Great Mathematicians

[Gauss]

The purpose of this thread is to provide short and interesting biographies of famous mathematicians.

• Aristotle (384 B.C.E – 322 B.C.E)
  
  Aristotle was born in Stargirus in northern Greece. His father was the personal physician of the King of Macedonia. Because his father died when Aristotle was young, Aristotle could not follow the custom of following his father’s profession. Aristotle became an orphan at a young age when his mother also died. His guardian who raised him taught him poetry, rhetoric, and Greek. At the age of 17, his guardian sent him to Athens to further his education. Aristotle joined Plato’s Acadmey where for 20 years he attended Plato’s lectures, later presenting his own lectures on rhetoric. When Plato died in 347 B.C.E, Aristotle was not chosen to succeed him because his views differed too much from those of Plato. Instead, Aristotle joined the court of King Hermeas where he remained for three years, and married the niece of the King. When the Persians defeated Hermeas, Aristotle moved to Mytilene and, at the invitation of King Philip of Macedonia, he tutored Alexander, Philip’s son, who later became Alexander the Great. Aristotle tutored Alexander for five years and after the death of King Philip, he returned to Athens and set up his own school, called the Lyceum.
  
  Aristotle’s followers were called the peripatetics, which means “to walk about,” because Aristotle often walked around as he discussed philosophical questions. Aristotle taught at the Lyceum for 13 years where he lectured to his advanced students in the morning and gave popular lectures to a broad audience in the evening. When Alexander the Great died in 323 B.C.E, a backlash against anything related to Alexander led to trumped-up charges of impiety against Aristotle. Aristotle fled to Chalcis to avoid prosecution. He only lived one year in Chalcis, dying of a stomach ailment in 322 B.C.E.
  
  Aristotle wrote three types of works: those written for a popular audience, compilations of scientific facts, and systematic treatises. The systematic treatises included works on logic, philosophy, psychology, physics, and natural history. Aristotle’s writings were preserved by a student and were hidden in a vault where a wealthy book collector discovered them about 200 years later. They were taken to Rome, where they were studied by scholars and issued in new editions, preserving them for posterity.

 

[Gauss]

• George Boole (1815 – 1864)
  
  George Boole, the son of a cobbler, was born in Lincoln, England, in November 1815. Because of his family’s difficult financial situation, Boole had to struggle to educate himself while supporting his family. Nevertheless, he became one of the most important mathematicians of the 1800s. Although he considered a career as a clergyman, he decided instead to go into teaching and soon afterward opened a school of his own. In his preparation for teaching mathematics, Boole – unsatisfied with textbooks of his day – decided to read the works of the great mathematicians. While reading papers of the great French mathematician Lagrange, Boole made discoveries in the calculus of variations, the branch of analysis dealing of finding curves and surfaces optimizing certain parameters.
  
  In 1848 Boole published The Mathematical Analysis of Logic, the first of his contributions to symbolic logic. In 1849 he was appointed professor of mathematics at Queen’s College in Cork, Ireland. In 1854 he published The Laws of Thought, his most famous work. In this book Boole introduced what is now called Boolean algebra in his honor. Boole wrote textbooks on differential equations and on difference equations that were used in Great Britain until the end of the nineteenth century. Boole married in 1855; his wife was the niece of the professor of Greek at Queen’s College. In 1864 Boole died from pneumonia, which he contracted as a result of keeping a lecture engagement even though he was soaking wet from a rainstorm.
  

 

[The SC]

al-Khwārizmī, in full Muḥammad ibn Mūsā al-Khwārizmī (born c. 780, Baghdad, Iraq—died c. 850), Muslim mathematician and astronomer whose major works introduced Hindu-Arabic numerals and the concepts of algebra into European mathematics. Latinized versions of his name and of his most famous book title live on in the terms algorithm and algebra.

Al-Khwārizmī lived in Baghdad, where he worked at the “House of Wisdom” (Dār al-Ḥikma) under the caliphate of al-Maʾmūn. (The House of Wisdom acquired and translated scientific and philosophic treatises, particularly Greek, as well as publishing original research.) Al-Kwārizmī’s work on elementary algebra, al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr waʾl-muqābala (“The Compendious Book on Calculation by Completion and Balancing”), was translated into Latin in the 12th century, from which the title and term Algebra derives. Algebra is a compilation of rules, together with demonstrations, for finding solutions of linear and quadratic equations based ... (150 of 459 words)

 

[The SC]

Damascus mathematician Abu al-Hasan Ahmad ibn Ibrahim al-Uqlidisi ("the Euclidian," fl. ca. 953 CE) further advanced the Indian mode of calculation. The Indian system had used a dustboard to perform and erase a series of calculations.

Al-Uqlidisi adapted the Indian system for pen and paper. Mathematicians could now “show their work,” sharing problems, equations, and methods for solving them across time and space. Mathematics advanced rapidly as a result of recording and publication.

Al-Battani (850-929 CE) contributed significant work developing trigonometry, computing the first table of cotangents. Al-Biruni (973-1050 CE) also advanced trigonometry, and used it to calculate the coordinates of cities to determine the qibla (direction of Makkah) from any location. Omar Khayyam (b. 1048 CE) classified and solved cubic equations.

By the 10th century, Muslim mathematicians had developed and applied the theory of trigonometric functions -- sine, cosine, and tangent -- as well as spherical trigonometry. They used symbols to describe the binomial theorem, and used decimals to express fractions that aided accurate solution of complex problems.

Major Arabic mathematical works were brought to Al-Andalus by the 9th century, along with important Greek translations and commentaries. Together with a translation of Euclid’s Elements, they became the two foundations of subsequent mathematical developments in Al-Andalus. It is clear from their own achievements that scholars in Al-Andalus followed advancements in other Muslim lands, and contributed their own.
Today, al-Khwarizmi’s work exists only as a Latin translation made in Toledo, Spain, by Gerard of Cremona (d. 1187 CE). Europeans did not gain access to the mathematical knowledge found in Spain and North Africa until the 12th and 13th centuries CE. It entered Europe both through scholarly and commercial means. Fibonacci (d. 1250 CE), an Italian mathematician who traveled between Europe and North Africa, transmitted mathematical knowledge from Muslim lands to Europe and made his own discoveries.

Mathematicians in Al-Andalus also did original work. Maslama al-Majriti (d. 1007 CE) was a mathematician and astronomer who translated Ptolemy’s Almagest, and corrected and added to al-Khwarizmi’s astronomical tables. Al-Majriti also used advanced techniques of surveying using triangulation.

Al-Zarqali, or Arzachel in Latin, was a mathematician and astronomer who worked in Córdoba during the 11th century. He was skilled at making instruments for the study of astronomy, and built a famous water clock that could tell the hours of the day and night, as well as the days of the lunar month. Al-Zarqali contributed to the famous Toledan Tables of astronomical data, and published an almanac that correlated the days of the month on different calendars such as the Coptic, Roman, lunar and Persian, gave the positions of the planets, and predicted solar and lunar eclipses. He created tables of latitude and longitude to aid navigation and cartography.

Another prominent Andalusian mathematician and astronomer in Seville was al-Bitruji (d. 1204 CE), known in Europe as Alpetragius. He developed a theory of the movement of stars described in The Book of Form. Ibn Bagunis of Toledo was a mathematician renowned for his work in geometry. Abraham bar Hiyya was a Jewish mathematician who assisted Plato of Tivoli with translation of important mathematical and astronomical works, including his own Liber Embadorum, in 1145 CE.Abu al-Hakam al-Kirmani was a prominent 12th century scholar of Al-Andalus, a scholar of geometry and logic.

No branch of mathematics is more visible in Muslim culture than geometry. Geometric design reached heights of skill and beauty that was applied to nearly every art form, from textiles to illustration to architectural decoration. Tessellated, or complex, overall patterns were used in Andalusian architecture to cover walls, ceilings, floors and arches. Some scholars of Islamic arts believe that these designs were much more than artisans’ work -- they consciously expressed the mathematical knowledge of the culture that produced them.

Recently, Paul J. Steinhardt of Princeton and Peter J. Lu of Harvard University discovered Medieval Islamic tessellations designed 500 years ago; they were unusually complex, with polygons of multiple shapes, overlaid by zigzag lines. These designs are known today as quasicrystalline because they have fivefold or tenfold rotational symmetry; that means they can be rotated around a point to five or ten positions and still look the same. Such designs can be infinitely extended without repeating. In the 1970s, Oxford mathematician Roger Penrose calculated the principles behind quasicrystalline symmetry. Steinhardt and Lu discovered that such patterns of stars and polygons have decorated mosques and palaces since the 15th century CE.

Mathematics

 

[Gauss]

Karl Friedrich Gauss (1777 – 1855)

Karl Friedrich Gauss, the son of a bricklayer, was a child prodigy. He demonstrated his potential at the age of 10, when he quickly solved a problem assigned by a teacher to keep the class busy. The teacher asked the students to find the sum of the first 100 positive integers. Gauss realized that this sum could be found by forming 50 pairs, each with the sum 101: 1 + 100, 2 + 99,……, 50 + 51. This brilliance attracted the sponsorship of patrons, including Duke Ferdinand of Brunswick, who made it possible for Gauss to attend Caroline College and the University of Göttingen. While a student he invented the method of least squares, which is used to estimate the most likely value of a variable from experimental results. In 1796 Gauss made a fundamental discovery in geometry, advancing a subject that had not advanced since ancient times. He showed that a 17-sided regular polygon could be drawn using just a ruler and compass.

In 1799 Gauss presented the first rigorous proof of the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n roots (counting multiplicities). Gauss achieved worldwide fame when he successfully calculated the orbit of the first asteroid discovered, Ceres, using scanty data.

Gauss was called the Prince of Mathematics by his contemporary mathematicians. Although Gauss is noted for his many discoveries in geometry, algebra, analysis, astronomy, and physics, he had a special interest in number theory, which can be seen from his statement “Mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics.” Gauss laid the foundations for modern number theory with the publication of his book Disquisitiones Arithmeticae in 1801.

 

[AUSTERLITZ]

Archimides (Syracuse,Greek-italian)
Newton(English)
Leibnitz.(German)
Euler(Swiss)
Ramanujan(Indian)
Pythagoras(Greek)
Descartes( french)
Aryabhatta(Indian)
Pascal(French)
Euclid(Greek-egyptian)
Brahmagupta(indian)
Turing(british)
Omar khayyam(persian)
Riemann(german)
Al khwarizmi(Arab)
Gauss(german)
Boole(British)
Poincare(French)
Hilbert(german)
Einstein(German)
Fibonacci(Italy)
Neumann(Hungary)
Lagrange(French/italy)
Cauchy(France)
Grothedieck(Germany)
Fermat(France)
Abel (Norway)
Galileo(Italy)

 

[temp1994]

From Indian side: pretty much whole of abstract mathematics ( most of what Europeans credit to Arabs ) was developed by Indian Mathematicians.

Aryabhata - Wikipedia, the free encyclopedia

"
Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost.

His major work,Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic,algebra,plane trigonometry, and spherical trigonometry. It also contains continued fractions,quadratic equations, sums-of-power series, and a table of sines.

The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary,Varahamihira, and later mathematicians and commentators, includingBrahmaguptaandBhaskara I. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise inAryabhatiya. It also contained a description of several astronomical instruments: the gnomon(shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra/chakra-yantra), a cylindrical stickyasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.[8]

A third text, which may have survived in the Arabic translation, is Al ntf orAl-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known.

Probably dating from the 9th century, it is mentioned by the Persianscholar and chronicler of India,Abū Rayhān al-Bīrūnī.

Aryabhatiya

Direct details of Aryabhata's work are known only from the Aryabhatiya. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra(or the treatise from the Ashmaka). It is also occasionally referred to as Arya-shatas-aShTa(literally, Aryabhata's 108), because there are 108 verses in the text. It is written in the very terse style typical of sutra literature, in which each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The text consists of the 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

1. Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present a cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha(c. 1st century BCE). There is also a table of sines (jya), given in a single verse. The duration of the planetary revolutions during amahayugais given as 4.32 million years.
2. Ganitapada(33 verses): covering mensuration(kṣetra vyāvahāra), arithmetic and geometric progressions,gnomon/ shadows (shanku-chhAyA), simple,quadratic,simultaneous, and indeterminate equations
3. Kalakriyapada(25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (adhikamAsa),kShaya-tithis, and a seven-day week with names for the days of week.
4. Golapada(50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic,celestial equator, node, shape of the earth, cause of day and night, rising ofzodiacal signson horizon, etc. In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (Bhashya, c. 600 CE) and by Nilakantha Somayaji in hisAryabhatiya Bhasya,(1465 CE).

Mathematics
Place value system and zero
The place-value system, first seen in the 3rd-century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol forzero, the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with nullcoefficients

However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form.

Approximation ofπ
Aryabhata worked on the approximation for pi(

), and may have come to the conclusion that

is irrational. In the second part of the Aryabhatiyam(gaṇitapāda10), he writes:

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇāmayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ."Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."
​

This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.

It is speculated that Aryabhata used the wordāsanna(approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by Lambert.

After Aryabhatiya was translated into Arabic(c. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.

Trigonometry
In Ganitapada 6, Aryabhata gives the area of a triangle as
​

tribhujasya phalashariram samadalakoti bhujardhasamvargah
that translates to: "for a triangle, the result of a perpendicular with the half-side is the area."

Aryabhata discussed the concept of sine in his work by the name of ardha-jya, which literally means "half-chord". For simplicity, people started calling it jya. When Arabic writers translated his works from Sanskrit into Arabic, they referred it asjiba. However, in Arabic writings, vowels are omitted, and it was abbreviated asjb. Later writers substituted it withjaib, meaning "pocket" or "fold (in a garment)". (In Arabic,jibais a meaningless word.) Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jaibwith its Latin counterpart,sinus, which means "cove" or "bay"; thence comes the English sine. Alphabetic code has been used by him to define a set of increments. If we use Aryabhata's table and calculate the value of sin(30) (corresponding to hasjha) which is 1719/3438 = 0.5; the value is correct. His alphabetic code is commonly known as the Aryabhata cipher.[18]

Indeterminate equations
A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + by = c, a topic that has come to be known as diophantine equations. This is an example from Bhāskara's commentary on Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7
That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose more ancient parts might date to 800 BCE. Aryabhata's method of solving such problems is called thekuṭṭaka(कुट्टक) method.Kuttaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the Aryabhata algorithm.[19]The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulbasutras.

Algebra
InAryabhatiya, Aryabhata provided elegant results for the summation ofseriesof squares and cubes:[20]

and

(see squared triangular number)
Astronomy
Aryabhata's system of astronomy was called theaudAyaka system, in which days are reckoned fromuday, dawn atlankaor "equator". Some of his later writings on astronomy, which apparently proposed a second model (orardha-rAtrikA, midnight) are lost but can be partly reconstructed from the discussion in Brahmagupta's khanDakhAdyaka. In some texts, he seems to ascribe the apparent motions of the heavens to the Earth's rotation. He may have believed that the planet's orbits as elliptical rather than circular.[21][22]

Motions of the solar system
Aryabhata correctly insisted that the earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of the earth, contrary to the then-prevailing view, that the sky rotated. This is indicated in the first chapter of theAryabhatiya, where he gives the number of rotations of the earth in ayuga,and made more explicit in hisgolachapter:[24]

In the same way that someone in a boat going forward sees an unmoving [object] going backward, so [someone] on the equator sees the unmoving stars going uniformly westward. The cause of rising and setting [is that] the sphere of the stars together with the planets [apparently?] turns due west at the equator, constantly pushed by the cosmic wind.

Aryabhata described a geocentric model of the solar system, in which the Sun and Moon are each carried byepicycles. They in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta(c. CE 425), the motions of the planets are each governed by two epicycles, a smaller manda(slow) and a largerśīghra(fast).[25]The order of the planets in terms of distance from earth is taken as: the Moon,Mercury,Venus, the Sun,Mars,Jupiter,Saturn, and the asterisms."[8]

The positions and periods of the planets was calculated relative to uniformly moving points. In the case of Mercury and Venus, they move around the Earth at the same mean speed as the Sun. In the case of Mars, Jupiter, and Saturn, they move around the Earth at specific speeds, representing each planet's motion through the zodiac. Most historians of astronomy consider that this two-epicycle model reflects elements of pre-PtolemaicGreek astronomy.[26]Another element in Aryabhata's model, theśīghrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying heliocentric model.[27]

Eclipses
Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by pseudo-planetary demonsRahuandKetu, he explains eclipses in terms of shadows cast by and falling on Earth. These will only occur when the earth-moon orbital plane intersects the earth-sun orbital plane, at points called lunar nodes. Thus, the lunar eclipse occurs when the moon enters into the Earth's shadow (verse gola.37). He discusses at length the size and extent of the Earth's shadow (verses gola.38–48) and then provides the computation and the size of the eclipsed part during an eclipse. Later Indian astronomers improved on the calculations, but Aryabhata's methods provided the core. His computational paradigm was so accurate that 18th-century scientist Guillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations of the duration of the lunar eclipseof 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.[8]

Sidereal periods
Considered in modern English units of time, Aryabhata calculated the sidereal rotation(the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1 seconds;[28]the modern value is 23:56:4.091. Similarly, his value for the length of thesidereal yearat 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days)[29]is an error of 3 minutes and 20 seconds over the length of a year (365.25636 days).

Heliocentrism
As mentioned, Aryabhata advocated an astronomical model in which the Earth turns on its own axis. His model also gave corrections (theśīgraanomaly) for the speeds of the planets in the sky in terms of the mean speed of the sun. Thus, it has been suggested that Aryabhata's calculations were based on an underlyingheliocentricmodel, in which the planets orbit the Sun,[30][31][32]though this has been rebutted.[33]It has also been suggested that aspects of Aryabhata's system may have been derived from an earlier, likely pre-PtolemaicGreek, heliocentric model of which Indian astronomers were unaware,[34]though the evidence is scant.[35]The general consensus is that a synodic anomaly (depending on the position of the sun) does not imply a physically heliocentric orbit (such corrections being also present in lateBabylonian astronomical texts), and that Aryabhata's system was not explicitly heliocentric.

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​

 

[temp1994]

​

Mahāvīra (mathematician) - Wikipedia, the free encyclopedia

"

Mahāvīra(orMahaviracharya, "Mahavira the Teacher") was a 9th-centuryJainmathematicianfromMysore,India. He was the author ofGaṇitasārasan̄graha (orGanita Sara Samgraha, c. 850), which revised the Brāhmasphuṭasiddhānta.He was patronised by the Rashtrakuta king Amoghavarsha.[4]He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics.He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems.He is highly respected among Indian mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle.Mahāvīra's eminence spread in all South India and his books proved inspirational to other mathematicians in Southern India.It was translated into Telugu language by Pavuluri Mallana asSaar Sangraha Ganitam.

He discovered algebraic identities like a3=a(a+b)(a-b) +b2(a-b) + b3.He also found out the formula for nCr as [n(n-1)(n-2)...(n-r+1)]/r(r-1)(r-2)...2*1.He devised formula which approximated area and perimeters of ellipses and found methods to calculate the square of a number and cube roots of a number.He asserted that the square root of a negative number did not exist.

Rules for decomposing fractions
Mahāvīra'sGaṇita-sāra-saṅgrahagave systematic rules for expressing a fraction as the sum of unit fractions. This follows the use of unit fractions in Indian mathematics in the Vedic period, and the Śulba Sūtras' giving an approximation of √2 equivalent to

.

In theGaṇita-sāra-saṅgraha(GSS), the second section of the chapter on arithmetic is namedkalā-savarṇa-vyavahāra (lit. "the operation of the reduction of fractions"). In this, thebhāgajātisection (verses 55–98) gives rules for the following:

• To express 1 as the sum ofnunit fractions (GSSkalāsavarṇa75, examples in 76):

rūpāṃśakarāśīnāṃ rūpādyās triguṇitā harāḥ kramaśaḥ /

dvidvitryaṃśābhyastāv ādimacaramau phale rūpe //

When the result is one, the denominators of the quantities having one as numerators are [the numbers] beginning with one and multiplied by three, in order. The first and the last are multiplied by two and two-thirds [respectively].​

​

 

[temp1994]

• To express 1 as the sum of an odd number of unit fractions (GSSkalāsavarṇa77):

• To express a unit fraction
  
  as the sum ofnother fractions with given numerators
  
  (GSSkalāsavarṇa78, examples in 79):

​

 

[temp1994]

• To express any fraction
  
  as a sum of unit fractions (GSSkalāsavarṇa80, examples in 81):

Choose an integerisuch that

is an integerr, then write

and repeat the process for the second term, recursively. (Note that ifiis always chosen to be thesmallestsuch integer, this is identical to the greedy algorithm for Egyptian fractions.)

• To express a unit fraction as the sum of two other unit fractions (GSSkalāsavarṇa85, example in 86):[13]

where

is to be chosen such that

is an integer (for which

must be a multiple of

).

​

 

[temp1994]

• To express a fraction
  
  as the sum of two other fractions with given numerators
  
  and
  
  (GSSkalāsavarṇa87, example in 88):[13]

where

is to be chosen such that

divides

Some further rules were given in theGaṇita-kaumudiof Nārāyaṇain the 14th century.​

"

​

 

[temp1994]

Bhāskara II - Wikipedia, the free encyclopedia​

"​

Lilavati[edit]
The first sectionLīlāvatī(also known aspāṭīgaṇitaoraṅkagaṇita) consists of 277 verses.It covers calculations, progressions,mensuration, permutations, and other topics.​

Bijaganita[edit]
The second sectionBījagaṇitahas 213 verses.It discusses zero, infinity, positive and negative numbers, and indeterminate equations including (the now called) Pell's equation, solving it using akuṭṭakamethod.In particular, he also solved the

case that was to elude Fermat and his European contemporaries centuries later.​

Grahaganita
In the third sectionGrahagaṇita, while treating the motion of planets, he considered their instantaneous speeds.He arrived at the approximation:​

for

close to

, or in modern notation:

.
In his words:​

bimbārdhasya koṭijyā guṇastrijyāhāraḥ phalaṃ dorjyāyorantaram​

This result had also been observed earlier by Muñjalācārya (or Mañjulācārya) in 932, in his astronomical work 'Laghu-mānasam, in the context of a table of sines.​

Bhāskara also stated that at its highest point a planet's instantaneous speed is zero.​

​

 

[C130]

Isaac Newton

best known for the laws of motion and gravity, but he was a brilliant mathematician as well

 

[temp1994]

Mathematics[edit]
Some of Bhaskara's contributions to mathematics include the following:​

• A proof of thePythagorean theoremby calculating the sameareain two different ways and then canceling out terms to geta2 + b2 = c2.

• InLilavati, solutions ofquadratic,cubicandquarticindeterminate equationsare explained.[11]

• Solutions of indeterminate quadratic equations (of the typeax2+b=y2).

• Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by theRenaissanceEuropean mathematicians of the 17th century

• A cyclicChakravala methodfor solving indeterminate equations of the formax2+bx+c=y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than thechakravalamethod.

• The first general method for finding the solutions of the problemx2 − ny2= 1 (so-called "Pell's equation") was given by Bhaskara II.[12]

• Solutions ofDiophantine equationsof the second order, such as 61x2+ 1 =y2. This very equation was posed as a problem in 1657 by theFrenchmathematicianPierre de Fermat, but its solution was unknown in Europe until the time ofEulerin the 18th century.[13]

• Solved quadratic equations with more than one unknown, and foundnegativeandirrationalsolutions.[citation needed]

• Preliminary concept ofmathematical analysis.

• Preliminary concept ofinfinitesimalcalculus, along with notable contributions towardsintegral calculus.[14]

• Conceiveddifferential calculus, after discovering thederivativeanddifferentialcoefficient.

• StatedRolle's theorem, a special case of one of the most important theorems in analysis, themean value theorem. Traces of the general mean value theorem are also found in his works.

• Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)

• InSiddhanta Shiromani, Bhaskara developedspherical trigonometryalong with a number of othertrigonometricresults. (See Trigonometry section below.)

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Arithmetic[edit]
Bhaskara'sarithmetictextLeelavaticovers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions,plane geometry,solid geometry, the shadow of thegnomon, methods to solveindeterminateequations, andcombinations.​

Lilavatiis divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and mensuration. More specifically the contents include:​

• Definitions.
• Properties ofzero(includingdivision, and rules of operations with zero).
• Further extensive numerical work, including use ofnegative numbersandsurds.
• Estimation ofπ.
• Arithmetical terms, methods ofmultiplication, andsquaring.
• Inverserule of three, and rules of 3, 5, 7, 9, and 11.
• Problems involvinginterestand interest computation.

• Indeterminate equations (Kuttaka), integer solutions (first and second order). His contributions to this topic are particularly important,[citation needed]since the rules he gives are (in effect) the same as those given by therenaissanceEuropean mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work ofAryabhataand subsequent mathematicians.

His work is outstanding for its systemisation, improved methods and the new topics that he has introduced. Furthermore theLilavaticontained excellent recreative problems and it is thought that Bhaskara's intention may have be.​

Algebra[edit]
HisBijaganita("Algebra") was a work in twelve chapters. It was the first text to recognize that a positive number has twosquare roots(a positive and negative square root).[15]His workBijaganitais effectively a treatise on algebra and contains the following topics:​

• Positive andnegative numbers.
• Zero.
• The 'unknown' (includes determining unknown quantities).
• Determining unknown quantities.
• Surds(includes evaluating surds).
• Kuttaka(for solvingindeterminate equationsandDiophantine equations).
• Simple equations (indeterminate of second, third and fourth degree).
• Simple equations with more than one unknown.
• Indeterminatequadratic equations(of the type ax2+ b = y2).
• Solutions of indeterminate equations of the second, third and fourth degree.
• Quadratic equations.
• Quadratic equations with more than one unknown.
• Operations with products of several unknowns.

Bhaskara derived a cyclic,chakravalamethodfor solving indeterminate quadratic equations of the form ax2+ bx + c = y.[16]Bhaskara's method for finding the solutions of the problem Nx2+ 1 = y2(the so-called "Pell's equation") is of considerable importance.[12]​

Trigonometry[edit]
TheSiddhānta Shiromani(written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also discoveredspherical trigonometry, along with other interestingtrigonometricalresults. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, discoveries first found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for

and

.​

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