Math & Physics

770 advances: 707 achieved, 34 at the frontier now, 29 ahead.

The frontier

• AlphaTensor matrix multiplication 2022 · frontier
  Discovered faster matrix multiplication algorithms, including 47 multiplications for 4×4 over GF(2) (Nature, Oct 2022).
• FunSearch cap set constructions 2023 · frontier
  LLM-guided search found larger cap sets in dimension 8 (Nature, Dec 2023).
• GraphCast weather forecasting 2023 · frontier
  ML 10-day forecasts outperforming HRES (Science, Nov 2023).
• Gravitational wave background 2023 · frontier
  Random gravitational-wave signal potentially detectable by gravitational wave experiments. 2023 – Experimental evidence of stochastic gravitational wave background
• Neutrino astronomy 2023 · frontier
  Observational astronomy that benefits from the direct, or indirect, detection of neutrinos. 2023 – First "image" of the Milky Way in neutrinos instead of light
• AI for science 2024 · frontier
  AI-driven materials, math, and lab discovery (GNoME 2023, AlphaProof 2024).
• AI theorem proving 2024 · frontier
  AlphaProof: IMO silver level, 2024; gold-level LLMs in 2025.
• GenCast probabilistic weather 2024 · frontier
  Beats ENS ensemble forecasts (Nature, Dec 2024).
• AlphaGeometry 2024 · frontier
  Olympiad geometry at near-gold level (Nature, Jan 2024).
• AlphaProof + AlphaGeometry 2: IMO silver 2024 · frontier
  28/42 at IMO 2024, with formal Lean proofs.
• DESI 3D map of the universe 2024 · frontier
  Reconstruction. 2024 – The Dark Energy Spectroscopic Instrument (DESI) released a precise 3D map of the cosmos, providing evidence that dark energy, the force accelerating the universe's expansion, may be weakening over time, challenging the cosmologi…
• Gold-medal level at IMO 2025 (Gemini Deep Think, OpenAI) 2025 · frontier
  35/42 in natural language; DeepMind officially certified.
• AlphaEvolve: 4×4 complex matrix mult in 48, kissing number 593 (11D) 2025 · frontier
  Improved 14 matrix multiplication targets and ~20% of 50+ open constructions.
• SQUID "bare loop" synthetic-area-spread sensor 2026 · frontier
  Adelaide team fixes zero-field detection in SQUID arrays without varied loop sizes (Phys. Rev. Applied).
• Erdős problem #728 resolved by AI + Lean (Aristotle) 2026 · frontier
  GPT-5.2 Pro proof repaired and Lean-verified by Harmonic's Aristotle.
• Erdős Sidon-set prize problem, LLM-formalized proof 2026 · frontier
  Counterexample {1,2,4,8,13}; Lean proof generated by ChatGPT (PNAS, May 2026).
• Erdős unit distance conjecture disproved by AI 2026 · frontier
  OpenAI general reasoning model constructed n^(1+δ) unit-distance configurations; externally verified (May 20, 2026).
• AlphaProof Nexus: 9 Erdős problems, 44 OEIS conjectures 2026 · frontier
  Autonomous Lean-verified proofs (arXiv 2605.22763).
• FrontierMath Erdős benchmark: 2/68 open problems solved 2026 · frontier
  A pre-release GPT-6 Astra resolved #74 and #126 under controlled conditions.
• Navier–Stokes: claimed forced finite-time blow-up (OpenAI) (claimed) 2026 · frontier
  OpenAI internal model claims finite-time blow-up for Navier-Stokes with a smooth external force (Clay alternatives C/D); Lean 4 formalization (paper/Lean mismatch disputed, arXiv 2610.08144); unforced problem remains open; not peer reviewed; Clay (Sep 11) says apparently settled, prize process not started.
• KLS conjecture proved (human-directed AI) 2026 · frontier
  31-year-old Kannan–Lovász–Simonovits conjecture resolved with agentic AI exploration; independently validated and machine-checked.
• 3D Ghys conjecture proved with Intern-S2 2026 · frontier
  Shanghai AI Lab model helped prove a 1995 dynamics conjecture; Peking University team formally verified it.
• Linear Hadwiger conjecture proved (GPT-6 Astra) (claimed) 2026 · frontier
  Norin & Steiner preprint credits GPT-6 Astra with the proof.
• Hilbert's tenth problem over Q: negative answer (OpenAI) (claimed) 2026 · frontier
  OpenAI manuscript claims no algorithm decides rational solvability of integer polynomials.
• OpenAI: local smoothing conjecture proof, ETH disproof, QFT result (claimed) 2026 · frontier
  OpenAI released AI-produced results incl. a local smoothing proof and a Lean-formalized natural-language disproof of the Exponential Time Hypothesis.
• Explicit counterexample to Tsirelson's problem (AI + Lean) (claimed) 2026 · frontier
  Binary linear system game with ~1.4M equations, found with GPT-6 Astra and Fable 5.1 (Gao, Yu & Zhi).
• Brennan's conjecture proved by OpenAI; experts confirm and shorten 2026 · frontier
  OpenAI preprints proved Brennan's conjecture and disproved Kraetzer's t^2/4 conjecture; Binder, Ivrii, Prause and Smirnov validated it and published a much shorter proof of a stronger statement.
• Nagata conjecture: OpenAI proof under expert review (claimed) 2026 · frontier
  OpenAI released a proof that d > m*sqrt(r) for r >= 10 very general points in P^2; Tevelev's arXiv account explains it and notes it is not yet peer reviewed.
• Erdős–Hajnal property proved for all paths P_t (with GPT-6 Astra) (claimed) 2026 · frontier
  Tung H. Nguyen (Oxford) credits GPT-6 Astra with key lemmas; a DeepMind internal model independently proved the P6 case with a Lean formalisation.
• Golomb–Welch conjecture (perfect Lee codes) claimed, ChatGPT-assisted (claimed) 2026 · frontier
  Chen, Huang and Luo claim Lee balls of radius >= 2 never tile Z^n for n >= 3, settling the ~1970 conjecture; ChatGPT helped explore strategies and check calculations.
• OpenAI: finite-time blow-up for the unforced 3D incompressible Euler equations (claimed) 2026 · frontier
  Companion result: smooth, compactly supported divergence-free initial velocity on R^3 whose unforced Euler solution develops a singularity in finite time; Lean 4 formalization in the same repo.
• OpenAI math batch: 721 manuscripts / 373 families / 17 fields (openai/math) (claimed) 2026 · frontier
  Current catalog 721 manuscripts / 373 result families across 17 fields (released Oct 6 as 722 / 372). Internal frontier model; drawn from an evaluation of ~4,000 problems; ~3 h ChatGPT-Pro-equivalent compute per result. Not Erdos problems or a benchmark: research manuscripts across 17 fields (number theory, algebraic geometry, PDE, TCS...) claiming results such as Milne's rationality conjecture, BSD in Selmer coranks 0/1, Hilbert 10 over Q, Kobayashi canonical ampleness, Brennan, Nagata. Oct 7: 3 withdrawn (sign error, incl. a "rational Hodge conjecture for products of K3 surfaces" paper), 14 revised; catalog now 721 manuscripts / 373 families (Oct 26). ~300 of 719 formalized in Lean per OpenAI (Zenodo audit could not reproduce that count); OpenAI says ~50% released unconfirmed. Overlap with human work exists (exact overlaps conjecture, solved concurrently by Kogler); no source gives a total of already-known results.
• OpenAI withdraws 3 math manuscripts a day after release (sign error) 2026 · frontier
  Sign error in "Algebraicity of Weil classes on split abelian eightfolds" invalidated it and two dependents (Kuga–Satake for K3; rational Hodge for products of K3 surfaces).
• Classification of Finite Simple Groups fully formalized in Lean by AI agent swarm (Axiom) 2026 · frontier
  Dozens of AI agents directed by Axiom mathematicians produced a sorry-free ~43.1M-line Lean proof of the "Enormous Theorem" (15,000 pages, 500+ papers) in 24 days, surfacing 34 noteworthy deviations from the literature and a gap in a published route to the Unbalanced Group Theorem, repaired with a new argument.
• Riemann hypothesis · future · progress: none
  Millennium Prize problem: all nontrivial zeros of the zeta function lie on the critical line. It underpins the distribution of primes.
• P vs NP · future · progress: none
  Millennium Prize problem: can every quickly checked solution also be quickly found? The answer shapes cryptography, AI and optimization.
• Navier–Stokes existence and smoothness (unforced) · future · progress: early
  Millennium Prize problem: do smooth 3D fluid flows stay smooth forever? The unforced question remains open.
• Yang–Mills existence and mass gap · future · progress: none
  Millennium Prize problem: give quantum Yang–Mills theory a rigorous footing and prove its lightest particle has positive mass.
• Hodge conjecture · future · progress: none
  Millennium Prize problem linking the topology of complex algebraic varieties to their algebraic subvarieties.
• Birch and Swinnerton-Dyer conjecture · future · progress: early
  Millennium Prize problem: the rank of an elliptic curve equals the order of vanishing of its L-function at 1.
• Nature of dark energy · future · progress: early
  What drives the accelerating expansion of the universe: a cosmological constant or something that evolves?
• Matter–antimatter asymmetry · future · progress: none
  Why the universe contains matter but almost no antimatter. The Standard Model's CP violation is far too small to explain it.
• Self-driving laboratories · future · progress: demonstrated
  Robotic labs where AI plans, runs, and analyzes experiments in closed loops, 24/7. This turns science from artisanal into industrial.
• AI-run mathematics · future · progress: demonstrated
  AI systems find, prove and formally check new theorems with people as reviewers. Expert verification, not generation, is now the bottleneck.
• "Solve Everything": AI-run science · future · progress: early
  Automated discovery across all fields: AI generates hypotheses, tests them in robotic labs, and formally verifies results. It feeds every capstone.
• Quantum simulation of quantum field theory and spacetime · future · progress: early
  Using quantum computers to simulate field theories and toy models of quantum gravity that no classical machine can, giving new physics a testbed.
• Antimatter production at gram scale · future · progress: early
  Producing and storing antimatter in quantities useful for propulsion and high-energy physics. Today's output is nanograms, so this is an efficiency gap of many orders of magnitude.
• Space gravitational-wave observatory (LISA) · future · progress: demonstrated
  LISA-class detectors listening to merging black holes across cosmic time. They are precision probes of spacetime's structure.
• Tabletop tests of quantum gravity · future · progress: early
  Entangling masses through gravity alone to show, for the first time, whether gravity is quantum.
• Engineered negative-energy (Casimir) cavities · future · progress: early
  Measuring and shaping the small negative energy densities quantum fields already allow between plates, the only known foothold toward exotic matter.
• Physics beyond the Standard Model · future · progress: early
  Identifying dark matter, dark energy, and the next layer of particle physics, and with them new possible handles on spacetime.
• A theory unifying GR and QM that permits spacetime engineering · future · progress: none
  An empirically confirmed theory of quantum gravity that also says how spacetime geometry can be engineered. Without it, FTL remains speculation.
• Exotic matter / negative energy density at macroscopic scale · future · progress: none
  Matter or field configurations violating the energy conditions over metres, not nanometres. Every known warp or wormhole solution requires it.
• Warp-metric energy reduced to physical scale · future · progress: early
  Reshaping the warp metric so it needs planetary rather than galactic mass-energy; published refinements have cut estimates by many orders of magnitude.
• Laboratory spacetime metric engineering · future · progress: none
  A measurable, deliberately engineered distortion of spacetime in the lab: the first "warp field" experiment.
• Traversable wormhole stabilization · future · progress: none
  Creating and holding open a wormhole throat large enough to pass signals, then matter (Morris–Thorne solution).
• Complete causal model of aging · future · progress: early
  A mechanistic map of how the hallmarks of aging interact, showing which interventions reset the system.
• Planetary climate engineering models · future · progress: demonstrated
  Climate and atmosphere models accurate enough to engineer a planet, tested first on Earth and Mars data.
• Atom-interferometer gravity sensors · future · progress: demonstrated
  Matter-wave interferometers measuring gravity and spacetime with record precision.
• Networked optical atomic clocks · future · progress: demonstrated
  Clocks so precise that they sense centimetre changes in gravitational potential, a new instrument for testing relativity.
• Dark matter identified · future · progress: none
  Detecting what dark matter is, the first new ingredient of the universe since the Standard Model.
• Next-generation collider (FCC / CEPC) · future · progress: early
  A ~90 km collider (CERN's FCC or China's CEPC) to study the Higgs with precision and then reach ~100 TeV, an order of magnitude beyond the LHC, using high-field superconducting magnets. Both are at the planning stage.
• Rigorous warp and wormhole mathematics · future · progress: early
  AI-assisted, formally verified analysis of which spacetime geometries are physically allowed and at what energy cost.

How we got here

• Event Horizon Telescope 2019 · achieved
  Project to create a global interferometric telescope array. 2019 – The Event Horizon Telescope Collaboration publishes the image of the black hole at the center of the M87 Galaxy.
• GW170817 2017 · achieved
  Gravitational wave signal detected by the LIGO observatory on 17 August 2017. 2017: Gravitational wave signal GW170817 is observed by the LIGO / Virgo collaboration.
• Prism 2016 · achieved
  Transparent optical element with flat, polished surfaces that refract light. 2016 - University of New South Wales engineers established a new world record for unfocused sunlight conversion to electricity with an efficiency increase to 34.5%.
• Sphere packing 2016 · achieved
  An arrangement of non-overlapping spheres within a containing space. 2016 – Maryna Viazovska solves the sphere packing problem in dimension 8. Subsequent work building on this leads to a solution for dimension 24.
• Topological order 2016 · achieved
  Order at absolute zero characterized by robust ground state degeneracy, quantized non-Abelian geometric phases, anyonic excitations, and long-range entanglement. 2016 – Topological order – topological phase transitions and order – recognized (Nobel prize, David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz)
• LIGO detects gravitational waves 2015 · achieved
  LIGO measured a length change one-thousandth of a proton's width to hear two black holes merge, a century after Einstein's prediction (Nobel Prize 2017).
• Lean theorem prover and Mathlib 2013 · achieved
  Lean (2013) and the Mathlib library (2017).
• Photonic matter 2012 · achieved
  Theoretical natural state of matter. 2012: Photonic molecules are discovered at MIT
• Higgs boson discovered (ATLAS and CMS) 2012 · achieved
  The last missing particle of the Standard Model, found 48 years after it was predicted (Nobel Prize 2013).
• Neanderthal genome project 2010 · achieved
  Effort to sequence the Neanderthal genome. 2010: The Neanderthal Genome Project presented preliminary genetic evidence that interbreeding likely occurred and that a small but significant portion of Neanderthal admixture is present in modern non-African populations.
• Quantum machine 2010 · achieved
  Human-made device whose collective motion follows the laws of quantum mechanics. 2009: The first quantum machine was created by Aaron D. O'Connell
• Angular resolution 2010 · achieved
  Ability of any image-forming device to distinguish small details of an object. Space based radiotelescope with the highest angular resolution (RadioAstron project).
• IceCube Neutrino Observatory 2010 · achieved
  A cubic kilometre of Antarctic ice instrumented as a telescope; it identified the first high-energy astrophysical neutrino sources.
• Anyon 2009 · achieved
  Type of particle that occurs only in two-dimensional systems. Six-photon graph state entanglement is used to simulate the fractional statistics of anyons living in artificial spin-lattice models.
• Quantum Darwinism 2008 · achieved
  Theory of the emergence of classicality by environment-induced selection of quantum states. The concept of Quantum Darwinism is supported.
• Polar code 2008 · achieved
  Type of error correcting code. 2008 – Erdal Arıkan introduces polar codes, the first practical construction of codes that achieves capacity for a wide array of channels
• Large Hadron Collider 2008 · achieved
  The world's most powerful collider (13.6 TeV), built in LEP's tunnel by thousands of scientists from over 100 countries.
• Spin–orbit interaction 2007 · achieved
  Relativistic interaction of a particle's spin with its motion inside a potential. Spin–orbit interaction of electrons are measured.
• E8 2007 · achieved
  248-dimensional exceptional simple Lie group. 2007 – a team of researchers throughout North America and Europe used networks of computers to map E8 (mathematics).
• Spin quantum number 2006 · achieved
  Quantum number parameterizing spin and angular momentum. Cristophe Boehme, University of Utah, demonstrates the feasibility of reading data using the nuclear spin on a silicon-phosphorus Kane quantum computer.
• Free-electron laser 2005 · achieved
  Type of laser. 2005 - The first X-ray free-electron laser in the soft X-ray regime, FLASH, became an operational user facility at DESY for X-ray diffraction experiments.
• Arithmetic progression 2004 · achieved
  Sequence of numbers with constant differences between consecutive numbers. 2004: Ben Green and Terence Tao announce their proof on arithmetic progressions in prime numbers known as the Green–Tao Theorem.
• Radiation damage 2003 · achieved
  Effects of ionizing radiation on living and nonliving things. 2003 - Raimond Ravelli et al. demonstrated X-ray radiation damage -induced phasing method for structure determination.
• Poincaré conjecture proved (Perelman) 2003 · achieved
  Grigori Perelman proved the Poincaré conjecture using Ricci flow with surgery; the only Millennium Prize problem solved so far (prize declined, 2010).
• Catalan's conjecture 2002 · achieved
  Theorem about consecutive powers. 2002 – Preda Mihăilescu proves Catalan's conjecture.
• 2dF Galaxy Redshift Survey 2001 · achieved
  Astronomical survey. 2001 – The 2dF Galaxy Redshift Survey (2dF) by an Australian/British team gave strong evidence that the matter density is near 25% of critical density.
• Quark–gluon plasma 2000 · achieved
  Phase of quantum chromodynamics characterised by an assembly of quarks and gluons at thermal and chemical equilibrium. 2000 – Quark–gluon plasma found
• Associahedron 2000 · achieved
  (𝑛−2)-dimensional convex polytope in which each vertex corresponds to a parenthesization of a word of 𝑛 letters and each edge corresponds to an application of the associativity rule. Jim Stasheff Associahedra later used in the definition of weak n -categories
• Slow light 1999 · achieved
  Propagation of an optical pulse at a very low group velocity. 1999 – Lene Vestergaard Hau: Slow light experimentally demonstrated
• Indian Ocean Dipole 1999 · achieved
  Irregular oscillation of sea-surface temperatures in the Indian Ocean. Indian Ocean Dipole is an unusual pattern in the ocean-atmosphere system of the equatorial Indian Ocean that influences the monsoon and can offset the adverse impact of El Nino.
• Supernova Cosmology Project 1998 · achieved
  Cosmology group. 1998 – The Supernova Cosmology Project and High-Z Supernova Search Team discover cosmic acceleration based on distances to Type Ia supernovae, providing the first direct evidence for a non-zero cosmological constant.
• Preadditive category 1998 · achieved
  Mathematical category whose hom sets form Abelian groups. Maxim Kontsevich Calabi–Yau categories: A linear category with a trace map for each object of the category and an associated symmetric (with respects to objects) nondegenerate pairing to the trace map.
• Spin foam 1998 · achieved
  In quantum gravity, a structure consisting of two-dimensional faces for path integrals. John Baez Spin foam models: A 2-dimensional cell complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams are functors between spin network categories.
• Collider Detector at Fermilab 1997 · achieved
  American experimental physics device (1985–2011). 1997: CDF and DØ experiments at Fermilab: Top quark.
• AdS/CFT correspondence 1997 · achieved
  Duality between theories of gravity on anti-de Sitter space and conformal field theories. 1997 – Juan Maldacena proposed the AdS/CFT correspondence
• Hubble Deep Field 1996 · achieved
  Multiple exposure image of deep space in the constellation Ursa Major. 1996 – The first Hubble Deep Field is released, providing a clear view of very distant galaxies when the universe was around one-third of its present age.
• Bailey–Borwein–Plouffe formula 1995 · achieved
  Formula for calculating π. 1995 – Simon Plouffe discovers Bailey–Borwein–Plouffe formula capable of finding the n th binary digit of π.
• Qubit 1995 · achieved
  Unit of quantum information, analogous to the classical bit. 1995 – Benjamin Schumacher coins the term qubit and proves the quantum noiseless coding theorem
• Bose–Einstein condensate 1995 · achieved
  State of matter of a dilute gas of bosons cooled to temperatures very near absolute zero. 1995 – Wolfgang Ketterle: Bose–Einstein condensate observed
• Cobordism hypothesis 1995 · achieved
  Classification of topological quantum field theories. John Baez – James Dolan Tangle hypothesis: The n -category of framed n -tangles in n + k dimensions is ( n + k )-equivalent to the free weak k -tuply monoidal n -category with duals on one object.
• Modularity theorem 1994 · achieved
  Theorem in mathematics. 1994 – Andrew Wiles proves part of the Taniyama–Shimura conjecture and thereby proves Fermat's Last Theorem.
• Homological mirror symmetry 1994 · achieved
  Equivalence between the derived Fukaya category of a symplectic manifold and the derived category of coherent sheaves on a complex manifold.
• ESRF synchrotron light source 1994 · achieved
  The first third-generation hard X-ray synchrotron, a shared European facility for protein, materials and chemistry structures.
• Quantum teleportation 1993 · achieved
  Scheme for the transfer of quantum information from one system to a remote one using entanglement. 1993 – Quantum teleportation of unknown states proposed
• Kelvin probe force microscope 1991 · achieved
  A noncontact variant of atomic force microscopy. 1991: Kelvin probe force microscope invented.
• Large Electron–Positron Collider (LEP) 1989 · achieved
  CERN's 27 km collider measured the Z and W bosons with precision and showed there are exactly three light neutrino families; its tunnel now houses the LHC.
• Great Attractor 1988 · achieved
  Gravity anomaly. 1988 – Measurements of galaxy large-scale flows provide evidence for the Great Attractor.
• CfA2 Great Wall 1988 · achieved
  One of the largest known superstructures in the Universe. 1988 – The CfA2 Great Wall is discovered in the CfA2 redshift survey.
• Topological string theory 1988 · achieved
  String theory with a topologically twisted 𝒩=(2,2) sigma-model action on the worldsheet and 6 target-space dimensions. Edward Witten Topological string theory
• Topological quantum field theory 1988 · achieved
  Quantum field theory whose correlation functions do not depend on the spacetime metric. Edward Witten Topological quantum field theory TQFT: A monoidal functor Z: nCob → Hilb satisfying some axioms
• Conformal field theory 1988 · achieved
  Quantum field theory that is invariant under conformal transformations. Graeme Segal Conformal field theory CFT: A symmetric monoidal functor Z: nCob C → Hilb satisfying some axioms
• Linear logic 1987 · achieved
  System of resource-aware logic. Jean-Yves Girard Linear logic: The internal logic of a linear category (an enriched category with its Hom-sets being linear spaces )
• Ribet's theorem 1986 · achieved
  Result concerning properties of Galois representations associated with modular forms. 1986 – Ken Ribet proves Ribet's theorem.
• EUV lithography research 1986 · achieved
  Kinoshita (NTT) demonstrated soft X-ray projection (1986); the US EUV LLC followed (1997).
• De Branges's theorem 1985 · achieved
  Theorem. 1985 – Louis de Branges de Bourcia proves the Bieberbach conjecture.
• Jones polynomial 1984 · achieved
  Mathematical invariant of a knot or link. 1984 – Vaughan Jones discovers the Jones polynomial in knot theory, which leads to other new knot polynomials as well as connections between knot theory and other fields.
• Nuclear winter 1983 · achieved
  Hypothetical climatic effect of nuclear war. 1983 – The TTAPS study in Science first introduces the possibility of a nuclear winter, and a co-author Carl Sagan publishes an article on the subject in Parade magazine.
• Faltings' theorem 1983 · achieved
  Theorem. 1983 – Gerd Faltings proves the Mordell conjecture and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat's Last Theorem.
• Classification of finite simple groups 1983 · achieved
  Theorem. 1983 – the classification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning thirty years, is completed,
• Homotopy 1983 · achieved
  Continuous deformation between two continuous maps. Alexander Grothendieck Elementary modelizers: Categories of presheaves that modelize homotopy types (thus generalizing the theory of simplicial sets ). Canonical modelizers are also used in pursuing stacks
• Fermilab Tevatron (top quark 1995) 1983 · achieved
  The first large superconducting-magnet synchrotron; it found the top quark in 1995, completing the third quark family.
• Aspect's experiment 1982 · achieved
  Quantum mechanics experiment. 1982 – Aspect experiment confirms violations of Bell's inequalities
• Fractional quantum Hall effect 1981 · achieved
  Physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantized plateaus at fractional values of e²/h. 1981 – Fractional quantum Hall effect discovered
• Hyperbolic group 1981 · achieved
  Mathematical concept. Mikhail Gromov develops the theory of hyperbolic groups, revolutionizing both infinite group theory and global differential geometry,
• Quantum Hall effect 1980 · achieved
  A quantum-mechanical version of the Hall effect. 1980 – Quantum Hall effect
• Combinatorial species 1978 · achieved
  Endofunctor on the category of finite sets and bijections, which can be used to analyse discrete structures in terms of generating functions. André Joyal Combinatorial species in enumerative combinatorics
• Wigner–Weyl transform 1978 · achieved
  Mapping between functions in the quantum phase space. Francois Bayen–Moshe Flato–Chris Fronsdal– André Lichnerowicz –Daniel Sternheimer Deformation quantization, later to be a part of categorical quantization
• Dissipative system 1977 · achieved
  Thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. 1977 – Ilya Prigogine wins the Nobel prize for his work on dissipative structures in thermodynamic systems far from equilibrium. The importation and dissipation of energy could reverse the 2nd law of thermodynamics
• Quantum Zeno effect 1977 · achieved
  Effect in quantum mechanics which disallows certain conditions in the decaying of a quantum state.
• Trellis modulation 1976 · achieved
  Modulation scheme in telecommunication. 1976 – Gottfried Ungerboeck gives the first paper on trellis modulation; a more detailed exposition in 1982 leads to a raising of analogue modem POTS speeds from 9.6 kbit/s to 33.6 kbit/s
• Four color theorem 1976 · achieved
  Statement in mathematics. Kenneth Appel and Wolfgang Haken prove the four colour theorem, the first theorem to be proved by computer.
• BTA-6 1975 · achieved
  Large telescope in Russia. 1975 — BTA-6 is the first major telescope to use an altazimuth mount, which is mechanically simpler but requires computer control for accurate pointing.
• Superposition principle 1975 · achieved
  Fundamental physics principle stating that physical solutions of linear systems are linear. R. P. Poplavskii publishes "Thermodynamical models of information processing" (in Russian) which shows the computational infeasibility of simulating quantum systems on classical computers, due to the superposition principle.
• Fractal 1975 · achieved
  Set (in mathematics) of non-integral dimension. 1975 – Benoit Mandelbrot, fractals theory,
• Automated Meteorological Data Acquisition System 1974 · achieved
  Snow depth observation data near GPS station.
• Hawking radiation 1974 · achieved
  Radiation with a blackbody spectrum radiated from an event horizon. 1974 – Stephen Hawking: Black hole radiation ( Hawking radiation ) predicted
• Hulse–Taylor binary pulsar (indirect gravitational waves) 1974 · achieved
  Propagating spacetime ripple. 1974: Russell Alan Hulse and Joseph Hooton Taylor, Jr. discover indirect evidence for gravitational wave radiation in the Hulse–Taylor binary
• Dennard scaling 1974 · achieved
  Shrinking MOSFETs keep power density constant; it ended about 2006.
• Complete metric space 1973 · achieved
  Metric space in which cauchy sequence converges to an element of the space. William Lawvere Finds that Cauchy completeness can be expressed for general enriched categories with the category of generalized metric spaces as a special case.
• Natural nuclear fission reactor 1972 · achieved
  Naturally occurring uranium deposit where self-sustaining nuclear chain reactions have occurred. 1976 – Alexander Shlyakhter uses samarium ratios from the Oklo prehistoric natural nuclear fission reactor in Gabon to show that some laws of physics have remained unchanged for over two billion years.
• Grothendieck universe 1972 · achieved
  Transitive set closed under the operations of parameterized union, power set and unordered pairs. Alexander Grothendieck Grothendieck universes for sets as part of foundations for categories
• Fujita scale 1971 · achieved
  Scale for rating tornado intensity. 1971 – Ted Fujita introduces the Fujita scale for rating tornadoes.
• Synchrotron radiation 1971 · achieved
  Electromagnetic radiation emitted when charged particles are accelerated radially. 1971 - Gerd Rosenbaum, Kenneth Holmes, and Jean Witz first discussed the potential of synchrotron X-ray diffraction for biological applications.
• Weather radar 1970 · achieved
  Radar used to locate and monitor meteorological conditions. 1970s Weather radars are becoming more standardized and organized into networks. The number of scanned angles was increased to get a three-dimensional view of the precipitation, which allowed studies of thunderstorms.
• Standard Model 1970 · achieved
  Theory of particle physics based on Yang–Mills theory with gauge group SU(3)×SU(2)×U(1) and spontaneous symmetry breaking. 1970–73 – Standard Model of elementary particles invented
• Quantum state 1970 · achieved
  State of an isolated quantum system, represented as an element of a projective Hilbert space. 1970 – Using quantum states to encode information is first proposed: Stephen Wiesner invents conjugate coding and applies it to design “money physically impossible to counterfeit” (still technologically unfeasible today).
• Room-temperature semiconductor laser 1970 · achieved
  Alferov and Hayashi/Panish heterostructure lasers.
• Saffir–Simpson hurricane wind scale 1969 · achieved
  System of classifying tropical cyclones into five categories based on measured wind speed and used to estimate potential damage. 1969 – Saffir–Simpson Hurricane Scale created, used to describe hurricane strength on a category range of 1 to 5. Popularized during Hurricane Gloria of 1985 by media.
• Horizon problem 1969 · achieved
  Cosmological fine-tuning problem. 1969 – Charles Misner formally presents the Big Bang horizon problem.
• Operad 1968 · achieved
  Prototypes of algebras that model properties such as commutativity, anticommutativity, or associativity. Michael Boardman and Rainer Vogt (1968), Peter May (1972) Operads: An abstraction of the family of composable functions of several variables together with an action of permutation of variables.
• Neutrino detector 1967 · achieved
  Physics apparatus which is designed to study neutrinos. 1967 — The first neutrino telescope opened in Africa.
• Langlands program 1967 · achieved
  Web of far-reaching and influential conjectures about connections between number theory and geometry. 1967 – Robert Langlands formulates the influential Langlands program of conjectures relating number theory and representation theory.
• Solar neutrino problem 1967 · achieved
  Major discrepancy between measurements of the numbers of neutrinos flowing through the Earth and theoretical models of the solar interior. 1967 – Solar neutrino problem found
• Spectral line 1967 · achieved
  Optical phenomenon. 1967 – John Bahcall, Wal Sargent, and Maarten Schmidt measure the fine-structure splitting of spectral lines in 3C191 and thereby show that the fine-structure constant does not vary significantly with time.
• Baryogenesis 1967 · achieved
  Hypothesized processes that could produce baryonic asymmetry, favoring matter (baryons) over antimatter (antibaryons). 1967 – Andrei Sakharov presents the requirements for baryogenesis, a baryon - antibaryon asymmetry in the universe.
• Model category 1967 · achieved
  Mathematical category with weak equivalences, fibrations and cofibrations.
• Kochen–Specker theorem 1967 · achieved
  Theorem that any hidden-variable theory describing quantum mechanics must be contextual. Simon Kochen–Ernst Specker Kochen–Specker theorem in quantum mechanics
• Chen's theorem 1966 · achieved
  Mathematics theorem in number theory, which was first stated and proved by Chen Jingrun. 1966 — Chen Jingrun proves Chen's theorem, a close approach to proving the Goldbach conjecture.
• Nonstandard analysis 1966 · achieved
  Alternative formulation of calculus using a logically rigorous notion of infinitesimal numbers. 1966 – Abraham Robinson presents non-standard analysis.
• Matrix exponential 1966 · achieved
  Matrix operation generalizing exponentiation of scalar numbers. 1966 – E. J. Putzer presents two methods for computing the exponential of a matrix in terms of a polynomial in that matrix.
• Seismic moment 1966 · achieved
  Product of the shear modulus, area of rupture along the fault, and the average slip. 1966 – Keiiti Aki discovers the seismic moment ( M 0 {\displaystyle M_{0}} )
• Jefimenko's equations 1966 · achieved
  Solution for electric field and magnetic field due to a distribution of moving electric charges and electric current in space. 1966 – Oleg Jefimenko introduces time-dependent (retarded) generalizations of Coulomb's law and the Biot–Savart law
• Crystalline cohomology 1966 · achieved
  Weil cohomology theory for schemes over a base field, whose values are modules over the ring of Witt vectors over the base field, that replaces Zariski open sets by infinitesimal thickenings of Zariski open sets with divided power structures. Alexander Grothendieck Crystals (a kind of sheaf used in crystalline cohomology )
• SLAC two-mile linac (quark evidence 1968) 1966 · achieved
  The 3 km Stanford linear accelerator; its 1968 deep inelastic scattering experiments gave the first evidence that protons contain quarks.
• Fuzzy set 1965 · achieved
  Sets whose elements have degrees of membership. 1965 – Iranian mathematician Lotfi Asker Zadeh founded fuzzy set theory as an extension of the classical notion of sets and he founded the field of fuzzy mathematics.
• Plasma cosmology 1965 · achieved
  1965 – Hannes Alfvén proposes the now-discounted concept of ambiplasma to explain baryon asymmetry and supports the idea of an infinite universe.
• Strict 2-category 1965 · achieved
  Generalization of category. Charles Ehresmann Defines both strict 2-categories and strict n -categories
• Enriched category 1965 · achieved
  Category whose hom sets have additional structure. Max Kelly – Samuel Eilenberg Enriched category theory: Categories C enriched over a category V are categories with Hom-sets Hom C not just a set or class but with the structure of objects in the category V.
• CP violation 1964 · achieved
  Violation of CP (charge-parity) symmetry in particle physics and cosmology. 1964 – CP violation discovered by James Cronin and Val Fitch.
• Bell's theorem 1964 · achieved
  In quantum physics, the theorem that locally causal hidden-variable theories cannot reproduce correlations predicted by quantum mechanics. 1964 – Bell's Theorem initiates quantitative study of quantum entanglement
• Grothendieck topology 1964 · achieved
  Structure on a category C which makes the objects of C act like the open sets of a topological space. Alexander Grothendieck Topologizes categories axiomatically by imposing a Grothendieck topology on categories which are then called sites. The purpose of sites is to define coverings on them so sheaves over sites can be defined.
• Plate tectonics 1963 · achieved
  Scientific theory that describes the large-scale motions of Earth's lithosphere. 1963: Lawrence Morley, Fred Vine, and Drummond Matthews: Paleomagnetic stripes in ocean crust as evidence of plate tectonics ( Vine–Matthews–Morley hypothesis ).
• Attractor 1963 · achieved
  Set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system.
• Forcing 1963 · achieved
  In set theory, a technique for enlarging models of axioms of set theory (e.g. ZFC) by adjoining new elements, often used for proving consistency and independence results. 1963 – Paul Cohen uses his technique of forcing to show that neither the continuum hypothesis nor the axiom of choice can be proven from the standard axioms of set theory.
• Monoidal category 1963 · achieved
  Category admitting tensor products. Saunders Mac Lane Monoidal categories, also called tensor categories: Strict 2-categories with one object made by a relabelling trick to categories with a tensor product of objects that is secretly the composition of morphisms in the 2-…
• Triangulated category 1963 · achieved
  Category with the additional structure of a translation functor and a class of distinguished triangles. Jean-Louis Verdier Triangulated categories and triangulated functors. Derived categories and derived functors are special cases of these
• Topos 1963 · achieved
  Category admitting all finite limits and power objects, that can be viewed as either as a mathematical universe (replacing the category of sets) or as a generalized space (considered as a generalization of the category of sheaves on a space). Alexander Grothendieck Grothendieck toposes, which are categories which are like universes (generalized spaces) of sets in which one can do mathematics
• Etale cohomology 1963 · achieved
  Sheaf cohomology on the étale site. Alexander Grothendieck Étale cohomology
• SQUID 1962 · achieved
  Very sensitive superconducting magnetometer used to measure extremely subtle magnetic fields. 1962 – SQUID superconducting quantum interference device invented
• Low-density parity-check code 1962 · achieved
  Linear error correcting code. 1962 – Robert G. Gallager proposes low-density parity-check codes; they are unused for 30 years due to technical limitations
• Poincaré conjecture 1961 · achieved
  Theorem in geometric topology that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. 1961 – Stephen Smale proves the Poincaré conjecture for all dimensions greater than or equal to 5.
• Compact space 1961 · achieved
  Topological space in which from every open cover of the space, a finite cover can be extracted. Richard Swan Shows there is a 1−1 correspondence between topological vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring C ( X ) of continuous functions on X ( Serre–Swan theorem )
• Lidar 1961 · achieved
  Laser ranging soon after the laser (1961–63).
• Birch and Swinnerton-Dyer conjecture 1960 · achieved
  Conjecture. Birch and Swinnerton-Dyer conjecture formulated through investigations on a computer.
• Three-body problem 1960 · achieved
  Classical mechanics problem of three massive point particles interacting via Newtonian gravity; special case of the 𝑛‐body problem for 𝑛=3. Using computational investigations of the 3-body problem, Minovitch formulates the gravity assist method.
• Scheme 1960 · achieved
  Object of algebraic geometry, commutative algebra and number theory; a ringed space locally isomorphic to the spectrum of a commutative ring. Bertrand Toën-Gabriele Vezzosi Homotopical algebraic geometry: The main idea is to extend schemes by formally replacing the rings with any kind of "homotopy-ring-like object".
• Derived category 1960 · achieved
  Homotopy category of chain complexes in an abelian category with inverses to quasi-isomorphisms adjoined. Jean-Louis Verdier Defines derived categories and redefines derived functors in terms of derived categories
• Representable functor 1960 · achieved
  Functor F: C → Set to the category of sets, naturally isomorphic to a hom-functor hom(X, –) for some object X in C. Alexander Grothendieck Representable functors
• Kan extension 1960 · achieved
  Category theory constructs. Daniel Kan Kan extensions
• Laser 1960 · achieved
  Maiman, 1960.
• Shrikhande graph 1959 · achieved
  Strongly regular graph with 16 vertices and 48 edges. Shrikhande graph – Graph invented by the Indian mathematician S.S. Shrikhande in 1959.
• Limit 1958 · achieved
  Category theory term. Daniel Kan Limits in category theory.
• Adjoint functor 1958 · achieved
  Relationship that two functors may have. Daniel Kan Daniel Kan introduces Adjoint functors. They are critical, for example, in the theory of sheaves.
• Monad 1958 · achieved
  Monoid in an endofunctor category. Roger Godement Monads in category theory (then called standard constructions and triples).
• Green–Kubo relations 1957 · achieved
  Equation relating transport coefficients to correlation functions. 1957 – Ryogo Kubo derives the first of the Green-Kubo relations for linear transport coefficients
• Fokker–Planck equation 1957 · achieved
  Partial differential equation. 1957 – Aleksandr Solomonovich Kompaneets derives his Compton scattering Fokker–Planck equation
• Passivation 1957 · achieved
  Physico-chemical processes contributing to protect a surface against oxidation. 1957 – Mohamed M. Atalla develops the process of silicon surface passivation by thermal oxidation at Bell Laboratories. The surface passivation process has since been critical to solar cell efficiency.
• Constructive proof 1957 · achieved
  Method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. 1957 – Stephen Smale provides the existence proof for crease-free sphere eversion.
• Cyclic code 1957 · achieved
  Block code where the circular shifts of each codeword gives another word that belongs to the code. 1957 – Eugene Prange first discusses cyclic codes
• BCS theory 1957 · achieved
  Theory describing superconductivity as a microscopic effect caused by a condensation of Cooper pairs into a boson-like state. 1957 – BCS theory explaining superconductivity
• Many-worlds interpretation 1957 · achieved
  Interpretation of quantum mechanics which denies the collapse of the wavefunction. 1957 – Many-worlds, also called the relative state formulation or the Everett interpretation.
• Grothendieck–Hirzebruch–Riemann–Roch theorem 1957 · achieved
  Result in algebraic geometry. Alexander Grothendieck Grothendieck–Hirzebruch–Riemann–Roch theorem for smooth schemes; the proof introduces K-theory.
• Abelian category 1957 · achieved
  Preadditive category with a zero object and all binary biproducts, kernels and cokernels in which all monomorphisms and epimorphisms are normal. Alexander Grothendieck Abelian categories in homological algebra that combine exactness and linearity.
• Exotic sphere 1956 · achieved
  Smooth manifold that is homeomorphic but not diffeomorphic to a sphere. 1956 – John Milnor discovers the existence of an Exotic sphere in seven dimensions, inaugurating the field of differential topology.
• Chomsky hierarchy 1956 · achieved
  Containment hierarchy of classes of formal grammars. 1956 – Noam Chomsky describes a hierarchy of formal languages.
• Parity violation (Wu experiment) 1956 · achieved
  Flip in the sign of one spatial coordinate, in classical and quantum physics. 1956–57 – Parity violation proved by Chien-Shiung Wu
• Convolutional code 1955 · achieved
  Type of error-correcting code using convolution. 1955 – Peter Elias introduces convolutional codes
• Uniform polyhedron 1955 · achieved
  Polyhedron which has regular polygons as faces and is vertex-transitive. 1955 – H. S. M. Coxeter et al. publish the complete list of uniform polyhedron,
• Raychaudhuri equation 1954 · achieved
  Result in general relativity. Raychaudhuri equation – Discovered by the Bengali physicist Amal Kumar Raychaudhuri in 1954. This was a key ingredient of the Penrose-Hawking singularity theorems of general relativity.
• CERN founded 1954 · achieved
  Twelve European states pooled resources into one laboratory for particle physics; CERN later invented the World Wide Web (1989).
• Maser 1953 · achieved
  Device that produces coherent electromagnetic waves through amplification by stimulated emission. 1948 – Invention of the maser and laser by Charles Townes
• Karnaugh map 1952 · achieved
  Method to simplify boolean algebra expressions; refinement of Edward Veitch's 1952 Veitch diagram. Karnaugh mapping technique (digital logic) Karnaugh
• Yoneda lemma 1952 · achieved
  In category theory, a theorem that embeds a locally small category into a category of functors. Nobuo Yoneda Yoneda publishes his famous lemma. Yoneda's Lemma allows one to consider objects in a (small) category as a presheaves. Yoneda lemma plays a critical role in the study of representable functors in algebraic geometry.
• Multivariate statistics 1952 · achieved
  Simultaneous observation and analysis of more than one outcome variable. Multivariate Statistical Theory, is one of the base foundational concept of statistics was developed by C. R. Rao in 1952 when he was in Indian Statistical Institute, Kolkata.
• Bubble chamber (Glaser) 1952 · achieved
  Donald Glaser's bubble chamber made particle tracks visible and photographable, revealing the zoo of hadrons behind the quark model.
• Field ion microscope 1951 · achieved
  Microscope used for imaging atom arrangements. 1951: Erwin Wilhelm Müller invents the field ion microscope and is the first to see atoms.
• Subduction 1951 · achieved
  Geological process at tectonic plate boundaries. 1951 – Alfred Rittmann links subduction, volcanism and the Wadati–Benioff zone
• Noctilucent cloud 1950~ · achieved
  Highly-elevated cloud-like ice crystals in the upper atmosphere of Earth visible during astronomical twilight. PMC: Noctilucent veils Noctilucent billows or whirls Noctilucent bands
• Simplicial set 1950 · achieved
  Construction in categorical homotopy theory; contravariant functor from the simplex category to the category of sets. Samuel Eilenberg –Joe Zilber Simplicial sets as a purely algebraic model of well behaved topological spaces. A simplicial set can also be seen as a presheaf on the simplex category.
• Hamming code 1950 · achieved
  Error-correcting code. 1947 – Richard W. Hamming invents Hamming codes for error detection and correction (to protect patent rights, the result is not published until 1950)
• Ramanujan theta function 1950~ · achieved
  Mathematic term. Ramanujan theta function, Ramanujan prime, Ramanujan summation, Ramanujan graph and Ramanujan's sum – Discovered by the Indian mathematician Srinivasa Ramanujan in the early 20th century.
• Kraft's inequality 1949 · achieved
  Necessary and sufficient condition for the existence of a prefix code for a given set of codeword lengths. 1949 – Leon G. Kraft discovers Kraft's inequality, which shows the limits of prefix codes
• Shannon–Fano coding 1949 · achieved
  Technique for constructing a prefix code. 1949 – Robert M. Fano publishes Transmission of Information. M.I.T. Press, Cambridge, Massachusetts – Shannon–Fano coding
• Nyquist–Shannon sampling theorem 1949 · achieved
  Theorem in signal processing describing discrete samples of a continuous signal. 1949 – Claude E. Shannon publishes Communication in the Presence of Noise – Nyquist–Shannon sampling theorem and Shannon–Hartley law
• Radiocarbon dating 1949 · achieved
  Technique based on carbon-14 decay to determine the age of organic materials. 1945: Willard Libby began his work on radiocarbon dating. He published his paper in 1946, a second paper in Science in 1947.
• Weil conjectures 1949 · achieved
  Theorem. André Weil Formulates the Weil conjectures on remarkable relations between the cohomological structure of algebraic varieties over C and the diophantine structure of algebraic varieties over finite fields.
• Atomic clock 1948 · achieved
  Extremely accurate reference clock used as a standard for timekeeping. 1948: The first atomic clock is developed at the National Bureau of Standards.
• Feynman diagram 1948 · achieved
  Pictorial representations of the behavior of subatomic particles. 1948 – Feynman diagrams
• Path integral 1948 · achieved
  Formal sum or integral over all histories of a quantum system. 1941 – Feynman path integral
• Cosmic microwave background 1948 · achieved
  Electromagnetic radiation as a remnant from an early stage of the universe in Big Bang cosmology. 1948 – George Gamow predicts the existence of the cosmic microwave background radiation by considering the behavior of primordial radiation in an expanding universe.
• Steady-state model 1948 · achieved
  Alternative to the Big Bang model of the evolution of the universe. 1948 – Hermann Bondi, Thomas Gold, and Fred Hoyle propose steady state cosmologies based on the perfect cosmological principle.
• Alpher–Bethe–Gamow theory 1948 · achieved
  Scientific theory. 1948 – Ralph Alpher, Hans Bethe ( "in absentia" ), and George Gamow examine element synthesis in a rapidly expanding and cooling universe, and suggest that the elements were produced by rapid neutron capture.
• Cox's theorem 1946 · achieved
  Theorem that derivates of the laws of probability from a certain set of postulates. 1946 – Cox's theorem derives the axioms of probability from simple logical assumptions,
• Astronomical interferometer 1946 · achieved
  Array of separate telescopes, mirror segments, or radio telescope antennas that work together as a single telescope. 1946 – Martin Ryle and Vonberg build the first two-element astronomical radio interferometer (see history of astronomical interferometry )
• Spectral sequence 1946 · achieved
  In homological algebra, a means of computing homology groups by taking successive approximations. Jean Leray Invents spectral sequences as a method for iteratively approximating cohomology groups by previous approximate cohomology groups. In the limiting case it gives the sought cohomology groups.
• BBGKY hierarchy 1945 · achieved
  Set of equations describing the dynamics of a system of many interacting particles. 1945–1946 – Nikolay Bogoliubov develops a general method for a microscopic derivation of kinetic equations for classical statistical systems using BBGKY hierarchy
• Morphism 1945 · achieved
  Map (arrow) between two objects of a category. for all X, Y in Ob( C ) the Hom-sets Hom C ( X, Y ) are finite-dimensional chain complexes of Z -graded modules
• Sheaf cohomology 1945 · achieved
  Right derived functors of the global sections functor Γ: AbSh → Ab. Jean Leray Defines Sheaf cohomology using his new concept of sheaf.
• Sheaf 1945 · achieved
  Collection of objects associated to subsets of a space in a manner admitting gluing and restriction. Jean Leray Starts sheaf theory: At this time a sheaf was a map that assigned a module or a ring to a closed subspace of a topological space. The first example was the sheaf assigning to a closed subspace its p-th cohomology group.
• Cohomology ring 1945 · achieved
  Concept in algebraic topology. Beno Eckmann Defines the cohomology ring building on Heinz Hopf 's work.
• Category 1945 · achieved
  Algebraic structure of objects and morphisms between objects, which can be associatively composed if the (co)domains agree. Categories of abstract algebraic structures including representation theory and universal algebra;
• Maksutov telescope 1944 · achieved
  Catadioptric telescope design. 1944 — Dmitri Dmitrievich Maksutov invents the Maksutov telescope.
• Itō calculus 1944 · achieved
  Calculus of stochastic differential equations. 1957 – Kiyosi Itô develops Itô calculus.
• Ising model 1944 · achieved
  Mathematical model of ferromagnetism in statistical mechanics. 1944 – Theory of magnetism in 2D: Ising model
• Gelfand–Naimark theorem 1943 · achieved
  Mathematics theorem in functional analysis.
• Gauss–Markov process 1942 · achieved
  Stochastic processes. 1942 – Joseph L. Doob states his theorem on Gauss–Markov processes
• Nuclear reaction 1942 · achieved
  Process in which two nuclei collide to produce one or more nuclides. 1942 – December 2 – Enrico Fermi and his team achieve the first controlled nuclear reaction at Chicago Pile-1 constructed at the University of Chicago in a squash court underneath Stagg Field.
• Nickel–Strunz classification 1941 · achieved
  Scheme for categorizing minerals. 1941 – Nickel-Strunz classification, Karl H. Strunz, Mineralogische Tabellen
• Exact sequence 1941 · achieved
  Sequence of homomorphisms such that each kernel equals the preceding image. Witold Hurewicz First fundamental theorem of homological algebra: Given a short exact sequence of spaces there exist a connecting homomorphism such that the long sequence of cohomology groups of the spaces is exact.
• Nuclear fission product 1940 · achieved
  Product of nuclear fission. 1940 – July – The paper explaining that Dr. Yoshio Nishina and Kenjiro Kimura discovered symmetric fission on the previously described test appeared in Nature.
• Spontaneous fission 1940 · achieved
  Form of radioactive decay found in very heavy chemical elements. On July 1, Georgy Flyorov and Konstantin Petrzhak publish the discovery of spontaneous fission, in uranium atoms insulated from cosmic rays 60 meters underground in the Dinamo station of the Moscow Metro.
• Isotope of neptunium 1940 · achieved
  On May 27, Edwin McMillan and Philip Abelson publish the discovery of neptunium at the Berkeley Radiation Laboratory. They use the 60-inch cyclotron produce a small sample of neptunium-239 via neutron bombardment of uranium-238.
• Continuum hypothesis 1940 · achieved
  Hypothesis that no set has a cardinality between that of the integers and that of the real numbers. 1940 – Kurt Gödel shows that neither the continuum hypothesis nor the axiom of choice can be disproven from the standard axioms of set theory.
• Isotope of plutonium 1940 · achieved
  1940: Pu-239 isotope (isotope of plutonium ) a form of matter existing with the capacity for use as a destructive element (because the isotope has an exponentially increasing spontaneous fissile decay ) within nuclear devices — Glenn Sea…
• Coherent sheaf 1940 · achieved
  Finite-type sheaf F of modules over a ringed space such that the kernel of a surjective morphism from a finite direct sum of the structure sheaf onto it is also of finite type. Jean-Pierre Serre Shows there is a 1−1 correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring ( Serre–Swan theorem ).
• Hopf algebra 1940 · achieved
  Bialgebra that admits an antipode. Heinz Hopf Hopf algebras.
• Injective module 1940 · achieved
  Module that makes certain exact sequences split. Reinhold Baer Injective modules.
• p–n junction (Ohl) 1940 · achieved
  Russell Ohl at Bell Labs; the first silicon solar cell effect.
• Rossby wave 1939 · achieved
  Type of inertial wave in the atmospheres and oceans of planets that largely owe their properties to rotation of the planet. 1939 – Rossby waves were first identified in the atmosphere by Carl-Gustaf Arvid Rossby who explained their motion. Rossby waves are a subset of inertial waves.
• Block design 1939 · achieved
  Set together with a family of subsets in math. Partial Balanced Incomplete Block Designs (BIBD) was developed by Raj Chandra Bose and Kesavan Raghavan Nair in 1939.
• Vlasov equation 1938 · achieved
  Partial differential equation describing the time evolution of plasma. 1938 – Anatoly Vlasov proposes the Vlasov equation for a correct dynamical description of ensembles of particles with collective long range interaction
• LU decomposition 1938 · achieved
  Matrix decomposition. 1938 – Tadeusz Banachiewicz introduces LU decomposition.
• Nuclear fission 1938 · achieved
  Nuclear reaction splitting the nucleus of an atom into multiple parts. 1904 – Frederick Soddy first proposes a bomb powered by nuclear fission to the Royal Engineers .
• Superfluidity 1938 · achieved
  Non-classical state of the matter. 1938 – Pyotr Kapitsa: Superfluidity discovered
• Transmission electron microscopy 1938 · achieved
  Analytic technique. 1938: James Hillier builds another TEM.
• Dirac large numbers hypothesis 1938 · achieved
  The hypothesis that the numerical similarity between the proton–electron gravitational force–electrical force ratio and the classical electron radius–age of the universe ratio is not a coincidence. 1938 – Paul Dirac suggests the large numbers hypothesis, that the gravitational constant may be small because it is decreasing slowly with time.
• Confidence interval 1937 · achieved
  Range of estimates for an unknown parameter. 1937 – Jerzy Neyman introduces the concept of confidence interval in statistical testing,
• Vinogradov's theorem 1937 · achieved
  Theorem stating that a sufficiently large odd integer is the sum of three primes. 1937 — I. M. Vinogradov proves Vinogradov's theorem that every sufficiently large odd integer is the sum of three primes, a close approach to proving Goldbach's weak conjecture.
• New Foundations 1937 · achieved
  Axiomatic set theory permitting set comprehension by stratified formulae, hence with a universal set, but in which the singleton map 𝑥↦{𝑥} fails to exist. Andre Boileau– André Joyal –John Zangwill LST, local set theory: Local set theory is a typed set theory whose underlying logic is higher-order intuitionistic logic.
• Stone's representation theorem for Boolean algebras 1936 · achieved
  Theorem. Marshall Stone Stone representation theorem for Boolean algebras initiates various Stone dualities.
• Deformation 1936 · achieved
  Transformation of a body from a reference configuration to a current configuration. A strain gauge is a device used to measure the strain of an object. As the object is deformed, the foil is deformed, causing its electrical resistance to change. The strain gauge was invented in 1936 by Edward E.
• Mahalanobis distance 1936 · achieved
  Distance measure. Mahalanobis distance – Introduced in 1936 by the Indian (Bengali) statistician Prasanta Chandra Mahalanobis (29 June 1893 – 28 June 1972), this distance measure, based upon the correlation between variables, is used to identify and analy…
• Zeroth law of thermodynamics 1935 · achieved
  Principle stating if two systems are in thermal equilibrium with another, they are with each other. 1935 – Ralph H. Fowler invents the title ' the zeroth law of thermodynamics ' to summarise postulates made by earlier physicists that thermal equilibrium between systems is a transitive relation
• Scanning electron microscope 1935 · achieved
  Type of electron microscope. Scanning electron microscope Knoll
• Homotopy group 1935 · achieved
  Algebraic construct classifying topological spaces. Witold Hurewicz Higher homotopy groups of a topological space.
• Magnitude 1935 · achieved
  Strength of seismic waves of an earthquake. The Richter magnitude scale, or local magnitude M L scale, assigns a number to quantify the amount of seismic energy released by an earthquake.
• Vacuum energy 1934 · achieved
  Background energy existing in space. 1934 – Georges Lemaître interprets the cosmological constant as due to a vacuum energy with an unusual perfect fluid equation of state.
• Ext functor 1934 · achieved
  Derived functors of the Hom functor. Reinhold Baer Ext groups, Ext functor (for abelian groups and with different notation).
• Probability axioms 1933 · achieved
  Axioms that are relevant to the probability theory. 1933 – Andrey Nikolaevich Kolmogorov publishes his book Basic notions of the calculus of probability ( Grundbegriffe der Wahrscheinlichkeitsrechnung ), which contains an axiomatization of probability based on measure theory.
• Cosmological principle 1933 · achieved
  Notion that the spatial distribution of matter in the universe is homogeneous and isotropic at large scales. 1933 – Edward Milne names and formalizes the cosmological principle.
• Singular homology 1933 · achieved
  Homology of the chain complex of singular chains (integer linear combinations of maps from simplexes into a given topological space). Solomon Lefschetz Singular homology of topological spaces.
• Discovery of the neutron 1932 · achieved
  Scientific background leading to the discovery of the neutron and other subatomic particles. On January 1, Harold Urey, Ferdinand Brickwedde, and George M Murphy publish the discovery of deuterium.
• Antimatter 1932 · achieved
  Substance made of the distinct antiparticles of common matter particles. 1932 – Carl David Anderson: Antimatter ( positrons ) discovered
• Onsager reciprocal relations 1931 · achieved
  Equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium. 1931 – Lars Onsager publishes his groundbreaking paper deriving the Onsager reciprocal relations
• Gödel's incompleteness theorems 1931 · achieved
  Theorem that a wide class of logical systems cannot be both consistent and complete. 1931 – Kurt Gödel proves his incompleteness theorem, which shows that every axiomatic system for mathematics is either incomplete or inconsistent.
• Electron microscope 1931 · achieved
  A magnifying system capable of showing the interior or surface of a microscopic object by means of directed electron beams. 1931: The electron microscope is invented by Ernst Ruska.
• Cohomology 1931 · achieved
  Sequences of abelian groups attached to a topological space. 1931 – Georges de Rham develops theorems in cohomology and characteristic classes,
• Lawrence's cyclotron (Berkeley) 1931 · achieved
  Ernest Lawrence's working cyclotron at Berkeley (1931) put particle acceleration on a lab bench; his Radiation Laboratory became the model for team-based big science.
• Semiconductor band model (Wilson) 1931 · achieved
  A. H. Wilson explained insulators, metals and semiconductors.
• Radiosonde 1930 · achieved
  Meteorological instrumentation. 1930, January 30 – Pavel Molchanov invents and launches the first radiosonde.
• Phase contrast microscopy 1930 · achieved
  Optical microscopy technique. 1930: The Phase-contrast microscopy is invented by Frits Zernike.
• Three utilities problem 1930 · achieved
  Mathematical problem. 1930 – Casimir Kuratowski shows that the three-cottage problem has no solution,
• Module 1930~ · achieved
  Generalization of vector space, with scalars in a ring instead of a field. Emmy Noether Module theory is developed by Noether and her students, and algebraic topology starts to be properly founded in abstract algebra rather than by ad hoc arguments.
• Lambda calculus 1930~ · achieved
  Formal system in mathematical logic. A programming language is a machine-readable artificial language. Programming languages can be used to create programs that specify the behavior of a machine, to express algorithms precisely, or as a mode of human communication.
• Discovery of Pluto 1930 · achieved
  Dwarf planet in the Solar System. 19 January 2006 First Pluto /Charon flyby (on 14 July 2015); continued on to 486958 Arrokoth flyby (on 1 January 2019)
• Arctic exploration 1930 · achieved
  Physical exploration of the Arctic region of the Earth. Soviet and Russian drifting ice stations are important contributors to exploration of the Arctic.
• Law of thermodynamics 1929 · achieved
  Observational basis of the study of thermodynamics. 1929: Lars Onsager 's reciprocal relations, a potential fourth law of thermodynamics
• Expansion of the universe 1929 · achieved
  Increase in distance between parts of the universe over time. 1929 – Edwin Hubble: Expansion of the universe confirmed
• Hubble–Lemaître law 1929 · achieved
  Observation in physical cosmology. 1929 – Edwin Hubble demonstrates the linear redshift-distance relationship and thus shows the expansion of the universe.
• Chain complex 1929 · achieved
  In homological algebra, a structure consisting of a sequence of modules and a sequence of homomorphisms between consecutive modules such that the image of each homomorphism is included in the kernel of the next. Walther Mayer Chain complexes.
• Associative algebra 1929 · achieved
  Algebra over a ring such that multiplication is associative.
• Particle accelerator 1929 · achieved
  Device to propel charged particles to high speeds. A cyclotron is a type of particle accelerator that accelerate charged particles using a high-frequency, alternating voltage. The cyclotron was invented in 1929 by Ernest O. Lawrence at the University of California at Berkeley.
• Klein–Nishina formula 1929 · achieved
  Electron-photon scattering cross section. Klein–Nishina formula
• Fluctuation-dissipation theorem 1928 · achieved
  Theorem. 1928 – Harry Nyquist derives the fluctuation-dissipation theorem, a relationship to explain Johnson noise in a resistor
• Johnson–Nyquist noise 1928 · achieved
  Thermal noise present in all electrical circuits. 1928 – John B. Johnson discovers Johnson noise in a resistor
• Near-field scanning optical microscopy 1928 · achieved
  Microscopy technique. 1928: Edward Hutchinson Synge publishes theory underlying the near-field scanning optical microscope
• Igneous rock 1928 · achieved
  One of the three main rock types; formed through the cooling and solidification of magma or lava. 1928 – N. L. Bowen publishes The Evolution of the Igneous Rocks, revolutionizing experimental igneous petrology
• Antiparticle 1928 · achieved
  Concept that pairs of particles exist that have equal mass, but opposite sign for properties such as charge, lepton and baryon number. 1928 – Paul Dirac proposes the antiparticle
• Heyting algebra 1928 · achieved
  Bounded lattice that models intuitionistic propositional logic. Arend Heyting Brouwer's intuitionistic logic made into formal mathematics, as logic in which the Heyting algebra replaces the Boolean algebra.
• Band theory of solids (Bloch) 1928 · achieved
  Electron waves in a periodic lattice.
• Density matrix 1927 · achieved
  Matrix describing a quantum system in a pure or mixed state, a statistical mixture of quantum states. 1927 – John von Neumann introduces the density matrix representation, establishing quantum statistical mechanics
• Von Neumann entropy 1927 · achieved
  Type of entropy in quantum statistical mechanics. 1927 – John von Neumann defines the von Neumann entropy, extending the Gibbs entropy to quantum mechanics
• Born rule 1927 · achieved
  In quantum mechanics, the rule that the probability of observing an eigenvalue is the square of the modulus of the inner product between the quantum state and the eigenstate corresponding to the eigenvalue. 1927 – Max Born: Born rule
• Uncertainty principle 1927 · achieved
  Fundamental principle in quantum physics. 1927 – Werner Heisenberg: Uncertainty principle
• Dirac equation 1927 · achieved
  Relativistic quantum mechanical wave equation. 1927 – Paul Dirac: Dirac equation
• Čech cohomology 1927 · achieved
  Cohomology theory based on the intersection properties of open covers of a topological space. Eduard Čech Čech cohomology, homotopy groups of a topological space.
• Hartree–Fock method 1927 · achieved
  Method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. 1927 – Hartree–Fock method developed for simulating a quantum many-body system in a stationary state.
• Fermi–Dirac statistics 1926 · achieved
  Behaviour of many identical fermions, particles with half-integer spin. 1926 – Fermi–Dirac Statistics
• Schrödinger equation 1926 · achieved
  Partial differential equation describing how the quantum state of a non-relativistic physical system changes with time. 1925: Erwin Schrödinger: Schrödinger equation ( quantum mechanics )
• Atomic radius 1926 · achieved
  Any measure of the size of atoms. 1926 - Victor Goldschmidt distinguished between atomic and ionic radii and postulated some rules for atom substitution in crystal structures.
• Groupoid 1926 · achieved
  Category where every morphism is invertible; generalization of a group. Heinrich Brandt Defines the notion of groupoid.
• Primary decomposition 1926 · achieved
  In algebra, expression of an ideal as the intersection of ideals of a specific type. 1926 – Primary decomposition algorithm presented by Grete Hermann
• Metallicity 1925 · achieved
  Fraction of mass of a star or other kind of astronomical object that is not in hydrogen or helium. 1925: Cecilia Payne-Gaposchkin: Discovery of the composition of the Sun and that hydrogen is the most abundant element in the Universe
• Stellar structure 1925 · achieved
  Internal structure of a star. 1925 – Stellar structure understood
• Matrix mechanics 1925 · achieved
  Formulation of quantum mechanics. 1925 – Werner Heisenberg: Matrix mechanics
• Quantum mechanics 1925 · achieved
  Planck 1900 to Heisenberg and Schrödinger, 1925–26.
• Langmuir probe 1924 · achieved
  Device used to measure plasma. 1924 — Irving Langmuir — Langmuir probe (to measure plasma parameters)
• Pauli exclusion principle 1924 · achieved
  Quantum mechanical principle that two identical fermions cannot occupy the same quantum state simultaneously. 1924: Wolfgang Pauli: quantum Pauli exclusion principle
• Matter wave 1924 · achieved
  Aspect of wave–particle duality. 1924 – Louis de Broglie: De Broglie wave
• Bose-Einstein statistics 1924 · achieved
  Statistical description for the behaviour of bosons. 1924 – Bose–Einstein statistics
• Electrolyte 1923 · achieved
  Medium which conducts ions but not electrons. 1923 – Debye and Erich Hückel publish a statistical treatment of the dissociation of electrolytes
• Stern–Gerlach experiment 1923 · achieved
  Physical experiment that demonstrated the quantization of angular momentum. 1923 – Stern–Gerlach experiment
• Künneth theorem 1923 · achieved
  Theorem. Hermann Künneth Künneth formula for homology of product of spaces.
• Friedmann–Lemaître–Robertson–Walker metric 1922 · achieved
  Metric properties of the spacetime based on Albert Einstein's metric tensor and field solution for the spacetime. 1922–37 – Friedmann–Lemaître–Robertson–Walker metric cosmological model
• Commutative ring 1921 · achieved
  Algebraic structure. 1921 – Emmy Noether introduces the first general definition of a commutative ring.
• Ewald's sphere 1921 · achieved
  Energy conservation during diffraction by atoms. 1921 - Paul Peter Ewald introduced a spherical construction for explaining the occurrence of diffraction spots, which is now called Ewald's sphere.
• Central limit theorem 1920 · achieved
  Key theorem in probability theory. 1920 – The central limit theorem in its modern form was formally stated.
• Milankovitch cycles 1920 · achieved
  Periodic changes in the orbital movements of the Earth, which cause cyclic variations in terrestrial climate. 1920 – Milutin Milanković proposes that long term climatic cycles may be due to changes in the eccentricity of the Earth's orbit and changes in the Earth's obliquity.
• Stellar nucleosynthesis 1920 · achieved
  Process by which the natural abundances of the chemical elements within stars change due to nuclear fusion reactions in the cores and their overlying mantles. 1920: Arthur Eddington: Stellar nucleosynthesis
• Great Debate 1920 · achieved
  Astronomy debate held on 26 April 1920 at the Smithsonian Museum of Natural History between Harlow Shapley and Heber Curtis about whether spiral nebulae were different galaxies or part of the Milky Way. 1920 – The Shapley-Curtis Debate, on the distances to spiral nebulae, takes place at the Smithsonian.
• Saha ionization equation 1920 · achieved
  Relation between ionization state of a gas to the temperature and pressure. Saha ionisation equation – The Saha equation, derived by the Bengali scientist Meghnad Saha (6 October 1893 – 16 February 1956) in 1920, conceptualises ionisations in context of stellar atmospheres.
• Brun's theorem 1919 · achieved
  Theorem that the sum of the reciprocals of the twin primes converges. 1919 – Viggo Brun defines Brun's constant B 2 for twin primes.
• Kaluza–Klein theory 1919 · achieved
  Field theory on a spacetime with small, compact extra dimensions beyond the large observed dimensions. 1919–1926 – Kaluza–Klein theory proposing unification of gravity and electromagnetism
• Mount Wilson Observatory 1919 · achieved
  Astronomical observatory in Los Angeles County, California. 1919 – Albert A. Michelson makes the first interferometric measurements of stellar diameters at Mount Wilson Observatory (see history of astronomical interferometry )
• Nuclear transmutation 1918 · achieved
  Conversion of one chemical element or an isotope into another chemical element. Atomic transmutation Rutherford
• Julia set 1918 · achieved
  Fractal set named after Gaston Maurice Julia. Mandelbrot, from studies of the Fatou, Julia and Mandelbrot sets, coined and popularized the term 'fractal' to describe these structures' self-similarity.
• Ramanujan–Petersson conjecture 1916 · achieved
  Conjecture. 1916 – Srinivasa Ramanujan introduces Ramanujan conjecture. This conjecture is later generalized by Hans Petersson.
• Schwarzschild metric 1916 · achieved
  Spherically symmetric static vacuum solution of Einstein’s field equations with zero cosmological constant. 1916 – Schwarzschild metric modeling gravity outside a large sphere
• Chart recorder 1915 · achieved
  Monitor and document variables like temperature, pressure, or humidity, providing a continuous visual record to ensure conditions meet specifications.. 1915 — J.C. Stevens — Chart recorder (first chart recorder for environmental monitoring)
• Schwarzschild radius 1915 · achieved
  Distance from a massive body where the escape velocity equals the speed of light. 1915: Karl Schwarzschild: discovery of the Schwarzschild radius leading to the identification of black holes
• Metamorphic facies 1915 · achieved
  Set of metamorphic mineral assemblages that are formed under similar pressures and temperatures. 1915 – Pentti Eskola develops the concept of metamorphic facies
• Noether's theorem 1915 · achieved
  Physical law that differentiable symmetries correspond to conservation laws. 1915 – Emmy Noether: Noether's theorem relates symmetries to conservation laws.
• General relativity 1915 · achieved
  Standard and classical physics theory of gravity and space. 1915 – Albert Einstein publishes the General Theory of Relativity , showing that an energy density warps spacetime .
• Atomic number 1913 · achieved
  Number of protons found in the nucleus of an atom. 1913: Henry Moseley: defined atomic number
• Bohr model 1913 · achieved
  Atomic model introduced by Niels Bohr in 1913. 1913 – Niels Bohr publishes the Bohr model of the atom, which explains the spectral lines, and definitively established the quantum mechanics behaviour of the matter.
• Debye model 1912 · achieved
  Mathematical model of a solid that treates the vibrations of the atomic lattice as phonons in a box. 1912 – Peter Debye gives an improved heat capacity estimate by allowing low-frequency phonons
• Isotope 1912 · achieved
  Nuclides having the same atomic number but different mass numbers. 1912 – J.J. Thomson discovers isotopes through experiments with neon.
• Brouwer fixed-point theorem 1912 · achieved
  Every continuous function on a compact set has a fixed point. 1912 – Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem,
• X-ray diffraction 1912 · achieved
  Physical phenomenon. 1912: Max von Laue: x-ray diffraction
• Bragg's law 1912 · achieved
  Gives the angles for coherent and incoherent scattering from a crystal lattice. 1912 - Bragg diffraction, expressed through Bragg's law, is first presented by Lawrence Bragg on 11 November 1912 to the Cambridge Philosophical Society.
• Cloud chamber 1911 · achieved
  Scientific device. 1911: The cloud chamber, the first particle detector, is invented by Charles Thomson Rees Wilson.
• Superconductivity 1911 · achieved
  Electrical conductivity with exactly zero resistance. Dutch physicist Heike Kamerlingh Onnes discovered Superconductivity.
• Highly composite number 1910 · achieved
  Positive integer with more divisors than any smaller positive integer has. 1910s — Srinivasa Aaiyangar Ramanujan develops over 3000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions.
• Axiomatic system 1909 · achieved
  Set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. 1909 – Constantin Carathéodory develops an axiomatic system of thermodynamics
• Oil drop experiment 1909 · achieved
  Experiment conducted by R. A. Millikan and H. Fletcher to measure elementary electric charge. 1909: Robert Andrews Millikan: conducts the oil drop experiment and determines the charge on an electron
• Waring's problem 1909 · achieved
  Problem in number theory. 1909 — David Hilbert proves Waring's problem.
• Student's t-distribution 1908 · achieved
  Probability distribution. 1908 – Student's t-distribution for the mean of small samples published in English (following earlier derivations in German).
• Minkowski spacetime 1908 · achieved
  Mathematical space setting which eases explanation of special relativity. 1908 – Hermann Minkowski: Minkowski space
• Heat capacity 1907 · achieved
  Thermal property describing the energy required to change a material's temperature. 1907 – Einstein uses quantum theory to estimate the heat capacity of an Einstein solid
• Photodetector 1907 · achieved
  Sensors of light or other electromagnetic energy. 1907: Rosenthal puts in his image telegraph for the first time a photocell.
• Pirani gauge 1906 · achieved
  Thermal conductivity gauge. 1906 — Marcello Pirani – Pirani gauge (to measure pressures in vacuum systems)
• Third law of thermodynamics 1906 · achieved
  Law of physics stating that the entropy of a perfect crystal at absolute zero is exactly equal to zero. 1906: Walther Nernst: Third law of thermodynamics
• Annus Mirabilis papers 1905 · achieved
  Papers of Albert Einstein published in the Annalen der Physik scientific journal in 1905. 1905 – Albert Einstein, in the first of his miracle year papers, argues that the reality of quanta would explain the photoelectric effect
• Mass–energy equivalence 1905 · achieved
  Physical law relating mass to energy. 1905 – Albert Einstein proposes mass–energy equivalence, deriving from his theory of special relativity, which he had developed earlier that year.
• Infrared spectroscopy 1905 · achieved
  Involves the interaction of infrared radiation with matter. Moon Partial failure Collect its surface thermography.
• Relativity 1905 · achieved
  Special relativity 1905; general relativity 1915.
• Plum pudding model 1904 · achieved
  Scientific model of the atom first proposed by J. J. Thomson in 1904. 1898: J.J. Thomson proposed the plum pudding model of an atom
• Lebesgue integration 1901 · achieved
  Method of integration. 1901 – Henri Lebesgue publishes on Lebesgue integration.
• Exterior derivative 1901 · achieved
  In differential geometry, a differential operation defined in differential forms that increases the form degree by 1. 1901 – Élie Cartan develops the exterior derivative,
• Planck's law 1900 · achieved
  Physical law that describes the amount of spectral radiance at a certain wavelength radiated by a black body cavity in thermal equilibrium. 1900 – Max Planck suggests that light may be emitted in discrete frequencies, giving his law of black-body radiation
• Great Paris Exhibition Telescope 1900 · achieved
  Future monumental telescope in Paris. 1900 — The largest refractor ever, Great Paris Exhibition Telescope of 1900 with an objective of 49.2 inch (1.25 m) diameter is temporarily exhibited at the Paris 1900 Exposition.
• Hilbert's problems 1900 · achieved
  Twenty-three problems in mathematics published in 1900. 1900 – David Hilbert states his list of 23 problems, which show where some further mathematical work is needed.
• Gamma ray 1900 · achieved
  Electromagnetic radiation of high frequency and high energy. 1900 – Paul Villard: Gamma rays
• Ultraviolet catastrophe 1900 · achieved
  Classical physics prediction that black body radiation grows unbounded with frequency. 1900 – Max Planck resolves the ultraviolet catastrophe by suggesting that black-body radiation consists of discrete packets, or quanta, of energy.
• Liénard–Wiechert potential 1900 · achieved
  Electromagnetic effect of point charges. 1900 – The Liénard–Wiechert potentials are introduced as time-dependent (retarded) electrodynamic potentials
• Old quantum theory 1900 · achieved
  Set of heuristic corrections to classical physics that served as a predecessor to modern quantum mechanics (1900–1925). Old quantum theory Planck
• Crescograph 1900 · achieved
  Measuring device. Crescograph – The crescograph, a device for measuring growth in plants, was invented in the early 20th century by the Bengali scientist Sir Jagadish Chandra Bose.
• Solid hydrogen 1899 · achieved
  Hydrogen in a solid state. 1899 – James Dewar collects solid hydrogen for the first time.
• Architectural acoustics (Sabine) 1898 · achieved
  The reverberation equation; Boston Symphony Hall (1900).
• Oscilloscope 1897 · achieved
  Used for measuring waves. Ferdinand Braun invents the "inertialess cathode ray oscillograph tube", a principle that remained unchanged in television picture tubes.
• Kelvin–Helmholtz mechanism 1897 · achieved
  Process of energy release of a contracting star or planet.
• Supersaturation 1896 · achieved
  State of a solution that contains more solute than can be dissolved at equilibrium. 1896 - Charles Wilson discovers that energetic particles produce droplet tracks in supersaturated gases.
• International Cloud Atlas 1896 · achieved
  Cloud atlas. 1896 – IMO publishes the first International cloud atlas.
• Zeeman effect 1896 · achieved
  Effect of splitting a spectral line into several components in the presence of a static magnetic field. 1896 – Pieter Zeeman: Zeeman effect
• Radioactive decay 1896 · achieved
  Emission of subatomic particles by the decay of an atomic nucleus. 1896 – Henri Becquerel: Radioactivity
• Simplicial homology 1895 · achieved
  Concept in algebraic topology. Henri Poincaré Simplicial homology.
• Curie’s principle 1894 · achieved
  Symmetry principle. 1894 - Pierre Curie described the now called Curie's principle for the symmetry properties of crystals.
• Fundamental group 1894 · achieved
  Mathematical group of the homotopy classes of loops in a topological space. Henri Poincaré Fundamental group of a topological space.
• Rangefinder 1894 · achieved
  Device for determining the distance to an object; device that measures distance from the observer to a target, in a process called ranging. A stadimeter, a type of optical rangefinder, is an optical device for estimating the range to an object of known height by measuring the angle between the top and bottom of the object as observed at the device.
• Polar stratospheric cloud 1893 · achieved
  Clouds occurring in the stratosphere in high-latitude regions. 1893 – Henrik Mohn reveals a discovery of nacreous clouds in what is now considered the stratosphere.
• Wien's displacement law 1893 · achieved
  Relation between the wavelength of the black-body radiation curve peak and the temperature. 1893 – Wilhelm Wien: Wien's displacement law for black-body radiation
• Generating primes 1890 · achieved
  Algorithms to generate prime numbers. F. W. Lawrence provides a high-level design for a mechanical, automated prime sieve in his paper "Factorisation of numbers".
• Resolution 1890 · achieved
  In algebra, an exact sequence which is used to describe the structure of an object. David Hilbert Resolution of modules and free resolution of modules.
• Activation energy 1889 · achieved
  Energy that must be input to a system to undergo a reaction or process. 1889 – Svante Arrhenius introduces the idea of activation energy for chemical reactions, giving the Arrhenius equation
• Nernst equation 1889 · achieved
  Physical law in electrochemistry. 1889 – Walther Nernst relates the voltage of electrochemical cells to their chemical thermodynamics via the Nernst equation
• Lorentz ether theory 1889 · achieved
  Defunct theory of electromagnetism. 1889, 1892 – Lorentz–FitzGerald contraction
• Le Chatelier's principle 1888 · achieved
  Principle to predict effects of a change in conditions on a chemical equilibrium. 1888 – Henri-Louis Le Chatelier states his principle that the response of a chemical system perturbed from equilibrium will be to counteract the perturbation
• Correlation 1888 · achieved
  Any statistical relationship, whether causal or not, between two random variables or bivariate data. 1888 – Francis Galton introduces the concept of correlation,
• Automorphism group 1888 · achieved
  Mathematical group formed from the automorphisms of an object. 1888 - Sophus Lie publishes work on transformation groups, serving as the foundation for the modern theory of Lie groups.
• Rydberg formula 1888 · achieved
  Formula for spectral line wavelengths in alkali metals. 1888 – Johannes Rydberg: Rydberg formula
• Hygrometer 1887 · achieved
  Instrument used for measuring the moisture content in the atmosphere. 1887 — Richard Assmann invents the psychrometer (Wet and Dry Bulb Thermometers)
• Electromagnetic radiation 1887 · achieved
  Form of energy emitted and absorbed by particles which are charged which shows wave-like behavior as it travels through space. 1887 – Heinrich Rudolf Hertz: Electromagnetic waves
• Michelson–Morley experiment 1887 · achieved
  Attempt to detect the relative motion of matter through the stationary luminiferous aether. 1887 – The Michelson–Morley experiment, intended to measure the relative motion of Earth through the (assumed) stationary luminiferous aether, got no results.
• Transverse wave 1886 · achieved
  Moving wave that consists of oscillations occurring perpendicular (right angled) to the direction of energy transfer (or the propagation of the wave). 1886 to 1888: After noticing how discharging an electric current into a coil produced a spark in a second nearby coil, Heinrich Hertz sees a way to build a test apparatus to solve von Helmholtz "Berlin Prize" problem.
• Nappe 1884 · achieved
  Type of rock. 1884 – Marcel Alexandre Bertrand, Nappe and Thrust fault theory
• Klein bottle 1882 · achieved
  Type of non-orientable surface. 1882 – Felix Klein discovers the Klein bottle,
• Eyepiece 1880 · achieved
  Type of lens attached to a variety of optical devices such as telescopes and microscopes; usually the lens that is closest to the eye when someone looks through the device, placed near the focal point of the objective to magnify the image. 1880 — Ernst Abbe designs the first orthoscopic eyepiece (Kellner's was solely achromatic rather than orthoscopic, despite his description).
• Piezoelectricity 1880 · achieved
  Electric charge generated in certain solids due to mechanical stress. 1880: Pierre Curie and Jacques Curie: Piezoelectricity.
• Statistical entropy (Boltzmann) 1877 · achieved
  Entropy in statistics. 1877: Ludwig Boltzmann: Statistical definition of entropy.
• Loschmidt's paradox 1876 · achieved
  In physics, the apparent contradiction that time-irreversible macrophysics arises from time-symmetric microphysics. 1876 – Loschmidt criticises Boltzmann's H theorem as being incompatible with microscopic reversibility ( Loschmidt's paradox ).
• Statistical ensemble 1876 · achieved
  Set of systems under identical condition. 1876 – Josiah Willard Gibbs publishes the first of two papers (the second appears in 1878) which discuss phase equilibria, statistical ensembles, the free energy as the driving force behind chemical reactions, and chemical thermodynami…
• Second law of thermodynamics 1874 · achieved
  Law of physics stating that systems spontaneously evolve towards states of higher entropy. 1874 – Thomson formally states the second law of thermodynamics
• McLeod gauge 1874 · achieved
  Type of measuring device. 1874 — Herbert McLeod invents the McLeod gauge
• Equation of state 1873 · achieved
  Equation describing the state of matter under a given set of physical conditions. 1873 - Johannes Diderik van der Waals formulates his equation of state
• Van der Waals' force 1873 · achieved
  Residual forces between molecules or atomic groups not arising from covalent or ionic bonds. 1873: Johannes Diderik van der Waals: was one of the first to postulate an intermolecular force: the van der Waals force.
• Crookes radiometer 1873 · achieved
  Device that rotates under influence of light. 1873: William Crookes, a chemist, invents the Crookes radiometer as the by-product of some chemical research.
• Boltzmann equation 1872 · achieved
  Equation of statistical mechanics. 1872 – Ludwig Boltzmann states the Boltzmann equation for the temporal development of distribution functions in phase space, and publishes his H-theorem.
• Dedekind cut 1872 · achieved
  Method of construction of the real numbers. 1872 – Richard Dedekind invents what is now called the Dedekind Cut for defining irrational numbers, and it is now used for defining surreal numbers.
• H-theorem 1872 · achieved
  Theorem stating that a quantity H (meant to represent entropy) increases monotonically with time, given an initial condition called molecular chaos (thus breaking time reversal symmetry). 1872 – Ludwig Boltzmann presents his H-theorem, and with it the formula Σ p i log p i for the entropy of a single gas particle
• Resistance thermometer 1871 · achieved
  Type of temperature sensor (thermometer). 1871 — William Siemens describes the Resistance thermometer at the Bakerian Lecture
• Diffuse sky radiation 1871 · achieved
  Solar radiation scattered and absorbed in Earth's atmosphere. 1871: Lord Rayleigh: Diffuse sky radiation ( Rayleigh scattering ) explains why sky appears blue.
• Rayleigh scattering 1871 · achieved
  Elastic scattering of light, or other electromagnetic radiation, by particles of size much smaller than the wavelength of the radiation. 1871 – Lord Rayleigh discusses the blue sky law and sunsets ( Rayleigh scattering )
• Virial theorem 1870 · achieved
  General equation that relates the time-averaged total kinetic energy of a stable system consisting of N particles, bound by potential forces, with that of the time-averaged total potential energy. 1870 – Clausius proves the scalar virial theorem
• Crookes tube 1869 · achieved
  Early type of cathode ray tube. 1869 – William Crookes invents the Crookes tube.
• Independence 1868 · achieved
  Concept in mathematical logic. 1868 – Eugenio Beltrami demonstrates the independence of Euclid ’s parallel postulate from the other axioms of Euclidean geometry.
• Maxwell's demon 1867 · achieved
  Thought experiment. 1867 – Maxwell asks whether Maxwell's demon could reverse irreversible processes
• Frequentist inference 1866 · achieved
  Statistical inference based on frequency and proportion in sample data. 1866 – John Venn 's Logic of Chance defends the frequency interpretation of probability.
• Macroscopic scale 1865 · achieved
  Length scale on which objects or phenomena are large enough to be visible almost practically with the naked eye. 1865 – Clausius introduces the modern macroscopic concept of entropy
• Entropy 1865 · achieved
  Physical property of the state of a system, measure of disorder. 1854 – Rankine introduces his thermodynamic function, later identified as entropy
• Maxwell's equations 1865 · achieved
  Unified electromagnetism, 1865.
• Pyrometer 1864 · achieved
  Remote-sensing thermometer used to measure high temperatures. 1864 — Henri Becquerel suggests an optical pyrometer
• A Dynamical Theory of the Electromagnetic Field 1864 · achieved
  1865 physics paper by James Maxwell. 1864 – James Clerk Maxwell: A Dynamical Theory of the Electromagnetic Field ( electromagnetic radiation )
• Centrifuge 1864 · achieved
  Device for rotating containers about a fixed axis and separating materials in said containers by mass inertia. 1864 – Antonin Prandtl [ wikidata ] invents first centrifuge to separate cream from milk.
• Heat death paradox 1862 · achieved
  Paradox relating to fate of universe. 1862 – Lord Kelvin solves the heat death paradox.
• Statistical mechanics 1860 · achieved
  Physics of large number of particles' statistical behavior. 1860 – James Clerk Maxwell: Introduced statistical mechanics with the Maxwell distribution
• Maxwell–Boltzmann distribution 1859 · achieved
  Specific probability distribution function, important in physics. 1859 – James Clerk Maxwell discovers the distribution law of molecular velocities.
• Riemann hypothesis 1859 · achieved
  Conjecture in mathematics linked to the repartition of prime numbers. 1859 – Bernhard Riemann formulates the Riemann hypothesis, which has strong implications about the distribution of prime numbers.
• Black body 1859 · achieved
  Idealized physical body that absorbs all incident electromagnetic radiation. 1859 – Gustav Kirchhoff shows that energy emission from a black body is a function of only temperature and frequency
• Möbius strip 1858 · achieved
  Two-dimensional surface with only one side and only one edge. 1858 – August Ferdinand Möbius invents the Möbius strip,
• Mean free path 1857 · achieved
  Average distance travelled by a moving particle between successive impacts. 1857 – Rudolf Clausius: Introduced the concept of mean free path
• Clausius theorem 1854 · achieved
  Theorem that for a thermodynamic system undergoing a thermodynamic cycle, the following inequality holds: ∮ δ𝑄/𝑇 ≤ 0, where δ𝑄 is heat absorbed by the system and 𝑇 is the temperature. 1854 – Clausius establishes the importance of dQ/T ( Clausius's theorem ), but does not yet name the quantity
• Spectroscopy 1854 · achieved
  Measurement and interpretation of interactions between matter and electromagnetic radiation of varying frequencies. 1854 – Gustav Robert Kirchhoff, physicist and one of the founders of spectroscopy, publishes Kirchhoff's Laws on the conservation of electric charge and energy, which are used to determine currents in each branch of a circuit.
• Valence 1852 · achieved
  Measure of an element's combining capacity with other atoms when it forms chemical compounds or molecules. 1852 – Edward Frankland develops the theory of chemical valence
• Stokes parameters 1852 · achieved
  1852 – George Gabriel Stokes defines the Stokes parameters of polarization
• Heat death of the universe 1852 · achieved
  Possible of the universe. 1852 – William Thomson, 1st Baron Kelvin puts forward the problem of the heat death of the universe
• Mathematical singularity 1850 · achieved
  In general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. 1850 – Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular points.
• Thermodynamics 1850 · achieved
  Carnot 1824, Clausius 1850, Kelvin.
• Vapor pressure 1849 · achieved
  Pressure exterted by a vapor in thermodynamic equilibrium. 1849 – William John Macquorn Rankine calculates the correct relationship between saturated vapour pressure and temperature using his hypothesis of molecular vortices
• Hypsometer 1849 · achieved
  Instrument for measuring height or elevation. 1849 — Henri Victor Regnault – Hypsometer
• Soliton 1849 · achieved
  Self-reinforcing solitary wave packet that maintains its shape while it propagates at a constant velocity. 1849 – George Gabriel Stokes shows that solitary waves can arise from a combination of periodic waves.
• First law of thermodynamics 1847 · achieved
  Statement of conservation of energy as it applies specifically to a thermodynamic system or process. 1847 – Hermann von Helmholtz publishes a definitive statement of the conservation of energy, the first law of thermodynamics.
• Barograph 1845 · achieved
  A barometer that records the barometric pressure over time in graphical form. 1845 — Francis Ronalds invents the first successful Barograph based on photography
• Faraday effect 1845 · achieved
  Physical law. 1845 – Michael Faraday: Faraday rotation (interaction of light and magnetic field)
• Pressure measurement 1844 · achieved
  Technique to measure pressure. 1844 — Lucien Vidi invents the aneroid Barograph
• Conservation of energy 1843 · achieved
  Law of physics and chemistry. 1843: James Prescott Joule: Law of Conservation of energy ( first law of thermodynamics ), also 1847 – Helmholtz, Conservation of energy.
• Quaternion 1843 · achieved
  Noncommutative extension of the real numbers. William Rowan Hamilton discovers quaternions.
• Redshift 1842 · achieved
  Relative increase in wavelength of electromagnetic radiation in the emission spectrum of astronomical objects. 1842 – Christian Doppler proposes the redshift and blueshift effects, based on an analog effect found in sound. Hippolyte Fizeau discovered independently the same phenomenon on electromagnetic waves in 1848.
• Laurent series 1841 · achieved
  Power series generalized to allow negative powers. 1841 – Karl Weierstrass discovers but does not publish the Laurent expansion theorem.
• Photoelectric effect 1839 · achieved
  Emission of electrons when light hits a material. 1839 – Alexandre Edmond Becquerel observes the photoelectric effect with an electrode in a conductive solution exposed to light.
• Cell theory 1839 · achieved
  Schleiden and Schwann: all living things are made of cells.
• Line of force 1838 · achieved
  Historical concept in electromagnetism. 1838 – Michael Faraday: Lines of force
• Cathode ray 1838 · achieved
  Stream of electrons observed in vacuum tubes. 1838 – Michael Faraday uses Volta's battery to discover cathode rays.
• Stellar parallax 1838 · achieved
  Apparent shift of position of a nearby star against the background of distant objects during Earth's orbital period. 1832–1838 – Following over 100 years of unsuccessful attempts, Thomas Henderson, Friedrich Bessel, and Otto Struve measure the parallax of a few nearby stars; these are the first measurements of any distances outside the Solar System.
• Angle trisection 1837 · achieved
  Construction of an angle equal to one third a given angle. 1837 – Pierre Wantzel proves that doubling the cube and trisecting the angle are impossible with only a compass and straightedge, as well as the full completion of the problem of constructibility of regular polygons
• Ice age 1837 · achieved
  Period of long-term reduction in temperature of Earth's surface and atmosphere. 1837 – Louis Agassiz begins his glaciation studies which eventually demonstrate that the Earth has had at least one ice age
• Dirichlet's theorem on arithmetic progressions 1835 · achieved
  Theorem that, for coprime 𝑎 and 𝑑, there are infinitely many primes congruent to 𝑎 modulo 𝑑. 1835 – Lejeune Dirichlet proves Dirichlet's theorem about prime numbers in arithmetical progressions.
• Ideal gas law 1834 · achieved
  Equation of state of a hypothetical ideal gas. 1834 – Benoît Paul Émile Clapeyron – Ideal gas law.
• Faraday's laws of electrolysis 1834 · achieved
  Physical law regarding electrolysis. 1834 – Michael Faraday publishes Faraday's laws of electrolysis.
• Hamiltonian mechanics 1833 · achieved
  Formulation of classical mechanics in terms of phase space and Hamiltonian function. 1833 – William Rowan Hamilton: Hamiltonian mechanics
• Lenz's law 1833 · achieved
  Electromagnetic phenomena where changing magnetic fields induce currents with opposing fields.
• Wheatstone bridge 1833 · achieved
  Measuring instrument in electricity. English physicist Samuel Hunter Christie invented the Wheatstone bridge (It is named after Charles Wheatstone who popularized it).
• Magnetometer 1833 · achieved
  Measuring instrument for magnetism. A spin-exchange relaxation-free (SERF) magnetometer achieves very high magnetic field sensitivity by monitoring a high density vapor of alkali metal atoms precessing in a near-zero magnetic field.
• Algebraic equation 1832 · achieved
  Equation of the form of equality of two polynomials. 1832 – Évariste Galois presents a general condition for the solvability of algebraic equations, thereby essentially founding group theory and Galois theory.
• Faraday's law of induction 1831 · achieved
  Basic law of electromagnetism of magnetic fields inducing a potential difference. English physicist Michael Faraday published the law of induction (Joseph Henry developed the same law independently).
• Point group 1830 · achieved
  Group of geometric symmetries (isometries) that keep at least one point fixed. 1830 - Johann F. C. Hessel published an independent geometrical derivation of the 32 point groups ( crystal classes ).
• Green's theorem 1828 · achieved
  Line integral around a closed curve to a double integral over its enclosed region. 1828 – George Green proves Green's theorem.
• Group 1827 · achieved
  Algebraic set with an invertible, associative internal operation admitting a neutral element. 1827: Évariste Galois development of group theory.
• Ohm's law 1827 · achieved
  Relationship between voltage and current across an ideal resistor. 1826 – Georg Simon Ohm states his Ohm's law of electrical resistance in the journals of Schweigger and Poggendorff, and also published in his landmark pamphlet Die galvanische Kette mathematisch bearbeitet in 1827.
• Electrical resistance and conductance 1827 · achieved
  Opposition to the passage of an electric current. German physicist Georg Ohm introduced the concept of electrical resistance.
• Ampère's circuital law 1826 · achieved
  Relates the integrated magnetic field around a closed loop to the electric current passing through the loop. 1826 – Ampère's circuital law
• Crystallographic restriction theorem 1826 · achieved
  Theorem about admissible crystal symmetries. 1826 - Moritz Ludwig Frankenheim derived the 32 crystal classes by using the crystallographic restriction, consistent with Haüy's laws, that only 2, 3, 4 and 6-fold rotational axes are permitted.
• Spherical aberration 1825 · achieved
  Optical aberration. 1825: Joseph Jackson Lister develops combined lenses that cancelled spherical and chromatic aberration.
• Stokes' theorem 1825 · achieved
  Theorem in vector calculus. 1825 – André-Marie Ampère discovers Stokes' theorem.
• Carnot cycle 1824 · achieved
  Theoretical thermodynamic cycle proposed by Nicolas Léonard Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. 1824: Carnot: described the Carnot cycle, the idealized heat engine.
• Ideal gas 1824 · achieved
  Theoretical gas composed of many randomly moving point particles whose only interactions are perfectly elastic collisions. 1824 – Nicolas Léonard Sadi Carnot: Ideal gas cycle analysis ( Carnot cycle ), internal combustion engine
• Sophie Germain's theorem 1823 · achieved
  On the divisibility of solutions to Fermat's Last Theorem for prime exponent. 1823 – Sophie Germain's Theorem is published in the second edition of Adrien-Marie Legendre 's Essai sur la théorie des nombres
• Dimension 1822 · achieved
  Maximum number of independent directions within a mathematical space. 1822 – Joseph Fourier formally introduces the use of dimensions for physical quantities in his Theorie Analytique de la Chaleur.
• Soddy's hexlet 1822 · achieved
  Chain of 6 spheres tangent to 3 given spheres. 1822—Irisawa Shintarō Hiroatsu analyzes Soddy's hexlet in a Sangaku.
• Cauchy's integral theorem 1822 · achieved
  Theorem. 1822— Augustin-Louis Cauchy presents Cauchy's integral theorem for integration around the boundary of a rectangle in the complex plane.
• Heat equation 1822 · achieved
  Partial differential equation for distribution of heat in a given region over time. 1822 – Joseph Fourier: Heat equation
• Cours d'Analyse 1821 · achieved
  Famous textbook on calculus. 1821— Augustin-Louis Cauchy publishes Cours d'Analyse, which purportedly contains an erroneous “proof” that the pointwise limit of continuous functions is continuous.
• Right-hand rule 1820 · achieved
  Common mnemonic for understanding orientation of axes in three-dimensional space. One week after Ørsted's discovery, French physicist André-Marie Ampère published his law. He also proposed the right-hand screw rule.
• Electromagnetism 1820 · achieved
  Ørsted 1820, Faraday's induction 1831.
• Intermediate value theorem 1817 · achieved
  Theorem. 1817— Bernard Bolzano presents the intermediate value theorem —a continuous function that is negative at one point and positive at another point must be zero for at least one point in between.
• Birefringence 1816 · achieved
  Optical phenomenon. 1816 – David Brewster discovers stress birefringence
• Generating function 1814 · achieved
  Formal power series with coefficients that encode information about a sequence.
• Catastrophism 1813 · achieved
  Scientific theory following which the Earth has largely been shaped by sudden, short-lived, violent events, possibly worldwide in scope. 1813 – Georges Cuvier publishes his Essay on the Theory of the Earth, proposing catastrophism on the basis of his work in biostratigraphy
• Avogadro's law 1811 · achieved
  Physical law that equal volumes of all gases at the same temperature and pressure have the same number of molecules. 1811 – Avogadro's law, a gas law relating volume and amount of substance.
• Gay-Lussac's law 1809 · achieved
  Relationship between pressure and temperature of a gas at constant volume.. 1809 – Gay-Lussac's law, a gas law relating temperature and pressure.
• Theodolite 1809 · achieved
  Surveying instrument that measures azimuth and elevation between points. 1809 – Thomas Forster observes with a theodolite the drift of small free pilot balloons filled with "inflammable gas".
• Polarization 1808 · achieved
  Property of waves that can oscillate with more than one orientation. 1808 – Étienne-Louis Malus discovers polarization by reflection
• Fourier series 1807 · achieved
  Decomposition of periodic functions into sums of simpler sinusoidal forms. 1807— Joseph Fourier announces his discoveries about the trigonometric decomposition of functions.
• Beaufort Scale 1806 · achieved
  Empirical measure describing wind speed based on observed conditions. 1806 – Francis Beaufort introduces his system for classifying wind speeds.
• Least squares method 1805 · achieved
  Approximation method in statistics. Publication Least squares Adrien-Marie Legendre publishes the least squares method in Nouvelles méthodes pour la détermination des orbites des comètes, presenting a method for estimating quantities from observations.
• Ruffini's rule 1804 · achieved
  Polynomial division computation method. Sharaf al-Dīn al-Tūsī (1135–1213) writes the Al-Mu'adalat ( Treatise on Equations ), which deals with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions.
• Double-slit experiment 1803 · achieved
  Experiment in quantum mechanics that shows wave–particle duality. 1803 – Thomas Young develops the Double-slit experiment and demonstrates the effect of interference.
• Ultraviolet radiation 1801 · achieved
  Form of electromagnetic radiation. 1801 – Johann Ritter discovers ultraviolet radiation from the Sun
• Wave–particle duality 1801 · achieved
  Conclusion that quantum objects behave at times like particles and at times like waves. 1801 – Thomas Young demonstrates the wave nature of light and the principle of interference
• Infrared radiation 1800 · achieved
  Electromagnetic radiation with longer wavelengths than those of visible light. 1800 – William Herschel discovers infrared radiation from the Sun.
• Abel–Ruffini theorem 1799 · achieved
  Theorem that a generic equation of degree 5 or higher cannot be solved by radicals. Paolo Ruffini partially proves the Abel–Ruffini theorem that quintic or higher equations cannot be solved by a general formula,
• Fundamental theorem of algebra 1799 · achieved
  Every polynomial has a real or complex root. Carl Friedrich Gauss proves the fundamental theorem of algebra (every polynomial equation has a solution among the complex numbers),
• Prime number theorem 1796 · achieved
  Theorem in number theory. 1796— Adrien-Marie Legendre conjectures the prime number theorem.
• Heptadecagon 1796 · achieved
  Polygon with 17 sides. 1796 – Carl Friedrich Gauss proves that the regular 17-gon can be constructed using only a compass and straightedge
• Nebular hypothesis 1796 · achieved
  Astronomical theory that the Solar System formed from nebulous material. 1796 – Pierre Laplace re-states the nebular hypothesis for the formation of the Solar System from a spinning nebula of gas and dust.
• Lagrangian mechanics 1788 · achieved
  Formulation of classical mechanics based on the Lagrangian function on the tangent bundle of configuration space. 1760 – Joseph-Louis Lagrange: Lagrangian mechanics
• Charles's law 1787 · achieved
  Charles's law is an experimental gas law which describes how gases tend to expand when heated.. 1787 – Charles's law (gas law, relating volume and temperature).
• Line chart 1786 · achieved
  Chart type. 1786 – William Playfair 's Commercial and Political Atlas introduces graphs and bar charts of data,
• Laplace transform 1785 · achieved
  The integral transform ∫₀^∞ d𝑠 𝑓(𝑡)⁢exp(−𝑠𝑡). French mathematician Pierre-Simon Laplace developed the Laplace transform to transform a linear differential equation into an algebraic equation. Later, his transform became a tool in circuit analysis.
• Coulomb's law 1785 · achieved
  A physical law that states the electric force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them.. French physicist Charles-Augustin de Coulomb formulated and published Coulomb's law in his paper Premier Mémoire sur l’Électricité et le Magnétisme.
• Caloric theory 1783 · achieved
  Obsolete scientific theory of heat flow. 1783 – In Lavoisier 's article "Reflexions sur le phlogistique", he deprecates the phlogiston theory and proposes a caloric theory of heat.
• Six's thermometer 1782 · achieved
  Meteorological instrument to measure Temperature. 1780 – James Six invents the Six's thermometer, a thermometer that records minimum and maximum temperatures. See ( Six's thermometer )
• Conservation of mass 1782 · achieved
  Scientific law that a closed system's mass remains constant. 1782 – Antoine Lavoisier: conservation of mass
• Heat transfer 1777 · achieved
  Transport of thermal energy in physical systems. 1777 – Carl Wilhelm Scheele distinguishes heat transfer by thermal radiation from that by convection and conduction
• Messier object 1771 · achieved
  Astronomical objects catalogued by Charles Messier. 1771: Charles Messier: publishes catalogue of astronomical objects ( Messier Objects ) now known to include galaxies, star clusters, and nebulae.
• Lagrange's four-square theorem 1770 · achieved
  Theorem. 1770 — Joseph Louis Lagrange proves the four-square theorem, that every positive integer is the sum of four squares of integers.
• Divergence theorem 1762 · achieved
  Generalization of the fundamental theorem in vector calculus. 1762— Joseph-Louis Lagrange discovers the divergence theorem.
• Atmosphere of Venus 1761 · achieved
  Atmosphere of planet Venus. 1761: Mikhail Lomonosov: discovery of the atmosphere of Venus.
• Bayes' theorem 1761 · achieved
  Theorem describing the probability of an event based on prior knowledge of conditions that might be related to the event. 1761— Thomas Bayes proves Bayes' theorem.
• Latent heat 1761 · achieved
  Released or absorbed energy during a constant-temperature process. 1750: Joseph Black: describes latent heat.
• Kite experiment 1752 · achieved
  Science experiment on lightning and electricity. 1752 – Benjamin Franklin: kite experiment
• Cramer's rule 1750 · achieved
  Theorem of solving a system of linear equations. Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer's rule and studies algebraic curves, matrices and determinants.
• D'Alembert's formula 1747 · achieved
  Mathematical solution. 1747— Jean le Rond d'Alembert solves the vibrating string problem (one-dimensional wave equation ).
• Goldbach's conjecture 1742 · achieved
  Conjecture that every even integer greater than 2 is the sum of two primes. 1742— Christian Goldbach conjectures that every even number greater than two can be expressed as the sum of two primes, now known as Goldbach's conjecture.
• Refraction 1740 · achieved
  In physics, the change in direction of a wave at the interface between two media. 1740 – Jean le Rond d'Alembert, in Mémoire sur la réfraction des corps solides, explains the process of refraction.
• Homogeneous differential equation 1739 · achieved
  Mathematical relation with derivatives. 1739—Leonhard Euler solves the general homogeneous linear ordinary differential equation with constant coefficients.
• Kinetic theory 1738 · achieved
  Mathematical model explaining macroscopic properties of gases in microscopic terms. 1738 – Daniel Bernoulli: First model of the kinetic theory of gases
• Seven Bridges of Königsberg 1736 · achieved
  Mathematical problem. 1736—Leonhard Euler solves the problem of the Seven Bridges of Königsberg, in effect creating graph theory.
• Atmospheric circulation 1735 · achieved
  Large-scale movement of air, a means by which thermal energy is distributed on Earth's surface. 1735 – The first ideal explanation of global circulation was the study of the Trade winds by George Hadley.
• Basel problem 1735 · achieved
  Mathematical problem. 1735—Leonhard Euler solves the Basel problem, relating an infinite series to π.
• Integrating factor 1734 · achieved
  Function that is commonly used to solve ordinary differential equations. 1734— Leonhard Euler introduces the integrating factor technique for solving first-order ordinary differential equations.
• Achromatic lens 1733 · achieved
  Optical instrument corrected for dispersion at 2 wavelengths. 1733 — Chester Moore Hall invents the achromatic lens.
• Normal distribution 1733 · achieved
  Probability distribution. 1733—Abraham de Moivre introduces the normal distribution to approximate the binomial distribution in probability.
• Parallel postulate 1733 · achieved
  Axiom in Euclidean geometry. 1733 – Giovanni Gerolamo Saccheri studies what geometry would be like if Euclid's fifth postulate were false,
• Optical aberration 1728 · achieved
  Phenomenon in astronomy. 1728 – James Bradley discovers the aberration of starlight and uses it to determine that the speed of light is about 283,000 km/ s
• Richardson extrapolation 1722 · achieved
  A sequence acceleration method, used to improve the rate of convergence. 1722— Takebe Kenko introduces Richardson extrapolation.
• De Moivre's formula 1722 · achieved
  Theorem. Abraham de Moivre states de Moivre's formula connecting trigonometric functions and complex numbers,
• Parabolic reflector 1720 · achieved
  Reflector / collector that has the shape of a paraboloid. 1720 — John Hadley develops ways of aspherizing spherical mirrors to make very accurate parabolic mirrors and produces a much improved Gregorian telescope
• Olbers's paradox 1720 · achieved
  Cosmological contradiction between the observed darkness of the night sky and a static universe model. 1720 – Edmund Halley puts forth an early form of Olbers' paradox (if the universe is infinite, every line of sight would end at a star, thus the night sky would be entirely bright).
• Proper motion 1718 · achieved
  Astronomical measure of the observed changes in the apparent places of stars in the sky. 1718 – Edmund Halley discovers proper motion of stars, dispelling the concept of the " fixed stars ".
• Human sex ratio 1710 · achieved
  Ratio of males to females in a population. 1710 – John Arbuthnot argues that the constancy of the ratio of male to female births is a sign of divine providence,
• Bernoulli number 1708 · achieved
  Rational number sequence 𝐵ₖ such that (𝑚+1)∑𝑛ᵐ=(ᵐ⁺¹₀)𝐵₀𝑛ᵐ⁺¹−(ᵐ⁺¹₁)𝐵₁𝑛ᵐ+(ᵐ⁺¹₂)𝐵₂𝑛ᵐ¯¹−(ᵐ⁺¹₃)𝐵₃𝑛ᵐ¯²+⋯. 1708— Seki Takakazu discovers Bernoulli numbers. Jacob Bernoulli, whom the numbers are named after, is believed to have independently discovered the numbers shortly after Takakazu.
• Absolute zero 1702 · achieved
  Coldest possible temperature. 1702 – Guillaume Amontons introduces the concept of absolute zero, based on observations of gases
• Newton's law of cooling 1701 · achieved
  Physical law. 1701 — Newton publishes anonymously a method of determining the rate of heat loss of a body and introduces a scale, which had 0 degrees represent the freezing point of water, and 12 degrees for human body temperature.
• L'Hôpital's rule 1696 · achieved
  Rule that uses derivatives to help evaluate limits involving indeterminate forms. 1696— Guillaume de l'Hôpital states his rule for the computation of certain limits.
• Phlogiston theory 1694 · achieved
  Superseded scientific theory about combustion. 1694–1734 – Georg Ernst Stahl names Becher's combustible earth as phlogiston and develops the theory
• Bentley's paradox 1692 · achieved
  Cosmological paradox involving gravity. 1692 – Richard Bentley puts forth Bentley's paradox, the idea that Newtonian gravity is unstable.
• Differential equation 1691 · achieved
  Mathematical equation involving derivatives of an unknown function. 1691—Gottfried Leibniz discovers the technique of separation of variables for ordinary differential equations.
• Fundamental interaction 1687 · achieved
  Any of the physical interactions or forces: gravitational, electromagnetic, strong nuclear, and weak nuclear. 1687: Sir Isaac Newton: classical mathematical description of the fundamental force of universal gravitation and the three physical laws of motion.
• Scientific law 1687 · achieved
  Statement based on repeated experimental observations that describes some aspects of the universe. 1687 – Isaac Newton 's laws describe large-scale motion throughout the universe.
• Newtonian mechanics 1687 · achieved
  Principia, 1687.
• Calculus 1687 · achieved
  Newton and Leibniz, 1660s–1680s.
• Trade wind 1686 · achieved
  Permanent east-to-west prevailing winds that flow in the Earth's equatorial region. 1686 – Edmund Halley presents a systematic study of the trade winds and monsoons and identifies solar heating as the cause of atmospheric motions.
• Aerial telescope 1684 · achieved
  Very long focal length refracting telescope without a tube, built in the 17th century; the objective was mounted on a pole, tree, etc. on a swivel ball-joint; the observer stood on the ground and maneuvered the eyepiece to aim at celestial objects. 1684 — Christiaan Huygens publishes "Astroscopia Compendiaria" in which he described the design of very long aerial telescopes.
• Resultant 1683 · achieved
  Polynomial expression of the coefficients of two polynomials, which is equal to zero if and only if the polynomials have a common root. 1683— Seki Takakazu discovers the resultant and determinant.
• Seki Takakazu determinants 1683 · achieved
  Determinants, independently of Leibniz.
• Huygens–Fresnel principle 1678 · achieved
  Method of analysis. 1678 – Christiaan Huygens states his principle of wavefront sources and demonstrates the refraction and diffraction of light rays.
• Vis viva 1676 · achieved
  Physical quantity, historically invented by G. Leibniz, equivalent to (in modern terminology) twice the kinetic energy. 1676–1689 – Gottfried Leibniz develops the concept of vis viva, a limited version of the conservation of energy
• Rømer's determination of the speed of light 1676 · achieved
  1676 experiment which showed that light has finite speed. 1676 – Ole Rømer: Rømer's determination of the speed of light traveling from the moons of Jupiter.
• Chain rule 1676 · achieved
  Method of differentiating composed functions. Leibniz derived the chain rule. The rule is used by AI to train neural networks, for example the backpropagation algorithm uses the chain rule.
• Taylor's theorem 1671 · achieved
  Approximation of a function by a truncated power series. 1671—James Gregory discovers Taylor's theorem.
• Fossil 1669 · achieved
  Preserved remains or traces of organisms from a past geological age. 1669: Nicholas Steno: proposes that fossils are organic remains embedded in layers of sediment, basis of stratigraphy.
• Curved mirror 1668 · achieved
  Mirror with a curved reflecting surface. 1668 — Isaac Newton produces the first functioning reflecting telescope using a spherical primary mirror and a flat diagonal secondary mirror. This design is termed the ' Newtonian '.
• Spontaneous generation 1668 · achieved
  Disproven theory of life arising from nonliving matter. 1668: Francesco Redi: disproved idea of spontaneous generation.
• Reflecting telescope 1668 · achieved
  Telescope design that collects and focuses light with a combination of mirrors. 1668: First functional reflecting telescope constructed by Isaac Newton.
• Combustion 1667 · achieved
  High-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. 1667 – J. J. Becher puts forward a theory of combustion involving combustible earth in his book Physica subterranea (see Phlogiston theory ).
• Fundamental theorem of calculus 1665 · achieved
  Calculus theorem describing the duality of differentiation and integration. 1665— Isaac Newton works on the fundamental theorem of calculus and develops his version of infinitesimal calculus.
• Diffraction 1665 · achieved
  Various phenomena that occur when a wave encounters an obstacle or a slit. 1665 – Francesco Maria Grimaldi highlights the phenomenon of diffraction
• Paraboloid 1663 · achieved
  Quadric surface of special kind. 1663 — Scottish mathematician James Gregory designs a reflecting telescope with paraboloid primary mirror and ellipsoid secondary mirror.
• Boyle's law 1662 · achieved
  Experimental gas law. 1660 – Robert Boyle experimentally discovers Boyle's law, relating the pressure and volume of a gas (published 1662)
• Fermat's principle 1662 · achieved
  Principle of least time. In optics, Hero formulated the principle of the shortest path of light: If a ray of light propagates from point A to point B within the same medium, the path-length followed is the shortest possible.
• Hooke's law 1660 · achieved
  Empirical physical law of mechanics that the force on a spring is proportional to its displacement. 1660 – Robert Hooke: Hooke's law
• Discovery of Titan (Huygens) 1655 · achieved
  Largest moon of Saturn. Saturn flyby (closest approach 124,000 km), close encounter of Titan and encounters with a dozen other moons. USA (NASA) Voyager 1
• Probability 1654 · achieved
  Measure of the expectation that an event will occur or a statement is true. 1654— Blaise Pascal and Pierre de Fermat create the theory of probability.
• Expectation 1654 · achieved
  Long-run average value of a random variable.
• Pascal's law 1653 · achieved
  Principle in fluid mechanics. 1660 – Blaise Pascal: Pascal's law
• Vacuum pump 1650 · achieved
  Equipment producing low air pressure. 1650: Vacuum pump: Invented by Otto von Guericke.
• Atmospheric pressure 1648 · achieved
  Static pressure exerted by the weight of the atmosphere. 1648 – Blaise Pascal rediscovers that atmospheric pressure decreases with height, and deduces that there is a vacuum above the atmosphere.
• Descartes' theorem 1643 · achieved
  Theorem. 1643—René Descartes develops Descartes' theorem.
• Barometer 1643 · achieved
  Torricelli, 1643.
• Projective geometry (Desargues) 1639 · achieved
  Perspective formalized as mathematics.
• Fermat's Last Theorem 1637 · achieved
  Theorem in number theory that there are no nontrivial integer solutions of xⁿ+yⁿ=zⁿ for integer n>2. c. 1000 – Abu-Mahmud Khujandi first states a special case of Fermat's Last Theorem.
• Cycloid 1634 · achieved
  Curve traced by a point on a circle as it rolls along a straight line without slipping. 1634— Gilles de Roberval shows that the area under a cycloid is three times the area of its generating circle.
• Galilean invariance 1632 · achieved
  Principle in classical mechanics stating that the laws of motion are the same in all inertial frames. 1632 – Galileo Galilei: The Galilean principle (the laws of motion are the same in all inertial frames )
• Refracting telescope 1621 · achieved
  Type of optical telescope. 1621: Cornelis Drebbel presents, in London, a compound microscope with a convex objective and a convex eyepiece (a "Keplerian" microscope).
• Scientific method 1620 · achieved
  Bacon's Novum Organum 1620, Galileo, the Royal Society 1660.
• Kepler–Poinsot polyhedron 1619 · achieved
  Regular star polyhedron. 1619 – Johannes Kepler discovers two of the Kepler-Poinsot polyhedra.
• Euler's number 1618 · achieved
  Mathematical constant; limit of (1 + 1/n)^n as n approaches infinity; transcendental number approximately equal 2.718281828. 1618—John Napier publishes the first references to e in a work on logarithms.
• Logarithms 1614 · achieved
  Napier, 1614.
• Inertia 1613 · achieved
  The tendency of an object to remain in its existing state of motion is called its inertia. 1613 – Galileo Galilei: Inertia
• Thermometer 1612 · achieved
  Santorio's scale thermoscope; Fahrenheit's mercury thermometer, 1714.
• Kepler conjecture 1611 · achieved
  Mathematical theorem about sphere packing. Kepler conjecture is almost all but certainly proved algorithmically by Thomas Hales in 1998.
• Galilean moons 1610 · achieved
  Four largest moons of Jupiter. 1610 – Galileo Galilei: discovered the Galilean moons of Jupiter
• Kepler's laws of planetary motion 1609 · achieved
  Scientific laws describing motion of planets around the Sun. 1609, 1619 – Kepler: Kepler's laws of planetary motion
• Optical telescope 1608 · achieved
  Telescope which is used to focus light from the visible part of the electromagnetic spectrum. 1608: Earliest record of an optical telescope.
• Telescope 1608 · achieved
  Lippershey patent 1608; Galileo's observations 1609.
• Fixed star 1600 · achieved
  Astronomical bodies that appear not to move relative to each other in the night sky. 1600 – William Gilbert rejects the idea of a limiting sphere of the fixed stars for which no proof has been offered.
• Continental drift 1596 · achieved
  Movement of Earth's continents relative to each other. 1596 – Abraham Ortelius, Flemish-Spanish cartographer, first envisages the continental drift theory.
• Thermoscope 1592 · achieved
  Old version of modern thermometer that measures the heat by both temperature and pressure.. 1592–1593 — Galileo Galilei builds a device showing variation of hotness known as the thermoscope using the contraction of air to draw water up a tube.
• Optical microscope 1590 · achieved
  Microscope that uses visible light. 1590: earliest date of a claimed Hans Martens/ Zacharias Janssen invention of the compound microscope (claim made in 1655).
• Celestial globe 1589 · achieved
  Type of star chart where the map is arranged on a globe. Seamless celestial globe – Considered one of the most remarkable feats in metallurgy, it was invented in India in between 1589 and 1590 CE.
• Tychonic system 1588 · achieved
  Model of the Solar System proposed in 1588 by the Danish astronomer Tycho Brahe. Geoheliocentricsm – A similar model was implicitly mentioned in the Hindu astronomical treatise Tantrasamgraha ( c. 1500 CE ) by Nilakantha Somayaji of the Kerala school of astronomy and mathematics.
• Equal temperament (Zhu Zaiyu, Stevin) 1584 · achieved
  The 12th root of 2 calculated to high precision.
• Taqi al-Din observatory and clocks 1577 · achieved
  Istanbul observatory; mechanical astronomical clocks.
• Mercator projection 1569 · achieved
  Cylindrical map projection invented by Gerardus Mercator in 1569. 1569: Mercator Projection map created by Gerardus Mercator
• Odds 1560 · achieved
  Ratio of the probability that an event will happen to the probability that it will not happen. 1560s (published 1663) – Cardano 's Liber de ludo aleae attempts to calculate probabilities of dice throws.
• Viète's formula 1550~ · achieved
  Infinite product converging to the inverse of π. 16th century: François Viète discovers Viète's formula for π. 1500: Scipione del Ferro solves the special cubic equation x 3 = p x + q {\displaystyle x^{3}=px+q}.
• Vieta's formulas 1550~ · achieved
  Relations between the coefficients and the roots of a polynomial. 16th century: François Viète discovers Vieta's formulas.
• Cubic function 1545 · achieved
  A polynomial function in a single variable where the highest exponent of the variable is 3. 1545 – Gerolamo Cardano published Cardano's method for finding the roots of a cubic polynomial
• Heliocentrism (Copernicus) 1543 · achieved
  De revolutionibus, 1543.
• Quartic equation 1540 · achieved
  Polynomial equation of degree four. 1540— Lodovico Ferrari solves the quartic equation.
• Weather lore 1510 · achieved
  Body of informal folklore related to the prediction of the weather. 1510 – Leonhard Reynmann, astronomer of Nuremberg, publishes ″Wetterbüchlein Von warer erkanntnus des wetters″, a collection of weather lore.
• Anemometer 1450 · achieved
  Meteorological instrumentation used for measuring the speed of wind. 1450 – Leone Battista Alberti developed a swinging-plate anemometer, and is known as the first anemometer.
• Cheugugi rain gauge 1441 · achieved
  A standardized rain gauge network under King Sejong.
• Ulugh Beg Observatory 1420 · achieved
  Samarkand observatory and star catalogue.
• Mathematical notation 1412 · achieved
  System of symbolic representations of mathematical objects and ideas. Arab mathematician Abū al-Hasan ibn Alī al-Qalasādī takes "the first steps toward the introduction of algebraic symbolism." He uses "short Arabic words, or just their initial letters, as mathematical symbols."
• Kerala school infinite series 1400 · achieved
  Madhava's series for sine, cosine, and π, ~1400.
• Generalized continued fraction 1380 · achieved
  Generalization of continued fractions in which the partial numerators and partial denominators can assume arbitrary complex values. 1380: Madhava of Sangamagrama discovers continued fractions and uses them to solve transcendental equations.
• Taylor series 1380 · achieved
  Expression of a function as an infinite sum. 1380: 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.
• Power series 1350~ · achieved
  Infinite sum of monomials. 14th century – Madhava discovers the power series expansion for sin ⁡ x {\displaystyle \sin x}, cos ⁡ x {\displaystyle \cos x}, arctan ⁡ x {\displaystyle \arctan x} and π / 4 {\displaystyle \pi /4} This theory is now well known in the…
• Jacob's staff 1350~ · achieved
  Calibrated wooden rod with sliding crosspiece used for measuring the altitude of celestial bodies or the height of distant objects. 14th century: Jacob's staff described by Levi ben Gerson
• Polynomial ring 1300~ · achieved
  Ring of polynomials (with one or several variables) with coefficients in a given ring. Chinese mathematician Zhu Shijie deals with polynomial algebra, solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, quintic and higher-order polynomial equations.
• Mechanical clock 1300 · achieved
  Verge-escapement clocks in European towers, ~1300.
• Factorization 1260 · achieved
  (mathematical) decomposition into a product. 1260 – Al-Farisi gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning factorization and combinatorial methods.
• Sample size determination 1250~ · achieved
  Act of choosing the number of observations to include in a statistical sample. 13th century – An important contribution of Ibn Adlan was on sample size for use of frequency analysis.
• Lens 1249 · achieved
  Optical device which transmits and refracts light. 1249 – Roger Bacon writes about convex lens spectacles for treating long-sightedness
• Snow gauge 1247 · achieved
  Instrument to gather and measure the amount of solid precipitation (snow). Snow gauge: The first use of snow gauges were precipitation gauges that was widely used in 1247 during the Southern Song dynasty to gather meteorological data.
• Tusi couple 1247 · achieved
  Nasir al-Din al-Tusi's geometric device, later used by Copernicus.
• Dresden Codex 1200 · achieved
  Maya book with Venus tables and eclipse predictions.
• Chakravala method 1114 · achieved
  Cyclic algorithm to solve indeterminate quadratic equations. Chakravala method – The Chakravala method, a cyclic algorithm to solve indeterminate quadratic equations is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE) although some attribute it to Jayadeva (c. 950~1000 CE).
• Early world maps 1100 · achieved
  List of early depictions of the world. 1100–1165: Muhammad al-Idrisi, aka Idris al-Saqalli aka al-sharif al-idrissi of Andalusia and Sicily. Known for having drawn some of the most advanced ancient world maps.
• Su Song's astronomical clock tower 1094 · achieved
  Water-driven clock tower with an escapement and chain drive.
• Magnetic declination 1088 · achieved
  Angle on the horizontal plane between magnetic north and true north. 1088 – Shen Kuo first recognizes magnetic declination.
• Omar Khayyam's cubic equations 1070 · achieved
  Geometric solution of cubic equations.
• True north 1050~ · achieved
  Direction on Earth's surface. 11th century: Shen Kuo discovers the concepts of true north and magnetic declination.
• Atmospheric refraction 1021 · achieved
  Deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere. 1021 – Ibn al-Haytham (Alhazen) writes on the atmospheric refraction of light, the cause of morning and evening twilight.
• Alhazen's problem 1021 · achieved
  Mathematical problem in geometrical optics. 1021 – Ibn al-Haytham formulated and solved Alhazen's problem geometrically.
• Book of Optics (Alhazen) 1021 · achieved
  Ibn al-Haytham's experimental optics.
• Parabola 1020 · achieved
  Plane curve: symmetrical conic section. 1020 – Abu al-Wafa' al-Buzjani gave the formula: sin (α + β) = sin α cos β + sin β cos α. Also discussed the quadrature of the parabola and the volume of the paraboloid.
• Law of sines 1000~ · achieved
  Property of all triangles on a Euclidean plane. ca. 1000 – Law of sines is discovered by Muslim mathematicians, but it is uncertain who discovers it first between Abu-Mahmud al-Khujandi, Abu Nasr Mansur, and Abu al-Wafa.
• Sextant 994 · achieved
  Angle measurement navigational instrument. Sextant and mural instrument: The first known mural sextant was constructed in Ray, Iran, by Abu-Mahmud al-Khujandi in 994.
• Snell's law 984 · achieved
  Formula deriving the angle of refraction of light between two mediums. Snell's law: The law was first accurately described by the Persian scientist Ibn Sahl at the Baghdad court in 984.
• Binomial coefficient 975 · achieved
  Family of positive integers that occur as coefficients in the binomial theorem. 975 — The earliest triangle of binomial coefficients (Pascal triangle) occur in the 10th century in commentaries on the Chandas Shastra.
• Monomial 953 · achieved
  Polynomial which has only one term. 953—Persia, Al-Karaji is the "first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations that are at the core of algebra today.
• Zero of a function 940 · achieved
  Element of the domain where function's value is zero. 940—Iran, Abu al-Wafa' al-Buzjani extracts roots using the Indian numeral system.
• Thabit number 895 · achieved
  Integer of the form 3 · 2ⁿ - 1 for non-negative n. 895 — Thabit ibn Qurra gives a theorem by which pairs of amicable numbers can be found, (i.e., two numbers such that each is the sum of the proper divisors of the other).
• Cubic equation 895 · achieved
  A polynomial equation in a single variable where the highest exponent of the variable is 3.. Thabit ibn Qurra: the only surviving fragment of his original work contains a chapter on the solution and properties of cubic equations.
• Algorism 850~ · achieved
  Mathematical technique for arithmetic. 9th century: 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 ( Addit…
• Sine quadrant 850~ · achieved
  Type of quadrant used by medieval Arabic astronomers. Sine quadrant: A type of quadrant used by medieval Arabic astronomers, it was described by Muhammad ibn Mūsā al-Khwārizmī in 9th century Baghdad.
• Astrolabe refinements 850 · achieved
  Islamic astronomers perfected the planispheric astrolabe.
• House of Wisdom 830 · achieved
  Abbasid center of translation and scholarship in Baghdad.
• Amicable numbers 826 · achieved
  Pair of integers related by their divisors. 826 – 901: Thabit ibn Qurra (Latinized: Thebit.) studied at Baghdad's House of Wisdom under the Banu Musa brothers. Discovered a theorem that enables pairs of amicable numbers to be found. [ citation needed ] Later, al-Baghdadi (b.
• Algebra 820 · achieved
  Al-Khwarizmi's al-Jabr, ~820. Golden Age of Islam.
• Planetarium 810 · achieved
  Theatre built primarily for presenting educational and entertaining shows about astronomy and the night sky. 810 – 887: Abbas ibn Firnas. Planetarium, artificial crystals. According to one account that was written seven centuries after his death, Ibn Firnas was injured during an elevated winged trial flight.
• Mathematics in medieval Islam 750~ · achieved
  The body of mathematics preserved and advanced under the Islamic civilization between circa 622 and 1600. 8th century – Al-Khalil, an Arab mathematician studying cryptology, wrote the Book of Cryptographic Messages.
• Fibonacci sequence 700 · achieved
  Entire infinite integer series where the next number is the sum of the two preceding it (0,1,1,2,3,5,8,13,21,...). Fibonacci numbers – This sequence was first described by Virahanka (c. 700 CE), Gopāla (c. 1135), and Hemachandra (c. 1150), as an outgrowth of the earlier writings on Sanskrit prosody by Pingala (c. 200 BCE).
• Mathematical interpolation 665 · achieved
  Method for constructing new data points between known data points. Finite difference interpolation – The Indian mathematician Brahmagupta presented what is possibly the first instance of finite difference interpolation around 665 CE.
• Brahmagupta's interpolation formula 628 · achieved
  Second-order polynomial interpolation formula. 628: Brahmagupta discovers second-order interpolation, in the form of Brahmagupta's interpolation formula.
• Brahmagupta's formula 628 · achieved
  Theorem. 628: Brahmagupta discovers Brahmagupta's formula, a generalization of Heron's formula to cyclic quadrilaterals.
• Brāhmasphuṭasiddhānta 628 · achieved
  Indian mathematics treatise. Brahmagupta writes the Brahmasphuta-siddhanta, where zero is clearly explained, and where the modern place-value Indian numeral system is fully developed.
• Zero and positional decimal 628 · achieved
  Brahmagupta formalized zero as a number, 628 AD.
• Positional notation 550 · achieved
  Method for representing or encoding numbers. 550— Hindu mathematicians give zero a numeral representation in the positional notation Indian numeral system.
• Candle clock 550~ · achieved
  Thin candle with consistently spaced markings (usually with numbers), that when burned, indicate the passage of periods of time. Candle clock: Candle clocks have been used in China since at least the 6th century AD. The earliest reference of a candle clock is in a poem by You Jiangu, from about AD 520.
• Diophantine equation 500 · achieved
  Polynomial equation with integer solutions. 500 — Aryabhata solves the general linear diophantine equation.
• Hindu–Arabic decimal place value 500 · achieved
  Place-value decimal numerals in India by ~5th century.
• Principal ideal domain 499 · achieved
  Algebraic structure. 499: Aryabhata discovers Bezout's identity, a foundational result to the theory of principal ideal domains.
• Aryabhata's astronomy and sine tables 499 · achieved
  The Aryabhatiya, 499 AD.
• Trigonometric function 450~ · achieved
  Function of an angle. Trigonometric functions – The trigonometric functions sine and versine originated in Indian astronomy along with the cosine and inversine, adapted from the full-chord Greek versions (to the modern half-chord versions).
• Bakhshali manuscript 400~ · achieved
  Group of mathematical manuscripts including an early use of the positional zero. c.
• Pappus's hexagon theorem 340~ · achieved
  Theorem that, if the vertices of a hexagon lie alternately on two lines, then the three pairs of opposite sides meet in three collinear points. c. 340—Greece, Pappus of Alexandria states his hexagon theorem and his centroid theorem.
• Chinese remainder theorem 300 · achieved
  Theorem for solving simultaneous congruences. 300–500—The Chinese remainder theorem is developed by Sun Tzu.
• Forma Urbis Romae 205 · achieved
  Marble map of ancient Rome (c.205-208). See Forma Urbis Romae (Severan Marble Plan), a carved marble ground plan of every architectural feature in ancient Rome.
• Greek alphabetic numerals 150 · achieved
  Numeration system used by the Koine Greek and earlier. 150: 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.
• Ptolemy's world map 150 · achieved
  2nd century Greco-Roman map of the world. Ptolemy's world map — It included 8,000 locations from Shetland islands to the Ethiopian plateau and from the Canary Islands to China.
• Zhang Heng seismoscope 132 · achieved
  The first seismoscope, 132 AD.
• Reflection 130 · achieved
  Change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. 130 AD – Claudius Ptolemy (in his work Optics ) wrote about the properties of light including: reflection, refraction, and color and tabulated angles of refraction for several media
• Water-powered armillary sphere 125 · achieved
  Zhang Heng's hydraulic armillary sphere.
• Mesoamerican Long Count calendar 36 BCE · achieved
  Base-20 day count; the earliest Long Count dates are from the 1st century BC.
• Maya zero and long count calendar 36 BCE · achieved
  The Long Count includes zero; earliest dated stela 36 BC.
• Julian calendar 45 BCE · achieved
  Calendar introduced by Julius Caesar in 45 BC. They made use of the Julian Calendar .
• The Nine Chapters on the Mathematical Art 100 BCE · achieved
  Han-era compendium including Gaussian elimination and negative numbers.
• Stellar precession 150 BCE~ · achieved
  Gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. 2nd century BC: Hipparchos discovers Axial precession.
• Negative real number 150 BCE · achieved
  Real number that is strictly less than zero. 150 BC – China, negative numbers appear in the Chinese text The Nine Chapters on the Mathematical Art.
• Horner's method 150 BCE · achieved
  Algorithm for polynomial evaluation. 150 BC – China, Horner's method appears in the Chinese text The Nine Chapters on the Mathematical Art.
• Gaussian elimination 150 BCE · achieved
  Algorithm for solving systems of linear equations. 150 BC – China, A method of Gaussian elimination appears in the Chinese text The Nine Chapters on the Mathematical Art.
• Magic square 190 BCE · achieved
  Sums of each row, column, and main diagonals are equal. 190 BC: Magic squares appear in China. The theory of magic squares can be considered the first example of a vector space.
• Epicycle 200 BCE · achieved
  Model of planetary motion in which planets moved in a small circle which in turn moves around the Earth. 200 BC: Apollonius of Perga develops epicycles. While an incorrect model, it was a precursor to the development of Fourier series.
• Apollonius' theorem 200 BCE · achieved
  Theorem. 200 BC: Apollonius of Perga discovers Apollonius's theorem.
• Conic section 225 BCE · achieved
  Curve obtained by intersecting a cone and a plane. 225 BC – Apollonius of Perga writes On Conic Sections and names the ellipse, parabola, and hyperbola,
• Indian mathematics 250 BCE~ · achieved
  Mathematics on the Indian subcontinent. 3rd century BC – Indian texts use the Sanskrit word " Shunya " to refer to the concept of "void" ( zero ).
• False position method 250 BCE · achieved
  Root-finding algorithm. Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu ( The Nine Chapters on the Mathematical Art ), which contains solutions of linear equations solved using the rule of double false position, geometric solut…
• Pascal's triangle 250 BCE~ · achieved
  Triangular array of the binomial coefficients in mathematics.
• Raised-relief map 250 BCE~ · achieved
  Three-dimensional object representing a real terrain. Raised-relief map (terrain model): The raised-relief map may have existed in China since the 3rd century BC, if the accounts in the Records of the Grand Historian (by Sima Qian, 91 BC) about Qin Shi Huang 's (r.
• Archimedean mechanics 250 BCE · achieved
  Lever, buoyancy, and early calculus-like methods.
• Fundamental theorem of arithmetic 300 BCE · achieved
  Theorem about prime factorization of a number. 300 BC: Euclid proves the Fundamental Theorem of Arithmetic.
• Euclid's theorem 300 BCE · achieved
  Theorem that the number of prime numbers is infinite. 300 BCE — Euclid proves the number of prime numbers is infinite.
• Rod calculus 300 BCE · achieved
  Calculating method used in ancient China. 300–500—In China, a description of rod calculus is written by Sun Tzu.
• Brahmi numerals 300 BCE~ · achieved
  Numeral system predating modern Hindu-Arabic numerals. c. 300 BC – India, Brahmi numerals (ancestor of the common modern base 10 numeral system )
• Combination 300 BCE · achieved
  Way of selecting things out of a group where order does not matter. 300 BC – India, Bhagavati Sutra, which contains the earliest information on combinations.
• Compass and straightedge construction 300 BCE~ · achieved
  Way of drawing geometric objects using only an idealized compass and straightedge. A geometric construction for the solution of the cubic is sought (doubling the cube problem). It is now well known that the general cubic has no such solution using Euclidean tools.
• Alchemy 300 BCE · achieved
  Greco-Egyptian, Chinese and Islamic transmutation and elixir traditions, from Zosimos to Newton's private notebooks.
• Euclidean geometry 300 BCE · achieved
  Euclid's Elements, ~300 BC.
• Archimedean property 350 BCE~ · achieved
  The absence of infinitesimals in a mathematical system. 4th century BC: Eudoxus of Cnidus states the Archimedean property.
• Logical negation 350 BCE~ · achieved
  Operation that takes a proposition p to another proposition "not p", written ¬p, which is interpreted intuitively as being true when p is false, and false when p is true; unary (single-argument) logical connective. 4th century BC: Greek philosophers study the properties of logical negation.
• Meteorology (Aristotle) 350 BCE · achieved
  Treatise by Aristotle. 350 BC – The Greek philosopher Aristotle writes Meteorology, a work which represents the sum of knowledge of the time about Earth sciences, including weather and climate.
• Logic 350 BCE · achieved
  Study of correct reasoning. 350 BC – Greece, Aristotle discusses logical reasoning in Organon.
• Infinite set 400 BCE~ · achieved
  Set with infinite cardinality. c. 400 BC – India, write the Surya Prajnapti, a mathematical text classifying all numbers into three sets: enumerable, innumerable, and infinite.
• Counting rods and decimal system 400 BCE · achieved
  Rod numerals enabled place-value decimal computation.
• Method of exhaustion 450 BCE~ · achieved
  Primitive way of calculating area. Late 5th century BC: Antiphon discovers the method of exhaustion, foreshadowing the concept of a limit.
• Apastamba Dharmasutra 450 BCE~ · achieved
  One of the oldest Dharma texts of Hinduism. 5th century BC – Apastamba, author of the Apastamba Sulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates the square root of 2 correct to five decimal places
• Lune 470 BCE · achieved
  Shape. 470 BC – 410 BC – Greece, Hippocrates of Chios utilizes lunes in an attempt to square the circle.
• Vedanga Jyotisha 500 BCE~ · achieved
  Work. final centuries BC – Indian astronomer Lagadha writes the Vedanga Jyotisha, a Vedic text on astronomy that describes rules for tracking the motions of the Sun and the Moon and uses geometry and trigonometry for astronomy.
• Luoshu Square 500 BCE~ · achieved
  Chinese name for a specific 3×3 magic square occurring in legendary accounts of the creation of Chinese writing and culture. second half of 1st millennium BC – The Luoshu Square, the unique normal magic square of order three, was discovered in China.
• Triangulation 550 BCE~ · achieved
  Determining the location of a point by forming triangles to it from known points. Triangulation — The use of triangles to estimate distances dates to antiquity.
• Greek natural philosophy 585 BCE~ · achieved
  Thales to Aristotle: nature explained without gods.
• Thales' theorem 600 BCE~ · achieved
  Theorem. 600 BC: Thales of Miletus is credited with proving Thales's theorem.
• Pythagorean triple 600 BCE~ · achieved
  Three positive integers, the squares of two of which sum to the square of the third. c. 600 BC – India, the other Vedic "Shulba Sutras" ("rule of chords" in Sanskrit ) use Pythagorean triples, contain a number of geometrical proofs, and approximate π at 3.16.
• Vyakarana 700 BCE~ · achieved
  The Vedic era study of grammar and linguistics, one of six Vedangas. 700 BC: Grammar is first studied in India (note that Sanskrit Vyākaraṇa predates Pāṇini ).
• Pell's equation 700 BCE~ · achieved
  Mathematical equation, specifically a kind of Diophantine equation. 700 BC: Pell's equations are first studied by Baudhayana in India, the first diophantine equations known to be studied.
• Shulba Sutras geometry 800 BCE~ · achieved
  Vedic geometry, including Pythagorean relations.
• Fraction 1,000 BCE~ · achieved
  Mathematical representation of a portion of a whole. Fraction and arithmetics with fractions — The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. The Egyptians used Egyptian fractions c.
• Babylonian astronomy 1,000 BCE~ · achieved
  Systematic astronomical records and predictive arithmetic.
• Star catalogue 1,100 BCE~ · achieved
  Astronomical catalogue that lists stars. 1100 BC Star catalogue — Three Stars Each is the earliest known catalogue in long-running tradition of Babylonian astronomy, likely drawing on Sumerian and/or Elamite constellations.
• Rhind Mathematical Papyrus 1,550 BCE~ · achieved
  Egyptian mathematics: fractions, geometry, and algebra problems.
• Square root 1,650 BCE~ · achieved
  Inverse operation of square for finding the original base number; that base number itself. Square root — The Rhind Mathematical Papyrus is a copy from 1650 BC of an earlier Berlin Papyrus and other texts – possibly the Kahun Papyrus – that shows how the Egyptians extracted square roots by an inverse proportion method.
• Ancient Egyptian multiplication 1,700 BCE~ · achieved
  Multiplication algorithm. c. 1700–2000 BC – Egyptians develop earliest known algorithms for multiplying two numbers
• Methods of computing square roots 1,800 BCE~ · achieved
  Algorithms for calculating square roots. By the 4th century: A square root finding algorithm with quartic convergence, known as the Bakhshali method (after the Bakhshali manuscript which records it), is discovered in India.
• YBC 7289 1,800 BCE~ · achieved
  Ancient clay tablet showing an approximation of √2. 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.
• Egyptian fraction 1,800 BCE~ · achieved
  Finite sum of distinct unit fractions. 1800 BC: The Middle Kingdom of Egypt develops Egyptian fraction notation.
• Moscow Mathematical Papyrus 1,800 BCE~ · achieved
  Ancient Egyptian mathematical manuscript. 1800 BC – Moscow Mathematical Papyrus, findings volume of a frustum
• Plimpton 322 (Babylonian Pythagorean triples) 1,800 BCE~ · achieved
  Babylonian clay tablet listing Pythagorean triples.
• Babylonian numerals 2,000 BCE~ · achieved
  Numeral system used in ancient Babylonia. 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. 13 200 would be written the same as 132 ).
• Papyrus Berlin 6619 2,000 BCE~ · achieved
  Mathematical papyrus. 2000 BC: Pythagorean triples are first discussed in Babylon and Egypt, and appear on later manuscripts such as the Berlin Papyrus 6619.
• Early geometry (Egypt and Babylon) 2,000 BCE~ · achieved
  Land survey and area computations.
• Quadratic equation 2,100 BCE~ · achieved
  Polynomial equation in a single variable where the highest exponent of the variable is 2. 2100 BC: Quadratic equations, in the form of problems relating the areas and sides of rectangles, are solved by Babylonians.
• Goniometer 2,200 BCE~ · achieved
  Measuring instrument. 2200 BC: Protractor, Phase IV, Lothal, Indus Valley (modern-day India), a Xancus shell cylinder with sawn grooves, at right angles, in its top and bottom surfaces, has been proposed as an angle marking tool.
• Calendar and astronomy 2,500 BCE~ · achieved
  Egyptian civil calendar; Babylonian astronomical records.
• Pi 2,560 BCE~ · achieved
  Constant ratio of the circumference of a circle to its diameter. Pi — Based on the measurements of the Great Pyramid of Giza (c. 2560 BC), some Egyptologists have claimed that the ancient Egyptians used an approximation of π as ⁠ 22 / 7 ⁠ from as early as the Old Kingdom.
• Deben 2,600 BCE~ · achieved
  Ancient Egyptian weight unit. 2600 BC: 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, excavated from the reign of Sneferu, though earlier usage has been…
• Weighing scale 2,600 BCE~ · achieved
  Instrument used in measuring the weight or mass of an object. Weighing scale — The oldest attested evidence for the existence of weighing scales dates to the Fourth Dynasty of Egypt, with Deben (unit) balance weights, from the reign of Sneferu (c.
• Ruler 2,650 BCE~ · achieved
  Tool used to draw or measure distances along a straight line. 2650 BC: The Ruler, or Measuring rod, in the subdivided Nippur, copper rod, of the Sumerian Civilisation (modern-day Iraq).
• Egyptian civil calendar 2,700 BCE~ · achieved
  A 365-day solar calendar, the ancestor of the Julian calendar.
• Egyptian numeral 3,000 BCE~ · achieved
  Numerals used in Ancient Eygpt. 3000 BC: The first deciphered numeral system is that of the Egyptian numerals, a sign-value system (as opposed to a place-value system).
• Numerals and arithmetic 3,000 BCE~ · achieved
  Sumerian sexagesimal numeration.
• Decimal 3,100 BCE~ · achieved
  Numeral system with ten as its base. c. 3100 BC – Egypt, the earliest known decimal system allows indefinite counting by way of introducing new symbols.
• Ishango bone 20,000 BCE~ · achieved
  Tally-marked baboon fibula from the Congo, ~20,000 years old, possibly an early counting record.
• Tally stick 42,000 BCE~ · achieved
  Ancient memory aid device used to record and document quantities. 44 kya – 42 kya: Tally sticks (see Lebombo bone ) in Eswatini
• Lebombo bone tally 42,000 BCE~ · achieved
  Notched baboon fibula from Eswatini, about 42,000 years old.