Aryabhata contributes ‘ZERO, Pi’ etc to Mathematics and calculates Eclipses in Astronomy

Aryabhata, born in 476 CE, was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy.
There is a general tendency to misspell his name as “Aryabhatta” by analogy with other names having the “bhatta” suffix, but all his astronomical text spells his name as Aryabhata.
He mentions in his work Aryabhatiya that it was composed 3,630 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476

Though his birthplace is uncertain, he went to Kusumapura (Pataliputra or modern day Patna) for advanced studies and lived there for sometime as the head of an institution (kulapati).
Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.

He wrote many books on mathematics, astronomy etc but most of them are lost today.
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.
And Arabic translation of Aryabhata’s work is Al ntf or Al-nanf and it claims that it is a translation by Aryabhata, but the original Sanskrit name of this work is not known.

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 mathematician Georges Ifrah explains that knowledge of zero was implicit in Aryabhata’s place-value system as a place holder for the powers of ten with null coefficients.
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 of π

Aryabhata worked on the approximation for pi (π), and may have come to the conclusion that is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes :

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

Translation : “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 calculates to 3.1416 close to the actual value Pi (3.14159).
Aryabhata used the word āsanna (approaching / approximating), to mean that not only is this an approximation but that the value is incommensurable (or irrational).
This is quite a sophisticated insight, because the irrationality of pi(π) was proved only in 1761 by Johann Heinrich Lambert.
After Aryabhatiya was translated into Arabic (during 820 CE) this approximation was mentioned in Al-Khwarizmi‘s book on algebra.

Contributions in Trigonometry

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

tribhujasya phalashariram samadalakoti bhujardhasamvargah

Translation : “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 (half-wave)“. For simplicity, people started calling it jya.
When Arabic writers translated his works from Sanskrit into Arabic, they referred it as jiba.
However, in Arabic writings, vowels are omitted, and it was abbreviated as jb.
Later writers substituted it with jaib, meaning “pocket” or “fold (in a garment)“.
Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jaib with 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.

Indeterminate or Diophantine Equations

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.
They were discussed extensively in ancient Vedic text Baudhayana Sulba Sutras, which date to 800 BCE.
Aryabhata’s method of solving such problems is called the kuṭṭaka (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.
The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulbasutras.

In his contribution towards Algebra, Aryabhata provided elegant results for the summation of series of squares and cubes in his book Aryabhatiya.

Aryabhata’s contributions in Astronomy

Aryabhata’s system of astronomy was called the audAyaka system, in which days are reckoned from sunrise, dawn at lanka or “equator“.
Some of his later writings on astronomy, which apparently proposed a second model (or ardha-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 and he may have believed that the planet’s orbits as elliptical rather than circular.

Motions of the Solar System

In the first chapter of his book Aryabhatia, he 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 in other parts of the world, that the sky rotated.
Here, he gives the number of rotations of the earth in a yuga, and made more explicit in his gola chapter.

Eclipses

Lunar and Solar eclipses were scientifically explained by Aryabhata by stating that the Moon and planets shine by reflected sunlight.
Instead of the prevailing cosmogony in which eclipses were caused by pseudo-planetary nodes Rahu and Ketu, he explains eclipses in terms of shadows cast by and falling on Earth. Thus, the lunar eclipse occurs when the moon enters into the Earth’s shadow and solar eclipse occurs when Moon intersects Sunrays from falling on Earth.
He discussed 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 eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.

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; whereas the modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days) is an error of 3 minutes and 20 seconds over the length of a year (365.25636 days).

Heliocentrism

Aryabhata advocated an astronomical model in which the Earth turns on its own axis. His model also gave corrections (the śīgra anomaly) for the speeds of the planets in the sky in terms of the mean speed of the sun.
Aryabhata’s calculations were based on an underlying heliocentric model, in which the planets orbit the Sun, though this has been rebutted.

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Bhaskaracharya Bhāskara II -introduced concept of ‘Infinity’

Bhaskaracharya (Bhāskara the teacher) was an Indian mathematician and astronomer of 12th century AD.
He is refered as Bhāskara II to avoid confusion with Bhāskara I (of 7th century AD).
He was born near Vijjadavida (Bijapur in modern Karnataka) and lived between 1114-1185 AD.

He represented the peaks of mathematical knowledge in the 12th century and was the head of the astronomical observatory at Ujjain, the leading mathematical centre of ancient India.
Bhaskara II’s family belonged to Deshastha Brahmin community, which served as court scholars at Kings forts.
He learned Mathematics from his father Maheswara, an astrologer.
He imparted his knowledge of mathematics to his son Lokasamudra, whose son had started a school to study the works of his grand father in 1207 AD.

His main work Siddhānta Shiromani, (Sanskrit for “Crown of treatises,“) is divided into four parts called Lilāvati(beautiful woman, named after his daughter Lilavati), Bijaganita, Grahaganita (mathematics of planets) and Golādhyāya (study of sphere/earth).
These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karna Kautoohala.
Bhāskara’s work on calculus predates Newton and Leibniz by over half a millennium.
He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhāskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.

Lilavati (meaning a beautiful woman) is based on Arithmetic. It is believed that Bhaskara named this book after his daughter Lilavati. Many of the problems in this book are addressed to his daughter. For example “Oh Lilavati, intelligent girl, if you understand addition & subtraction, tell me the sum of the amounts 2, 5, 32, 193, 18, 10 & 100, as well as [the remainder of] those when subtracted from 10000.” The book contains thirteen chapters, mainly definitions, arithmetical terms, interest computation, arithmetical & geometric progressions. Many of the methods in the book on computing numbers such as multiplications, squares & progressions were based on common objects like kings & elephants, which a common man could understand.

Bijaganita is on Algebra & contains 12 chapters.
“A positive number has two square-roots (a negative root & a positive root)“. This was published in this text for the very first time. It contains concepts of positive & negative numbers, zero, the ‘unknown‘ (includes determining unknown quantities), surds, simple equations & quadratic equations.

Bhaskara was the first to introduce the concept of Infinity : If any finite number is divided by zero, the result is infinity.
Also the fact that if any finite number is added to infinity then the sum is infinity. He developed a proof of the Pythogorean theorem by calculating the same area in two different ways & then cancelling out two terms to get a² + b² = c².
He is also known for his calculation of the time required (365.2588 days) by the Earth to orbit the Sun which differs from the modern day calculation of 365.2563 days, by just 3.5 minutes!
The law of Gravitation had been proved by Bhaskara 500 years before it was rediscovered by Newton.

Bhaskaracharya’s contributions

Mathematics
Some of Bhaskara’s contributions to mathematics include the following:
A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a² + b² = c².
In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained.
Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century
A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.
The first general method for finding the solutions of the problem x² − ny² = 1 (so-called “Pell’s equation“) was given by Bhaskara II.
Solutions of Diophantine equations of the second order, such as 61x² + 1 = y². This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century.
Solved quadratic equations with more than one unknown, and found negative and irrational solutions.
Preliminary concept of mathematical analysis.
Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.
Conceived differential calculus, after discovering the derivative and differential coefficient.
Stated Rolle’s theorem, a special case of one of the most important theorems in analysis, the mean 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.)
In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below.)

Arithmetic
Bhaskara’s arithmetic text Leelavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations.
Lilavati is 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 of zero (including division, and rules of operations with zero).
Further extensive numerical work, including use of negative numbers and surds.
Estimation of π.
Arithmetical terms, methods of multiplication, and squaring.
Inverse rule of three, and rules of 3, 5, 7, 9, and 11.
Problems involving interest and interest computation.
Indeterminate equations (Kuttaka), integer solutions (first and second order). His contributions to this topic are particularly important, since the rules he gives are (in effect) the same as those given by the renaissance European 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 of Aryabhata and subsequent mathematicians.
His work is outstanding for its systemisation, improved methods and the new topics that he has introduced. Furthermore the Lilavati contained excellent recreative problems and it is thought that Bhaskara’s intention may have be.

Algebra
His Bijaganita (“Algebra”) was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root).
His work Bijaganita is effectively a treatise on algebra and contains the following topics:
Positive and negative numbers.
Zero.
The ‘unknown’ (includes determining unknown quantities).
Determining unknown quantities.
Surds (includes evaluating surds).
Kuttaka (for solving indeterminate equations and Diophantine equations).
Simple equations (indeterminate of second, third and fourth degree).
Simple equations with more than one unknown.
Indeterminate quadratic 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, chakravala method for solving indeterminate quadratic equations of the form ax² + bx + c = y.
Bhaskara’s method for finding the solutions of the problem Nx² + 1 = y² (the so-called “Pell’s equation“) is of considerable importance.

Trigonometry
The Siddhanta Shiromani (written in 1150) demonstrates Bhaskara’s knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also discovered spherical trigonometry, along with other interesting trigonometrical results. 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 the now well known results for sin(a + b) and sin(a – b)

Calculus
His work, the Siddhanta Shiromani, is an astronomical treatise and contains many theories not found in earlier works. Preliminary concepts of infinitesimal calculus and mathematical analysis, along with a number of results in trigonometry, differential calculus and integral calculus that are found in the work are of particular interest.
Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement. Bhaskara also goes deeper into the ‘differential calculus‘ and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of ‘infinitesimals‘.
There is evidence of an early form of Rolle’s theorem in his work

• If  f (a) = f (b) = 0, then f ‘ (x) = 0 for some x with a<x<b then
• He gave the result that if x =(approx) y then sin(y) – sin(x) =(approx) (y-x) cos(y), thereby finding the derivative of sine, although he never developed the notion of derivatives.

Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse.
In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti, or a 1⁄33750 of a second, and his measure of velocity was expressed in this infinitesimal unit of time.
He was aware that when a variable attains the maximum value, its differential vanishes.
He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.
In this result, there are traces of the general mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle’s theorem. The mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara’s Lilavati.
Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara’s work and further advanced the development of calculus in India.

Astronomy

Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year, the time that is required for the Earth to orbit the Sun, as 365.2588 days which is same as in Suryasiddhanta. The modern accepted measurement is 365.2563 days, a difference of just 3.5 minutes !
His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere.
The twelve chapters of the first part cover topics such as:
Mean longitudes of the planets.
True longitudes of the planets.
The three problems of diurnal rotation.
Syzygies.
Lunar eclipses.
Solar eclipses.
Latitudes of the planets.
Sunrise equation
The Moon’s crescent.
Conjunctions of the planets with each other.
Conjunctions of the planets with the fixed stars.
The paths of the Sun and Moon.
The second part contains thirteen chapters on the sphere. It covers topics such as:
Praise of study of the sphere.
Nature of the sphere.
Cosmography and geography.
Planetary mean motion.
Eccentric epicyclic model of the planets.
The armillary sphere.
Spherical trigonometry.
Ellipse calculations.
First visibilities of the planets.
Calculating the lunar crescent.
Astronomical instruments.
The seasons.
Problems of astronomical calculations.

Engineering

The earliest reference to a perpetual motion machine date back to 1150, when Bhāskara II described a wheel that he claimed would run forever.
Bhāskara II used a measuring device known as Yasti-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.

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Surya Siddhanta, >2 Million Years old Book is First on Astronomy

Surya Siddhanta is the first among the traditions or doctrines (siddhanta) in archaeo-astronomy of the Vedic era.
Infact, it is the oldest ever book in world which describes earth as sphere but not flat, gravity being reason for objects falling on earth etc.

This is the knowledge that the Sun god gave to an Asura called Maya in Treta Yuga.
This Maya is father-in-law of Ravana, the villain of first ever epic poem, Ramayana.
Going by calculations of Yugas, first version ofSurya Siddhanta must have been known around 2 million years ago.
However, the present version available is believed to be more than 2500 years old, which still makes it the oldest book on earth in Astronomy.

This book covers kinds of time, length of the year of gods and demons, day and night of god Brahma, the elapsed period since creation, how planets move eastwards and sidereal revolution. The lengths of the Earth’s diameter, circumference are also given. Eclipses and color of the eclipsed portion of the moon is mentioned.
This explains the archeo-astronomical basis for the sequence of days of the week named after the Sun, Moon, etc. Musings that there is no above and below and that movement of the starry sphere is left to right for Asuras (demons) makes interesting reading.
Citation of the Surya Siddhanta is also found in the works of Aryabhata.
The work as preserved and edited by Burgess (1860) dates to the Middle Ages.
Utpala, a 10th-century commentator of Varahamihira, quotes six shlokas of the Surya Siddhanta of his day, not one of which is to be found in the text now known as the Surya Siddhanta. The present version was modified by Bhaskaracharya during the Middle Ages.
The present Surya Siddhanta may nevertheless be considered a direct descendant of the text available to Varahamihira (who lived between 505–587 CE)

Table of contents in Surya Siddhanta are :-

• The Mean Motions of the Planets
• True Places of the Planets
• Direction, Place and Time
• The Moon and Eclipses
• The Sun and Eclipses
• The Projection of Eclipses
• Planetary Conjunctions
• Of the Stars
• Risings and Settings
• The Moon’s Risings and Settings
• Certain Malignant Aspects of the Sun and Moon
• Cosmogony, Geography, and Dimensions of the Creation
• The Gnomon
• The Movement of the Heavens and Human Activity

Methods for accurately calculating the shadow cast by a gnomon are discussed in both Chapters 3 and 13.

Few excerpts from Surya Siddhanta

• The average length of the tropical year as 365.2421756 days, which is only 1.4 seconds shorter than the modern value of 365.2421904 days !
• The average length of the sidereal year, the actual length of the Earth’s revolution around the Sun, as 365.2563627 days, which is virtually the same as the modern value of 365.25636305 days. This remained the most accurate estimate for the length of the sidereal year anywhere in the world for over a thousand years!
• Not content to limit measurements to Earth, the Surya Siddhanta also states the motion, and diameters of the planets! For instance the estimate for the diameter of Mercury is 3,008 miles, an error of less than 1% from the currently accepted diameter of 3,032 miles. It also estimates the diameter of Saturn as 73,882 miles, which again has an error of less than 1% from the currently accepted diameter of 74,580.
• Aside from inventing the decimal system, zero and standard notation (giving the ancient Indians the ability to calculate trillions when the rest of the world struggled with 120) the Surya Siddhanta also contains the roots of Trigonometry.
• It uses sine (jya), cosine (kojya or “perpendicular sine”) and inverse sine (otkram jya) for the first time!
• “Objects fall on earth due to a force of attraction by the earth. therefore, the earth, the planets, constellations, the moon and the sun are held in orbit due to this attraction”. (this was also discussed in Prasnopanishad
  It was not until the late 17th century in 1687, that Isaac Newton rediscovered the Law of Gravity.
• The Surya Siddhanta also goes into a detailed discussion about time cycles and that time flows differently in differently circumstances, the roots of relativity. Here we have a perfect example of Indian philosophy’s belief that science and religion are not mutually exclusive. Unlike, Abrahamic religions, one does not have to dig and try all ways to force scientific truth from scriptures. By contrast it is stated in cold hard numbers by the Sun God, Surya.
• This work shows that spirituality is all about the search for Truth (Satya) and that Science is as valid a path to God as living in a monastery. It is the search for ones own personal Truth that will lead one ultimately to God.

The astronomical time cycles contained in the text were remarkably accurate at the time.

• That which begins with respirations (prana) is called real…. Six respirations make a vinadi, sixty of these a nadi
• And sixty nadis make a sidereal day and night. Of thirty of these sidereal days is composed a month; a civil (savana) month consists of as many sunrises
• A lunar month, of as many lunar days (tithi); a solar (saura) month is determined by the entrance of the sun into a sign of the zodiac; twelve months make a year. This is called a day of the gods. (Day at North Pole)
• The day and night of the gods and of the demons are mutually opposed to one another. Six times sixty of them are a year of the gods, and likewise of the demons. (Day and Night being six months each at South Pole)
• Twelve thousand of these divine years are denominated a chaturyuga; of ten thousand times four hundred and thirty-two solar years
• Is composed that chaturyuga, with its dawn and twilight. The difference of the kritayuga and the other yugas, as measured by the difference in the number of the feet of Virtue in each, is as follows:
  a. The tenth part of a chaturyuga, multiplied successively by four, three, two, and one, gives the length of the krita and the other yugas: the sixth part of each belongs to its dawn and twilight.
  b. One and seventy chaturyugas make a manu; at its end is a twilight which has the number of years of a kritayuga, and which is a deluge.
  c. In a kalpa are reckoned fourteen manus with their respective twilights; at the commencement of the kalpa is a fifteenth dawn, having the length of a kritayuga.
  d. The kalpa, thus composed of a thousand chaturyugas, and which brings about the destruction of all that exists, is a day of Brahma; his night is of the same length.
  e. His extreme age is a hundred, according to this valuation of a day and a night. The half of his life is past; of the remainder, this is the first kalpa.
  f. And of this kalpa, six manus are past, with their respective twilights; and of the Manu son of Vivasvant, twenty-seven chaturyugas are past;
  g. Of the present, the twenty-eighth, chaturyuga, this kritayuga is past..

Planetary Diameters in Surya Siddhanta

Surya Siddhanta also estimates the diameters of the planets. The estimate for the diameter of Mercury is 3,008 miles, an error of less than 1% from the currently accepted diameter of 3,032 miles.
It also estimates the diameter of Saturn as 73,882 miles, which again has an error of less than 1% from the currently accepted diameter of 74,580.
Its estimate for the diameter of Mars is 3,772 miles, which has an error within 11% of the currently accepted diameter of 4,218 miles.
It also estimated the diameter of Venus as 4,011 miles and Jupiter as 41,624 miles, which are roughly half the currently accepted values, 7,523 miles and 88,748 miles, respectively.

Trigonometry in Surya Siddhanta

Surya Siddhanta contains the roots of modern trigonometry. It uses sine (jya), cosine (kojya or “perpendicular sine”) and inverse sine (otkram jya) for the first time, and also contains the earliest use of the tangent and secant when discussing the shadow cast by a gnomon in verses 21–22 of Chapter 3:

Of [the sun’s meridian zenith distance] find the jya (“base sine”) and kojya (cosine or “perpendicular sine”). If then the jya and radius be multiplied respectively by the measure of the gnomon in digits, and divided by the kojya, the results are the shadow and hypotenuse at mid-day.
In modern notation, this gives the shadow of the gnomon at midday as :

Even today many astrologers in India use Surya Siddhanta as base to compute their Panchangs (Almanacs) in many languages.

Download SuryaSiddhanta pdf

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