Famous Scientists

  • Home
  • Top 100 Scientists
  • List of Scientists
  • Blog

Diophantus

Diophantus

Lived c. 210 – c. 295 AD.

Diophantus is known as the father of algebra. Roughly five centuries after Euclid’s era, he solved hundreds of algebraic equations in his great work Arithmetica, and was the first person to use algebraic notation and symbolism.

Today we usually indicate the unknown quantity in algebraic equations with the letter x. In the oldest copies of Arithmetica the unknown quantity is indicated by a character similar to an accented Greek letter sigma: ς’.

Arithmetica inspired some of the world’s greatest mathematicians including Leonhard Euler and Pierre de Fermat to make significant new discoveries.

Advertisements

The Life of Diophantus

Diophantus (pronounced dy-o-Fant-us) flourished during the third century AD in the Greco-Roman city of Alexandria in Egypt.

Like other educated people in the Eastern Mediterranean at that time he was a Greek speaker. We do not know what he looked like. The years of his birth and death are highly uncertain.

The little we know about Diophantus’ life comes from a word puzzle reputed to be his epitaph. Presumably it was written by a friend who knew Diophantus’ life story and who wished to give him a fittingly algebraic memorial. The epitaph is known to us through the Greek author Metrodorus who recorded it in his anthology of puzzles in about the sixth century.

A 1941 translation of the epitaph by Ivor Thomas says:

This tomb holds Diophantus. Ah, what a marvel.
And the tomb tells scientifically the measure of his life.
God vouchsafed that he should be a boy for the sixth part of his life;
When a twelfth was added, his cheeks acquired a beard;
He kindled for him the light of marriage after a seventh,
And in the fifth year after his marriage He granted him a son.
Alas! late-begotten and miserable child, when he had reached the measure of half his father’s life, the chill grave took him.
After consoling his grief by this science of numbers for four years, he reached the end of his life.

We can solve the epitaph as an algebraic equation:

diophantus epitaph

And we find x = 84, from which it follows:

Diophantus’ boyhood lasted 14 years.
When he was 21, his beard grew.
He married at 33.
His son was born when Diophantus was 38.
His son died age 42, when Diophantus was 80.
Diophantus died at age 84.

Lifetimes of Selected Greek Mathematicians

diophantus-life-scholars

A Brief History of Algebra before Diophantus

Algebra has a long history. Between 2000 – 1600 BC the Babylonians produced rather sophisticated algebra some of which survives on clay tablets. Babylon’s mathematicians were not concerned with exact numerical solutions to problems – they were happy with good approximations from reference tables they compiled. They could solve quadratic equations using geometric drawings of areas and lengths of squares.

The Rhind Mathematical Papyrus dating to about 1550 BC contains Ancient Egyptian algebra, such as: What must the number (1 + ½ + ¼) be multiplied by to give the answer 10?

The Nine Chapters on the Mathematical Art from China was composed in stages over a timespan possibly stretching between 1000 BC – 200 AD. In the eighth chapter, agricultural problems give rise to linear equations solved using rows of numbers similar to matrices.

Modern historians of mathematics sometimes wrangle over Book 2 of Euclid’s Elements from about 300 BC, debating whether it contains algebra written in geometric language. Certainly the great 11th century Persian mathematician Omar Khayyam had no doubts:

Omar Khayyam“Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved by Propositions 5 and 6 of Book 2 of Euclid’s Elements.”

Omar Khayyam
 

Khayyam was arguing for algebra to be seen as a legitimate branch of mathematics. Many mathematicians felt uneasy about algebra because it lacked the compelling logical rigor Euclid’s Elements had brought to geometry. Khayyam’s work with cubic equations gave him certainty that algebra and geometry are linked.

The Mathematics of Diophantus

Introduction to Arithmetica

Diophantus tells us at the beginning of his classic work Arithmetica that he has written it as a textbook to help his friend Dionysius (and others presumably) to solve mathematics problems. Arithmetica tackles the construction and solution of equations to find one or more unknowns. All copies of Arithmetica in Diophantus’ time were handwritten. Copies were made by scribes for over a thousand years until the first copies were printed in Europe.

Diophantus“Knowing, my most esteemed friend Dionysius, that you are anxious to learn how to investigate problems in numbers, I have tried, beginning from the foundations on which the science is built up, to set forth to you the nature and power subsisting in numbers.

“Perhaps the subject will appear rather difficult, because is not yet familiar (beginners are, as a rule, too ready to despair of success); but you, with the impulse of your enthusiasm and the benefit of my teaching, will find it easy to master; for eagerness to learn, when seconded by instruction, ensures rapid progress.”

Diophantus
Arithmetica, c. 250 AD
 

Arithmetica may have been conceived in a similar way to Euclid’s Elements. Euclid compiled and, where necessary, improved on the work of mathematicians such as Eudoxus and the Pythagoreans. Any other books dating from classical times with similar themes to Arithmetica have been lost.

Volumes of Arithmetica

Diophantus composed Arithmetica in thirteen volumes of which six survived in Greek. Four exist as Arabic translations.

  • Volumes 1, 2, and 3 survive in Greek from Byzantium.
  • Volumes 4, 5, 6, and 7 exist as Arabic translations of Greek from Baghdad. They were translated in the ninth century by the Byzantine Christian scholar Kostas Luka known in Arabic as Qusta ibn Luqa. The books were discovered in 1971 in Meshed, Iran, where they had been misfiled for centuries in the Astan Quds Library as the work of Qusta ibn Luqa rather than Diophantus.
  • Three other volumes of Arithmetica exist in Greek, but their volume numbers are uncertain – they could be any three of volumes 8, 9, 10, 11, 12, or 13.

The Arabic translations contain more commentary on the solutions than the Greek versions. It is possible that the Arabic editions were copied from the lost edition adapted by Hypatia for the students at her school.

The Mathematics in Arithmetica

Diophantus begins with definitions and rules. For example, he defines the results of multiplication of quantities with different signs and tells his readers he will indicate subtraction with a symbol. He says:

Diophantus“A minus multiplied by a minus makes a plus; a minus multiplied by a plus makes a minus; and the sign of a minus is a truncated Ψ turned upside down, thus diophantus-minus-sign.”

Diophantus
Arithmetica, c. 250 AD
 

Book 1, Problem 1

The Problem
In the very first problem in the very first book of Arithmetica Diophantus asks his readers to divide a given number into two numbers that have a given difference.

The number he gives his readers is 100 and the given difference is 40.

The Solution
Diophantus writes [we use modern notation]:

2x + 40 = 100

Hence x = 30

Therefore the two numbers are 30 and 70.

Things Get Harder

Diophantus continues in this way, describing hundreds of problems which he translates into solvable equations. The level of difficulty rises as he introduces quadratics, cubics, and equations in higher powers of x.

Quadratics, Cubics, and More

At high school, we learn about equations of general form ax2 + bx + c = 0; these are quadratic equations.

Cubic equations are of general form ax3 + bx2 + cx + d = 0. Cubic equations are harder to solve than quadratics.

Diophantus ignores negative and irrational solutions to equations.

A major criticism of Diophantus is that he rarely offers his readers general methods to solve a particular class of problem such as quadratics.

The level of mathematical sophistication in Arithmetica is sometimes disconcertingly high. It required the genius of 18th century mathematician Joseph Lagrange to finally prove that every number can be written as the sum of four squares – a result Diophantus was well aware of.

Diophantine Equations

In Arithmetica, Diophantus launched the study of indeterminate equations – these are polynomial equations in which the number of unknowns exceeds the number of equations given.

The solutions to Diophantus’ indeterminate equations were always positive rational numbers. (Diophantus was interested only in single number solutions, so he did not, for example, seek two numbers as solutions to quadratic equations.)

Nowadays we define a Diophantine Equation as an indeterminate equation whose solutions must be integers.

Consider this Diophantine Equation:

Fermat's last theorem

If n=2, we have Pythagoras’s theorem, which has an infinite number of whole number solutions, the most famous example of which is the 3-4-5 triangle: x=3, y=4, z=5.

Fermat’s Last Theorem claims that if n is a whole number bigger than 2, the equation has no whole number solutions for x, y and z. Fermat claimed to have proved this for all values of n, but famously said the margin of his book was too small to write his proof. Not surprisingly, the book was Diophantus’ Arithmetica.

After Arithmetica

About four centuries after Diophantus wrote Arithmetica, the great seventh century Indian mathematician Brahmagupta found the general solution for linear and quadratic equations.

After another two centuries, the great Persian mathematician al-Khwarizmi presented systematic solutions of linear and quadratic equations. This was soon followed by Qusta ibn Luqa’s ninth century translation of Arithmetica into Arabic.

Brahmagupta and al-Khwarizmi’s works were less ambitious than Diophantus’ in that they dealt with equations in x and x2. Diophantus frequently dealt with cubic and higher power equations, up to x9. Also, neither of them used the symbolic algebra Diophantus had pioneered. Brahmagupta and al-Khwarizmi’s crucial contribution is the concept of the general solution of an equation. General solutions are not offered in any of the surviving books of Arithmetica.

The fusion of al-Khwarizmi’s highly systematic algebra with Diophantus’ intriguing problems and solutions led to a great flowering of algebra in Persia and the Islamic world. In the eleventh century, Omar Khayyam showed how the intersections of conic sections such as parabolas and circles can yield geometric solutions of cubic equations.

Following the Renaissance, European mathematicians of the highest rank were captivated by the mathematics of Arithmetica.

In 1535, Niccolo Tartaglia found general solutions for all cubic equations.

In the 1600s, Arithmetica inspired a great number of Pierre de Fermat’s new ideas. He worked on Arithmetica for pleasure, much as a modern person might work on a crossword puzzle or a game of Sudoku. When new ideas came to him, he scribbled them in the margin of the book. These ideas, including Fermat’s Last Theorem, transformed number theory.

In the 1700s, about 1,500 years after Diophantus wrote Arithmetica Leonhard Euler took great delight and inspiration from attacking its trickier problems. Euler’s words provide us with a fitting final tribute to Diophantus:

Leonhard Euler“Diophantus himself, it is true, gives only the most special solutions of all the questions which he treats… Nevertheless, the actual methods which he uses for solving any of his problems are as general as those which are in use today; nay, we are obliged to admit that there is hardly any method yet invented in this kind of analysis of which there are not sufficiently distinct traces to be discovered in Diophantus.”

Leonhard Euler
Novi Commentarii Academiae Petropolitanae, 1761, Translated by Sir Thomas Heath
 
Advertisements

Author of this page: The Doc
© All rights reserved.

Cite this Page

Please use the following MLA compliant citation:

"Diophantus." Famous Scientists. famousscientists.org. 8 Jul. 2018. Web.  
<www.famousscientists.org/diophantus/>.

Published by FamousScientists.org

Further Reading
Sir Thomas Heath
Diophantus of Alexandria; A study in the history of Greek algebra
Cambridge University Press, 1910

Morris Kline
Mathematical Thought from Ancient to Modern Times
Oxford University Press, New York, 1972

Norbert Schappacher
Diophantus of Alexandria : a Text and its History

More from FamousScientists.org:
  • Euclid
    Euclid
  • Omar Khayyam
    Omar Khayyam
  • Hypatia
    Hypatia
  • Brahmagupta
    Brahmagupta
Advertisements

Search Famous Scientists

Scientist of the Week

  • Linda Buck: Discovered how we smell things

Recent Scientists of the Week

  • Jan Ingenhousz: Discovered photosynthesis
  • Barry Marshall: Overturned the Medical Establishment
  • Linus Pauling: Maverick Giant of Chemistry
  • William Röntgen: The Discovery of X-rays
  • Howard Florey: Brought penicillin to the world
  • Henrietta Leavitt: The key to the size of the universe
  • Archimedes: A mind beyond his time
  • Stanley Milgram: The infamous Obedience Experiments
  • C. V. Raman: Color change allows harm-free health check of living cells
  • Rosalind Franklin: Shape-shifting DNA
  • Robert Boyle: A new science is born: chemistry
  • Carl Woese: Rewrote Earth’s history of life
  • Alfred Wegener: Shunned after he discovered that continents move
  • Henri Poincaré: Is the solar system stable?
  • Polly Matzinger: The dog whisperer who rewrote our immune system’s rules
  • Otto Guericke: In the 1600s found that space is a vacuum
  • Alister Hardy: Aquatic ape theory: our species evolved in water
  • Elizebeth Friedman: Became the world’s most famous codebreaker
  • Evangelista Torricelli: We live at the bottom of a tremendously heavy sea of air
  • Eudoxus: The first mathematical model of the universe
  • James Black: Revolutionized drug design with the Beta-blocker
  • Inge Lehmann: Discovered our planet’s solid inner core
  • Chen-Ning Yang: Shattered a fundamental belief of physicists
  • Robert Hooke: Unveiled the spectacular microscopic world
  • Barbara McClintock: A Nobel Prize after years of rejection
  • Pythagoras: The cult of numbers and the need for proof
  • J. J. Thomson: Discovered the electron
  • Johannes Kepler: Solved the mystery of the planets
  • Dmitri Mendeleev: Discovered 8 new chemical elements by thinking
  • Maurice Hilleman: Record breaking inventor of over 40 vaccines
  • Marie Curie: Won – uniquely – both the chemistry & physics Nobel Prizes
  • Jacques Cousteau: Marine pioneer, inventor, Oscar winner
  • Niels Bohr: Founded the bizarre science of quantum mechanics
  • Srinivasa Ramanujan: Untrained genius of mathematics
  • Milutin Milankovic: Proved Earth’s climate is regulated by its orbit
  • Antoine Lavoisier: The giant of chemistry who was executed
  • Emmy Noether: The greatest of female mathematicians, she unlocked a secret of the universe
  • Wilder Penfield: Pioneer of brain surgery; mapped the brain’s functions
  • Charles Nicolle: Eradicated typhus epidemics
  • Samuel Morse: The telegraph and Morse code
  • Jane Goodall: Major discoveries in chimpanzee behavior
  • John Philoponus: 6th century anticipation of Galileo and Newton
  • William Perkin: Youthful curiosity brought the color purple to all
  • Democritus: Atomic theory BC and a universe of diverse inhabited worlds
  • Susumu Tonegawa: Discovered how our bodies make millions of different antibodies
  • Cecilia Payne: Discovered that stars are almost entirely hydrogen and helium

Top 100 Scientists

  • Our Top 100 Scientists

Our Most Popular Scientists

  • Astronomers
  • Biologists & Health Scientists
  • Chemists
  • Geologists and Paleontologists
  • Mathematicians
  • Physicists
  • Scientists in Ancient Times

List of Scientists

  • Alphabetical List

Recent Posts

  • Perfect Numbers and our Tiny Universe
  • What Happens when the Universe chooses its own Units?
  • Hipparchus and the 2000 Year-Old Clue
  • Darwin Pleaded for Cheaper Origin of Species
  • You Will Die For Showing I’m Wrong!
  • Getting Through Hard Times – The Triumph of Stoic Philosophy
  • Johannes Kepler, God, and the Solar System
  • Charles Babbage and the Vengeance of Organ-Grinders
  • Howard Robertson – the Man who Proved Einstein Wrong
  • Susskind, Alice, and Wave-Particle Gullibility




Alphabetical List of Scientists

Louis Agassiz | Maria Gaetana Agnesi | Al-BattaniAbu Nasr Al-Farabi | Alhazen | Jim Al-Khalili | Muhammad ibn Musa al-Khwarizmi | Mihailo Petrovic Alas | Angel Alcala | Salim Ali | Luis Alvarez | Andre Marie Ampère | Anaximander | Carl Anderson | Mary Anning | Virginia Apgar | Archimedes | Agnes Arber | Aristarchus | Aristotle | Svante Arrhenius | Oswald Avery | Amedeo Avogadro | Avicenna

Charles Babbage | Francis Bacon | Alexander Bain | John Logie Baird | Joseph Banks | Ramon Barba | John Bardeen | Charles Barkla | Ibn Battuta | William Bayliss | George Beadle | Arnold Orville Beckman | Henri Becquerel | Emil Adolf Behring | Alexander Graham Bell | Emile Berliner | Claude Bernard | Timothy John Berners-Lee | Daniel Bernoulli | Jacob Berzelius | Henry Bessemer | Hans Bethe | Homi Jehangir Bhabha | Alfred Binet | Clarence Birdseye | Kristian Birkeland | James Black | Elizabeth Blackwell | Alfred Blalock | Katharine Burr Blodgett | Franz Boas | David Bohm | Aage Bohr | Niels Bohr | Ludwig Boltzmann | Max Born | Carl Bosch | Robert Bosch | Jagadish Chandra Bose | Satyendra Nath Bose | Walther Wilhelm Georg Bothe | Robert Boyle | Lawrence Bragg | Tycho Brahe | Brahmagupta | Hennig Brand | Georg Brandt | Wernher Von Braun | J Harlen Bretz | Louis de Broglie | Alexander Brongniart | Robert Brown | Michael E. Brown | Lester R. Brown | Eduard Buchner | Linda Buck | William Buckland | Georges-Louis Leclerc, Comte de Buffon | Robert Bunsen | Luther Burbank | Jocelyn Bell Burnell | Macfarlane Burnet | Thomas Burnet

Benjamin Cabrera | Santiago Ramon y Cajal | Rachel Carson | George Washington Carver | Henry Cavendish | Anders Celsius | James Chadwick | Subrahmanyan Chandrasekhar | Erwin Chargaff | Noam Chomsky | Steven Chu | Leland Clark | John Cockcroft | Arthur Compton | Nicolaus Copernicus | Gerty Theresa Cori | Charles-Augustin de Coulomb | Jacques Cousteau | Brian Cox | Francis Crick | James Croll | Nicholas Culpeper | Marie Curie | Pierre Curie | Georges Cuvier | Adalbert Czerny

Gottlieb Daimler | John Dalton | James Dwight Dana | Charles Darwin | Humphry Davy | Peter Debye | Max Delbruck | Jean Andre Deluc | Democritus | René Descartes | Rudolf Christian Karl Diesel | Diophantus | Paul Dirac | Prokop Divis | Theodosius Dobzhansky | Frank Drake | K. Eric Drexler

John Eccles | Arthur Eddington | Thomas Edison | Paul Ehrlich | Albert Einstein | Gertrude Elion | Empedocles | Eratosthenes | Euclid | Eudoxus | Leonhard Euler

Michael Faraday | Pierre de Fermat | Enrico Fermi | Richard Feynman | Fibonacci – Leonardo of Pisa | Emil Fischer | Ronald Fisher | Alexander Fleming | John Ambrose Fleming | Howard Florey | Henry Ford | Lee De Forest | Dian Fossey | Leon Foucault | Benjamin Franklin | Rosalind Franklin | Sigmund Freud | Elizebeth Smith Friedman

Galen | Galileo Galilei | Francis Galton | Luigi Galvani | George Gamow | Martin Gardner | Carl Friedrich Gauss | Murray Gell-Mann | Sophie Germain | Willard Gibbs | William Gilbert | Sheldon Lee Glashow | Robert Goddard | Maria Goeppert-Mayer | Thomas Gold | Jane Goodall | Stephen Jay Gould | Otto von Guericke

Fritz Haber | Ernst Haeckel | Otto Hahn | Albrecht von Haller | Edmund Halley | Alister Hardy | Thomas Harriot | William Harvey | Stephen Hawking | Otto Haxel | Werner Heisenberg | Hermann von Helmholtz | Jan Baptist von Helmont | Joseph Henry | Caroline Herschel | John Herschel | William Herschel | Gustav Ludwig Hertz | Heinrich Hertz | Karl F. Herzfeld | George de Hevesy | Antony Hewish | David Hilbert | Maurice Hilleman | Hipparchus | Hippocrates | Shintaro Hirase | Dorothy Hodgkin | Robert Hooke | Frederick Gowland Hopkins | William Hopkins | Grace Murray Hopper | Frank Hornby | Jack Horner | Bernardo Houssay | Fred Hoyle | Edwin Hubble | Alexander von Humboldt | Zora Neale Hurston | James Hutton | Christiaan Huygens | Hypatia

Ernesto Illy | Jan Ingenhousz | Ernst Ising | Keisuke Ito

Mae Carol Jemison | Edward Jenner | J. Hans D. Jensen | Irene Joliot-Curie | James Prescott Joule | Percy Lavon Julian

Michio Kaku | Heike Kamerlingh Onnes | Pyotr Kapitsa | Friedrich August Kekulé | Frances Kelsey | Pearl Kendrick | Johannes Kepler | Abdul Qadeer Khan | Omar Khayyam | Alfred Kinsey | Gustav Kirchoff | Martin Klaproth | Robert Koch | Emil Kraepelin | Thomas Kuhn | Stephanie Kwolek

Joseph-Louis Lagrange | Jean-Baptiste Lamarck | Hedy Lamarr | Edwin Herbert Land | Karl Landsteiner | Pierre-Simon Laplace | Max von Laue | Antoine Lavoisier | Ernest Lawrence | Henrietta Leavitt | Antonie van Leeuwenhoek | Inge Lehmann | Gottfried Leibniz | Georges Lemaître | Leonardo da Vinci | Niccolo Leoniceno | Aldo Leopold | Rita Levi-Montalcini | Claude Levi-Strauss | Willard Frank Libby | Justus von Liebig | Carolus Linnaeus | Joseph Lister | John Locke | Hendrik Antoon Lorentz | Konrad Lorenz | Ada Lovelace | Percival Lowell | Lucretius | Charles Lyell | Trofim Lysenko

Ernst Mach | Marcello Malpighi | Jane Marcet | Guglielmo Marconi | Lynn Margulis | Barry Marshall | Polly Matzinger | Matthew Maury | James Clerk Maxwell | Ernst Mayr | Barbara McClintock | Lise Meitner | Gregor Mendel | Dmitri Mendeleev | Franz Mesmer | Antonio Meucci | John Michell | Albert Abraham Michelson | Thomas Midgeley Jr. | Milutin Milankovic | Maria Mitchell | Mario Molina | Thomas Hunt Morgan | Samuel Morse | Henry Moseley

Ukichiro Nakaya | John Napier | Giulio Natta | John Needham | John von Neumann | Thomas Newcomen | Isaac Newton | Charles Nicolle | Florence Nightingale | Tim Noakes | Alfred Nobel | Emmy Noether | Christiane Nusslein-Volhard | Bill Nye

Hans Christian Oersted | Georg Ohm | J. Robert Oppenheimer | Wilhelm Ostwald | William Oughtred

Blaise Pascal | Louis Pasteur | Wolfgang Ernst Pauli | Linus Pauling | Randy Pausch | Ivan Pavlov | Cecilia Payne-Gaposchkin | Wilder Penfield | Marguerite Perey | William Perkin | John Philoponus | Jean Piaget | Philippe Pinel | Max Planck | Pliny the Elder | Henri Poincaré | Karl Popper | Beatrix Potter | Joseph Priestley | Proclus | Claudius Ptolemy | Pythagoras

Adolphe Quetelet | Harriet Quimby | Thabit ibn Qurra

C. V. Raman | Srinivasa Ramanujan | William Ramsay | John Ray | Prafulla Chandra Ray | Francesco Redi | Sally Ride | Bernhard Riemann | Wilhelm Röntgen | Hermann Rorschach | Ronald Ross | Ibn Rushd | Ernest Rutherford

Carl Sagan | Abdus Salam | Jonas Salk | Frederick Sanger | Alberto Santos-Dumont | Walter Schottky | Erwin Schrödinger | Theodor Schwann | Glenn Seaborg | Hans Selye | Charles Sherrington | Gene Shoemaker | Ernst Werner von Siemens | George Gaylord Simpson | B. F. Skinner | William Smith | Frederick Soddy | Mary Somerville | Arnold Sommerfeld | Hermann Staudinger | Nicolas Steno | Nettie Stevens | William John Swainson | Leo Szilard

Niccolo Tartaglia | Edward Teller | Nikola Tesla | Thales of Miletus | Theon of Alexandria | Benjamin Thompson | J. J. Thomson | William Thomson | Henry David Thoreau | Kip S. Thorne | Clyde Tombaugh | Susumu Tonegawa | Evangelista Torricelli | Charles Townes | Youyou Tu | Alan Turing | Neil deGrasse Tyson

Harold Urey

Craig Venter | Vladimir Vernadsky | Andreas Vesalius | Rudolf Virchow | Artturi Virtanen | Alessandro Volta

Selman Waksman | George Wald | Alfred Russel Wallace | John Wallis | Ernest Walton | James Watson | James Watt | Alfred Wegener | John Archibald Wheeler | Maurice Wilkins | Thomas Willis | E. O. Wilson | Sven Wingqvist | Sergei Winogradsky | Carl Woese | Friedrich Wöhler | Wilbur and Orville Wright | Wilhelm Wundt

Chen-Ning Yang

Ahmed Zewail

Return to top of page

Famous Scientists - Privacy - Contact - About - Content & Imagery © 2023