29 Historic Math Books You Can Read for Free

The scribe Ahmes says himself in the opening lines that he copied this roll from an older text, so one of the oldest known mathematics books is also a copy. It holds 84 problems, running from sharing loaves among workers to the slope of a pyramid. Because the Egyptians wrote every fraction except 2/3 as a sum of unit fractions of the form 1/n, a long table at the start of the roll gives 2 divided by the odd numbers ready-made in that form.

The roll survives in pieces today, and most of it can be seen with photographs on the museum’s collection page.

China’s oldest mathematics handbook, compiled over several generations and given its final form in the Han period, made up of 246 problems. In the eighth chapter, systems of linear equations are solved on a table of numbers with the method we now call Gaussian elimination, and negative numbers are told apart with red and black rods. Where the Greek tradition goes from axiom to theorem, this book says everything as problem, answer and method, and I don’t know a better source for seeing that second face of mathematics.

The copy made by hand for the imperial library in 1773 can be read online, with the volumes on linear equations scanned in full.

Seven works of Archimedes are copied onto parchment in Constantinople in the 10th century. In the 13th century a monk scrapes and washes the leaves, writes a prayer book over them, and Archimedes’ lines stay buried under those prayers for centuries. After the book finds its buyer in 1998, the Walters Museum in Baltimore spends ten years photographing it in 12 different bands of light and brings the erased text back line by line. The Method and the Stomachion, found in no other copy, come out of these pages.

It is also the only book on the list written in Istanbul. The images, processed layers and transcriptions are all open under a free licence.

The word algebra comes from the al-jabr in this book’s title. In 9th-century Baghdad, al-Khwarizmi sorts quadratic equations into six types and solves each one both with a rule written out in words and by drawing squares and rectangles. This Oxford copy is the only manuscript behind the book’s first printed edition in 1831.

The library’s digital archive holds only two pages of it, not the whole book, but those two pages carry the geometric solutions of x² + 21 = 10x and x² = 3x + 4. The library catalogue leads to these pages.

Leonardo of Pisa spends his youth in North Africa beside his merchant father and in 1202 carries the numerals and methods of reckoning he learned there into Europe with this book. The famous rabbit problem is in it too, and in the Florence copy a small column of numbers runs down the right margin of the page, starting 1, 2, 3, 5, 8 and going all the way to 377.

This is one of the 19 known manuscript copies of the book, and in 2020 the Museo Galileo and the Florence National Library put it online from first page to last.

The first mathematics book printed in Europe. It comes out in Treviso on 10 December 1478, its author puts his name nowhere, and he says right at the start that he wrote it not for scholars but for young people learning trade. It teaches adding, subtracting, multiplying and dividing with the new numerals through everyday problems, from buying cloth and spices to splitting a partnership’s profits, and it has multiplications done by the lattice method too.

Copies are extremely rare, and this one is bought at auction by the publisher George Arthur Plimpton and later left to Columbia. The pages of the book can be viewed in Columbia’s digital collections.

The first printed edition of Euclid’s Elements, four years after the Treviso Arithmetic. The text rests on a Latin translation made from the Arabic, and the real difficulty is printing hundreds of geometric figures next to the words, which Ratdolt solves with fine woodcut figures set into the margins.

The University of Utah scans this copy with a 39-megapixel Hasselblad at 600 ppi, and the whole 1482 edition is open page by page.

Pacioli’s Summa gathers arithmetic, algebra, geometry and commercial reckoning into one volume, and it holds the first printed account of double-entry bookkeeping. That is why accountants count a Franciscan friar as their founder. There is even an illustrated table in it showing how to count to 10,000 on your fingers.

A few years later in Milan, Pacioli teaches geometry to Leonardo da Vinci, and the two work together on a book about the golden ratio. The 1494 copy scanned by the Wellcome is free.

Dürer writes this book not in Latin but in German, for painters, goldsmiths, stonemasons and carpenters. In the fourth book he draws polyhedra unfolded flat on paper, as nets that close up when you cut and fold them, and these are the first known polyhedral nets. At the end of the book come the famous woodcuts of an artist drawing in perspective with a thread and a frame.

I’d send anyone curious about where the strange polyhedron in Melencolia I comes from to this book. The Getty copy is scanned from cover to cover.

The general solutions of cubic and quartic equations are printed for the first time in this book. Tartaglia gives Cardano the solution of the cubic in 1539 and makes him swear not to publish it, and when Cardano learns that Scipione del Ferro had found it earlier, he breaks his oath and one of the ugliest fights in the history of mathematics begins.

The book has the problem of splitting 10 into two parts whose product is 40, and the answer comes out as 5 + √−15 and 5 − √−15. Cardano finds these numbers useless and puts them aside, but this is the first calculation done with complex numbers. The pages of the 1545 first edition can be read as scans.

The equals sign appears here for the first time. Recorde explains that he chose two parallel lines because no two things can be more equal, and his lines are much longer than ours, nearly the length of a word. The book moves through dialogues between a master and his scholar and is one of the first algebra books in English.

A year after the book comes out, Recorde dies in the debtors’ prison he was sent to over damages he couldn’t pay in a lawsuit against him. The whole 348-image copy is open in the public domain.

The first complete English translation of Euclid, with a long preface by John Dee. The Euclid of Megara in the title is actually the wrong man, the Renaissance kept mixing up Euclid the mathematician of Alexandria with Euclid the philosopher of Megara. What makes the book a legend is the fold-up paper models in Book XI, lift them off the page and pyramids and prisms stand up, and the printer pastes them by hand into every copy.

I wrote about Billingsley’s paper models at length in a separate piece. The scan of the Smithsonian copy includes photographs of the pop-ups.

Napier spends about 20 years computing his tables, and with this book he turns multiplication into addition and division into subtraction. Kepler gets so much out of the tables that in 1619 he writes Napier a letter of thanks, not knowing Napier had died two years earlier. Three years after the tables, Napier also describes the calculating rods called Napier’s bones.

The 1614 edition is scanned among the Smithsonian’s books on logarithms.

Kepler’s third law, that the square of a planet’s orbital period is proportional to the cube of its mean distance from the Sun, sits in the fifth book among pages devoted to the music the planets make. In the second book he studies systematically the ways of tiling the plane with regular polygons and draws the star polyhedra.

The man who asked why a snowflake has six points writing out notes for the planets in the same book is, to me, the best picture of Kepler there is. The Library of Congress scan downloads as a PDF too, and the Smithsonian copy is in the archive as well.

This is the edition in whose margin Fermat wrote that he had found a marvellous proof but the margin was too narrow to hold it. Fermat’s own copy is lost, and the notes survive only because his son Samuel adds them to a new edition in 1670. The theorem stays open for 357 years, until Andrew Wiles proves it in 1994.

The Kyoto copy isn’t Fermat’s, but it is the same edition, so the margins are just as narrow. The scan of the 1621 edition is open in high resolution.

Descartes publishes La Géométrie as one of three essays following his Discourse on Method. The way to turn geometry problems into algebra, the habit of writing unknowns as x, y and z and knowns as a, b and c, and exponent notation like x³ all come out of these hundred-odd pages. The book reads as if it were made difficult on purpose, and Descartes leaves many proofs to the reader as exercises.

David Eugene Smith and Marcia Latham’s 1925 translation is printed side by side with a facsimile of the original, so you see Descartes’ own pages too.

Galileo writes this book under house arrest and can’t print it in Italy, the manuscript is smuggled out, and the book comes off the Elzevier press in Leiden. On the first day Galileo notices that there are as many square numbers as natural numbers, because the two can be paired one to one, and decides that saying more or fewer about infinite sets makes no sense. Cantor comes back to that question 240 years later.

On the second day he explains why the bones of a giant animal have to thicken out of proportion, the square-cube law. NYPL’s 1638 copy is scanned, and the 1914 English translation is in the archive.

This is Newton’s own copy. Blank leaves are sewn in between the pages of the first edition, and while preparing the second edition Newton crosses out printed lines, writes corrections in the margins and adds new paragraphs on the blank leaves. You read the book that first set down the laws of motion and universal gravitation while watching it change in its author’s hands.

It is one of the first documents Cambridge puts out when it opens its Newton archive in 2011, and the copy full of Newton’s notes is open in high resolution.

When Jacob Bernoulli dies in 1705, twenty years of work on probability are left unfinished, and his nephew Nicolaus publishes the book in 1713. The first proof of the law of large numbers, the idea that if you toss a coin often enough the share of tails gets close to half, is in the fourth part, and the Bernoulli numbers appear for the first time in the second. The fourth part ends unfinished, where Bernoulli left it.

This record leads to scans at two different libraries, and the 1899 German translation is in full view.

With this book Euler puts the function at the centre of analysis. He defines the exponential and logarithmic functions, sine and cosine in their modern sense, and the relation e^ix = cos x + i sin x is in this book too. To my eye the chapters on infinite series and products read better than many analysis books written today.

The second volume is given over to curves and surfaces, and the plates at the end draw dozens of curves. The Latin scan is in the Euler Archive.

Agnesi is the eldest of at least 21 children, and she grows this book out of notes she kept for teaching her siblings mathematics. It is one of the first complete textbooks to teach analysis in Italian rather than Latin. One curve in it is called versiera in Italian, it gets read as witch when it is translated into English, and the curve is still known as the witch of Agnesi.

The Cambridge mathematician John Colson learns Italian late in life just to translate it. Both volumes of the 1748 edition are open access, and Colson’s translation, printed in 1801, is in the archive.

Gauss publishes this book at 24. It contains the proof of quadratic reciprocity and the demonstration that the regular 17-gon can be drawn with compass and straightedge, which Gauss finds when he is 18. With this book number theory stops being a pile of methods and becomes a discipline.

The new mathematics strains the typesetters so much that a four-page errata list is added at the end, and the pages with the worst errors are reprinted and slipped into the bound copies afterwards. The scan of the Smithsonian copy runs to 714 pages.

Ada Lovelace translates Menabrea’s French paper on Babbage’s Analytical Engine and adds notes about three times longer than the paper itself. In Note G she shows step by step, in a table, how the engine would compute the Bernoulli numbers, and that table is counted as the first published computer program. Lovelace signs her notes only A.A.L.

The engine is never built, but the full text with the notes is free.

Byrne teaches the first six books of Euclid with red, yellow, blue and black shapes in place of letters, and the proofs flow across the page with almost no words. Colour printing is so expensive that the book doesn’t sell in its day, and its publisher William Pickering goes bankrupt a few years later, with the book thought to have played a part.

Today it looks like a piece of design printed seventy years before the Bauhaus, and a paper engineer has since folded Byrne’s figures into three dimensions. Toronto’s scan of the colour copy is open from cover to cover.

Boole turns logic into algebra and writes, with a single equation like x(1 − x) = 0, that nothing can be both itself and not itself at once. A teacher in Cork with no university degree takes logic out of philosophy and carries it into mathematics with this book.

When Shannon applies this algebra to the relay circuits of telephone exchanges 83 years later, the cornerstone of today’s computers falls into place. The full text downloads as PDF and TeX.

A short novel about a square living in a two-dimensional world who one day meets a three-dimensional sphere. It is still the best way in for thinking about a fourth dimension, and at the same time it makes fun of the rigid class and gender lines of Victorian England. Abbott signs the first edition with the pen name A Square.

The only novel on the list, and the full text is on Gutenberg.

Hilbert rebuilds Euclidean geometry on undefined terms like point, line and plane and twenty axioms. Hidden assumptions Euclid used unnoticed for two thousand years, such as a line that crosses one side of a triangle having to cross another side too, are written down openly as axioms here for the first time.

Reading Euclid in this book after Byrne’s coloured pages is like putting two different worlds side by side. The full text of the Townsend translation is free.

Hardy writes this book to put the teaching of analysis in English universities on solid ground, and it has hardly been out of print since its first edition. He takes everything from real numbers to limits, from derivatives to infinite series, with proofs, and some of the exercises are old Cambridge examination questions. Decades later he writes A Mathematician’s Apology.

On Gutenberg you can download even the PDF and the LaTeX source of the third edition, which means you can retypeset the book on your own computer.

Whitehead and Russell set out to derive all of mathematics from logic. The proposition 1 + 1 = 2 is only approached on page 379 of the first volume, the arithmetic proof arrives in the second volume, and the authors note under it, with a dry joke, that the proposition is occasionally useful.

When Gödel shows in 1931 that such a system can never be complete, the project’s grand aim is closed off, but the book stays on as the language of logic. All three volumes are open in full view.