Click on the below to see a list of primary sources from math history. They are organized according to the chapters of Math History: A Long-Form Mathematics Textbook. Clicking on the local link will pull up a flipbook-style reader. For books under copyright (which is mostly a new translation of an old text), a link is provided to purchase the book.
If you know of a good primary source that I do not include here, please let me know! You can email me at LongFormMath@gmail.com.
Chapter 1: Number Systems
- Here are pictures of the Lebombo bone.
- Here are pictures of the Ishango bone.
- Here are pictures of Quipus.
- Here are pictures of old abaci.
- 1573 Ming dynasty style suanpan
- Qing Dynasty ring abacus
- Russian schoty abacus
Chapter 2: Ancient Methods
- Pictures of the Rhind papyrus can be found on the British Museum’s website, and fragments of it that I found online are below.
- Pictures of the Moscow papyrus are harder to find. Problem 14 is pictured below. If you know of other images please let me know and I’ll add them below.
- This is an English translation with commentary of Book on Numbers and Computation, also known as the Suàn shù shū, which was one of the earliest surviving books in Chinese math history.
Chapter 3: Transition to Proofs
- Much of what we know about ancient Greek math is from the writings of Proclus, which includes the so-called “Eudemian Summary.” Most of these references are found in the following text, called Commentary on the First Book of Euclid’s Elements, which does not focus solely on Euclid.
- We discussed the Vedic Śulbasūtras in this chapter, and how their constructions of fire altars illustrate some of their math knowledge.
- Link to view a book locally on the geometry in the Śulbasūtras, with detailed look at the Śulbasūtra text.
- Link to view the Baudhāyana Śulbasūtra in Sanskrit online.
- Link to view the Baudhāyana Śulbasūtra in Sanskrit locally.
- Link to view the Āpastamba Śulbasūtra in Sanskrit online.
- Link to view the Āpastamba Śulbasūtra in Sanskrit locally.
- Link to view the Kātyāyana Śulbasūtra in Sanskrit online.
- Link to view the Kātyāyana Śulbasūtra in Sanskrit locally.
Chapter 4: Euclidean Geometry
- Euclid’s Elements.
- This text contains the original Greek with an English translation on the side. This is J.L. Heiberg’s edition. Link to view it online.
- This text contains the original Greek with an English translation on the side. This is J.L. Heiberg’s edition. Link to view it locally.
- Link to peruse a good online edition. This is the version I regularly consulted while writing this chapter.
- Link to buy an English translation on Amazon.
- Link to buy Oliver Byrne’s beautiful edition of the first 6 books on Amazon.
- Much of what we know about ancient Greek math is from the writings of Proclus, which includes the so-called “Eudemian Summary.” Most of these references are found in the following text, called Commentary on the First Book of Euclid’s Elements, which includes a brief discussion on mathematicians other than Euclid, including Thales and Pythagoras.
Chapter 5: Number Theory
- The following are the key moments in Fermat’s little theorem.
- Local link of Fermat’s (circa August 1640) letter to Bernard Frenicle in which he first explores the little theorem. This includes the original French and an English translation.
- Local link of Fermat’s October 18, 1640, letter to Bernard Frenicle in which he states (but does not prove) his little theorem. This includes the original French and an English translation.
- Online link of Euler’s paper proving the little theorem.
- Local link of Euler’s paper proving the little theorem.
- Carl Friedrich Gauss’s 1801 masterpiece, Disquisitiones Arithmeticae, introduced modular arithmetic and formalized the concept of congruences, laying the foundation for modern number theory. This groundbreaking work includes the famous Law of Quadratic Reciprocity and explores the properties of modular systems.
- Pafnuty Chebyshev’s 1850 paper, Mémoire sur les nombres premiers (“Memoir on Prime Numbers”), laid the groundwork for analytic number theory. In this work, Chebyshev established bounds on the prime-counting function and developed tools that led to the eventual proof of the Prime Number Theorem.
- Fermat conjectured that every integer is the sum of four squares. Euler proved this conjeture using innovative algebraic techniques.
- Charles Hermite’s 1873 paper, Sur la fonction exponentielle (“On the Exponential Function”), provided the first proof of the transcendence of e. Indeed, it was the first proof of the trascendence of any specific number.
- Euclid’s Elements.
- This text contains the original Greek with an English translation on the side. This is J.L. Heiberg’s edition. Link to view it online.
- This text contains the original Greek with an English translation on the side. This is J.L. Heiberg’s edition. Link to view it locally.
- Link to peruse a good online edition. This is the version I regularly consulted while writing this chapter.
- Link to buy an English translation on Amazon.
- Link to buy Oliver Byrne’s beautiful edition of the first 6 books on Amazon.
Chapter 6: Algebra
- Girolamo Cardano’s 1545 masterpiece, Ars Magna (“The Great Art”), introduced general solutions to cubic and quartic equations, laying the groundwork for modern algebra. It also marked one of the earliest uses of complex numbers in mathematical problem-solving.
- Niels Henrik Abel’s 1824 paper, Mémoire sur les équations algébriques, où on démontre l’impossibilité de la résolution de l’équation générale du cinquième degré (“Memoir on Algebraic Equations, in which the Impossibility of the Resolution of the General Equation of the Fifth Degree is Demonstrated”), proved that no general solution by radicals exists for quintic equations. This groundbreaking work solved a 250-year-old question.
- The complete mathematical writings of Évariste Galois, including his famous letter to Auguste Chevalier written the night before his fatal duel (as scans, in the original French, and translated to English), are compiled in The Mathematical Writings of Évariste Galois by Peter M. Neumann. This collection provides translations, commentary, and critical context for Galois’s groundbreaking work in group theory and algebra.
- René Descartes’s 1637 La Géométrie (The Geometry) is a foundational work that introduced analytic geometry, bridging algebra and geometry. The edition below includes both the original French text and an English translation, woven together in the document.
- Leonhard Euler’s Elements of Algebra is a foundational text that systematically introduces the basics of algebra, including equations, number theory, and the foundations of modern algebraic thought. This work remains one of the most influential algebra textbooks in the history. In each document, the English translation precedes the original German.
Chapter 7: Early Calculus
- The Archimedes Palimpset is a document containing a copy of at least seven treatises by Archimedes. These treatises are The Equilibrium of Planes, Spiral Lines, The Measurement of the Circle, Sphere and Cylinder, On Floating Bodies, The Method of Mechanical Theorems, and the Stomachion. The Archimedes Palimpsest Project includes scans of this document in the Digital tab.
- Link to the Archimedes Palimpsest Project.
- Link to a list of English translations of Archimedes’ works which are freely available on the Internet.
- Link to a 6-page English translation of The Method of Mechanical Theorems.
- Link to buy The Works of Archimedes, by Thomas Heath, which is a comprehensive translation of most of Archimedes works with detailed commentary.
- Jyeṣṭhadeva’s book Yuktibhāṣā, written in 1530 in the language Malayalam, is a critical piece of evidence for what we know about Madhava of Sangamagrama and the Kerala school. The original source below is thanks to the Sayahna Foundation.
Chapter 8: Modern Analysis
- The first publication on the calculus is Gottfried Wilhelm Leibniz’s 1684 paper, Nova Methodus pro Maximis et Minimis (“A New Method for Maxima and Minima”). It presents Leibniz’s differential calculus, providing methods for determining maxima, minima, and tangents.
- Georg Cantor’s 1874 paper, Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen (“On a Property of the Class of All Real Algebraic Numbers”), introduced the groundbreaking proof that the real numbers are uncountable and that the algebraic numbers are countable. In fewer than 5 pages, this paper laid the foundation for modern set theory and the study of infinity.
- Link to Ian Bruce’s website where he includes his English translation of many important mathematical documents from the 17th and 18th centuries. For each document, his translation is first, followed by the original text (usually in Latin). These include some important works in analysis, such as:
- Isaac Newton’s Philosophiæ Naturalis Principia Mathematica and other writings. These foundational works laid the groundwork for calculus and the laws of motion.
- Gottfried Wilhelm Leibniz’s writings on calculus. These include seminal papers that introduced differential and integral calculus and established its notation.
- Brook Taylor’s Methodus Incrementorum Directa et Inversa. This work introduced Taylor series, a fundamental concept in analysis.
- Euler’s Mechanica, Volumes 1 and 2. These books applied calculus to mechanics, advancing the understanding of motion and dynamics.
- Euler’s Foundations of Differential Calculus. This foundational text formalized many concepts in differential calculus.
- Euler’s Foundations of Integral Calculus. This work explored integral calculus and its applications in-depth.
- Euler’s Introduction to the Analysis of the Infinite, Volume 1. This influential book introduced the concept of functions and laid important groundwork for modern analysis.
- Some more links for Euler’s work in analysis, including the Basel problem:
- Isaac Newton’s Mathematical Papers is a comprehensive collection of all known mathematical works by Newton, edited and annotated by D. T. Whiteside. They are available for purchase.
- Karl Weierstrass’s Abhandlungen Aus Der Functionenlehre (1886) explores advanced topics in the theory of functions, reflecting his foundational contributions to mathematical analysis. It is in German.
- Richard Dedekind’s 1872 essay, Stetigkeit und Irrationale Zahlen (“Continuity and Irrational Numbers”), introduced the concept of Dedekind cuts to rigorously define real numbers. This groundbreaking work provided a purely arithmetic foundation for the real number system, resolving ambiguities about irrational numbers.
- Karl Weierstrass’s 1856 paper Theorie der Abel’schen Functionen (Treatises from the Theory of Functions) presents groundbreaking research on the theory of Abelian functions, which has had a lasting impact on complex analysis.
Chapter 9: Topology
- The polyhedron formula was proved by Euler. I only could find the original Latin version of the paper.
- Henri Poincaré’s Oeuvres (collected works) is massive. And it is written in French with not much of it translated to English in non-copyrighted forms.
- Bernhard Riemann introduced the idea of a manifold. The following is an English translation of that paper.
- Grigori Perelman proved the Poincaré conjecture in three papers, posted on the arXiv in 2002 and 2003. They are below.
Chapter 10: Combinatorics
- Euler’s 1741 paper on the Königsberg bridge problem is often considered the beginning of graph theory and topology.
- Link to view the original Latin locally.
- Link to view the original Latin version online.
- Link to the book Graph Theory, 1736-1936. which begins with a translation and commentary of Euler’s paper. These pages are available on Google Books as of this writing.
- The earliest known reference to the four colour problem occurs in a letter dated October 23, 1852, from Augustus De Morgan to William Rowan Hamilton. In this letter, De Morgan described how one of his students had asked him whether every map can be coloured with only four colours.
- Blaise Pascal’s 1654 treatise, Traité du Triangle Arithmétique (“Treatise on the Arithmetic Triangle”), systematically studied the triangular arrangement of binomial coefficients. This paper explored properties such as symmetry, recursive relations, and applications to probability and algebra.
- Kenneth Appel and Wolfgang Haken’s announcement that they had proved the four-color theorem.
- Link to view the annoucement locally.
- Local link to Appel and Haken’s 1989 detailed but self-contained exposition of their proof, accessible to a general mathematical audience.
- Online link to Appel and Haken’s 1989 detailed but self-contained exposition of their proof, accessible to a general mathematical audience.
- Arthur Cayley’s 1875 paper, On the Analytical Forms Called Trees, with Application to the Theory of Chemical Combinations, introduces the concept of trees in graph theory and applies it to model chemical compounds, particularly hydrocarbons. It also included an enumeration of trees using generating functions.
- Gottfried Wilhelm Leibniz’s 1666 paper, Dissertatio de Arte Combinatoria (“Dissertation on the Art of Combinations”), is a foundational work in combinatorial mathematics and symbolic logic. In this dissertation, Leibniz explores the systematic combination of elements to form complex concepts, laying the groundwork for his later development of a universal characteristic language.
- Jacob Bernoulli’s 1713 masterpiece, Ars Conjectandi (“The Art of Conjecturing”), is an important work on combinatorics, including a systematic study of combinations, permutation and integer partitions. It introduced the Law of Large Numbers, formalized key concepts in combinatorics, and applied probability to practical problems.
Appendix A: Topics in Applied Math
- The correspondence between Blaise Pascal and Pierre de Fermat in 1654 marks the beginning of modern probability theory. Their letters discuss the “Problem of Points,” a fundamental question about dividing stakes in an unfinished game of chance. This exchange laid the groundwork for formalizing probability and decision-making under uncertainty. The following is their entire conversation translated to English.
- Jacob Bernoulli’s 1713 masterpiece, Ars Conjectandi (“The Art of Conjecturing”), is one of the foundational works in probability theory. It introduced the Law of Large Numbers, formalized key concepts in combinatorics, and applied probability to practical problems.
- Link to Ian Bruce’s website where he includes his English translations of many important mathematical documents from the 17th and 18th centuries. For each document, his translation is first, followed by the original text (usually in Latin). These include some important works in applied math, such as:
- Joseph-Louis Lagrange’s Mécanique Analytique. This groundbreaking work reformulated classical mechanics using calculus, laying the foundation for modern mechanics.
- Euler’s Mechanica, Volumes 1 and 2. These books applied calculus to mechanics, advancing the understanding of motion and dynamics.
- Jacob Bernoulli’s works on probability and mechanics. These include foundational contributions to the Bernoulli distribution and probability theory.
- Daniel Bernoulli’s writings on fluid mechanics and probability. His work on the Bernoulli principle and hydrodynamics had significant applications in physics.
- Christiaan Huygens’ Horologium Oscillatorium. This seminal text discusses pendulum motion, the center of oscillation, and dynamics.
Appendix B: Mathematician Pronunciation Guide
Appendix C: Mathematical Quotes
Appendix D: History of Math Words
Appendix E: History of Math Symbols
Appendix F: Biographical Sketches
- Leonhard Euler’s Letters to a German Princess, Volumes 1 and 2, are a series of 234 letters in which Euler explains complex scientific and mathematical concepts in an accessible manner.