The globe of the Earth stands supportless in space... Just as the [spherical] bulb of a Kadamba flower is covered all around by blossoms, just so is the globe of the Earth surrounded by all creatures, terrestrial as well as aquatic.
Aryabhata, (Aryabhatiya, Gola 6-7), quoted in C. K. Raju. Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe. Pearson Education India, 2007
When sixty times sixty years and three quarter yugas (of the current yuga) had elapsed, twenty three years had then passed since my birth.
quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022ISBN 9798885750189
This verse says that Aryabhata was exactly 23 years of age on the date which, according to the calendar being followed today, corresponds to March 21, 499 CE
Just as a man in a boat moving forward sees the stationary objects (on either side of the river) as moving backward, just so are the stationary stars seen by people at Lanka as moving exactly towards the west. (It so appears as if ) the entire structure of the asterisms together with the planets were moving exactly towards the west of Lanka, being constantly driven by the provector wind, to cause their rising and setting.
quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022ISBN 9798885750189
100 plus 4, multiplied by 8, and added to 62,000: this is the nearly approximate measure of the circumference of a circle whose diameter is 20,000.
According to the verse, Aryabhata’s value of π equals ((100+4)×8 + 62000)/20000, which is 3.1416. Ganitapada 10
quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022ISBN 9798885750189
Translates to: 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. Thus according to the rule ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.
His value of π is a very close approximation to the modern value and the most accurate among those of the ancients. There are reasons to believe that he devised a particular method for finding this value....
His value of π is a very close approximation to the modern value and the most accurate among those of the ancients. There are reasons to believe that he devised a particular method for finding this value. It is shown with sufficient grounds that he himself used it, and several later Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is ofGreek origin is critically examined and is found to be without foundation. He discovered this value independently and also realised that π is anirrational number. He had theIndian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit of discovering this exact value of π may be ascribed to the celebrated mathematician, Aryabhata I.
P Jha, Aryabhata I and the value of π, Math. Ed. (Siwan) 16 (3) (1982), 54-59, quoted in: J J O'Connor and E F RobertsonAryabhata the Elder, School of Mathematics and Statistics University of St Andrews, Scotland
Aryabhata is also known as Aryabhata I to distinguish him from the later mathematician of the same name who lived about 400 years later.
He is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimateknowledge ofmathematics,kinematics andspherics, handed over the three sciences to the learnedworld.
... it is extremely likely that Aryabhata knew the sign forzero and the numerals of theplace value system. This supposition is based on the following two facts: first, theinvention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations onsquare andcubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.
G Ifrah, in "A universal history of numbers : From prehistory to the invention of the computer" quoyed in in: J J O'Connor andE F Robertson "Aryabhata the Elder".
He gave more elegant rules for the sum of the squares and cubes of an initial segment of the positive integers. The sixth part of the product of three quantities consisting of the number of terms, the number of terms plus one, and twice the number of terms plus one is the sum of the squares. The square of the sum of the series is the sum of the cubes.
His work, calledAryabhatiya, is composed of three parts, in only the first of which use is made of a special notation of numbers. It is analphabetical system in which the twenty-five consonants represent 1-25, respectively; other letters stand for 30, 40, …., 100 etc. The other mathematical parts of Aryabhatiya consists of rules without examples. Another alphabetic system prevailed inSouthern India, the numbers 1-19 being designated byconsonants, etc.
The greatest of Hindu astronomers and mathematicians, Aryabhata, discussed in verse such poetic subjects as quadratic equations, sines, and the value of π; he explained eclipses, solstices and equinoxes, announced the sphericity of the earth and its diurnal revolution on its axis, and wrote, in daring anticipation of Renaissance science: “The sphere of the stars is stationary, and the earth, by its revolution, produces the daily rising and setting of planets and stars.”
The Hindus were not so successful in geometry. In the measurement and construction of altars the priests formulated the Pythagorean theorem (by which the square of the hypotenuse of a right-angled triangle equals the sum of the squares of the other sides) several hundred years before the birth of Christ. Aryabhata, probably influenced by the Greeks, found the area of a triangle, a trapezium and a circle, and calculated the value of π (the relation of diameter to circumference in a circle) at 3.1416—a figure not equaled in accuracy until the days of Purbach (1423-61) in Europe. Bhaskara crudely anticipated the differential calculus, Aryabhata drew up a table of sines, and the Surya Siddhanta provided a system of trigonometry more advanced than anything known to the Greeks.
Aryabhata is acknowledged as one of the astuteastronomers of early India. His school ofastronomy is well known and widespread all overIndia, especially in the South...Of late there is a tendency to spell his name as “Aryabhatta”. While Aryabhata himself mentionsKali 3600 to be the date of his composing the work, some say that Kali 3600 is the date of his birth. A view has been broached that Aryabhata hailed fromKerala.
Of late, there has been a tendency to spell the name as “Aryabhatta” with the suffix “bhatta”. Twoartificial satellites sent up into space byIndian scientists are given the names “Aryabhatta I” and “Aryabhatta II”. Some modern writers also make use of this spelling.
In, P.105-106.
He mentions his name at three places only as “Aryabhata”, towards the beginning and ending Verses of his workAryabhatiya,
In, p. 106.
He gives a clue to his date of birth in his Aryabhatiya... The date works out to the end of theKali year 3600, corresponding to theSaka year 421, the date being 21 March 499 ...and that he composed the Aryabhatiya when he was 25 years old, i.e. in Saka 444 or AD 522. Page 4
In, P.108-109.
The second reason adduced, viz., that Aryabhata should have hailed fromKerala is fragile. Besides the Aryabhatan system being prevalent in this land,-“all” commentaries on Aryabhaflya have been produced by Kerala astronomers really does not stand scrutiny.
His fame rests mainly on hisAryabhatiya, but from the writings ofVarahamihira (Sixth century AD),Bhaskara I, andBrahmagupta (seventh century) it is clear that earlier he composed theAryabhata Siddhantha (voluminous) is not extant. It is also calledArdharatrika Siddhanta, because in it the civil days were reckoned from one midnight to the next; 34 verses onastronomical instruments from this have been quoted by Ramakrishna Aradya.
Aryabhatiya, an improved work, is product of matureintellect, which he wrote when he was 23 years old. Unlike in theAryabhata siddhanta, the civil days are reckoned from onesunrise to the next, a practice which is still prevalent among the followers ofHindu calendar.
The Aryabhatiya consists of four sections:1.Dasagitika (10+3 couplets inGiti meter); 2.Ganitapada (33 verses onmathematics); 3.Kala-kriyapada 25 verses ontime-reckoning), and 4.Golapada (50 verses onspherical astronomy)
AnArabic translation of the Aryabhatiya entitledZij-al-Arjabar (800 AD) is attributed toAhwazi.
Use of better planetary parameters, the innovations in astronomical methods, and the concise style of exposition of Aryabhatiya makes it an excellent text book onAstronomy. As opposed to the geostationary theory, Aryahabata held the view that theearth rotates on its axis. His estimate of the period of thesiderealrotation of earth was 23 hours 56 min, and 4.1 s is close to the actual value.
Use of better planetary parameters, the innovations in astronomical methods, and the concise style of exposition of Aryabhatiya makes it an excellent text book onAstronomy. As opposed to the geostationary theory, Aryahabata held the view that theearth rotates on its axis. His estimate of the period of thesiderealrotation of earth was 23 hours 56 min, and 4.1 s is close to the actual value.
He was the father of theIndianepicyclicastronomy which resulted in the planetary theory that determines more accurately the true positions and distances of theplanets (including theSun and theMoon)...was also the first to producecelestial latitudes...proposed the scientific cause ofeclipses as against the mythological demonRahu [Moon's node]. His ideas resulted in the new school ofIndian Astronomy – the Āryabhata SchoolĀryapakșa based on the text of Āryabhatīțya.
The peculiar system of alphabetic numerals evolved by him with 33 consonants of theSanskrit alphabet (Nagari script) denoted various numbers in conjunction with vowels which themselves did not represent any numerical value. For examplekhyughr (=khu+yu+ghr) is denoted by which is the number of revolutions of theSun in ayuga (epoch)
The development of Indiantrigonometry, based onsine as againstchord of theGreeks, a necessity for astronomical calculations with his own concisenotation which expresses the full sine table in just one couplet for easy remembrance. One of the two methods suggested by him for thesine table is based on the property that the second order sine differences were proportional to sines themselves.
Ingeometry his greatest achievement was an accurate value ofπ. His rule is stated as:, which implies the approximation 3.1416 which is correct to the last decimal place.
The apparent motion of theSun along the ecliptic (red) - With Kala-kriya he turned toastronomy — in particular, treatingplanetary motion along the ecliptic.
Takao Hayashi in:Aryabhata I, Encyclopædia Britannica
...he flourished in Kusumapura—nearPatalipurta (Patna), then the capital of theGupta dynasty — where he composed at least two works,Aryabhatiya (c. 499) and the now lost Aryabhatasiddhanta. Aryabhatasiddhanta circulated mainly in the northwest of India and, through theSāsānian dynasty (224–651) ofIran, had a profound influence on the development ofIslamic astronomy. Its contents are preserved to some extent in the works ofVarahamihira (flourished c. 550),Bhaskara I (flourished c. 629),Brahmagupta (598–c. 665), and others. It is one of the earliest astronomical works to assign the start of each day to midnight.
Aryabhatiya...written in verse couplets ...contains astronomical tables and Aryabhata’s system of phonemic number notation, the work is characteristically divided into three sections:Ganita (“Mathematics”),Kala-kriya (“Time Calculations”), andGola (“Sphere”).
InGanita, he names the first 10 decimal places and givesalgorithms for obtaining square and cubic roots, utilizing thedecimal number system. Then he treats geometric measurements — employing 62,832/20,000 (= 3.1416) for π—and develops properties of similarright-angled triangles and of two intersecting circles.
With Kala-kriya he turned toastronomy — in particular, treatingplanetary motion along the ecliptic. The topics include definitions of various units of time, eccentric and epicyclic models of planetary motion (see Hipparchus for earlier Greek models), planetary longitude corrections for different terrestrial locations, and a theory of “lords of the hours and days” (an astrological concept used for determining propitious times for action).
