# Timeline of scientific discoveries

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The timeline below shows the date of publication of possible major scientific breakthroughs, theories and discoveries, along with the discoverer. For the purposes of this article, we do not regard mere speculation as discovery, although imperfect reasoned arguments, arguments based on elegance/simplicity, and numerically/experimentally verified conjectures qualify (as otherwise no scientific discovery before the late 19th **century** would count). We begin our timeline at the Bronze Age, as it is difficult to estimate the timeline before this point, such as of the discovery of counting, natural numbers and arithmetic.

To avoid overlap with Timeline of historic inventions, we do not **list** examples of documentation for manufactured substances and devices unless they reveal a more fundamental leap in the theoretical ideas in a field.

## Bronze Age[edit]

Many early innovations of the Bronze Age were requirements resulting from the increase in trade, and this also applies to the scientific advances of this period. For context, the major civilizations of this period are Egypt, Mesopotamia, and the Indus Valley, with Greece rising in importance towards the end of the third millennium **BC**. It is to be noted that the Indus Valley script remains undeciphered and there are very little surviving fragments of its writing, thus any inference about scientific discoveries in the region must be made based only on archaeological digs.

### Mathematics[edit]

#### Numbers, measurement and arithmetic[edit]

- Around 3000
**BC**: Units of measurement are developed in the major Bronze Age civilisations: Egypt, Mesopotamia, Elam and the Indus Valley. The Indus Valley may have been the major innovator on this, as the first measurement devices (rulers, protractors, weighing scales) were invented in Lothal in Gujarat, India.^{[1]}^{[2]}^{[3]}^{[4]} - 1800
**BC**: Fractions were first studied by the Egyptians in their study of Egyptian fractions.

#### Geometry and trigonometry[edit]

- 2100
**BC**: The concept of area is first recognised in Babylonian clay tablets,^{[5]}and 3-dimensional volume is discussed in an Egyptian papyrus. This begins the study of geometry. - Early 2nd millennium
**BC**: Similar triangles and side-ratios are studied in Egypt (e.g. in the Rhind Mathematical Papyrus, a copy of an older Middle Kingdom text) for the construction of pyramids, paving the way for the field of trigonometry.^{[6]}

#### Algebra[edit]

- 2100
**BC**: Quadratic equations, in the form of problems relating the areas and sides of rectangles, are solved by Babylonians.^{[7]}

#### Number theory and discrete mathematics[edit]

- 2000
**BC**: Pythagorean triples are first discussed in Babylon and Egypt, and appear on later manuscripts such as the Berlin Papyrus 6619.^{[8]}

#### Numerical mathematics and algorithms[edit]

- 2000
**BC**: Multiplication tables in Babylon.^{[9]} - 1800
**BC**– 1600**BC**: A numerical approximation for the square root of two, accurate to 6 decimal places, is recorded on YBC 7289, a Babylonian clay tablet believed to belong to a student.^{[10]} - 19th to 17th
**century**BCE: A Babylonian tablet uses 25⁄8 as an approximation for π, which has an error of 0.5%.^{[11]}^{[12]}^{[13]} - Early 2nd millennium BCE: The Rhind Mathematical Papyrus (a copy of an older Middle Kingdom text) contains the first documented instance of inscribing a polygon (in this case, an octagon) into a circle to estimate the value of π.
^{[14]}^{[15]}

#### Notation and conventions[edit]

- 3000
**BC**: The first deciphered numeral system is that of the Egyptian numerals, a sign-value system (as opposed to a place-value system).^{[16]} - 2000
**BC**: Primitive positional notation for numerals is seen in the Babylonian cuneiform numerals.^{[17]}However, the lack of clarity around the notion of zero made their system highly ambiguous (e.g. 13200 would be written the same as 132).^{[18]}

### Astronomy[edit]

- Early 2nd millennium
**BC**: The periodicity of planetary phenomenon is recognised by Babylonian astronomers.

### Biology and anatomy[edit]

- Early 2nd millennium
**BC**: Ancient Egyptians study anatomy, as recorded in the Edwin Smith Papyrus. They identified the heart and its vessels, liver, spleen, kidneys, hypothalamus, uterus, and bladder, and correctly identified that blood vessels emanated from the heart (however, they also believed that tears, urine, and semen, but not saliva and sweat, originated in the heart, see Cardiocentric hypothesis).^{[19]}

## Iron Age[edit]

### Mathematics[edit]

#### Geometry and trigonometry[edit]

- c. 700
**BC**: The Pythagoras theorem is discovered by Baudhayana in the Hindu Shulba Sutras in Upanishadic India.^{[20]}However, Indian mathematics, especially North Indian mathematics, generally did not have a tradition of communicating proofs, and it is not fully certain that Baudhayana or Apastamba knew of a proof.

#### Number theory and discrete mathematics[edit]

- c. 700
**BC**: Pell's equations are first studied by Baudhayana in India, the first diophantine equations known to be studied.^{[21]}

#### Geometry and trigonometry[edit]

- c. 600
**BC**: Thales of Miletus discovers Thales's theorem.

### Biology and anatomy[edit]

- 600
**BC**– 200**BC**: The Sushruta Samhita (3.V) shows an understanding of musculoskeletal structure (including joints, ligaments and muscles and their functions).^{[22]} - 600
**BC**– 200**BC**: The Sushruta Samhita refers to the cardiovascular system as a closed circuit.^{[23]} - 600
**BC**– 200**BC**: The Sushruta Samhita (3.IX) identifies the existence of nerves.^{[22]}

### Social science[edit]

#### Linguistics[edit]

## 500 **BC** – 1 **BC**[edit]

The Greeks make numerous advances in mathematics and astronomy through the Archaic, Classical and Hellenistic periods.

### Mathematics[edit]

#### Logic and proof[edit]

- 4th
**century****BC**: Greek philosophers study the properties of logical negation. - 4th
**century****BC**: The first true formal system is constructed by Pāṇini in his Sanskrit grammar.^{[24]}^{[25]} - c. 300
**BC**: Greek mathematician Euclid in the*Elements*describes a primitive form of formal proof and axiomatic systems. However, modern mathematicians generally believe that his axioms were highly incomplete, and that his definitions were not really used in his proofs.

#### Numbers, measurement and arithmetic[edit]

- 4th
**century****BC**: Eudoxus of Cnidus states the Archimedean property.^{[26]} - 4th-3rd
**century****BC**: In Mauryan India, The Jain mathematical text Surya Prajnapati draws a distinction between countable and uncountable infinities.^{[27]} - 3rd
**century****BC**: Pingala in Mauryan India studies binary numbers, making him the first to study the radix (numerical base) in history.^{[28]}

#### Algebra[edit]

- 5th
**century****BC**: Possible date of the discovery of the triangular numbers (i.e. the sum of consecutive integers), by the Pythagoreans.^{[29]} - c. 300
**BC**: Finite geometric progressions are studied by Euclid in Ptolemaic Egypt.^{[30]} - 3rd
**century****BC**: Archimedes relates problems in geometric series to those in arithmetic series, foreshadowing the logarithm.^{[31]} - 190
**BC**: Magic squares appear in China. The theory of magic squares can be considered the first example of a vector space. - 165-142
**BC**: Zhang Cang in Northern China is credited with the development of Gaussian elimination.^{[32]}

#### Number theory and discrete mathematics[edit]

- c. 500
**BC**: Hippasus, a Pythagorean, discovers irrational numbers.^{[33]}^{[34]} - 4th
**century****BC**: Thaetetus shows that square roots are either integer or irrational. - 4th
**century****BC**: Thaetetus enumerates the Platonic solids, an early work in graph theory. - 3rd
**century****BC**: Pingala in Mauryan India describes the Fibonacci sequence.^{[35]}^{[36]} - c. 300
**BC**: Euclid proves the infinitude of primes.^{[37]} - c. 300
**BC**: Euclid proves the Fundamental Theorem of Arithmetic. - c. 300
**BC**: Euclid discovers the Euclidean algorithm. - 3rd
**century****BC**: Pingala in Mauryan India discovers the binomial coefficients in a combinatorial context and the additive formula for generating them ,^{[38]}^{[39]}i.e. a prose description of Pascal's triangle, and derived formulae relating to the sums and alternating sums of binomial coefficients. It has been suggested that he may have also discovered the binomial theorem in this context.^{[40]} - 3rd
**century****BC**: Eratosthenes discovers the Sieve of Eratosthenes.^{[41]}

#### Geometry and trigonometry[edit]

- 5th
**century****BC**: The Greeks start experimenting with straightedge-and-compass constructions.^{[42]} - 4th
**century****BC**: Menaechmus discovers conic sections.^{[43]} - 4th
**century****BC**: Menaechmus develops co-ordinate geometry.^{[44]} - c. 300
**BC**: Euclid publishes the*Elements*, a compendium on classical Euclidean geometry, including: elementary theorems on circles, definitions of the centers of a triangle, the tangent-secant theorem, the law of sines and the law of cosines.^{[45]} - 3rd
**century****BC**: Archimedes derives a formula for the volume of a sphere in*The Method of Mechanical Theorems*.^{[46]} - 3rd
**century****BC**: Archimedes calculates areas and volumes relating to conic sections, such as the area bounded between a parabola and a chord, and various volumes of revolution.^{[47]} - 3rd
**century****BC**: Archimedes discovers the sum/difference identity for trigonometric functions in the form of the "Theorem of Broken Chords".^{[45]} - c. 200
**BC**: Apollonius of Perga discovers Apollonius's theorem. - c. 200
**BC**: Apollonius of Perga assigns equations to curves.

#### Analysis[edit]

- Late 5th
**century****BC**: Antiphon discovers the method of exhaustion, foreshadowing the concept of a limit. - 3rd
**century****BC**: Archimedes makes use of infinitesimals.^{[48]} - 3rd
**century****BC**: Archimedes further develops the method of exhaustion into an early description of integration.^{[49]}^{[50]} - 3rd
**century****BC**: Archimedes calculates tangents to non-trigonometric curves.^{[51]}

#### Numerical mathematics and algorithms[edit]

- 3rd
**century****BC**: Archimedes uses the method of exhaustion to construct a strict inequality bounding the value of π within an interval of 0.002.

### Physics[edit]

#### Astronomy[edit]

- 5th
**century****BC**: The earliest documented mention of a spherical Earth comes from the Greeks in the 5th**century****BC**.^{[52]}It is known that the Indians modeled the Earth as spherical by 300**BC**^{[53]} - 500
**BC**: Anaxagoras identifies moonlight as reflected sunlight.^{[54]} - 260
**BC**: Aristarchus of Samos proposes a basic heliocentric model of the universe.^{[55]} - c. 200
**BC**: Apollonius of Perga develops epicycles. While an incorrect model, it was a precursor to the development of Fourier series. - 2nd
**century****BC**: Hipparchos discovers the apsidal precession of the Moon's orbit.^{[56]} - 2nd
**century****BC**: Hipparchos discovers Axial precession.

#### Mechanics[edit]

- 3rd
**century****BC**: Archimedes develops the field of statics, introducing notions such as the center of gravity, mechanical equilibrium, the study of levers, and hydrostatics. - 350-50
**BC**: Clay tablets from (possibly Hellenistic-era) Babylon describe the mean speed theorem.^{[57]}

#### Optics[edit]

- 4th
**century****BC**: Mozi in China gives a description of the camera obscura phenomenon. - c. 300
**BC**: Euclid's*Optics*introduces the field of geometric optics, making basic considerations on the sizes of images.

#### Thermal physics[edit]

- 460
**BC**: Empedocles describes thermal expansion.^{[58]}

### Biology and anatomy[edit]

- 4th
**century****BC**: Around the time of Aristotle, a more empirically founded system of anatomy is established, based on animal dissection. In particular, Praxagoras makes the distinction between arteries and veins. - 4th
**century****BC**: Aristotle differentiates between near-sighted and far-sightedness.^{[59]}Graeco-Roman physician Galen would later use the term "myopia" for near-sightedness.

### Social science[edit]

#### Economics[edit]

- Late 4th
**century****BC**: Kautilya establishes the field of economics with the Arthashastra (literally "Science of wealth"), a prescriptive treatise on economics and statecraft for Mauryan India.^{[60]}

#### Linguistics[edit]

- 4th
**century****BC**: Pāṇini develops a full-fledged formal grammar (for Sanskrit).

### Astronomical and geospatial measurements[edit]

- 3rd
**century****BC**: Eratosthenes measures the circumference of the Earth.^{[61]} - 2nd
**century****BC**: Hipparchos measures the sizes of and distances to the moon and sun.^{[62]}

## 1 AD – 500 AD[edit]

Mathematics and astronomy flourish during the Golden Age of India (4th to 6th centuries AD) under the Gupta Empire. Meanwhile, Greece and its colonies have entered the Roman period in the last few decades of the preceding millennium, and Greek science is negatively impacted by the Fall of the Western Roman Empire and the economic decline that follows.

### Mathematics[edit]

#### Numbers, measurement and arithmetic[edit]

- 210 AD: Negative numbers are accepted as numeric by the late Han-era Chinese text
*The Nine Chapters on the Mathematical Art*.^{[63]}Later, Liu Hui of Cao Wei (during the Three Kingdoms period) writes down laws regarding the arithmetic of negative numbers.^{[64]}

#### Algebra[edit]

- 499 AD: Aryabhata discovers the formula for the square-pyramidal numbers (the sums of consecutive square numbers).
^{[65]} - 499 AD: Aryabhata discovers the formula for the simplicial numbers (the sums of consecutive cube numbers).
^{[65]}

#### Number theory and discrete mathematics[edit]

- 3rd
**century**AD: Diophantus discusses linear diophantine equations. - 499 AD: Aryabhata discovers Bezout's identity, a foundational result to the theory of principal ideal domains.
^{[66]} - 499 AD: Aryabhata develops Kuṭṭaka, an algorithm very similar to the Extended Euclidean algorithm.
^{[66]}

#### Geometry and trigonometry[edit]

- c. 60 AD: Heron's formula is discovered by Hero of Alexandria.
^{[67]} - c. 100 AD: Menelaus of Alexandria describes spherical triangles, a precursor to non-Euclidean geometry.
^{[68]} - 4th to 5th centuries: The modern fundamental trigonometric functions, sine and cosine, are described in the Siddhantas of India.
^{[69]}This formulation of trigonometry is an improvement over the earlier Greek functions, in that it lends itself more seamlessly to polar co-ordinates and the later complex interpretation of the trigonometric functions.

#### Numerical mathematics and algorithms[edit]

- By the 4th
**century**AD: a square root finding algorithm with quartic convergence, known as the Bakhshali method (after the Bakhshali manuscript which records it), is discovered in India.^{[70]} - 499 AD: Aryabhata describes a numerical algorithm for finding cube roots.
^{[71]}^{[72]} - 499 AD: Aryabhata develops an algorithm to solve the Chinese remainder theorem.
^{[73]} - 1st to 4th
**century**AD: A precursor to long division, known as "galley division" is developed at some point. Its discovery is generally believed to have originated in India around the 4th**century**AD,^{[74]}although Singaporean mathematician Lam Lay Yong claims that the method is found in the Chinese text*The Nine Chapters on the Mathematical Art*, from the 1st**century**AD.^{[75]}

#### Notation and conventions[edit]

- c. 150 AD: The Almagest of Ptolemy contains evidence of the Hellenistic zero. Unlike the earlier Babylonian zero, the Hellenistic zero could be used alone, or at the end of a number. However, it was usually used in the fractional part of a numeral, and was not regarded as a true arithmetical number itself.
- 3rd
**century**AD: Diophantus uses a primitive form of algebraic symbolism, which is quickly forgotten.^{[76]} - By the 4th
**century**AD: The present Hindu–Arabic numeral system with place-value numerals develops in Gupta-era India, and is attested in the Bakhshali Manuscript of Gandhara.^{[77]}The superiority of the system over existing place-value and sign-value systems arises from its treatment of zero as an ordinary numeral. - By the 5th
**century**AD: The decimal separator is developed in India,^{[78]}as recorded in al-Uqlidisi's later commentary on Indian mathematics.^{[79]} - By 499 AD: Aryabhata's work shows the use of the modern fraction notation, known as bhinnarasi.
^{[80]}

### Physics[edit]

#### Astronomy[edit]

- c. 150 AD: Ptolemy's Almagest contains practical formulae to calculate latitudes and day lengths.
- 2nd
**century**AD: Ptolemy formalises the epicycles of Apollonius. - By the 5th
**century**AD: The elliptical orbits of planets are discovered in India by at least the time of Aryabhata, and are used for the calculations of orbital periods and eclipse timings.^{[81]} - 499 AD: Historians speculate that Aryabhata may have used an underlying heliocentric model for his astronomical calculations, which would make it the first computational heliocentric model in history (as opposed to Aristarchus's model in form).
^{[82]}^{[83]}^{[84]}This claim is based on his description of the planetary period about the sun (*śīghrocca*), but has been met with criticism.^{[85]}

#### Optics[edit]

- 2nd
**century**- Ptolemy publishes his Optics, discussing colour, reflection, and refraction of light, and including the first known table of refractive angles.

### Biology and human anatomy[edit]

- 2nd
**century**AD: Galen studies the anatomy of pigs.^{[86]}

### Astronomical and geospatial measurements[edit]

- 499 AD: Aryabhata creates a particularly accurate eclipse chart. As an example of its accuracy, 18th
**century**scientist Guillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations (based on Aryabhata's computational paradigm) 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.^{[87]}

## 500 AD – 1000 AD[edit]

The Golden Age of Indian mathematics and astronomy continues after the end of the Gupta empire, especially in Southern India during the era of the Rashtrakuta, Western Chalukya and Vijayanagara empires of Karnataka, which variously patronised Hindu and Jain mathematicians. In addition, the Middle East enters the Islamic Golden Age through contact with other civilisations, and China enters a golden period during the Tang and Song dynasties.

### Mathematics[edit]

#### Numbers, measurement and arithmetic[edit]

- 628 AD: Brahmagupta states the arithmetic rules for addition, subtraction, and multiplication with zero, as well as the multiplication of negative numbers, extending the basic rules for the latter found in the earlier The Nine Chapters on the Mathematical Art.
^{[88]}

#### Algebra[edit]

- 628 AD: Brahmagupta provides an explicit solution to the quadratic equation.
^{[89]} - 9th
**century**AD: Jain mathematician Mahāvīra writes down a factorisation for the difference of cubes.^{[90]}

#### Number theory and discrete mathematics[edit]

- 628 AD: Brahmagupta writes down Brahmagupta's identity, an important lemma in the theory of Pell's equation.
- 628 AD: Brahmagupta produces an infinite (but not exhaustive) number of solutions to Pell's equation.
- c. 850 AD: Mahāvīra derives the expression for the binomial coefficient in terms of factorials, .
^{[39]} - c. 975 AD: Halayudha organizes the binomial coefficients into a triangle, i.e. Pascal's triangle.
^{[39]}

#### Geometry and trigonometry[edit]

- 628 AD: Brahmagupta discovers Brahmagupta's formula, a generalization of Heron's formula to cyclic quadrilaterals.

#### Analysis[edit]

- 10th
**century**AD: Manjula in India discovers the derivative, deducing that the derivative of the sine function is the cosine.^{[91]}

#### Numerical mathematics and algorithms[edit]

- 628 AD: Brahmagupta discovers second-order interpolation, in the form of Brahmagupta's interpolation formula.
- 629 AD: Bhāskara I produces the first approximation of a transcendental function with a rational function, in the sine approximation formula that bears his name.
- 816 AD: Jain mathematician Virasena describes the integer logarithm.
^{[92]} - 9th
**century**AD: Algorisms (arithmetical algorithms on numbers written in place-value system) are described by al-Khwarizmi in his*kitāb al-ḥisāb al-hindī*(*Book of Indian computation*) and*kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī*(*Addition and subtraction in Indian arithmetic*). - 9th
**century**AD: Mahāvīra discovers the first algorithm for writing fractions as Egyptian fractions,^{[93]}which is in fact a slightly more general form of the Greedy algorithm for Egyptian fractions.

#### Notation and conventions[edit]

- 628 AD: Brahmagupta invents a symbolic mathematical notation, which is then adopted by mathematicians through India and the Near East, and eventually Europe.

### Physics[edit]

#### Astronomy[edit]

- 6th
**century**AD: Varahamira in the Gupta empire is the first to describe comets as astronomical phenomena, and as periodic in nature.^{[94]}

#### Mechanics[edit]

- c. 525 AD: John Philoponus in Byzantine Egypt describes the notion of inertia, and states that the motion of a falling object does not depend on its weight.
^{[95]}His radical rejection of Aristotlean orthodoxy lead him to be ignored in his time.

#### Optics[edit]

- 984 AD: Ibn Sahl discovers Snell's law.
^{[96]}^{[97]}

### Astronomical and geospatial measurements[edit]

- 10th
**century**AD: Kashmiri^{[98]}^{[99]}^{[100]}^{[101]}astronomer Bhaṭṭotpala lists names and estimates periods of certain comets.^{[94]}

## 1000 AD – 1500 AD[edit]

### Mathematics[edit]

#### Algebra[edit]

- 11th
**century**: Alhazen discovers the formula for the simplicial numbers defined as the sums of consecutive quartic powers.

#### Number theory and discrete mathematics[edit]

- c. 1000 AD: al-Karaji uses mathematical induction.
^{[102]} - 12th
**century**AD: Bhāskara II develops the Chakravala method, solving Pell's equation.^{[103]}

#### Geometry and trigonometry[edit]

- 15th
**century**: Parameshvara discovers a formula for the circumradius of a quadrilateral.^{[104]}

#### Analysis[edit]

- 1380 AD: Madhava of Sangamagrama develops the Taylor series, and derives the Taylor series representation for the sine, cosine and arctangent functions, and uses it to produce the Leibniz series for π.
^{[105]} - 1380 AD: Madhava of Sangamagrama discusses error terms in infinite series in the context of his infinite series for π.
^{[106]} - 1380 AD: Madhava of Sangamagrama discovers continued fractions and uses them to solve transcendental equations.
^{[107]} - 1380 AD: The Kerala school develops convergence tests for infinite series.
^{[105]} - c. 1500 AD: Nilakantha Somayaji discovers an infinite series for π.
^{[108]}^{[109]}

#### Numerical mathematics and algorithms[edit]

- 12th
**century**AD: al-Tusi develops a numerical algorithm to solve cubic equations. - 1380 AD: Madhava of Sangamagrama solves transcendental equations by iteration.
^{[107]} - 1380 AD: Madhava of Sangamagrama discovers the most precise estimate of π in the medieval world through his infinite series, a strict inequality with uncertainty 3e-13.
- 1480 AD: Madhava of Sangamagrama found pi and that it was infinite.

### Physics[edit]

#### Astronomy[edit]

- 1058 AD: al-Zarqālī in Islamic Spain discovers the apsidal precession of the sun.
- c. 1500 AD: Nilakantha Somayaji develops a model similar to the Tychonic system. His model has been described as mathematically more efficient than the Tychonic system due to correctly considering the equation of the centre and latitudinal motion of Mercury and Venus.
^{[91]}^{[110]}

#### Mechanics[edit]

- 12th
**century**AD: Jewish polymath Baruch ben Malka in Iraq formulates a qualitative form of Newton's second law for constant forces.^{[111]}^{[112]}

#### Optics[edit]

- 11th
**century**: Alhazen systematically studies optics and refraction, which would later be important in making the connection between geometric (ray) optics and wave theory. - 11th
**century**: Shen Kuo discovers atmospheric refraction and provides the correct explanation of rainbow phenomenon - c1290 - Eyeglasses are invented in Northern Italy,
^{[113]}possibly Pisa, demonstrating knowledge of human biology^{[citation needed]}and optics, to offer bespoke works that compensate for an individual human disability.

### Astronomical and geospatial measurements[edit]

- 11th
**century**: Shen Kuo discovers the concepts of true north and magnetic declination. - 11th
**century**: Shen Kuo develops the field of geomorphology and natural climate change.

### Social science[edit]

#### Economics[edit]

- 1295 AD: Scottish priest Duns Scotus writes about the mutual beneficence of trade.
^{[114]} - 14th
**century**AD: French priest Jean Buridan provides a basic explanation of the price system.

### Philosophy of science[edit]

- 1220s - Robert Grosseteste writes on optics, and the production of lenses, while asserting models should be developed from observations, and predictions of those models verified through observation, in a precursor to the scientific method.
^{[115]} - 1267 - Roger Bacon publishes his Opus Majus, compiling translated Classical Greek, and Arabic works on mathematics, optics, and alchemy into a volume, and details his methods for evaluating the theories, particularly those of Ptolemy's 2nd
**century**Optics, and his findings on the production of lenses, asserting “*theories supplied by reason should be verified by sensory data, aided by instruments, and corroborated by trustworthy witnesses*", in a precursor to the peer reviewed scientific method.

## 16th **century**[edit]

The Scientific Revolution occurs in Europe around this period, greatly accelerating the progress of science and contributing to the rationalization of the natural sciences.

### Mathematics[edit]

#### Numbers, measurement and arithmetic[edit]

- 1545: Gerolamo Cardano discovers complex numbers.
^{[116]} - 1572: Rafael Bombelli provides rules for complex arithmetic.
^{[117]}

#### Algebra[edit]

- c. 1500: Scipione del Ferro solves the special cubic equation .
^{[118]}^{[119]} - 16th
**century**: Gerolamo Cardano solves the general cubic equation (by reducing them to the case with zero quadratic term). - 16th
**century**: Lodovico Ferrari solves the general quartic equation (by reducing it to the case with zero quartic term). - 16th
**century**: François Viète discovers Vieta's formulas.

#### Probability and statistics[edit]

- 1564: Gerolamo Cardano is the first to produce a systematic treatment of probability.
^{[120]}

#### Numerical mathematics and algorithms[edit]

- 16th
**century**: François Viète discovers Viète's formula for π.^{[121]}

#### Notation and conventions[edit]

Various pieces of modern symbolic notation were introduced in this period, notably:

- 1556: Niccolò Tartaglia introduces parenthesis.
- 1557: Robert Recorde introduces the equal sign.
^{[122]}^{[123]} - 1591: François Viète's New algebra shows the modern notational algebraic manipulation.

### Physics[edit]

#### Astronomy[edit]

- 1543: Nicolaus Copernicus develops a heliocentric model, rejecting Aristotle's earth-centric view, would be the first quantitative heliocentric model in history.
- Late 16th
**century**: Tycho Brahe proves that comets are astronomical (and not atmospheric) phenomena.

### Biology and anatomy[edit]

- 1543 – Vesalius: pioneering research into human anatomy

### Social science[edit]

#### Economics[edit]

- 1517: Nicolaus Copernicus develops the quantity theory of money and states the earliest known form of Gresham's law: ("Bad money drowns out good").
^{[124]}

## 17th **century**[edit]

- 1600 – William Gilbert: Earth's magnetic field
- 1608 – Earliest record of an optical telescope
- 1609 – Johannes Kepler: first two laws of planetary motion
- 1610 – Galileo Galilei:
*Sidereus Nuncius*: telescopic observations - 1614 – John Napier: use of logarithms for calculation
^{[125]} - 1619 – Johannes Kepler: third law of planetary motion
- 1620 – Appearance of the first compound microscopes in Europe
- 1628 – Willebrord Snellius: the law of refraction also known as Snell's law
- 1628 – William Harvey: blood circulation
- 1638 – Galileo Galilei: laws of falling body
- 1643 – Evangelista Torricelli invents the mercury barometer
- 1662 – Robert Boyle: Boyle's law of ideal gas
- 1665 – Philosophical Transactions of the Royal Society first peer reviewed scientific journal published.
- 1665 – Robert Hooke: discovers the cell
- 1668 – Francesco Redi: disproved idea of spontaneous generation
- 1669 – Nicholas Steno: Proposes that fossils are organic remains embedded in layers of sediment, basis of stratigraphy
- 1669 – Jan Swammerdam: epigenesis in insects
- 1672 – Sir Isaac Newton: discovers that white light is a spectrum of a mixture of distinct coloured rays
- 1673 – Christiaan Huygens: first study of oscillating system and design of pendulum clocks
- 1675 – Leibniz, Newton: infinitesimal calculus
- 1675 – Anton van Leeuwenhoek: observes microorganisms using a refined simple microscope
- 1676 – Ole Rømer: first measurement of the speed of light
- 1687 – Sir Isaac Newton: classical mathematical description of the fundamental force of universal gravitation and the three physical laws of motion

## 18th **century**[edit]

- 1735 – Carl Linnaeus described a new system for classifying plants in Systema Naturae
- 1745 – Ewald Jürgen Georg von Kleist first capacitor, the Leyden jar
- 1749 to 1789 - Buffon wrote Histoire naturelle
- 1750 – Joseph Black: describes latent heat
- 1751 – Benjamin Franklin: Lightning is electrical
- 1755 – Immanuel Kant: Gaseous Hypothesis in Universal Natural History and Theory of Heaven
- 1761 – Mikhail Lomonosov: discovery of the atmosphere of Venus
- 1763 – Thomas Bayes: publishes the first version of Bayes' theorem, paving the way for Bayesian probability
- 1771 – Charles Messier: Publishes catalogue of astronomical objects (
*Messier Objects*) now known to include galaxies, star clusters, and nebulae. - 1778 – Antoine Lavoisier (and Joseph Priestley): discovery of oxygen leading to end of Phlogiston theory
- 1781 – William Herschel announces discovery of Uranus, expanding the known boundaries of the solar system for the first time in modern history
- 1785 – William Withering: publishes the first definitive account of the use of foxglove (digitalis) for treating dropsy
- 1787 – Jacques Charles: Charles's law of ideal gas
- 1789 – Antoine Lavoisier: law of conservation of mass, basis for chemistry, and the beginning of modern chemistry
- 1796 – Georges Cuvier: Establishes extinction as a fact
- 1796 – Edward Jenner: small pox historical accounting
- 1796 – Hanaoka Seishū: develops general anaesthesia
- 1800 – Alessandro Volta: discovers electrochemical series and invents the battery

## 19th **century**[edit]

- 1802 – Jean-Baptiste Lamarck: teleological evolution
- 1805 – John Dalton: Atomic Theory in (Chemistry)
- 1820 – Hans Christian Ørsted discovers that a current passed through a wire will deflect the needle of a compass, establishing a deep relationship between electricity and magnetism (electromagnetism).
- 1820 - Michael Faraday and James Stoddart discover alloying iron with chromium produces a stainless steel resistant to oxidising elements (rust).
- 1821 – Thomas Johann Seebeck is the first to observe a property of semiconductors
- 1824 – Carnot: described the Carnot cycle, the idealized heat engine
- 1824 - Joseph Aspdin develops Portland cement (concrete), by heating ground limestone, clay and gypsum, in a kiln.
- 1827 – Evariste Galois development of group theory
- 1827 – Georg Ohm: Ohm's law (Electricity)
- 1827 – Amedeo Avogadro: Avogadro's law (Gas law)
- 1828 – Friedrich Wöhler synthesized urea, refuting vitalism
- 1830 – Nikolai Lobachevsky created Non-Euclidean geometry
- 1831 – Michael Faraday discovers electromagnetic induction
- 1833 – Anselme Payen isolates first enzyme, diastase
- 1837 - Charles Babbage proposes a design for the construction of a Turing complete, general purpose Computer, to be called the Analytical Engine.
- 1838 – Matthias Schleiden: all plants are made of cells
- 1838 – Friedrich Bessel: first successful measure of stellar parallax (to star 61 Cygni)
- 1842 – Christian Doppler: Doppler effect
- 1843 – James Prescott Joule: Law of Conservation of energy (First law of thermodynamics), also 1847 – Helmholtz, Conservation of energy
- 1846 – Johann Gottfried Galle and Heinrich Louis d'Arrest: discovery of Neptune
- 1847 - George Boole: publishes
*The Mathematical Analysis of Logic*, defining Boolean algebra; refined in his 1854 The Laws of Thought. - 1848 – Lord Kelvin: absolute zero
- 1856 - Robert Forester Mushet develops a process for the decarbonisation, and re-carbonisation of iron, thorough the addition of a calculated quantity of spiegeleisen, to produce cheap, consistently high quality steel.
- 1858 – Rudolf Virchow: cells can only arise from pre-existing cells
- 1859 – Charles Darwin and Alfred Wallace: Theory of evolution by natural selection
- 1861 – Louis Pasteur: Germ theory
- 1861 – John Tyndall: Experiments in Radiant Energy that reinforced the Greenhouse Effect
- 1864 – James Clerk Maxwell: Theory of electromagnetism
- 1865 – Gregor Mendel: Mendel's laws of inheritance, basis for genetics
- 1865 – Rudolf Clausius: Definition of entropy
- 1868 - Robert Forester Mushet discovers alloying steel with tungsten produces a harder, more durable alloy.
- 1869 – Dmitri Mendeleev: Periodic table
- 1871 – Lord Rayleigh: Diffuse sky radiation (Rayleigh scattering) explains why sky appears blue
- 1873 – Johannes Diderik van der Waals: was one of the first to postulate an intermolecular force: the van der Waals force.
- 1873 – Frederick Guthrie discovers thermionic emission.
- 1873 – Willoughby Smith discovers photoconductivity.
- 1875 – William Crookes invented the Crookes tube and studied cathode rays
- 1876 – Josiah Willard Gibbs founded chemical thermodynamics, the phase rule
- 1877 – Ludwig Boltzmann: Statistical definition of entropy
- 1880s - John Hopkinson develops Three-phase electrical supplies, mathematically proves how multiple AC dynamos can be connected in parallel, improves permanent magnets, and dynamo efficiency, by the addition of tungsten, and describes how temperature effects magnetism (Hopkinson effect).
- 1880 – Pierre Curie and Jacques Curie: Piezoelectricity
- 1884 – Jacobus Henricus van 't Hoff: discovered the laws of chemical dynamics and osmotic pressure in solutions (in his work "Etudes de dynamique chimique").
- 1887 – Albert A. Michelson and Edward W. Morley: lack of evidence for the aether
- 1888 – Friedrich Reinitzer discovers liquid crystals
- 1892 – Dmitri Ivanovsky discovers viruses
- 1895 – Wilhelm Conrad Röntgen discovers x-rays
- 1896 – Henri Becquerel discovers radioactivity
- 1896 – Svante Arrhenius derives the basic principles of the greenhouse effect
- 1897 – J.J. Thomson discovers the electron in cathode rays
- 1898 – Martinus Beijerinck: concluded a virus infectious—replicating in the host—and thus not a mere toxin and gave it the name "virus"
- 1898 – J.J. Thomson proposed the plum pudding model of an atom
- 1898 - Marie Curie discovered radium and polonium

## 20th **century**[edit]

- 1900 – Max Planck: explains the emission spectrum of a black body
- 1905 – Albert Einstein: theory of special relativity, explanation of Brownian motion, and photoelectric effect
- 1906 – Walther Nernst: Third law of thermodynamics
- 1907 – Alfred Bertheim: Arsphenamine, the first modern chemotherapeutic agent
- 1909 – Fritz Haber: Haber Process for industrial production of ammonia
- 1909 – Robert Andrews Millikan: conducts the oil drop experiment and determines the charge on an electron
- 1910 – Williamina Fleming: the first white dwarf, 40 Eridani B
- 1911 – Ernest Rutherford: Atomic nucleus
- 1911 – Heike Kamerlingh Onnes: Superconductivity
- 1912 – Alfred Wegener: Continental drift
- 1912 – Max von Laue : x-ray diffraction
- 1912 – Vesto Slipher : galactic redshifts
- 1912 – Henrietta Swan Leavitt: Cepheid variable period-luminosity relation
- 1913 – Henry Moseley: defined atomic number
- 1913 – Niels Bohr: Model of the atom
- 1915 – Albert Einstein: theory of general relativity – also David Hilbert
- 1915 – Karl Schwarzschild: discovery of the Schwarzschild radius leading to the identification of black holes
- 1918 – Emmy Noether: Noether's theorem – conditions under which the conservation laws are valid
- 1920 – Arthur Eddington: Stellar nucleosynthesis
- 1922 – Frederick Banting, Charles Best, James Collip, John Macleod: isolation and production of insulin to control diabetes
- 1924 – Wolfgang Pauli: quantum Pauli exclusion principle
- 1924 – Edwin Hubble: the discovery that the Milky Way is just one of many galaxies
- 1925 – Erwin Schrödinger: Schrödinger equation (Quantum mechanics)
- 1925 – Cecilia Payne-Gaposchkin: Discovery of the composition of the Sun and that hydrogen is the most abundant element in the Universe
- 1927 – Werner Heisenberg: Uncertainty principle (Quantum mechanics)
- 1927 – Georges Lemaître: Theory of the Big Bang
- 1928 – Paul Dirac: Dirac equation (Quantum mechanics)
- 1929 – Edwin Hubble: Hubble's law of the expanding universe
- 1929 – Alexander Fleming: Penicillin, the first beta-lactam antibiotic
- 1929 – Lars Onsager's reciprocal relations, a potential fourth law of thermodynamics
- 1930 – Subrahmanyan Chandrasekhar discovers his eponymous limit of the maximum mass of a white dwarf star
- 1931 – Kurt Gödel: incompleteness theorems prove formal axiomatic systems are incomplete
- 1932 – James Chadwick: Discovery of the neutron
- 1932 – Karl Guthe Jansky discovers the first astronomical radio source, Sagittarius A
- 1932 – Ernest Walton and John Cockcroft: Nuclear fission by proton bombardment
- 1934 – Enrico Fermi: Nuclear fission by neutron irradiation
- 1934 – Clive McCay: Calorie restriction extends the maximum lifespan of another species
- 1938 – Otto Hahn, Lise Meitner and Fritz Strassmann: Nuclear fission of heavy nuclei
- 1938 – Isidor Rabi: Nuclear magnetic resonance
- 1943 – Oswald Avery proves that DNA is the genetic material of the chromosome
- 1945 – Howard Florey Mass production of penicillin
- 1947 – William Shockley, John Bardeen and Walter Brattain invent the first transistor
- 1948 – Claude Elwood Shannon: 'A mathematical theory of communication' a seminal paper in Information theory.
- 1948 – Richard Feynman, Julian Schwinger, Sin-Itiro Tomonaga and Freeman Dyson: Quantum electrodynamics
- 1951 – George Otto Gey propagates first cancer cell line, HeLa
- 1952 – Jonas Salk: developed and tested first polio vaccine
- 1952 – Stanley Miller: demonstrated that the building blocks of life could arise from primeval soup in the conditions present during early earth (Miller-Urey experiment)
- 1952 – Frederick Sanger: demonstrated that proteins are sequences of amino acids
- 1953 – James Watson, Francis Crick, Maurice Wilkins and Rosalind Franklin: helical structure of DNA, basis for molecular biology
- 1957 - Chien Shiung Wu: demonstrated that parity, and thus charge conjugation and time-reversals, are violated for weak interactions
- 1962 – Riccardo Giacconi and his team discover the first cosmic x-ray source, Scorpius X-1
- 1963 – Lawrence Morley, Fred Vine, and Drummond Matthews: Paleomagnetic stripes in ocean crust as evidence of plate tectonics (Vine–Matthews–Morley hypothesis).
- 1964 – Murray Gell-Mann and George Zweig: postulates quarks leading to the standard model
- 1964 – Arno Penzias and Robert Woodrow Wilson: detection of CMBR providing experimental evidence for the Big Bang
- 1965 – Leonard Hayflick: normal cells divide only a certain number of times: the Hayflick limit
- 1967 – Jocelyn Bell Burnell and Antony Hewish discover first pulsar
- 1967 – Vela nuclear test detection satellites discover the first gamma-ray burst
- 1970 - James H. Ellis proposed the possibility of "non-secret encryption", more commonly termed public-key cryptography, a concept that would be implemented by his GCHQ colleague Clifford Cocks in 1973, in what would become known as the RSA algorithm, with key exchange added by a third colleague Malcolm J. Williamson, in 1975.
- 1971 – Place cells in the brain are discovered by John O'Keefe
- 1974 – Russell Alan Hulse and Joseph Hooton Taylor, Jr. discover indirect evidence for gravitational wave radiation in the Hulse–Taylor binary
- 1977 – Frederick Sanger sequences the first DNA genome of an organism using Sanger sequencing
- 1980 – Klaus von Klitzing discovered the quantum Hall effect
- 1982 – Donald C. Backer et al. discover the first millisecond pulsar
- 1983 – Kary Mullis invents the polymerase chain reaction, a key discovery in molecular biology
- 1986 – Karl Müller and Johannes Bednorz: Discovery of High-temperature superconductivity
- 1988 – Bart van Wees and colleagues at TU Deflt and Philips Research discovered the quantized conductance in a two-dimensional electron gas.
- 1992 – Aleksander Wolszczan and Dale Frail observe the first pulsar planets (this was the first confirmed discovery of planets outside the Solar System)
- 1994 – Andrew Wiles proves Fermat's Last Theorem
- 1995 – Michel Mayor and Didier Queloz definitively observe the first extrasolar planet around a main sequence star
- 1995 – Eric Cornell, Carl Wieman and Wolfgang Ketterle attained the first Bose-Einstein Condensate with atomic gases, so called fifth state of matter at an extremely low temperature.
- 1996 – Roslin Institute: Dolly the sheep was cloned.
^{[126]} - 1997 – CDF and DØ experiments at Fermilab: Top quark.
- 1998 – Supernova Cosmology Project and the High-Z Supernova Search Team: discovery of the accelerated expansion of the Universe and dark energy
- 2000 – The Tau neutrino is discovered by the DONUT collaboration

## 21st **century**[edit]

- 2001 – The first draft of the Human Genome Project is published.
- 2003 – Grigori Perelman presents proof of the Poincaré Conjecture.
- 2004 – Andre Geim and Konstantin Novoselov isolated graphene, a monolayer of carbon atoms, and studied its quantum electrical properties.
- 2005 – Grid cells in the brain are discovered by Edvard Moser and May-Britt Moser.
- 2010 – The first self-replicating, synthetic bacterial cells are constructed.
^{[127]} - 2010 – The Neanderthal Genome Project presented preliminary genetic evidence that interbreeding did likely take place and that a small but significant portion of Neanderthal admixture is present in modern non-African populations.
^{[citation needed]} - 2012 – Higgs boson is discovered at CERN (confirmed to 99.999% certainty)
- 2012 – Photonic molecules are discovered at MIT
- 2014 – Exotic hadrons are discovered at the LHCb
- 2016 – The LIGO team detected gravitational waves from a black hole merger
- 2017 – Gravitational wave signal GW170817 was observed by the LIGO/Virgo collaboration. This was the first instance of a gravitational wave event that was observed to have a simultaneous electromagnetic signal when space telescopes like Hubble observed lights coming from the event, thereby marking a significant breakthrough for multi-messenger astronomy.
^{[128]}^{[129]}^{[130]} - 2019 – The first ever image of a black hole was captured, using eight different telescopes taking simultaneous pictures, timed with extremely precise atomic clocks.
- 2020 – NASA and SOFIA (Stratospheric Observatory for Infrared Astronomy) discovered about 12oz of surface water in one of the moon's largest visible craters.
^{[131]}

## References[edit]

**^**Whitelaw, p. 14.**^**S. R. Rao (1985).*Lothal*. Archaeological Survey of India. pp. 40–41.**^**Rao (July 1992). "A Navigational Instrument of the Harappan Sailors" (PDF).*Marine Archaeology*.**3**: 61–66. Notes: protractor described as "compass" in article.**^**Petruso, Karl M (1981). "Early Weights and Weighing in Egypt and the Indus Valley".*M Bulletin*.**79**: 44–51. JSTOR 4171634.**^**Friberg, Jöran (2009). "A Geometric Algorithm with Solutions to Quadratic Equations in a Sumerian Juridical Document from Ur III Umma".*Cuneiform Digital Library Journal*.**3**.**^**Maor, Eli (1998).*Trigonometric Delights*. Princeton University Press. p. 20. ISBN 978-0-691-09541-7.**^**Cite error: The named reference`Friberg 2009`

was invoked but never defined (see the help page).**^**Richard J. Gillings,*Mathematics in the Time of the Pharaohs*, Dover, New York, 1982, 161.**^**Jane Qiu (7 January 2014). "Ancient times table hidden in Chinese bamboo strips".*Nature News*. doi:10.1038/nature.2014.14482. S2CID 130132289.**^**Beery, Janet L.; Swetz, Frank J. (July 2012), "The best known old Babylonian tablet?",*Convergence*, Mathematical Association of America, doi:10.4169/loci003889**^**Romano, David Gilman (1993).*Athletics and Mathematics in Archaic Corinth: The Origins of the Greek Stadion*. American Philosophical Society. p. 78. ISBN 9780871692061.A group of mathematical clay tablets from the Old Babylonian Period, excavated at Susa in 1936, and published by E.M. Bruins in 1950, provide the information that the Babylonian approximation of π was 3 1/8 or 3.125.

**^**Bruins, E. M. (1950). "Quelques textes mathématiques de la Mission de Suse" (PDF).**^**Bruins, E. M.; Rutten, M. (1961).*Textes mathématiques de Suse*. Mémoires de la Mission archéologique en Iran.**XXXIV**.**^**Imhausen, Annette (2007). Katz, Victor J. (ed.).*The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook*. Princeton University Press. ISBN 978-0-691-11485-9.**^**Rossi (2007).*Corinna Architecture and Mathematics in Ancient Egypt*. Cambridge University Press. ISBN 978-0-521-69053-9.**^**"Egyptian numerals". Retrieved 25 September 2013.**^**Stephen Chrisomalis (2010).*Numerical Notation: A Comparative History*. p. 248. ISBN 9780521878180.**^**Lamb, Evelyn (31 August 2014), "Look, Ma, No Zero!",*Scientific American*, Roots of Unity**^**Porter, Roy (17 October 1999).*The Greatest Benefit to Mankind: A Medical History of Humanity (The Norton History of Science)*. W. W. Norton. pp. 49–50. ISBN 9780393319804. Retrieved 17 November 2013.**^**Thibaut, George (1875). "On the Śulvasútras".*The Journal of the Asiatic Society of Bengal*.**44**: 227–275.**^**Seshadri, Conjeevaram (2010). Seshadri, C. S (ed.).*Studies in the History of Indian Mathematics*. New Delhi: Hindustan Book Agency. pp. 152–153. doi:10.1007/978-93-86279-49-1. ISBN 978-93-80250-06-9.- ^
^{a}^{b}Bhishagratna, Kaviraj KL (1907).*An English Translation of the Sushruta Samhita in Three Volumes*. Archived from the original on 4 November 2008. Alt URL **^**Patwardhan, Kishor (2012). "The history of the discovery of blood circulation: Unrecognized contributions of Ayurveda masters".*Advances in Physiology Education*.**36**(2): 77–82. doi:10.1152/advan.00123.2011. PMID 22665419.**^**Bhate, S. and Kak, S. (1993) Panini and Computer Science. Annals of the Bhandarkar Oriental Research Institute, vol. 72, pp. 79-94.**^**Kadvany, John (2007), "Positional Value and Linguistic Recursion",*Journal of Indian Philosophy*,**35**(5–6): 487–520, CiteSeerX 10.1.1.565.2083, doi:10.1007/s10781-007-9025-5, S2CID 52885600.**^**Knopp, Konrad (1951).*Theory and Application of Infinite Series*(English 2nd ed.). London and Glasgow: Blackie & Son, Ltd. p. 7. ISBN 0-486-66165-2.**^**Ian Stewart (2017).*Infinity: a Very Short Introduction*. Oxford University Press. p. 117. ISBN 978-0-19-875523-4. Archived from the original on 3 April 2017.**^**Van Nooten, B. (1 March 1993). "Binary numbers in Indian antiquity".*Journal of Indian Philosophy*.**21**(1): 31–50. doi:10.1007/BF01092744. S2CID 171039636.**^**Eves, Howard. "Webpage cites AN INTRODUCTION TO THE HISTORY OF MATHEMATICS". Mathcentral. Retrieved 28 March 2015.**^**Heath, Thomas L. (1956).*The Thirteen Books of Euclid's Elements*(2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.**^**Ian Bruce (2000) "Napier’s Logarithms", American Journal of Physics 68(2):148**^**Needham, Joseph (1986).*Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth*(Vol. 3), p 24. Taipei: Caves Books, Ltd.**^**Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum".*The Annals of Mathematics*.**^**James R. Choike (1980). "The Pentagram and the Discovery of an Irrational Number".*The Two-Year College Mathematics Journal*..**^**Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India",*Historia Mathematica*,**12**(3): 229–44, doi:10.1016/0315-0860(85)90021-7**^**Knuth, Donald (1968),*The Art of Computer Programming*,**1**, Addison Wesley, p. 100, ISBN 978-81-7758-754-8,Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns... both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed)...

**^**Ore, Oystein (1988) [1948],*Number Theory and its History*, Dover, p. 65**^**A. W. F. Edwards.*Pascal's arithmetical triangle: the story of a mathematical idea.*JHU Press, 2002. Pages 30–31.- ^
^{a}^{b}^{c}Edwards, A. W. F. (2013), "The arithmetical triangle", in Wilson, Robin; Watkins, John J. (eds.),*Combinatorics: Ancient and Modern*, Oxford University Press, pp. 166–180 **^**Amulya Kumar Bag (6 January 1966). "Binomial theorem in Ancient India" (PDF).*Indian J. Hist. Sci.*: 68–74.**^**Hoche, Richard, ed. (1866),*Nicomachi Geraseni Pythagorei Introductionis arithmeticae libri II*, Leipzig: B.G. Teubner, p. 31**^**Bold, Benjamin.*Famous Problems of Geometry and How to Solve Them*, Dover Publications, 1982 (orig. 1969).**^**Boyer 1991, "The Age of Plato and Aristotle" p. 93. "It was consequently a signal achievement on the part of Menaechmus when he disclosed that curves having the desired property were near at hand. In fact, there was a family of appropriate curves obtained from a single source – the cutting of a right circular cone by a plane perpendicular to an element of the cone. That is, Menaechmus is reputed to have discovered the curves that were later known as the ellipse, the parabola, and the hyperbola. [...] Yet the first discovery of the ellipse seems to have been made by Menaechmus as a mere by-product in a search in which it was the parabola and hyperbola that proffered the properties needed in the solution of the Delian problem."**^**Boyer 1991, "The Age of Plato and Aristotle" pp. 94–95. "Menaechmus apparently derived these properties of the conic sections and others as well. Since this material has a strong resemblance to the use of coordinates, as illustrated above, it has sometimes been maintained that Menaechmus had analytic geometry. Such a judgment is warranted only in part, for certainly Menaechmus was unaware that any equation in two unknown quantities determines a curve. In fact, the general concept of an equation in unknown quantities was alien to Greek thought. It was shortcomings in algebraic notations that, more than anything else, operated against the Greek achievement of a full-fledged coordinate geometry."- ^
^{a}^{b}Boyer 1991, "Greek Trigonometry and Mensuration" pp. 158–159. "Trigonometry, like other branches of mathematics, was not the work of any one man, or nation. Theorems on ratios of the sides of similar triangles had been known to, and used by, the ancient Egyptians and Babylonians. In view of the pre-Hellenic lack of the concept of angle measure, such a study might better be called "trilaterometry", or the measure of three sided polygons (trilaterals), than "trigonometry", the measure of parts of a triangle. With the Greeks we first find a systematic study of relationships between angles (or arcs) in a circle and the lengths of chords subtending these. Properties of chords, as measures of central and inscribed angles in circles, were familiar to the Greeks of Hippocrates' day, and it is likely that Eudoxus had used ratios and angle measures in determining the size of the earth and the relative distances of the sun and the moon. In the works of Euclid there is no trigonometry in the strict sense of the word, but there are theorems equivalent to specific trigonometric laws or formulas. Propositions II.12 and 13 of the*Elements*, for example, are the laws of cosines for obtuse and acute angles respectively, stated in geometric rather than trigonometric language and proved by a method similar to that used by Euclid in connection with the Pythagorean theorem. Theorems on the lengths of chords are essentially applications of the modern law of sines. We have seen that Archimedes' theorem on the broken chord can readily be translated into trigonometric language analogous to formulas for sines of sums and differences of angles." **^**Archimedes (1912),*The method of Archimedes recently discovered by Heiberg; a supplement to the Works of Archimedes*, Cambridge University Press**^**Eves, Howard (1963),*A Survey of Geometry (Volume One)*, Boston: Allyn and Bacon**^**Archimedes,*The Method of Mechanical Theorems*; see Archimedes Palimpsest**^**O'Connor, J.J. & Robertson, E.F. (February 1996). "A history of calculus". University of St Andrews. Retrieved 7 August 2007.**^**K., Bidwell, James (30 November 1993). "Archimedes and Pi-Revisited".*School Science and Mathematics*.**94**(3).**^**Boyer 1991, "Archimedes of Syracuse" p. 127. "Greek mathematics sometimes has been described as essentially static, with little regard for the notion of variability; but Archimedes, in his study of the spiral, seems to have found the tangent to a curve through kinematic considerations akin to differential calculus. Thinking of a point on the spiral 1=*r*=*aθ*as subjected to a double motion — a uniform radial motion away from the origin of coordinates and a circular motion about the origin — he seems to have found (through the parallelogram of velocities) the direction of motion (hence of the tangent to the curve) by noting the resultant of the two component motions. This appears to be the first instance in which a tangent was found to a curve other than a circle.

Archimedes' study of the spiral, a curve that he ascribed to his friend Conon of Alexandria, was part of the Greek search for the solution of the three famous problems."**^**Dicks, D.R. (1970).*Early Greek Astronomy to Aristotle*. Ithaca, N.Y.: Cornell University Press. pp. 68. ISBN 978-0-8014-0561-7.**^**E. At. Schwanbeck (1877).*Ancient India as described by Megasthenês and Arrian; being a translation of the fragments of the Indika of Megasthenês collected by Dr. Schwanbeck, and of the first part of the Indika of Arrian*. p. 101.**^**Warmflash, David (20 June 2019). "An Ancient Greek Philosopher Was Exiled for Claiming the Moon Was a Rock, Not a God".*Smithsonian Mag*. Retrieved 10 March 2020.**^**Draper, John William (2007) [1874]. "History of the Conflict Between Religion and Science". In Joshi, S. T. (ed.).*The Agnostic Reader*. Prometheus. pp. 172–173. ISBN 978-1-59102-533-7.**^**Jones, A., Alexander (September 1991). "The Adaptation of Babylonian Methods in Greek Numerical Astronomy" (PDF).*Isis*.**82**(3): 440–453. Bibcode:1991Isis...82..441J. doi:10.1086/355836.**^**Ossendrijver, Mathieu (29 January 2016). "Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph".*Science*.**351**(6272): 482–484. Bibcode:2016Sci...351..482O. doi:10.1126/science.aad8085. PMID 26823423. S2CID 206644971.**^**Valleriani, Matteo (3 June 2010).*Galileo Engineer*. Springer Science and Business Media.**^**Spaide RF, Ohno-Matsui KM, Yannuzzi LA, eds. (2013).*Pathologic Myopia*. Springer Science & Business Media. p. 2. ISBN 978-1461483380.**^**Mabbett, I. W. (1964). "The Date of the Arthaśāstra".*Journal of the American Oriental Society*. American Oriental Society.**84**(2): 162–169. doi:10.2307/597102. ISSN 0003-0279. JSTOR 597102.**^**D. Rawlins: "Methods for Measuring the Earth's Size by Determining the Curvature of the Sea" and "Racking the Stade for Eratosthenes", appendices to "The Eratosthenes–Strabo Nile Map. Is It the Earliest Surviving Instance of Spherical Cartography? Did It Supply the 5000 Stades Arc for Eratosthenes' Experiment?",*Archive for History of Exact Sciences*, v.26, 211–219, 1982**^**Bowen A.C., Goldstein B.R. (1991). "Hipparchus' Treatment of Early Greek Astronomy: The Case of Eudoxus and the Length of Daytime Author(s)".*Proceedings of the American Philosophical Society***135(2)**: 233–254.**^**Struik, pp. 32–33 . "In these matrices we find negative numbers, which appear here for the first time in history."**^**Luke Hodgkin (2005).*A History of Mathematics: From Mesopotamia to Modernity*. Oxford University Press. p. 88. ISBN 978-0-19-152383-0.Liu is explicit on this; at the point where the

*Nine Chapters*give a detailed and helpful 'Sign Rule'- ^
^{a}^{b}Boyer 1991, "The Mathematics of the Hindus" p. 207. "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." - ^
^{a}^{b}Bibhutibhushan Datta and Avadhesh Narayan Singh (1962).*History of Hindu Mathematics A source Book Part II*. Asia Publishing House. p. 92. **^**Heath, Thomas L. (1921).*A History of Greek Mathematics (Vol II)*. Oxford University Press. pp. 321–323.**^**Boyer 1991, "Greek Trigonometry and Mensuration" p. 163. "In Book I of this treatise Menelaus establishes a basis for spherical triangles analogous to that of Euclid I for plane triangles. Included is a theorem without Euclidean analogue – that two spherical triangles are congruent if corresponding angles are equal (Menelaus did not distinguish between congruent and symmetric spherical triangles); and the theorem*A*+*B*+*C*> 180° is established. The second book of the*Sphaerica*describes the application of spherical geometry to astronomical phenomena and is of little mathematical interest. Book III, the last, contains the well known "theorem of Menelaus" as part of what is essentially spherical trigonometry in the typical Greek form – a geometry or trigonometry of chords in a circle. In the circle in Fig. 10.4 we should write that chord AB is twice the sine of half the central angle AOB (multiplied by the radius of the circle). Menelaus and his Greek successors instead referred to AB simply as the chord corresponding to the arc AB. If BOB' is a diameter of the circle, then chord A' is twice the cosine of half the angle AOB (multiplied by the radius of the circle)."**^**Boyer 1991, p.^{[page needed]}.**^**Bailey, David; Borwein, Jonathan (2012). "Ancient Indian Square Roots: An Exercise in Forensic Paleo-Mathematics" (PDF).*American Mathematical Monthly*.**119**(8). pp. 646–657. Retrieved 14 September 2017.**^**37461 Aryabhata at the*Encyclopædia Britannica***^**Parakh, Abhishek (2006). "Aryabhata's Root Extraction Methods". arXiv:math/0608793.**^**Kak 1986.**^**Cajori, Florian (1928).*A History of Elementary Mathematics*.*Science*.**5**. The Open Court Company, Publishers. pp. 516–7. doi:10.1126/science.5.117.516. ISBN 978-1-60206-991-6. PMID 17758371.It will be remembered that the scratch method did not spring into existence in the form taught by the writers of the sixteenth

**century**. On the contrary, it is simply the graphical representation of the method employed by the Hindus, who calculated with a coarse pencil on a small dust-covered tablet. The erasing of a figure by the Hindus is here represented by the scratching of a figure.**^**Lay-Yong, Lam (1966). "On the Chinese Origin of the Galley Method of Arithmetical Division".*The British Journal for the History of Science*.**3**: 66–69. doi:10.1017/S0007087400000200.**^**Kurt Vogel, "Diophantus of Alexandria." in Complete Dictionary of Scientific Biography, Encyclopedia.com, 2008. Quote:*The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word ‘equals’, Diophantus took a fundamental step from verbal algebra towards symbolic algebra.***^**Pearce, Ian (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. Retrieved 24 July 2007.**^**Reimer, L., and Reimer, W.*Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians, Vol. 2*. 1995. pp. 22-22. Parsippany, NJ: Pearson ducation, Inc. as Dale Seymor Publications. ISBN 0-86651-823-1.**^**Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Katz, Victor J. (ed.).*The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook*. Princeton University Press. p. 530. ISBN 978-0-691-11485-9.**^**Miller, Jeff (22 December 2014). "Earliest Uses of Various Mathematical Symbols". Archived from the original on 20 February 2016. Retrieved 15 February 2016.**^**Hayashi (2008),*Aryabhata I*.^{[full citation needed]}**^**The concept of Indian heliocentrism has been advocated by B. L. van der Waerden,*Das heliozentrische System in der griechischen, persischen und indischen Astronomie.*Naturforschenden Gesellschaft in Zürich. Zürich:Kommissionsverlag Leeman AG, 1970.**^**B.L. van der Waerden, "The Heliocentric System in Greek, Persian and Hindu Astronomy", in David A. King and George Saliba, ed.,*From Deferent to Equant: A Volume of Studies in the History of Science in the Ancient and Medieval Near East in Honor of E. S. Kennedy*, Annals of the New York Academy of Science, 500 (1987), pp. 529–534.**^**Hugh Thurston (1996).*Early Astronomy*. Springer. p. 188. ISBN 0-387-94822-8.**^**Noel Swerdlow, "Review: A Lost Monument of Indian Astronomy,"*Isis*, 64 (1973): 239–243.**^**Pasipoularides, Ares (1 March 2014). "Galen, father of systematic medicine. An essay on the evolution of modern medicine and cardiology".*International Journal of Cardiology*.**172**(1): 47–58. doi:10.1016/j.ijcard.2013.12.166. PMID 24461486.**^**Ansari, S.M.R. (March 1977). "Aryabhata I, His Life and His Contributions".*Bulletin of the Astronomical Society of India*.**5**(1): 10–18. Bibcode:1977BASI....5...10A. hdl:2248/502.**^**Henry Thomas Colebrooke.*Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara*, London 1817, p. 339 (online)**^**Plofker 2007, pp. 428–434.**^**Tabak, John (2009),*Algebra: Sets, Symbols, and the Language of Thought*, Infobase Publishing, p. 42, ISBN 978-0-8160-6875-3- ^
^{a}^{b}Joseph, G. G. (2000),*The Crest of the Peacock: The Non-European Roots of Mathematics*, Princeton, NJ: Princeton University Press, 416 pages, ISBN 978-0-691-00659-8 **^**Gupta, R. C. (2000), "History of Mathematics in India", in Hoiberg, Dale; Ramchandani, Indu (eds.),*Students' Britannica India: Select essays*, Popular Prakashan, p. 329**^**Kusuba 2004, pp. 497–516.- ^
^{a}^{b}Kelley, David H. & Milone, Eugene F. (2011).*Exploring Ancient Skies: A Survey of Ancient and Cultural Astronomy*(2nd ed.). Springer Science+Business Media. p. 293. doi:10.1007/978-1-4419-7624-6. ISBN 978-1-4419-7624-6. OCLC 710113366. **^**Morris R. Cohen and I. E. Drabkin (eds. 1958),*A Source Book in Greek Science*(p. 220), with several changes. Cambridge, MA: Harvard University Press, as referenced by David C. Lindberg (1992),*The Beginnings of Western Science: The European Scientific Tradition in Philosophical, Religious, and Institutional Context, 600 B.C. to A.D. 1450*, University of Chicago Press, p. 305, ISBN 0-226-48231-6- Note the influence of Philoponus' statement on Galileo's
*Two New Sciences*(1638)

- Note the influence of Philoponus' statement on Galileo's
**^**http://spie.org/etop/2007/etop07fundamentalsII.pdf," R. Rashed credited Ibn Sahl with discovering the law of refraction [23], usually called Snell’s law and also Snell and Descartes’ law."**^**Smith, A. Mark (2015).*From Sight to Light: The Passage from Ancient to Modern Optics*. University of Chicago Press. p. 178. ISBN 9780226174761.**^**Bina Chatterjee (introduction by),*The Khandakhadyaka of Brahmagupta*, Motilal Banarsidass (1970), p. 13**^**Lallanji Gopal,*History of Agriculture in India, Up to C. 1200 A.D.*, Concept Publishing Company (2008), p. 603**^**Kosla Vepa,*Astronomical Dating of*, Indic Studies Foundation (2008), p. 372**Events**& Select Vignettes from Indian History**^**Dwijendra Narayan Jha (edited by),*The feudal order: state, society, and ideology in early medieval India*, Manohar Publishers & Distributors (2000), p. 276**^**Katz 1998, p. 255.**^**Florian Cajori (1918), Origin of the Name "Mathematical Induction",*The American Mathematical Monthly***25**(5), p. 197-201.**^**Radha Charan Gupta (1977) "Parameshvara's rule for the circumradius of a cyclic quadrilateral", Historia Mathematica 4: 67–74- ^
^{a}^{b}Katz 1995. **^**J J O'Connor and E F Robertson (2000). "Madhava of Sangamagramma".*MacTutor History of Mathematics archive*. School of Mathematics and Statistics, University of St Andrews, Scotland. Archived from the original on 14 May 2006. Retrieved 8 September 2007.- ^
^{a}^{b}Ian G. Pearce (2002). Madhava of Sangamagramma.*MacTutor History of Mathematics archive*. University of St Andrews. **^**Roy 1990, pp. 101–102.**^**Brink, David (2015). "Nilakantha's accelerated series for π".*Acta Arithmetica*.**171**(4): 293–308. doi:10.4064/aa171-4-1.**^**Ramasubramanian, K.; Srinivas, M. D.; Sriram, M. S. (1994). "Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion".*Current Science*.**66**: 784–790.**^**Crombie, Alistair Cameron,*Augustine to Galileo 2*, p. 67.**^**Pines, Shlomo (1970). "Abu'l-Barakāt al-Baghdādī , Hibat Allah".*Dictionary of Scientific Biography*.**1**. New York: Charles Scribner's Sons. pp. 26–28. ISBN 0-684-10114-9.

(cf. Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory",*Journal of the History of Ideas***64**(4), p. 521-546 [528].)**^**"The invention of spectacles".*The College of Optometrists*. The College of Optometrists. Retrieved 9 May 2020.**^**Mochrie, Robert (2005).*Justice in Exchange: The Economic Philosophy of John Duns Scotus***^**"Robert Grosseteste".*Stanford Encyclopaedia of Philosophy*. Stanford.edu. Retrieved 6 May 2020.**^**Kline, Morris.*A history of mathematical thought, volume 1*. p. 253.**^**Katz, Victor J. (2004), "9.1.4",*A History of Mathematics, Brief Version*, Addison-Wesley, ISBN 978-0-321-16193-2**^**Burton, David.*The History of Mathematics: An Introduction*(7th (2010) ed.). New York: McGraw-Hill.**^**Bruno, Leonard C (2003) [1999].*Math and mathematicians: the history of math discoveries around the world*. Baker, Lawrence W. Detroit, Mich.: U X L. p. 60. ISBN 0787638137. OCLC 41497065.**^**Westfall, Richard S. "Cardano, Girolamo".*The Galileo Project*. rice.edu. Archived from the original on 28 July 2012. Retrieved 2012-07-19.**^**Beckmann, Petr (1971).*A history of π*(2nd ed.). Boulder, CO: The Golem Press. pp. 94–95. ISBN 978-0-88029-418-8. MR 0449960.**^**Jourdain, Philip E. B. (1913).*The Nature of Mathematics*.**^**Robert Recorde,*The Whetstone of Witte*(London, England: John Kyngstone, 1557), p. 236 (although the pages of this book are not numbered). From the chapter titled "The rule of equation, commonly called Algebers Rule" (p. 236): "Howbeit, for easie alteration of*equations*. I will propounde a fewe examples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in worke use, a paire of paralleles, or Gemowe [twin, from*gemew*, from the French*gemeau*(twin / twins), from the Latin*gemellus*(little twin)] lines of one lengthe, thus: = , bicause noe .2. thynges, can be moare equalle." (However, for easy manipulation of*equations*, I will present a few examples in order that the extraction of roots may be more readily done. And to avoid the tedious repetition of these words "is equal to", I will substitute, as I often do when working, a pair of parallels or twin lines of the same length, thus: = , because no two things can be more equal.)**^**Volckart, Oliver (1997). "Early beginnings of the quantity theory of money and their context in Polish and Prussian monetary policies, c. 1520–1550".*The Economic History Review*. Wiley-Blackwell.**50**(3): 430–49. doi:10.1111/1468-0289.00063. ISSN 0013-0117. JSTOR 2599810.**^**"John Napier and logarithms". Ualr.edu. Retrieved 12 August 2011.**^**"The Roslin Institute (University of Edinburgh) – Public Interest: Dolly the Sheep".*www.roslin.ed.ac.uk*. Retrieved 14 January 2017.**^**"JCVI: First Self-Replicating, Synthetic Bacterial Cell Constructed by J. Craig Venter Institute Researchers". jcvi.org. Retrieved 12 August 2018.**^**Landau, Elizabeth; Chou, Felicia; Washington, Dewayne; Porter, Molly (16 October 2017). "NASA Missions Catch First Light from a Gravitational-Wave Event".*NASA*. Retrieved 17 October 2017.**^**"Neutron star discovery marks breakthrough for 'multi-messenger astronomy'". csmonitor.com. 16 October 2017. Retrieved 17 October 2017.**^**"Hubble makes milestone observation of gravitational-wave source". slashgear.com. 16 October 2017. Retrieved 17 October 2017.**^**"NASA's SOFIA Discovers Water on Sunlit Surface of Moon".*AP NEWS*. 26 October 2020. Retrieved 3 November 2020.

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