keerthana 26.9
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Name
While there is a tendency to misspell his name as "Aryabhatta" by analogy with other
names having the " bhatta" suffix, his name is properly spelled Aryabhata: everyastronomical text spells his name thus,[1] including Brahmagupta's references to him "in
more than a hundred places by name".[2] Furthermore, in most instances "Aryabhatta"
does not fit the metre either.[1]
Birth
Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali
Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was
born in 476 CE.[1]
Aryabhata provides no information about his place of birth. The only information comesfrom Bhāskara I, who describes Aryabhata as āśmakīya, "one belonging to the aśmaka
country." While aśmaka was originally situated in the northwest of India, it is widely
attested that, during the Buddha's time, a branch of the Aśmaka people settled in the
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region between the Narmada and Godavari rivers, in the South Gujarat–North
Maharashtra region of central India. Aryabhata is believed to have been born there.[1][3]
However, early Buddhist texts describe Ashmaka as being further south, in dakshinapath
or the Deccan, while other texts describe the Ashmakas as having fought Alexander ,
which would put them further north.[3] Bold text
Work
It is fairly certain that, at some point, he went to Kusumapura for advanced studies andthat he lived there for some time.[4] Both Hindu and Buddhist tradition, as well as
Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.[1] A verse
mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura, and,
because the university of Nalanda was in Pataliputra at the time and had an astronomical
observatory, it is speculated that Aryabhata might have been the head of the Nalandauniversity as well.[1] Aryabhata is also reputed to have set up an observatory at the Sun
temple in Taregana, Bihar.[5]
Other hypotheses
It was suggested that Aryabhata may have been from Kerala, but K. V. Sarma, anauthority on Kerala's astronomical tradition, disagreed[1] and pointed out several errors in
this hypothesis.[6]
Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is
an abstraction, standing for a point on the equator at the same longitude as his Ujjayini.[7]
Works
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 hassurvived 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 thewritings of Aryabhata's contemporary, Varahamihira, and later mathematicians and
commentators, including Brahmagupta and Bhaskara I. This work appears to be based onthe older Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. 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), acylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and
water clocks of at least two types, bow-shaped and cylindrical.[3]
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A third text, which may have survived in the Arabic translation, is Al ntf or Al-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 Persian scholar andchronicler of India, Abū Rayhān al-Bīrūnī [3]
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 for zero, the French mathematicianGeorges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value
system as a place holder for the powers of ten with null coefficients [8]
However, Aryabhata did not use the brahmi numerals. Continuing the Sanskritic tradition
from Vedic times, he used letters of the alphabet to denote numbers, expressingquantities, such as the table of sines in a mnemonic form.[9]
Approximation of pi
Aryabhata worked on the approximation for pi (π), and may have come to the conclusion
that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:
caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇāmayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ."Add four to 100, multiply by eight, and then add 62,000. By this rule the circumferenceof a circle with a diameter of 20,000 can be approached."[10]
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 word āsanna (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 inEurope only in 1761 by Lambert.[11]
After Aryabhatiya was translated into Arabic (ca. 820 CE) this approximation was
mentioned in Al-Khwarizmi's book on algebra.[3]
Mensuration and trigonometry
In Ganitapada 6, Aryabhata gives the area of a triangle as
tribhujasya phalashariram samadalakoti bhujardhasamvargah
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that translates to: "for a triangle, the result of a perpendicular with the half-side is the
area."[12]
Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally,it means "half-chord". For simplicity, people started calling it jya. When Arabic writers
translated his works from Sanskrit into Arabic, they referred it as jiba. However, inArabic writings, vowels are omitted, and it was abbreviated as jb. Later writers
substituted it with jiab, meaning "cove" or "bay." (In Arabic, jiba is a meaningless word.)Later in the 12th century, when Gherardo of Cremona translated these writings from
Arabic into Latin, he replaced the Arabic jiab with its Latin counterpart, sinus, which
means "cove" or "bay". And after that, the sinus became sine in English.[13]
determinate 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 + b = cy, a topic that has come to be
known as diophantine equations. This is an example from Bhāskara's commentary onAryabhatiya:
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. Ingeneral, 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 the kuṭṭakamethod. Kuttaka means "pulverizing" or "breaking into small pieces", and the method
involves a recursive algorithm for writing the original factors in smaller numbers. Todaythis 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.[14]
The diophantine equations are of interest in cryptology, and the RSA Conference, 2006,focused on the kuttaka method and earlier work in the Sulvasutras.
Algebra
In Aryabhatiya Aryabhata provided elegant results for the summation of series of squares
and cubes.
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Full name Pythagoras (Πυθαγόρας)
Bornc. 570 BC
Samos Island
Diedc. 495 BC (aged around 75)
Metapontum
Era Ancient philosophy
Region Western philosophy
School Pythagoreanism
Main
interests
Metaphysics, Music, Mathematics, Ethics,
Politics
Notable
ideas
Musica universalis, Golden ratio, Pythagorean
tuning, Pythagorean theorem
Biographical sources
Accurate facts about the life of Pythagoras are so few, and most information concerning
him is of so late a date, and so untrustworthy, that it is impossible to provide more than avague outline of his life. The lack of information by contemporary writers, together with
the secrecy which surrounded the Pythagorean brotherhood, meant that invention took the
place of facts. The stories which were created were eagerly sought by the Neoplatonist writers who provide most of the details about Pythagoras, but who were uncritical
concerning anything which related to the gods or which was considered divine.[3] Thus
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many myths were created – such as that Apollo was his father; that Pythagoras gleamed
with a supernatural brightness; that he had a golden thigh; that Abaris came flying to him
on a golden arrow; that he was seen in different places at one and the same time.[4] Withthe exception of a few remarks by Xenophanes, Heraclitus, Herodotus, Plato, Aristotle,
and Isocrates, we are mainly dependent on Diogenes Laërtius, Porphyry, and Iamblichus
for the biographical details. Aristotle had written a separate work on the Pythagoreans,which unfortunately has not survived.[5] His disciples Dicaearchus, Aristoxenus, and
Heraclides Ponticus had written on the same subject. These writers, late as they are, were
among the best sources from whom Porphyry and Iamblichus drew, besides the legendaryaccounts and their own inventions. Hence historians are often reduced to considering the
statements based on their inherent probability, but even then, if all the credible stories
concerning Pythagoras were supposed true, his range of activity would be impossibly
vast.[6]
Life
Bust of Pythagoras, Vatican
Herodotus, Isocrates, and other early writers all agree that Pythagoras was born on
Samos, the Greek island in the eastern Aegean, and we also learn that Pythagoras was the
son of Mnesarchus.[7] His father was a gem-engraver or a merchant. His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying, "He spoke
(agor-) the truth no less than did the Pythian ( Pyth-)," and Iamblichus tells the story that
the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and beneficial to humankind.[8] A late source gives his mother's name as
Pythais.[9] As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the
reign of Polycrates, at the age of 40, which would give a date of birth around 570 BC.[10]
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It was natural for the ancient biographers to inquire as to origins of Pythagoras'
remarkable system. In the absence of reliable information, however, a huge range of
teachers were assigned to Pythagoras. Some made his training almost entirely Greek,others exclusively Egyptian and Oriental. We find mentioned as his instructors
Creophylus,[11] Hermodamas,[12] Bias,[11] Thales,[11] Anaximander ,[13] and Perceives of
Syros.[14]
The Egyptians are said to have taught him geometry, the Phoenicians arithmetic,the Chaldeans astronomy, the Malians the principles of religion and practical maxims for
the conduct of life.[15] Of the various claims regarding his Greek teachers, Perceives is
mentioned most often.
It was the standard belief in antiquity that Pythagoras had undertaken extensive travels,and had visited not only Egypt, but Arabia, Phoenicia, Judaea, Babylon, and even India,
for the purpose of collecting all available knowledge, and especially to learn information
concerning the secret or mystic cults of the gods.[16] The journey to Babylon is possible,and not very unlikely. That Pythagoras visited Egypt, may be more probable, and many
ancient writers asserted this.[17] Enough of Egypt was known to attract the curiosity of an
inquiring Greek, and contact between Samos and other parts of Greece with Egypt ismentioned.[18]
It is not easy to say how much Pythagoras learned from the Egyptian priests, or indeed,
whether he learned anything at all from them. There was nothing in the symbolism which
the Pythagoreans adopted which showed the distinct traces of Egypt. The secret religiousrites of the Pythagoreans exhibited nothing but what might have been adopted in the spirit
of Greek religion, by those who knew nothing of Egyptian mysteries. The philosophy and
the institutions of Pythagoras might easily have been developed by a Greek mind exposed
to the ordinary influences of the age. Even the ancient authorities note the similarities between the religious and ascetic peculiarities of Pythagoras with the Orphic or Cretan
mysteries,
[19]
or the Delphic oracle.
[20]
There is little direct evidence as to the kind and amount of knowledge which Pythagoras
acquired, or as to his definite philosophical views. Everything of the kind mentioned byPlato and Aristotle is attributed not to Pythagoras, but to the Pythagoreans. Heraclitus
stated that he was a man of extensive learning;[21] and Xenophanes claimed that he
believed in the transmigration of souls.[22] Xenophanes mentions the story of hisinterceding on behalf of a dog that was being beaten, professing to recognize in its cries
the voice of a departed friend. Pythagoras is supposed to have claimed that he had been
Euphorbia, the son of Pan thus, in the Trojan war , as well as various other characters, atradesman, a courtesan, etc.[23]
Many mathematical and scientific discoveries were attributed to Pythagoras, including
his famous theorem,[24] as well as discoveries in the field of music,[25] astronomy,[26] and
medicine.[27] But it was the religious element which made the profoundest impressionupon his contemporaries. Thus the people of Croton were supposed to have identified
him with the Hyperborean Apollo,[28] and he was said to have practiced divination and
prophecy.[29] In the visits to various places in Greece - Delos, Sparta, Phil’s, Crete, etc.
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which are ascribed to him, he usually appears either in his religious or priestly guise, or
else as a lawgiver.[30]
Croton on the southern coast of Italy
After his travels, Pythagoras moved (around 530 BC) to Croton, in Italy (Magna
Graecia). Possibly the tyranny of Polycrates in Samos made it difficult for him to achieve
his schemes there. His later admirers claimed that Pythagoras was so overburdened with public duties in Samos, because of the high estimation in which he was held by his
fellow-citizens, that he moved to Croton.[31] On his arrival in Croton, he quickly attained
extensive influence, and many people began to follow him. Later biographers tell
fantastical stories of the effects of his eloquent speech in leading the people of Croton toabandon their luxurious and corrupt way of life and devote themselves to the purer
system which he came to introduce.[32]
His followers established a select brotherhood or club for the purpose of pursuing the
religious and ascetic practices developed by their master. The accounts agree that whatwas done and taught among the members was kept a profound secret. The esoteric
teachings may have concerned the secret religious doctrines and usages, which wereundoubtedly prominent in the Pythagorean system, and may have been connected withthe worship of Apollo.[33] Temperance of all kinds seems to have been strictly urged.
There is disagreement among the biographers as to whether Pythagoras forbade all
animal food,[34] or only certain types.[35] The club was in practice at once "a philosophicalschool, a religious brotherhood, and a political association." [36]
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Pythagoras, depicted on a 3rd-century coin
Such an aristocratic and exclusive club could easily have made many people in Croton
jealous and hostile, and this seems to have led to its destruction. The circumstances,
however, are uncertain. Conflict seems to have broken out between the towns of Sybaris
and Croton. The forces of Croton were headed by the Pythagorean Milo, and it is likelythat the members of the brotherhood took a prominent part. After the decisive victory by
Croton, a proposal for establishing a more democratic constitution, was unsuccessfullyresisted by the Pythagoreans. Their enemies, headed by Ceylon and Niño, the former of
whom is said to have been irritated by his exclusion from the brotherhood, roused the
populace against them. An attack was made upon them while assembled either in the
house of Milo, or in some other meeting-place. The building was set on fire, and many of the assembled members perished; only the younger and more active escaping.[37] Similar
commotions ensued in the other cities of Magna Graecia in which Pythagorean clubs had
been formed.
As an active and organized brotherhood the Pythagorean order was everywheresuppressed, and did not again revive. Still the Pythagoreans continued to exist as a sect,
the members of which kept up among themselves their religious observances and
scientific pursuits, while individuals, as in the case of Architect, acquired now and thengreat political influence. Concerning the fate of Pythagoras himself, the accounts varied.
Some say that he perished in the temple with his disciples,[38] others that he fled first to
Tarentum, and that, being driven from there, he escaped to Metapontum, and therestarved himself to death.[39] His tomb was shown at Metapontum in the time of Cicero.[40]
According to some accounts Pythagoras married Thaana, a lady of Croton. Their children
are variously stated to have included a son, Deluges, and three daughters, Demo,
Aragonite, and Myna.
Writings
No texts by Pythagoras are known to have survived, although forgeries under his name
— a few of which remain extant — did circulate in antiquity. Critical ancient sources like
Aristotle and Aristoxenus cast doubt on these writings. Ancient Pythagoreans usuallyquoted their master's doctrines with the phrase autos epee ("he himself said") —
emphasizing the essentially oral nature of his teaching.
Mathematics
The so-called Pythagoreans, who were the first to take up mathematics, not onlyadvanced this subject, but saturated with it, they fancied that the principles of
mathematics were the principles of all things.
— Aristotle, Metaphysics 1-5 , cc. 350 BC
Pythagorean theorem
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The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and
b) equals the area of the square on the hypotenuse (c).
A visual proof of the Pythagorean theorem
Since the fourth century AD, Pythagoras has commonly been given credit for discovering
the Pythagorean theorem, a theorem in geometry that states that in a right-angled trianglethe square of the hypotenuse (the side opposite the right angle), c, is equal to the sum of
the squares of the other two sides, b and a —that is, a2 + b2 = c2.
While the theorem that now bears his name was known and previously utilized by theBabylonians and Indians, he, or his students, are often said to have constructed the first
proof. It must, however, be stressed that the way in which the Babylonians handled
Pythagorean numbers implies that they knew that the principle was generally applicable,and knew some kind of proof, which has not yet been found in the (still largely
unpublished) cuneiform sources.[41] Because of the secretive nature of his school and the
custom of its students to attribute everything to their teacher, there is no evidence thatPythagoras himself worked on or proved this theorem. For that matter, there is no
evidence that he worked on any mathematical or meta-mathematical problems. Some
attribute it as a carefully constructed myth by followers of Plato over two centuries after
the death of Pythagoras, mainly to bolster the case for Platonic meta-physics, whichresonate well with the ideas they attributed to Pythagoras. This attribution has stuck down
the centuries up to modern times.[42] The earliest known mention of Pythagoras's name in
connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch.
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Musical theories and investigations
Medieval woodcut showing Pythagoras with bells in Pythagorean tuning
See also: Pythagorean tuning
According to legend, the way Pythagoras discovered that musical notes could betranslated into mathematical equations was when one day he passed blacksmiths at work,
and thought that the sounds emanating from their anvils being hit were beautiful and
harmonious and decided that whatever scientific law caused this to happen must bemathematical and could be applied to music. He went to the blacksmiths to learn how this
had happened by looking at their tools, he discovered that it was because the anvils were
"simple ratios of each other, one was half the size of the first, another was 2/3 the size,
and so on."
Pythagoreans elaborated on a theory of numbers, the exact meaning of which is still
debated among scholars. Another belief attributed to Pythagoras was that of the"harmony of the spheres". Thus the planets and stars moved according to mathematical
equations, which corresponded to musical notes and thus produced a symphony.[43]
Detracts
Pythagoras was also credited with devising the detracts, the triangular figure of four
rows, which add up to the perfect number, ten. As a mystical symbol, it was veryimportant to the worship of the Pythagoreans, who would swear oaths by it:
And the inventions were so admirable, and so divinized by those who understood them,that the members used them as forms of oath: "By him who handed to our generation thedetracts, source of the roots of ever-flowing nature." —Iamblichus, Vet. Pyth., 29
Pythagoreans
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See also: Pythagoreanism
Pythagoras, the man in the center with the book, teaching music, in The School of Athens
by Raphael
Both Plato and Isocrates affirm that, above all else, Pythagoras was famous for leaving
behind him a way of life.[50] Both Iamblichus and Porphyry give detailed accounts of theorganization of the school, although the primary interest of both writers is not historical
accuracy, but rather to present Pythagoras as a divine figure, sent by the gods to benefit
humankind.[51]
Pythagoras set up an organization which was in some ways a school, in some ways a brotherhood, and in some ways a monastery. It was based upon the religious teachings of
Pythagoras and was very secretive. The adherents were bound by a vow to Pythagoras
and each other, for the purpose of pursuing the religious and ascetic observances, and of studying his religious and philosophical theories. The claim that they put all their
property into a common stock is perhaps only a later inference from certain Pythagorean
maxims and practices.[52] On the other hand, it seems certain that there were many women among the adherents of Pythagoras.[53]
As to the internal arrangements of the sect, we are informed that what was done and
taught among the members was kept a profound secret towards all. Porphyry stated that
this silence was "of no ordinary kind." Candidates had to pass through a period of probation, in which their powers of maintaining silence (echemythia) were especially
tested, as well as their general temper, disposition, and mental capacity.[54] There were
also gradations among the members themselves. It was an old Pythagorean maxim, thatevery thing was not to be told to every body.[55] Thus the Pythagoreans were divided into
an inner circle called the mathematikoi ("learners") and an outer circle called theakousmatikoi ("listeners").[56] Iamblichus describes them in terms of esoterikoi andexoterikoi (or alternatively Pythagoreioi and Pythagoristai),[57] according to the degree of intimacy which they enjoyed with Pythagoras. Porphyry wrote "the mathematikoi learned
the more detailed and exactly elaborated version of this knowledge, the akousmatikoi
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(were) those who had heard only the summary headings of his (Pythagoras's) writings,
without the more exact exposition."
There were ascetic practices (many of which had, perhaps, a symbolic meaning) in theway of life of the sect.[58] Some represent Pythagoras as forbidding all animal food. This
may have been due to the doctrine of metempsychosis.[59]
Other authorities contradict thestatement. According to Aristoxenus,[60] he allowed the use of all kinds of animal food
except the flesh of oxen used for ploughing, and rams.[61] There is a similar discrepancyas to the prohibition of fish and beans.[62] But temperance of all kinds seems to have been
urged. It is also stated that they had common meals, resembling the Spartan system, at
which they met in companies of ten.[63]
Considerable importance seems to have been attached to music and gymnastics in thedaily exercises of the disciples. Their whole discipline is represented as encouraging a
lofty serenity and self-possession, of which, there were various anecdotes in antiquity.[64]
Iamblichus (apparently on the authority of Aristoxenus)[65] gives a long description of the
daily routine of the members, which suggests many similarities with Sparta. Themembers of the sect showed a devoted attachment to each other, to the exclusion of those
who did not belong to their ranks.[66] There were even stories of secret symbols, by whichmembers of the sect could recognize each other, even if they had never met before.[67]
Influence
Influence on Plato
Pythagoras, depicted as a medieval scholar in the Nuremberg Chronicle
Pythagoras, or in a broader sense, the Pythagoreans, allegedly exercised an important
influence on the work of Plato. According to R. M. Hare, this influence consists of three points: (1) The platonic Republic might be related to the idea of "a tightly organized
community of like-minded thinkers", like the one established by Pythagoras in Croton.
(2) There is evidence that Plato possibly took from Pythagoras the idea that mathematicsand, generally speaking, abstract thinking is a secure basis for philosophical thinking as
well as "for substantial theses in science and morals". (3) Plato and Pythagoras shared a
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"mystical approach to the soul and its place in the material world". It is probable that both
were influenced by Orphism.[68]
Aristotle claimed that the philosophy of Plato closely followed the teachings of thePythagoreans,[69] and Cicero repeats this claim: Platonem ferunt didicisse Pythagorea
omnia ("They say Plato learned all things Pythagorean").[70]
Bertrand Russell, in his History of Western Philosophy, contended that the influence of Pythagoras on Plato and
others was so great that he should be considered the most influential of all Western philosophers.
Influence on esoteric groups
Pythagoras started a secret society called the Pythagorean brotherhood devoted to the
study of mathematics. This had a great effect on future esoteric traditions, such asRosicrucianism and Freemasonry, both of which were occult groups dedicated to the
study of mathematics and both of which claimed to have evolved out of the Pythagorean
brotherhood.[citation needed ] The mystical and occult qualities of Pythagorean mathematics arediscussed in a chapter of Manly P. Hall's The Secret Teachings of All Ages entitled
"Pythagorean Mathematics".
Pythagorean theory was tremendously influential on later numerology, which was
extremely popular throughout the Middle East in the ancient world. The 8th-centuryMuslim alchemist Jabir ibn Hayyan grounded his work in an elaborate numerology
greatly influenced by Pythagorean theory.[citation needed ] Today, Pythagoras is revered as a
prophet by the Ahl al-Tawhid or Druze faith along with his fellow Greek, Plato.
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Early years
Old Swiss 10 Franc banknote honoring Euler
Euler was born on April 15, 1707 in Basel to Paul Euler, a pastor of the Reformed
Church. His mother was Marguerite Brucker, a pastor's daughter. He had two younger
sisters named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard, the
Eulers moved from Basel to the town of Riehen, where Euler spent most of hischildhood. Paul Euler was a friend of the Bernoulli family — Johann Bernoulli, who was
then regarded as Europe's foremost mathematician, would eventually be the most
important influence on young Leonhard. Euler's early formal education started in Basel,where he was sent to live with his maternal grandmother. At the age of thirteen he
enrolled at the University of Basel, and in 1723, received his Master of Philosophy with a
dissertation that compared the philosophies of Descartes and Newton. At this time, hewas receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered
his new pupil's incredible talent for mathematics.[6] Euler was at this point studying
theology, Greek , and Hebrew at his father's urging, in order to become a pastor, butBernoulli convinced Paul Euler that Leonhard was destined to become a great
mathematician. In 1726, Euler completed a dissertation on the propagation of sound withthe title De Sono[7]. At that time, he was pursuing an (ultimately unsuccessful) attempt to
obtain a position at the University of Basel. In 1727, he entered the Paris Academy Prize
Problem competition, where the problem that year was to find the best way to place the
masts on a ship. He won second place, losing only to Pierre Bouguer —who is now
known as "the father of naval architecture". Euler subsequently won this coveted annual prize twelve times in his career .[8]
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St. Petersburg
Around this time Johann Bernoulli's two sons, Daniel and Nicolas, were working at the
Imperial Russian Academy of Sciences in St Petersburg. On July 10, 1726, Nicolas diedof appendicitis after spending a year in Russia, and when Daniel assumed his brother's
position in the mathematics/physics division, he recommended that the post in physiologythat he had vacated be filled by his friend Euler. In November 1726 Euler eagerlyaccepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully
applied for a physics professorship at the University of Basel.[9]
1957 stamp of the former Soviet Union commemorating the 250th birthday of Euler. Thetext says: 250 years from the birth of the great mathematician, academician Leonhard
Euler.
Euler arrived in the Russian capital on 17 May 1727. He was promoted from his junior
post in the medical department of the academy to a position in the mathematicsdepartment. He lodged with Daniel Bernoulli with whom he often worked in close
collaboration. Euler mastered Russian and settled into life in St Petersburg. He also took
on an additional job as a medic in the Russian Navy.[10]
The Academy at St. Petersburg, established by Peter the Great, was intended to improveeducation in Russia and to close the scientific gap with Western Europe. As a result, it
was made especially attractive to foreign scholars like Euler. The academy possessed
ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy in
order to lessen the faculty's teaching burden, and the academy emphasized research and
offered to its faculty both the time and the freedom to pursue scientific questions.[8]
The Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility then gained
power upon the ascension of the twelve-year-old Peter II. The nobility were suspicious of
the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues.
Conditions improved slightly upon the death of Peter II, and Euler swiftly rose through
the ranks in the academy and was made professor of physics in 1731. Two years later,
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Daniel Bernoulli, who was fed up with the censorship and hostility he faced at St.
Petersburg, left for Basel. Euler succeeded him as the head of the mathematics
department.[11]
On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell,
a painter from the Academy Gymnasium.[12]
The young couple bought a house by the Neva River . Of their thirteen children, only five survived childhood. [13]
Berlin
Stamp of the former German Democratic Republic honoring Euler on the 200th
anniversary of his death. In the middle, it shows his polyhedral formula V + F − E = 2.
Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the
Great of Prussia. He lived for twenty-five years in Berlin, where he wrote over 380
articles. In Berlin, he published the two works which he would be most renowned for: the Introductio in analysin infinitorum, a text on functions published in 1748, and the Institutiones calculi differentialis,[14] published in 1755 on differential calculus.[15] In
1755, he was elected a foreign member of the Royal Swedish Academy of Sciences.
In addition, Euler was asked to tutor the Princess of Anhalt-Dessau[who?], Frederick's
niece. Euler wrote over 200 letters to her [when?], which were later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. This work contained Euler's exposition on various
subjects pertaining to physics and mathematics, as well as offering valuable insights intoEuler's personality and religious beliefs. This book became more widely read than any of
his mathematical works, and it was published across Europe and in the United States. The
popularity of the 'Letters' testifies to Euler's ability to communicate scientific matterseffectively to a lay audience, a rare ability for a dedicated research scientist.[15]
Despite Euler's immense contribution to the Academy's prestige, he was eventually
forced to leave Berlin. This was partly because of a conflict of personality with Frederick,
who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy. Voltaire was among those in
Frederick's employ, and the Frenchman enjoyed a prominent position in the king's social
circle. Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the direct opposite of Voltaire. Euler had limited
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training in rhetoric, and tended to debate matters that he knew little about, making him a
frequent target of Voltaire's wit.[15] Frederick also expressed disappointment with Euler's
practical engineering abilities:
“I wanted to have a water jet in my garden: Euler calculated the force of the
wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci. My mill was carried
out geometrically and could not raise a mouthful of water closer than fifty
paces to the reservoir. Vanity of vanities! Vanity of geometry![16] ”
A 1753 portrait by Emanuel Handmann. This portrayal suggests problems of the right
eyelid, and possible strabismus. The left eye appears healthy; it was later affected by acataract.[17]
Eyesight deterioration
Euler's eyesight worsened throughout his mathematical career. Three years after suffering
a near-fatal fever in 1735 he became nearly blind in his right eye, but Euler rather blamed
his condition on the painstaking work on cartography he performed for the St. PetersburgAcademy. Euler's sight in that eye worsened throughout his stay in Germany, so much so
that Frederick referred to him as "Cyclops". Euler later suffered a cataract in his good left
eye, rendering him almost totally blind a few weeks after its discovery in 1766. Even so,
his condition appeared to have little effect on his productivity, as he compensated for itwith his mental calculation skills and photographic memory. For example, Euler could
repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page
in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Euler's productivity on many areas of study actually increased. He produced
on average one mathematical paper every week in the year 1775.[3]
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Return to Russia
The situation in Russia had improved greatly since the accession to the throne of
Catherine the Great, and in 1766 Euler accepted an invitation to return to the St.Petersburg Academy and spent the rest of his life in Russia. His second stay in the
country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, andalmost his life. In 1773, he lost his wife Katharina after 40 years of marriage. Three yearsafter his wife's death Euler married her half sister, Salome Abigail Gsell (1723–1794).[18]
This marriage would last until his death.
On 18 September 1783, after a lunch with his family, during a conversation with Anders
Lexell about the newly discovered Uranus and its orbit Euler suffered a brain hemorrhage and died few hours later.[19] A short obituary for the Russian Academy of Sciences was
written by Jacob von Shtelin and a more detailed eulogy[20] was written and delivered at a
memorial meeting by Russian mathematician Nicolas Fuss, one of the Euler's disciples.In the eulogy written for the French Academy by the French mathematician and
philosopher Marquis de Condorcet, he commented,
“…il cessa de calculer et de vivre — … he ceased to calculate and to live.[21]
”
He was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky
Island. In 1785 Russian Academy of Sciences put a marble bust of Leonhard Euler on a pedestal next to the Director's seat. In 1837 Russian Academy of Sciences put a
headstone on his grave, which in 1956, for the 250th anniversary of Euler, was moved
together with his remains to the necropolis of 18th century at Alexander Nevsky Lavra.[22]
Euler's grave at the Alexander Nevsky Lavra
Contributions to mathematics and physics
Part of a series of articles on
The mathematical
constant e
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Natural logarithm · Exponentialfunction
Applications in: compoundinterest · Euler's identity &
Euler's formula · half-lives &
exponential growth/decay
Defining e: proof that e isirrational · representations of e ·
Lindemann–Weierstrasstheorem
People John Napier ·
Leonhard Euler
Schanuel's conjecture
Euler worked in almost all areas of mathematics: geometry, infinitesimal calculus,
trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and
other areas of physics. He is a seminal figure in the history of mathematics; if printed, his
works, many of which are of fundamental interest, would occupy between 60 and 80quarto volumes.[3] Euler's name is associated with a large number of topics.
Mathematical notation
Euler introduced and popularized several notational conventions through his numerousand widely circulated textbooks. Most notably, he introduced the concept of a function[2]
and was the first to write f ( x) to denote the function f applied to the argument x. He also
introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Euler's number ), the Greek letter Σ for
summations and the letter i to denote the imaginary unit.[23] The use of the Greek letter π
to denote the ratio of a circle's circumference to its diameter was also popularized byEuler, although it did not originate with him. [24]
Analysis
The development of infinitesimal calculus was at the forefront of 18th century
mathematical research, and the Bernoullis —family friends of Euler—were responsible
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for much of the early progress in the field. Thanks to their influence, studying calculus
became the major focus of Euler's work. While some of Euler's proofs are not acceptable
by modern standards of mathematical rigour ,[25] his ideas led to many great advances.Euler is well-known in analysis for his frequent use and development of power series, the
expression of functions as sums of infinitely many terms, such as
Notably, Euler directly proved the power series expansions for e and the inverse tangentfunction. (Indirect proof via the inverse power series technique was given by Newton and
Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the
famous Basel problem in 1735 (he provided a more elaborate argument in 1741):[25]
A geometric interpretation of Euler's formula
Euler introduced the use of the exponential function and logarithms in analytic proofs. He
discovered ways to express various logarithmic functions using power series, and he
successfully defined logarithms for negative and complex numbers, thus greatlyexpanding the scope of mathematical applications of logarithms.[23] He also defined the
exponential function for complex numbers, and discovered its relation to the
trigonometric functions. For any real number φ, Euler's formula states that the complexexponential function satisfies
A special case of the above formula is known as Euler's identity,
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called "the most remarkable formula in mathematics" by Richard Feynman, for its single
uses of the notions of addition, multiplication, exponentiation, and equality, and the
single uses of the important constants 0, 1, e, i and π.[26] In 1988, readers of theMathematical Intelligencer voted it "the Most Beautiful Mathematical Formula Ever".[27]
In total, Euler was responsible for three of the top five formulae in that poll .[27]
De Moivre's formula is a direct consequence of Euler's formula.
In addition, Euler elaborated the theory of higher transcendental functions by introducingthe gamma function and introduced a new method for solving quartic equations. He also
found a way to calculate integrals with complex limits, foreshadowing the development
of modern complex analysis, and invented the calculus of variations including its best-
known result, the Euler–Lagrange equation.
Euler also pioneered the use of analytic methods to solve number theory problems. In
doing so, he united two disparate branches of mathematics and introduced a new field of
study, analytic number theory. In breaking ground for this new field, Euler created thetheory of hypergeometric series, q-series, hyperbolic trigonometric functions and the
analytic theory of continued fractions. For example, he proved the infinitude of primes
using the divergence of the harmonic series, and he used analytic methods to gain some
understanding of the way prime numbers are distributed. Euler's work in this area led tothe development of the prime number theorem.[28]
Number theory
Euler's interest in number theory can be traced to the influence of Christian Goldbach, his
friend in the St. Petersburg Academy. A lot of Euler's early work on number theory was
based on the works of Pierre de Fermat. Euler developed some of Fermat's ideas, anddisproved some of his conjectures.
Euler linked the nature of prime distribution with ideas in analysis. He proved that thesum of the reciprocals of the primes diverges. In doing so, he discovered the connection
between the Riemann zeta function and the prime numbers; this is known as the Euler
product formula for the Riemann zeta function.
Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and he made distinct contributions to Lagrange's four-square theorem. He
also invented the totient function φ(n) which is the number of positive integers less than
or equal to the integer n that are coprime to n. Using properties of this function, hegeneralized Fermat's little theorem to what is now known as Euler's theorem. Hecontributed significantly to the theory of perfect numbers, which had fascinated
mathematicians since Euclid. Euler also made progress toward the prime number
theorem, and he conjectured the law of quadratic reciprocity. The two concepts areregarded as fundamental theorems of number theory, and his ideas paved the way for the
work of Carl Friedrich Gauss.[29]
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By 1772 Euler had proved that 231 − 1 = 2,147,483,647 is a Mersenne prime. It may have
remained the largest known prime until 1867.[30]
Graph theory
See also: Seven Bridges of Königsberg
Map of Königsberg in Euler's time showing the actual layout of the seven bridges,
highlighting the river Pregel and the bridges.
In 1736, Euler solved the problem known as the Seven Bridges of Königsberg.[31] The
city of Königsberg, Prussia was set on the Pregel River, and included two large islands
which were connected to each other and the mainland by seven bridges. The problem is
to decide whether it is possible to follow a path that crosses each bridge exactly once andreturns to the starting point. It is not possible: there is no Eulerian circuit. This solution is
considered to be the first theorem of graph theory, specifically of planar graph theory.[31]
Euler also discovered the formula V − E + F = 2 relating the number of vertices, edges,and faces of a convex polyhedron,[32] and hence of a planar graph. The constant in this
formula is now known as the Euler characteristic for the graph (or other mathematical
object), and is related to the genus of the object.[33] The study and generalization of this
formula, specifically by Cauchy[34] and L'Huillier ,[35] is at the origin of topology.
Applied mathematics
Some of Euler's greatest successes were in solving real-world problems analytically, and
in describing numerous applications of the Bernoulli numbers, Fourier series, Venn
diagrams, Euler numbers, the constants e and π, continued fractions and integrals. Heintegrated Leibniz's differential calculus with Newton's Method of Fluxions, and
developed tools that made it easier to apply calculus to physical problems. He made great
strides in improving the numerical approximation of integrals, inventing what are nowknown as the Euler approximations. The most notable of these approximations are Euler's
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method and the Euler–Maclaurin formula. He also facilitated the use of differential
equations, in particular introducing the Euler–Mascheroni constant:
One of Euler's more unusual interests was the application of mathematical ideas in music.
In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate
musical theory as part of mathematics. This part of his work, however, did not receivewide attention and was once described as too mathematical for musicians and too musical
for mathematicians.[36]
Physics and astronomy
Classical mechanics
Newton's Second Law
History of classical mechanics ·
Timeline of classical mechanics
[show]Branches
[show]Formulations
[show]Fundamentalconcepts
[show]Core topics
[hide]Scientists
Isaac Newton · Jeremiah
Horrocks · Leonhard
Euler · Jean le Rond
d'Alembert · Alexis
Clairaut · Joseph Louis
Lagrange · Pierre-Simon
Laplace · William Rowan
Hamilton · Siméon-Denis
Poisson
v • d • e
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Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering. Aside from successfully applying his analytic tools to problems in classical
mechanics, Euler also applied these techniques to celestial problems. His work inastronomy was recognized by a number of Paris Academy Prizes over the course of hiscareer. His accomplishments include determining with great accuracy the orbits of
comets and other celestial bodies, understanding the nature of comets, and calculating the
parallax of the sun. His calculations also contributed to the development of accuratelongitude tables.[37]
In addition, Euler made important contributions in optics. He disagreed with Newton's
corpuscular theory of light in the Opticks, which was then the prevailing theory. His
1740s papers on optics helped ensure that the wave theory of light proposed by ChristianHuygens would become the dominant mode of thought, at least until the development of
the quantum theory of light.[38]
Logic
He is also credited with using closed curves to illustrate syllogistic reasoning (1768).These diagrams have become known as Euler diagrams.[39]
Personal philosophy and religious beliefs
Euler and his friend Daniel Bernoulli were opponents of Leibniz's monadism and the
philosophy of Christian Wolff . Euler insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were
unable to provide. Euler's religious leanings might also have had a bearing on his dislike
of the doctrine; he went so far as to label Wolff's ideas as "heathen and atheistic".[40]
Much of what is known of Euler's religious beliefs can be deduced from his Letters to a
German Princess and an earlier work, Rettung der Göttlichen Offenbahrung Gegen die
Einwürfe der Freygeister ( Defense of the Divine Revelation against the Objections of the Freethinkers). These works show that Euler was a devout Christian who believed theBible to be inspired; the Rettung was primarily an argument for the divine inspiration of
scripture.[5]
There is a famous anecdote inspired by Euler's arguments with secular philosophers over
religion, which is set during Euler's second stint at the St. Petersburg academy. TheFrench philosopher Denis Diderot was visiting Russia on Catherine the Great's invitation.
However, the Empress was alarmed that the philosopher's arguments for atheism were
influencing members of her court, and so Euler was asked to confront the Frenchman.Diderot was later informed that a learned mathematician had produced a proof of the
existence of God: he agreed to view the proof as it was presented in court. Euler
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appeared, advanced toward Diderot, and in a tone of perfect conviction announced, "Sir,
, hence God exists—reply!". Diderot, to whom (says the story) allmathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court.
Embarrassed, he asked to leave Russia, a request that was graciously granted by the
Empress. However amusing the anecdote may be, it is apocryphal, given that Diderot wasa capable mathematician who had published mathematical treatises.[41]
Selected bibliography
The title page of Euler's Methodus inveniendi lineas curvas.
Euler has an extensive bibliography. His best known books include:
• Elements of Algebra. This elementary algebra text starts with a discussion of the
nature of numbers and gives a comprehensive introduction to algebra, including
formulae for solutions of polynomial equations.
• Introductio in analysin infinitorum (1748). English translation Introduction to
Analysis of the Infinite by John Blanton (Book I, ISBN 0-387-96824-5, Springer-
Verlag 1988; Book II, ISBN 0-387-97132-7, Springer-Verlag 1989).
•
Two influential textbooks on calculus: Institutiones calculi differentialis (1755)and Institutionum calculi integralis (1768–1770).
• Lettres à une Princesse d'Allemagne (Letters to a German Princess) (1768–1772).
Available online (in French). English translation, with notes, and a life of Euler,available online from Google Books: Volume 1, Volume 2
• Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti (1744). The Latin title
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translates as a method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense.[42]
A definitive collection of Euler's works, entitled Opera Omnia, has been published since1911 by the Euler Commission of the Swiss Academy of Sciences
Born15 April 1707
Basel, Switzerland
Died
18 September 1783
(aged 76)
[OS: 7 September 1783]
St. Petersburg, Russia
Residence Prussia, RussiaSwitzerland
Nationality Swiss
FieldsMathematician and
Physicist
InstitutionsImperial Russian
Academy of Sciences
Berlin Academy
Alma mater University of Basel
DoctoraladvisorJohann Bernoulli
Doctoral
students
Nicolas Fuss
Johann Hennert
Joseph Louis Lagrange
Stepan Rumovsky
Known for See full list
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Signature
Notes
He is the father of the mathematician JohannEuler
He is listed by academic genealogy
authorities as the equivalent to the doctoral
advisor of Joseph Louis Lagrange.