நான் யார்?: Hindu India-introductory course on contributions to the world

Research credit: @TrueIndology, Joseph T. Noony, Manasataramgini

Algebra:

Bakshali manuscript-archaelogical evidence, also alludes to earlier algebra knowledge.
Quadratic equations: Baudhayana, Manava, Apasthamba Sutras. ax2 = c, ax2 + bx = c.

Al Khwarizmi,Celebrated Arab mathematician. Credits India-Hisab Al-Hind (Calculations of India) & Jiz Al Sind Hind (Astronomical tables of Sind & Hind)
Al Jahiz & Al Andalusi: Ancient Arabs who spoke highly of Hindu civilization & India. Al Jahiz - 800 CE, The life and works of Jāḥiẓ. Jahiz, Charles Prett, University of California Press, 1969, Al Andalusi-1068 CE, Said Al Andalusi's book Ṭabaqāt al-ʼUmam. Chapter 5'.
Binary system of weights: An article about Chahnudaro excavations published in American oriental Society by its excavator cum Archaeologist Ernest Mackay
Trigonometry,Tantrasamgraha
Power Series,Tantrasamgraha
Kth power of series: Narayana Pandit of Kashi (~1370CE): Derived 'Varasankalita'-formula for the sum of kth power of a series. It was a critical input for Kerala mathematicians who went on to invent Calculus. Now credited to Bernoulli.
Calculus: Yuktibhasa. Invented by Madhava-Kerala school of mathematics. Arguably transmitted to Europe. Critical Issues in Mathematics Education edited by Bharath Sriraman, Paul Ernest, Brian Greer. Transmission of Calculus: CKR. Mathematics in India: M. D. Srinivas

Infinite series & error term by Madhava. (Later developed in Europe by James Gregory). “On an Untapped Source of Medieval Keralese Mathematics” by CT Rajagopal & Ms Rangachari

Different types of infinity (Samkhyat-enumerable, Asamkhyat-innumerable {Paritasamkyat, Yuktasamkyat, Asamkhyatasamkhyat}, Anant-infinite {Paritanant, Yuktanant, Anantanant}) by Jain mathematicians, later attributed to mathematician Georg Cantor.

Differential Calculus: The Principles of Differential Calculus were earliest stated in India by Hindu Mathematicians in Sanskrit Mañjula(932 CE) in his work Laghumānasa states the Differential Formula of the Sine Function in the following form : Sinθ′−Sinθ = Cosθ(θ′−θ) where (θ′−θ) tends to 0.
Hindu numerals: 876 CE inscription-Chaturbhuj Temple of Gwalior records a garden 270 hastas wide. Fibonacci 1175-1250 explained the superiority of ‘Modus Indorum’ to Roman numerals, giving credit to the Indians.
Pi: Yuktibhasa-Upto 100 decimals, Yajnavalkya~2000BCE- 3.125, Baudhayana~1800BCE- 3.088, Manava~1500BCE- 3.1604, 3.1416 of Aryabhata (499 ACE), 3.1415926535897 of Madhava (1370 ACE) and 355/113 of Nilakanta & recognition of irrationality (1480 ACE), 3.14159265358979324 in Karanapaddhati by Putumana Somayaji.
Trigonometric Series,Yuktibhasa
Taylor Series,Yuktibhasa
Gastric Surgery: Terracota model-150 CE
Plastic surgery: Susruta Samhita, theAṣṭāṅgahṛdayasaṃhitā, Practice observed by Joseph Constantine Carpue (‘the father of modern plastic surgery’) & transmitted to Europe.
Surgery,Susruta Samhita
Malaria origin: Susruta. Source: The mosquito, A human history of of deadliest predator by Timothy C Winegard
Brain surgery,Indus valley Civilization(c.2300 BCE) archaeology.Surgical hole drilled through skull
Pipes: Mark Kenoyer, Ancient Cities of the Indus Valley Civilization (Oxford University Press, 1998)
Flush toilet: Archaeology: Lothal, ancient India (Indus Valley Civilization, Gujarat).
Chess: Archaeology: 2300 BCE, Lothal, ancient India (Indus Valley Civilization, Gujarat). Chaturanga, transmitted to Persia as Shatranj by king Deva Sharma per Persian sources.
Spoked wheel (argued as studded wheel by dissenters): 'The world's oldest spoked wheel chariot was found in India dating back to ~2200 BCE. "Like spokes where none are last in order, Mightiest are the Sons of Pṛśni"RV V.58.5. The word āra appears ONLY in late books like V, IX & X while archaeology says it was invented~2500BCE'
Geometry,Fire altars: Satapatha Brahmana (2000 BCE) & Taittiriya Samhita (3000 BCE)
Sanskrit jyā & koti-jyā: sine & cosine. 'A history of Mathematics" by Victor Katz.
Relative motion: "Similar to a person in a boat moving forward who sees the stationary objects on the bank of the river as moving backwards, the stationary stars at Lanka (equator) are viewed as moving westwards"-Aryabhatta. Reused by Galileo. https://twitter.com/JoeAgneya/status/964885226512986112
Pythagoras theorem,Baudhayana’s Sulba Sutras: 800 BCE
Baudhayana (Pythagorean) triples: Sulba Sutras: (5,12,13), (8,15,17), (12,16,20), (12,35,37).
Tiled homes with designs still in use.
Square root:

Sqrt(2)=1.412156 correct to 5 decimals in the Sulba Sutras
Baudhayana, Manava and Apastamba Sutras: sqrt(10) correct to 6 decimal places, oldest formula for square roots.

Fibonacci series,Pingala series: Pingala-~400 BCE (Chandaḥśāstra) Virahanka~700ACE Gopāla (c.1135) Hemachandra (c.1150). Fibonacci calls it the Indian Series in his book, Liber Abaci.
Pascal series, Meruprastara: Triangle of binomial coefficients, an extention of pingala series: Pingala, Varahamihira~499AD, Mahavira-850AD- Multiplication formula for binomial coefficients, Halayudha-975AD, Utpala~1000AD-Equated additive & Multiplicative formulas:
Zero & negative numbers,The recognition of negative numbers and the rules for using zero(Brahmagupta). 0=Sunya (Samskritam)=Sifr (Arabic)=Cifra(German)=Cipher. Ref: https://books.google.com/books?id=47PHOZBdCLYC&printsec=frontcover&dq=the+puzzle+instinct+india&hl=en&sa=X&ved=0ahUKEwjcjOe4rpziAhWrwlQKHYbQAkEQ6AEIKjAA#v=onepage&q=india&f=false
Ahimsa-non cruelty towards animals: Samkya-Kapila Kardama: “The word ajā not only means ‘goat’. It also means ‘seed’. If you, purohitas, wish to sacrifice, you may sacrifice a seed” (MB XII.324), “No law is higher than non-violence” -Kapila to the sage Syumareshmi (seeing an animal tied up for sacrifice)-( Mahabharata XII.260.17), “Fearlessness to all living beings from my side! Svaha!”-(Baudhayana Grihya Sutra, 4.16.4), Ashta-anga Yoga’s 1st Anga’s 1st Yama: Ahimsa, Anushashana Parva of the Mahabharatha
Yoga,Yoga-Asana: posture popular in the west. Ashta-anga Yoga Sutras by Patanjali.
Samskritam-PAnini Backus Naur form: PAnini (5th century) who used the concept of recursion to formulate Samskritam grammatical rules that is simulated by Finite State Algorithms & considered the first modern generative grammar or context free grammar. This is considered the greatest linguistic achievement in human history. Until the 1900s, every European linguist studied PAnini's grammar as part of their syllabus. Context free grammar is the basis of modern computer languages and is called the Backus-Naur form. Oldest of the Indo-European & Indo-Aryan languages. Its influence is felt on all European languages including English. Influenced the field of morphology which studies how words are constructed.
Logic: Mary Everest Boole, niece of George Everest & wife of George Boole after whom Mt.Everest is named acknowledges Hindu influences on Charles Babbage, George Boole & Augustus De Morgan. Per Greek & Persian sources, Callisthenes, the Greek historian accompanying Alexander, took Indian logic texts to Greece. https://www.scribd.com/document/61916350/Indian-Thought-and-Western-Science-in-the-Nineteenth-Century Medatithi Gautama: father of logic & founder of NyAya, the NyAya Sutras Five membered syllogism (vs Greek three membered syllogism).
Empirism-Scientific method,Kapila Kardama: father of empirism. Samkya.
Slavery abolition: Megastheses in his Indica calls India a land without slaves, when the rest of the world had slaves. Hindu king Shivaji was the first king in the world to issue an anti-slavery proclamation. Source: Xenophobia in Seventeenth-Century India(Leiden university press, 2014) by Dutch historian Gijs Kruitzer
Atomic theory,Earliest concept in Vaiseshika authored by Kanada Kashyapa
Gravity,Earliest mention in Vaiseshika. Later in Surya Siddhantha and by Bhaskara & Brahmagupta. Called Gurutva-similar etymology.
Observer/Observed: Concepts of observer & observed. Observed: Dravya (substance), Guna (properties), Karma. Observer: Saamaanya (universal), Vishesha (Particularity), Samavaaya (Inherence). Samavaaya (meeting of observer & observed)-a Quantum mechanics concept. Abhava (voidness)-7th Padaartha
University system: Origin of the university system. Famous universities: Nalanda, Takshashila, Ujjain, Valabhi, Mithila, Telhara, Sharadha Peeth, Pushpagiri, Odantapuri, Vikramashila, Somapura, Bikrampur, Morena Golden Triangle (Mitavali, Padavali, Bateshwar), Kanthaloor Sala, Jagaddala, Nadia, Ratnagiri. In 637 AD, Chinese pilgrim Xuanzang visited Nalanda university. He saw that it had 10,000 students from all over the world and around 2000 teachers (Hindu Brahmanas, Buddhists & Jains). Kumarila Bhatta & Aryabhatta were from Nalanda. Source: The life of Xuanzang Book III (pp.112). Mahayana, Vedas, Shastras, Sankhya, Medicine, Linguistics, Etymology, Economics, Weaponry and Archery were taught to the students.
Ancient meridian: Avanti meridian-The first zero degree prime meridian of the earth for 100s of years. Ujjain astronomers: Surya Siddhanta author, Aryabhata, Varahamihira, Bhaskara I, Brahmagupta, Vateshwara, Bhaskara II.
360 degrees origin: Dirghatamas Auchatya, Shatapatha Brahmana, Vedanga Jyotisha
The first observation and recording of Heliocentric periods of Mars, Jupiter, Saturn and Venus,Surya Siddhanta and Aryabhatiya
Rotation of the earth: ( Aryabhata), Aitreya Brahmana III-14
The unification of physical laws and mathematics,Bhaskara the IInd. He calculated the instantaneous velocity of planets
Celestial navigation: Vasco da Gama returned to Europe with the rapalagai (with an Indian from Kenya, Kanha who knew how to use it). The 'rapalagai' of clear south Indian make was a wooden board with a graduated string to measure the pole star altitude & hence the latitude. Europeans began to master celestial navigation, invented the chronometer & conquered the seas.
Indian astronomical tables->Toledo tables->Alfonsine tables: Accurate tables predicting celestial body movement.
Elliptical orbits & accurate Heliocentric periods,Aryabhata 499CE
Ellipse Equations,Mahavira 850CE
Govindaswami's interpolation: Now called Newton-Gauss interpolation 2nd order.
Uniform Mean planetary motion & instantaneous velocity,Bhaskara II 1150CE
Partial heliocentric model: Neelakanta 1501 CE. Paramesvara studied under Madhava, Rudra and Narayana. He understood the factors causing difference between calculated & observed positions. After verifying the work of earlier teachers though observations of eclipses & planetary conjunctions, he composed Drgganita-NeelakanthaIndian planetary model and its revision by Nilakantha: K Ramasubramaniam
First unified model for inner & outer planets,Neelakanta 1501 CE
Correct Equation of center & calculation of eccentric orbits,Neelakanta 1501 CE
The earth is not encircled by the orbits of Mercury & Venus: Parameshvara 1410CE. Calculated positions are found to deviate from the observed. The shastras declare the observed to be true. Hence the twice born adept at astronomy must strive to obtain positions by observation-Parameshvara Drgganita 1.2-4. The planets have been observed by me for fifty five years and they differ from positions derived from Parahita ganitam-Nilakantha quotes his teacher Paramesvara. Since Saka 1315(AD 1393) I have observed a number of eclipses; all these occurred at times prior to calculated ones. This necessitated corrections in the computational process-Paramesvara(Siddhantadipika). Eclipse observations of Paramesvara: K Chandrahari
Angular motion of planets: Jyestadeva 1530CE. Yuktibhasa of Jyestadeva: KV Sharma
The motion of planets varies each second,-Sankara Varier~1530CE
Heliocentric model,Brahmagupta & Bhaskara
Tychonic model: Neelakantha Samayaji-erroneous model-all planets except earth revolve around the son. Tycho Brache, post Copernicus-The Tychonic Model.
Kepler requesting observations from India: 1626 Kepler’s letter to Jesuit mathematician Paul Guldin: “If I could only obtain soon the observations on the eclipse from India or from any other place…” C. Baumgardt, Kepler: Life and Letters, Philosophical Library https://www.nytimes.com/1990/01/23/science/after-400-years-a-challenge-to-kepler-he-fabricated-his-data-scholar-says.html
Martial Arts/King Fu: Shaolin temple art shows Indian monk Buddhabhadra, the founder of Shaolin temple (in 496 CE) teaching martial arts to his Chinese disciple, Sengchou.Bodhidharma, the prince of Kanchipuram, was associated with Shaolin after 527 CE. Source: The Bodhidharma Anthology (translation) The Earliest Records of Zen, Jeffrey L. Broughton; The Shaolin Monastery: History, Religion, and the Chinese Martial Arts, University of Hawaii press 2008
Acupressure: 108 Pressure points: Varma Kalai-still practiced in India for healing-originating from martial arts. Developed into Acupressure & Acupuncture.
Buttons: In his book An encyclopedia of the history of technology(Routledge Press 1990), noted Historian Ian McNeil writes:The button, in fact, was originally used more as an ornament than as a fastening, the earliest known being found at Mohenjo-daro in the Indus valley. It is made of a curved shell and about 5000 years oldSource/link of the book(available online for access)https://polifilosofie.files.wordpress.com/2012/12/encyclopedia-of-the-history-of-technology.pdf
Metallurgy: Rust-free iron pillar-400 ACE, metallurgy technique still unknownUkku (Wootz) Steel-considered the finest steel in the world & exported globally, 500 BCE onwards (Seric iron).
War ethics: Dharmayuddha (from Mahabharata, Bhishma Parva 6.1 Vulgate)

No war during night
Only one to one combat. A warrior has to give his opponent a notice before striking him
Unprepared and panic struck warriors cannot be struck
Surrendered and retreating warriors should never be harmed
Killing unarmed and unconscious warrior is prohibited
Killing non combatant humans and animals is prohibited
Women, farmers, prisoners of war cannot be killed
Land cannot be pillaged9)A warrior whose weapon is unfit or who does not have an armor cannot be harmed

Women in medicine: "A dhanvantari from Sarnath from around the reign of emperor harShavardhana. It is rather remarkable in depicting surgeons & physician performing various procedures. On the left you can see a surgeon performing a colorectal procedure probably for arsharoga. Notable is the evidence for females in the medical profession of the age"
Germ theory: Until 1850s, western medicine was based on miasma (noxious bad air) theory, bloodletting & cannibalism before adopting the germ theory. Ancient Ayurveda had recognized micro-organisms as the originators of disease. https://swarajyamag.com/books/smallpox. See Brighu Samhita-III/20:5-7
Inoculation/Vaccination,Cowpox based inoculation used by Indians for smallpox. Studied by Edward Jenner who was called the vaccination ‘pioneer’.
Brighu Samhita: Air as life sustenance, heart function, blood circulation. Brighu Samhita II/6:8-9, II/10:1-4
Crystallized sugar: Called Shakkar in India, originally invented in the Gupta rule. A Dictionary of the Economic Products of India, Volume 6, Part 2-Sir George Watt. Khanda (in Samskritam) became Kal Khandu in Tamizh & candy in English. Khanda->Candy, Shakkar->Sugar, Shakkarai->Jaggery. Etymology.
Freedom of religion, tolerance: Rig Veda: Ekam sat vipraha bahudha vadanti => Truth is one, the wise call it by many names.Samuddharita Samaya Chatustaya: patronizing all four main sects(=Shavism, Vaishnavism, Jainism, Buddhism) equally-sed by many kings /chieftains in India from Kashmir to Kanyakumari from 3rd century to 13th century as a matter of pride.Multiple sophisticated philosophies that encompass every religious belief accepting different ways of acquiring knowledge: Charvaka (atheists), Shramanas-Buddhists/Jains (non-theists), theist (Nyaya, Vaisheshika, Samkya, Yoga, Purva-Mimamsa, Uttara-Mimamsa/Vedanta) with varying shades of monism & duality.
Debate rules: Vaada-Vidhi: Rules of debate. Rich culture of debate between various philosophies.Sam Vaadam: Cultured debate requiring Purva Paksha (presenting the opponent’s point of view to the opponent’s satisfaction) & Uttara Paksha (counter argument). Victory is not a concern, only the truth is of concern.Jalpa Vaadam: Debate aimed at victory following debate norms.Vithanda Vaadam: Debate for victory without following existing norms.
Storytelling,Panchatantra: The most translated fable in the world.Katha Sarith Sagara: ocean of stories.Inspired the Arabian nights(eg: UpakoshA from BrihatKathAmanjari).Kalidasa: sophisticated plays in Samskritam.
Dance,Natya. Sophisticated dance techniques that showcase different Rasas (emotions).
Shampoo: Earliest historical reference to Shampoo comes from Ancient India (c.50 BC). Word 'Shampoo' has Sanskritic origin. It comes from Hindi word "Champi" (चंपी) ultimately from Sanskrit "chapayati" (चपयति). When the British colonised India, they learnt everyday bathing and head wash from their Indian servants who called it Champi. The word Champi was spelt by British as "Shampoo" and that is how it entered English lexicon.
Perfumes: An ancient terracotta perfume still from Indus valley civilization has been carbon dated to 3000 BC. Brhatsamhita mentions different types of perfumes and the ways of making them. Gandhayukti (cosmetics) was listed as Kamashastra art. Periplus says perfumes were exported to Europe from Ancient India.
Firecrackers (Ulkah) from Agnichurna (Saltpetre)
Gunpowder
Toothpaste, surgical instruments
Cheese