timeline of scientific discoveries

The timeline below shows the date of publication of possible major scientific breakthroughs, theories and discoveries, along with the discoverer. This article discounts 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). The timeline begins at the Bronze Age, as it is difficult to give even estimates for the timing of events prior to this, such as of the discovery of counting, natural numbers and arithmetic.

To avoid overlap with timeline of historic inventions, the timeline does 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

Many early innovations of the Bronze Age were prompted by 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

Numbers, measurement and arithmetic

* Around 3000 BC: Units of measurement are developed in the Americas as well as the major Bronze Age civilisations: Egypt, Mesopotamia, Elam and the Indus Valley.
** The oldest extant record of a unit of length, being the 2650 BCE, Sumerian city, Nippur (present day Iraq), cubit-rod ruler.
** The oldest attested evidence for the existence of units of weight, and weighing scales date to the Fourth Dynasty of Egypt, with Deben (unit) balance weights, from the reign of Sneferu (c. 2600 BC) excavated, though earlier usage has been proposed.
The Indus Valley may have been the major innovator on this, as the first measurement devices were invented in Lothal in Gujarat, India.
* 1800 BC: Fractions were first studied by the Egyptians in their study of Egyptian fractions.

Geometry and trigonometry

* 2100 BC: The concept of area is first recognised in Babylonian clay tablets, 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.

Algebra

* 2100 BC: Quadratic equations, in the form of problems relating the areas and sides of rectangles, are solved by Babylonians.

Number theory and discrete mathematics

* 2000 BC: Pythagorean triples are first discussed in Babylon and Egypt, and appear on later manuscripts such as the Berlin Papyrus 6619.

Numerical mathematics and algorithms

* 2000 BC: Multiplication tables in Babylon.
* 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.
* 19th to 17th century BCE: A Babylonian tablet uses = 3.125 as an approximation for , which has an error of 0.5%.
* 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 .

Notation and conventions

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

Astronomy

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

Biology and anatomy

* 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).

Iron Age

Mathematics

Geometry and trigonometry

* c. 700 BC: Pythagoras's theorem is discovered by Baudhayana in the Hindu Shulba Sutras in Upanishadic India. 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

* c. 700 BC: Pell's equations are first studied by Baudhayana in India, the first diophantine equations known to be studied.

Geometry and trigonometry

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

Biology and anatomy

* 600 BC – 200 BC: The Sushruta Samhita shows an understanding of musculoskeletal structure (including joints, ligaments and muscles and their functions) (3.V).Alt URL
/ref> It refers to the cardiovascular system as a closed circuit. In (3.IX) it identifies the existence of nerves.

Social science

Linguistics

* c. 700 BC: Grammar is first studied in India (note that Sanskrit Vyākaraṇa predates Pāṇini).

500 BC – 1 BC

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

Mathematics

Logic and proof

* 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.
* 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

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

Algebra

* 5th century BC: Possible date of the discovery of the triangular numbers (i.e. the sum of consecutive integers), by the Pythagoreans.
* c. 300 BC: Finite geometric progressions are studied by Euclid in Ptolemaic Egypt.
* 3rd century BC: Archimedes relates problems in geometric series to those in arithmetic series, foreshadowing the logarithm.
* 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.

Number theory and discrete mathematics

* c. 500 BC: Hippasus, a Pythagorean, discovers irrational numbers.
* 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.
* c. 300 BC: Euclid proves the infinitude of primes.
* 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 $\tbinom=\tbinom+\tbinom$,A. W. F. Edwards. ''Pascal's arithmetical triangle: the story of a mathematical idea.'' JHU Press, 2002. Pages 30–31. 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.
* 3rd century BC: Eratosthenes discovers the Sieve of Eratosthenes.

Geometry and trigonometry

* 5th century BC: The Greeks start experimenting with straightedge-and-compass constructions.Bold, Benjamin. ''Famous Problems of Geometry and How to Solve Them'', Dover Publications, 1982 (orig. 1969).
* 4th century BC: Menaechmus discovers conic sections.
* 4th century BC: Menaechmus develops co-ordinate geometry.
* 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.. "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."
* 3rd century BC: Archimedes derives a formula for the volume of a sphere in '' The Method of Mechanical Theorems''.
* 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.
* 3rd century BC: Archimedes discovers the sum/difference identity for trigonometric functions in the form of the "Theorem of Broken Chords".
* c. 200 BC: Apollonius of Perga discovers Apollonius's theorem.
* c. 200 BC: Apollonius of Perga assigns equations to curves.

Analysis

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

Numerical mathematics and algorithms

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

Physics

Astronomy

* 5th century BC: The earliest documented mention of a spherical Earth comes from the Greeks in the 5th century BC. It is known that the Indians modeled the Earth as spherical by 300 BC
* 500 BC: Anaxagoras identifies moonlight as reflected sunlight.
* 260 BC: Aristarchus of Samos proposes a basic heliocentric model of the universe.
* 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.
* 2nd century BC: Hipparchos discovers Axial precession.

Mechanics

* 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.

Optics

* 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

* 460 BC: Empedocles describes thermal expansion.

= Matter

=

* 600BC: Maharshi Kanada gives the ideal of the smallest units of matter. According to him, matter consisted of indestructible minutes particles called ''paramanus'', which are now called as atoms.

Biology and anatomy

* 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. Graeco-Roman physician Galen would later use the term "myopia" for near-sightedness.

Social science

Economics

* Late 4th century BC: Chanakya (also known as Kautilya) establishes the field of economics with the Arthashastra (literally "Science of wealth"), a prescriptive treatise on economics and statecraft for Mauryan India.

Linguistics

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

Astronomical and geospatial measurements

* 3rd century BC: Eratosthenes measures the circumference of the Earth.
* 2nd century BC: Hipparchos measures the sizes of and distances to the moon and sun.

1 AD – 500 AD

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

Numbers, measurement and arithmetic

* 210 AD: Negative numbers are accepted as numeric by the late Han-era Chinese text ''The Nine Chapters on the Mathematical Art''. Later, Liu Hui of Cao Wei period) writes down laws regarding the arithmetic of negative numbers.
* 499 AD: Aryabhata gives a new symbol for zero and uses it for the decimal system

Algebra

* 499 AD: Aryabhata discovers the formula for the square-pyramidal numbers (the sums of consecutive square numbers).. "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."
* 499 AD: Aryabhata discovers the formula for the simplicial numbers (the sums of consecutive cube numbers).

Number theory and discrete mathematics

* 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.
* 499 AD: Aryabhata develops Kuṭṭaka, an algorithm very similar to the Extended Euclidean algorithm.

Geometry and trigonometry

* c. 60 AD: Heron's formula is discovered by Hero of Alexandria.
* c. 100 AD: Menelaus of Alexandria describes spherical triangles, a precursor to non-Euclidean geometry.. "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)."
* 4th to 5th centuries: The modern fundamental trigonometric functions, sine and cosine, are described in the Siddhantas of India. 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

* 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.
* 499 AD: Aryabhata describes a numerical algorithm for finding cube roots.
* 499 AD: Aryabhata develops an algorithm to solve the Chinese remainder theorem.
* 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, 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.

Notation and conventions

* 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.
* 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. 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, as recorded in al-Uqlidisi's later commentary on Indian mathematics.
* By 499 AD: Aryabhata's work shows the use of the modern fraction notation, known as bhinnarasi.

Physics

Astronomy

* 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.
* 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). This claim is based on his description of the planetary period about the sun (''śīghrocca''), but has been met with criticism.

Optics

* 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

* 2nd century AD: Galen studies the anatomy of pigs.

Astronomical and geospatial measurements

* 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.

500 AD – 1000 AD

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

Numbers, measurement and arithmetic

* 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.

Algebra

* 628 AD: Brahmagupta provides an explicit solution to the quadratic equation.
* 9th century AD: Jain mathematician Mahāvīra writes down a factorisation for the difference of cubes.

Number theory and discrete mathematics

* 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, $\tbinom=\tfrac$.
* c. 975 AD: Halayudha organizes the binomial coefficients into a triangle, i.e. Pascal's triangle.

Geometry and trigonometry

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

Analysis

* 10th century AD: Manjula in India discovers the derivative, deducing that the derivative of the sine function is the cosine.

Numerical mathematics and algorithms

* 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.
* 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, which is in fact a slightly more general form of the Greedy algorithm for Egyptian fractions.

Notation and conventions

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

Physics

Astronomy

* 6th century AD: Varahamira in the Gupta empire is the first to describe comets as astronomical phenomena, and as periodic in nature.

Mechanics

* 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. His radical rejection of Aristotlean orthodoxy lead him to be ignored in his time.

Optics

* 984 AD: Ibn Sahl discovers Snell's law.

Astronomical and geospatial measurements

* 10th century AD: Kashmiri astronomer Bhaṭṭotpala lists names and estimates periods of certain comets.

1000 AD – 1500 AD

Mathematics

Algebra

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

Number theory and discrete mathematics

* c. 1000 AD: al-Karaji uses mathematical induction.
* 12th century AD: Bhāskara II develops the Chakravala method, solving Pell's equation.

Geometry and trigonometry

* 15th century: Parameshvara discovers a formula for the circumradius of a quadrilateral.

Analysis

* 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 .
* 1380 AD: Madhava of Sangamagrama discusses error terms in infinite series in the context of his infinite series for .
* 1380 AD: Madhava of Sangamagrama discovers continued fractions and uses them to solve transcendental equations.Ian G. Pearce (2002)
Madhava of Sangamagramma
''MacTutor History of Mathematics archive''. University of St Andrews.
* 1380 AD: The Kerala school develops convergence tests for infinite series.
* c. 1500 AD: Nilakantha Somayaji discovers an infinite series for .

Numerical mathematics and algorithms

* 12th century AD: al-Tusi develops a numerical algorithm to solve cubic equations.
* 1380 AD: Madhava of Sangamagrama solves transcendental equations by iteration.
* 1380 AD: Madhava of Sangamagrama discovers the most precise estimate of 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

Astronomy

* 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.

Mechanics

* 12th century AD: Jewish polymath Baruch ben Malka in Iraq formulates a qualitative form of Newton's second law for constant forces.

Optics

* 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, possibly Pisa, demonstrating knowledge of human biology and optics, to offer bespoke works that compensate for an individual human disability.

Astronomical and geospatial measurements

* 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

Economics

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

Philosophy of science

* 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.
* 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

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

Numbers, measurement and arithmetic

* 1545: Gerolamo Cardano discovers complex numbers.
* 1572: Rafael Bombelli provides rules for complex arithmetic.

Algebra

* c. 1500: Scipione del Ferro solves the special cubic equation $x^3 = px + q$.
* 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

* 1564: Gerolamo Cardano is the first to produce a systematic treatment of probability.

Numerical mathematics and algorithms

* 16th century: François Viète discovers Viète's formula for .

Notation and conventions

Various pieces of modern symbolic notation were introduced in this period, notably:
* 1556: Niccolò Tartaglia introduces parenthesis.
* 1557: Robert Recorde introduces the equal sign.
* 1591: François Viète's New algebra shows the modern notational algebraic manipulation.

Physics

Astronomy

* 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

* 1543 – Vesalius: pioneering research into human anatomy

Social science

Economics

* 1517: Nicolaus Copernicus develops the quantity theory of money and states the earliest known form of Gresham's law: ("Bad money drowns out good").

17th century

* 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
* 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 bodies
* 1643 – Evangelista Torricelli invents the mercury barometer
* 1662 – Robert Boyle: Boyle's law of ideal gases
* 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 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 mixture of distinct coloured rays (the spectrum)
* 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

* 1735 – Carl Linnaeus described a new system for classifying plants in Systema Naturae
* 1745 – Ewald Jürgen Georg von Kleist first capacitor, the Leyden jar
* 1749–1789 - 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