the circle is equal to 3 R 2 , or the arithmetical square between the inscribed and circumscribed squares. The ratio between the diameter and circumference, he believed, is not a function of the area of the circle. Cyrus Pitt Grosvenor was an American anti-slavery Baptist minister.

In his retirement, he worked on the problem of squaring the circle. He described his method in a pamphlet titled The circle squared , New York: Square the diameter of the circle; multiply the square by 2; extract the square root of the product; from the root subtract the diameter of the circle; square the remainder; multiply this square by four fifths; subtract the square from the diameter of the circle, i. Augustus De Morgan was born in Madura India , but his family moved to England when he was seven months old.

He lost the sight of his right eye shortly after birth. He was an extremely prolific writer. He wrote more than 1, articles for more than 15 periodicals. De Morgan also wrote textbooks on many subjects, including logic, probability, calculus and algebra. In , he was a co-founder of the London Mathematical Society and became its first President. His book A Budget of Paradoxes of pages, which was edited and published by his wife in , is an entertaining text.

In this work, De Morgan reviewed the works of 42 of these writers, bringing the subject down to He once remarked that it is easier to square the circle then to get round a mathematician. Asaph Hall was an astronomer. He published the results of an experiment in random sampling that Hall had convinced his friend, Captain O. This work is considered as a very early documentation use of random sampling which Nicholas Constantine Metropolis named as the Monte Carlo method during the Manhattan Project of World War II.

Charles Hermite in was appointed to a professorship at the Sorbonne, where he trained a whole generation of well-known French mathematicians. He was strongly attracted to number theory and analysis, and his favorite subject was elliptic functions, where these two fields touch in many remarkable ways.

His proof of the transcendence of e was high point in his career. For this, he used mechanical desk calculator and worked for almost 15 years. For a long time, this remained the most fabulous piece of calculation ever performed. The digits are large wooden characters attached to the dome-like ceiling. He published a table of primes up to 60 , and found the natural logarithms of 2, 3, 5 and 10 to places. For this, he used the formula John A.

Parker in his book The Quadrature of the Circle. His book also contains practical questions on the quadrature applied to the astronomical circles. Pliny Earle Chase was a scientist, mathematician, and educator who mainly contributed to the fields of astronomy, electromagnetism and cryptography. Carl Louis Ferdinand von Lindemann worked on non-Euclidean geometry. His result showed at last that the age-old problem of squaring the circle by a ruler-and-compass construction is impossible.

The interested reader is referred to the comparatively easy version given by Hobson. Nonetheless, there are still some amateur mathematicians who do not understand the significance of this result, and futilely look for techniques to square the circle. He also worked on projective geometry, Abelian functions and developed a method of solving equations of any degree using transcendental functions.

He compiled the bibliography entitled What is the Value of Pi. It contains titles and gives the result of 63 authors. A writer announced in the New York Tribune the rediscovery of a long-lost secret that gives 3. This announcement caused considerable discussion, and even near the beginning of the twentieth century 3. The author of the bill was a physician, Edwin J. Goodman , M.

Taylor I. Record, representative from Posey County. Edwin offered this contribution as a free gift for the sole use of the State of Indiana the others would evidently have to pay royalties. The bill was sent to the Senate for approval. Although it is an impressive observation, but suspiciously good. In fact, statisticians Sir Maurice George Kendall and Patrick Alfred Pierce Moran FRS have commented that one can do better to cut out a large circle and use a tape to measure to find its circumference and diameter.

Of course, he was not being serious. The most common type of mnemonic is the word-length mnemonic in which the number of letters in each word corresponds to a digit, for example, How I wish I could calculate pi by C. Adam C. Orr in Literary Digest , vol. However, there is a problem with this type of mnemonic, namely, how to represent the digit zero. Several people have come up with ingenious methods of overcoming this, most commonly using a ten-letter word to represent zero. In other cases, a certain piece of punctuation is used to indicate a naught.

Michael Keith with such similar understanding in his work Circle digits : a self-referential story , Mathematical Intelligencer , vol. His work on real analysis was very influential in England. Srinivasa Ramanujan was a famous mathematical prodigy. He collaborated with Hardy for five years, proving significant theorems about the number of partitions of integers.

Ramanujan also made important contributions to number theory and also worked on continued fractions, infinite series and elliptic functions. In , he became the youngest Fellow of the Royal Society. Ramanujan considered mathematics and religion to be linked. He was endowed with an astounding memory and remembered the idiosyncrasies of the first 10 , integers to such an extent that each number became like a personal friend to him.

One of the remarkable formulas for its elegance and inherent mathematical depth is. Each additional term of the series adds roughly 8 digits. He also developed the series. Hughes in his work A triangle that gives the area and circumference of any circle , and the diameter of a circle equal in area to any given square , Nature 93, , doi Paul for many years had bequeathed to the university a series of 60 drawings from and explanatory notes concerning the three classical problems of antiquity.

Alexander Osipovich Gelfond was a Soviet mathematician. Helen Abbot Merrill earned her Ph. She served as an associate editor of The American Mathematical Monthly during , and was a vice-president from to of the Mathematical Association of America. Edmund Georg Hermann Yehezkel Landau was a child prodigy. In , he gave a simpler proof of the prime number theorem. His masterpiece of was a treatise Handbuch der Lehre von der Verteilung der Primzahlen a two volume work giving the first systematic presentation of analytic number theory.

Landau wrote over papers on number theory, which had a major influence on the development of the subject. He started criticizing privately, and often publicly, their results. One cannot believe this definition was used, at least as an excuse, for a racial attack on Landau. He published this book on his own expense and distributed to colleges and public libraries throughout the United States without charge.

He stated his work to be the first basic mathematical principle ever developed in USA.

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Keymaker-CORE kickass. Digital Assets PDF thepiratebay Merriam PDF thepiratebay Like the algebra of Diophantus , the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.

The four fundamental operations addition, subtraction, multiplication, and division were known to many cultures before Brahmagupta. Brahmagupta describes multiplication in the following way:. The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is repeatedly multiplied by them and the products are added together.

It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier. Indian arithmetic was known in Medieval Europe as modus Indorum meaning "method of the Indians". The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find the cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots. Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers.

The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed]. Here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice.

He first describes addition and subtraction,. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added.

The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero. But his description of division by zero differs from our modern understanding:. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative.

A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero.

That of which [the square] is the square is [its] square-root. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey.

Also, if m and x are rational, so are d , a , b and c. A Pythagorean triple can therefore be obtained from a , b and c by multiplying each of them by the least common multiple of their denominators. The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces.

The nature of squares: The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas.

The key to his solution was the identity, [24]. Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,.

The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral. Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case.

Brahmagupta dedicated a substantial portion of his work to geometry. One theorem gives the lengths of the two segments a triangle's base is divided into by its altitude:. The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments. The perpendicular [altitude] is the square-root from the square of a side diminished by the square of its segment.

He further gives a theorem on rational triangles. A triangle with rational sides a , b , c and rational area is of the form:. The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal. The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular [altitudes].

He continues to give formulas for the lengths and areas of geometric figures, such as the circumradius of an isosceles trapezoid and a scalene quadrilateral, and the lengths of diagonals in a scalene cyclic quadrilateral. This leads up to Brahmagupta's famous theorem ,. Imaging two triangles within [a cyclic quadrilateral] with unequal sides, the two diagonals are the two bases. Their two segments are separately the upper and lower segments [formed] at the intersection of the diagonals.

The two [lower segments] of the two diagonals are two sides in a triangle; the base [of the quadrilateral is the base of the triangle]. Its perpendicular is the lower portion of the [central] perpendicular; the upper portion of the [central] perpendicular is half of the sum of the [sides] perpendiculars diminished by the lower [portion of the central perpendicular].

The diameter and the square of the radius [each] multiplied by 3 are [respectively] the practical circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of those two multiplied by ten. In some of the verses before verse 40, Brahmagupta gives constructions of various figures with arbitrary sides.

He essentially manipulated right triangles to produce isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with three equal sides, and a scalene cyclic quadrilateral. After giving the value of pi, he deals with the geometry of plane figures and solids, such as finding volumes and surface areas or empty spaces dug out of solids.

He finds the volume of rectangular prisms, pyramids, and the frustum of a square pyramid. He further finds the average depth of a series of pits. For the volume of a frustum of a pyramid, he gives the "pragmatic" value as the depth times the square of the mean of the edges of the top and bottom faces, and he gives the "superficial" volume as the depth times their mean area. The sines: The Progenitors, twins; Ursa Major, twins, the Vedas; the gods, fires, six; flavors, dice, the gods; the moon, five, the sky, the moon; the moon, arrows, suns [ Here Brahmagupta uses names of objects to represent the digits of place-value numerals, as was common with numerical data in Sanskrit treatises.

Progenitors represents the 14 Progenitors "Manu" in Indian cosmology or 14, "twins" means 2, "Ursa Major" represents the seven stars of Ursa Major or 7, "Vedas" refers to the 4 Vedas or 4, dice represents the number of sides of the traditional die or 6, and so on.

In Brahmagupta devised and used a special case of the Newton—Stirling interpolation formula of the second-order to interpolate new values of the sine function from other values already tabulated. The division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself.

In Brahmagupta's case, the disagreements stemmed largely from the choice of astronomical parameters and theories. Some of the important contributions made by Brahmagupta in astronomy are his methods for calculating the position of heavenly bodies over time ephemerides , their rising and setting, conjunctions , and the calculation of solar and lunar eclipses.

If the moon were above the sun, how would the power of waxing and waning, etc. The near half would always be bright. In the same way that the half seen by the sun of a pot standing in sunlight is bright, and the unseen half dark, so is [the illumination] of the moon [if it is] beneath the sun. The brightness is increased in the direction of the sun. At the end of a bright [i. Hence, the elevation of the horns [of the crescent can be derived] from calculation.

He explains that since the Moon is closer to the Earth than the Sun, the degree of the illuminated part of the Moon depends on the relative positions of the Sun and the Moon, and this can be computed from the size of the angle between the two bodies.

Further work exploring the longitudes of the planets, diurnal rotation, lunar and solar eclipses, risings and settings, the moon's crescent and conjunctions of the planets, are discussed in his treatise Khandakhadyaka. From Wikipedia, the free encyclopedia. This is the latest accepted revision , reviewed on 21 June Indian mathematician and astronomer — Zero Modern number system Brahmagupta's theorem Brahmagupta's identity Brahmagupta's problem Brahmagupta—Fibonacci identity Brahmagupta's interpolation formula Brahmagupta's formula.

Main article: Brahmagupta's formula. Main article: Brahmagupta theorem.

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Bhillamala was the capital of the Gurjaradesa , the second-largest kingdom of Western India, comprising southern Rajasthan and northern Gujarat in modern-day India. It was also a centre of learning for mathematics and astronomy.

He became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy during this period. He studied the five traditional Siddhantas on Indian astronomy as well as the work of other astronomers including Aryabhata I , Latadeva, Pradyumna, Varahamihira , Simha, Srisena, Vijayanandin and Vishnuchandra.

Scholars state that he incorporated a great deal of originality into his revision, adding a considerable amount of new material. A good deal of it is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself.

Later, Brahmagupta moved to Ujjaini , Avanti , [7] a major centre for astronomy in central India. Prithudaka Svamin wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations. Lalla and Bhattotpala in the 8th and 9th centuries wrote commentaries on the Khanda-khadyaka.

The kingdom of Bhillamala seems to have been annihilated but Ujjain repulsed the attacks. The court of Caliph Al-Mansur — received an embassy from Sindh, including an astrologer called Kanaka, who brought possibly memorised astronomical texts, including those of Brahmagupta. Brahmagupta's texts were translated into Arabic by Muhammad al-Fazari , an astronomer in Al-Mansur's court under the names Sindhind and Arakhand.

An immediate outcome was the spread of the decimal number system used in the texts. The mathematician Al-Khwarizmi — CE wrote a text called al-Jam wal-tafriq bi hisal-al-Hind Addition and Subtraction in Indian Arithmetic , which was translated into Latin in the 13th century as Algorithmi de numero indorum.

Through these texts, the decimal number system and Brahmagupta's algorithms for arithmetic have spread throughout the world. Al-Khwarizmi also wrote his own version of Sindhind , drawing on Al-Fazari's version and incorporating Ptolemaic elements. Indian astronomic material circulated widely for centuries, even passing into medieval Latin texts.

The historian of science George Sarton called Brahmagupta "one of the greatest scientists of his race and the greatest of his time. The difference between rupas , when inverted and divided by the difference of the [coefficients] of the [unknowns], is the unknown in the equation. The rupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted. He further gave two equivalent solutions to the general quadratic equation.

Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminished that by half the unknown [and] divide [the remainder] by its square. He went on to solve systems of simultaneous indeterminate equations stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable's coefficient.

In particular, he recommended using "the pulverizer" to solve equations with multiple unknowns. Subtract the colors different from the first color. If there are many [colors], the pulverizer [is to be used]. Like the algebra of Diophantus , the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar.

Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms. The four fundamental operations addition, subtraction, multiplication, and division were known to many cultures before Brahmagupta. Brahmagupta describes multiplication in the following way:. The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is repeatedly multiplied by them and the products are added together.

It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier. Indian arithmetic was known in Medieval Europe as modus Indorum meaning "method of the Indians". The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find the cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots.

Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers. The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed]. Here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice.

He first describes addition and subtraction,. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.

But his description of division by zero differs from our modern understanding:. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor].

The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey.

Also, if m and x are rational, so are d , a , b and c. A Pythagorean triple can therefore be obtained from a , b and c by multiplying each of them by the least common multiple of their denominators. The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces.

The nature of squares: The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas.

The key to his solution was the identity, [24]. Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,.

The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral. Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case. Brahmagupta dedicated a substantial portion of his work to geometry.

One theorem gives the lengths of the two segments a triangle's base is divided into by its altitude:. The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments. The perpendicular [altitude] is the square-root from the square of a side diminished by the square of its segment.

He further gives a theorem on rational triangles. A triangle with rational sides a , b , c and rational area is of the form:. The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal.

The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular [altitudes]. He continues to give formulas for the lengths and areas of geometric figures, such as the circumradius of an isosceles trapezoid and a scalene quadrilateral, and the lengths of diagonals in a scalene cyclic quadrilateral.

This leads up to Brahmagupta's famous theorem ,. Imaging two triangles within [a cyclic quadrilateral] with unequal sides, the two diagonals are the two bases. Their two segments are separately the upper and lower segments [formed] at the intersection of the diagonals. The two [lower segments] of the two diagonals are two sides in a triangle; the base [of the quadrilateral is the base of the triangle].

Its perpendicular is the lower portion of the [central] perpendicular; the upper portion of the [central] perpendicular is half of the sum of the [sides] perpendiculars diminished by the lower [portion of the central perpendicular].

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