A History of Mathematics

by Uta C. Merzbach, Carl B. Boyer

For more than forty years, A History of Mathematics has been the reference of choice for those looking to learn about the fascinating history of humankind’s relationship with numbers, shapes, and patterns. This revised edition features up-to-date coverage of topics such as Fermat’s Last Theorem and the Poincaré Conjecture, in addition to recent advances in areas such as finite group theory and computer-aided proofs.

Whether you're interested in the age of Plato and Aristotle or Poincaré and Hilbert, whether you want to know more about the Pythagorean theorem or the golden mean, A History of Mathematics is an essential reference that will help you explore the incredible history of mathematics and the men and women who created it.

Genres: Mathematics, History, Science, Nonfiction, Reference, Philosophy, Education

668 pages, Paperback

First published January 1, 1968

About the author

Uta C. Merzbach is Curator Emeritus of Mathematics at the Smithsonian Institution and Director of the LHM Institute.

Reviews

[Ted · Rating 5 out of 5]

Anyone looking for a good work on the history of mathematics could certainly consider Boyer's seminal work, which is nowadays readily available in second and third editions.

I've used the book for decades as a welcome reference work.

One of the editions shown on GR warns that familiarity with college level math is recommended for readers. Don't worry too much about that. Of course there are some equations, some small amount of rather technical discussion here and there. But it isn't a book whose aim is to teach math, and any problems encountered can be simply passed over; or, nowadays, one can Google the term of confusion and find lots of additional info to help one get past a road block. The book is, after all, a history book, and was definitely meant to relate history, not math itself. There are way more words than equations!!

There are exercises and problems presented at the end of chapters, which the author has divided into three categories: The first few are general questions which ask the reader to organize the information in the chapter into the historical framework; then there are questions requiring proofs or mathematical operations to get answers; and finally more advanced exercises. (In my first edition there are no answers to these provided.)

Each chapter also ends with a short Bibliography, mentioning 10-20 books and articles for further exploration. An index is at book's end.

[William Schram · Rating 5 out of 5]

This book reminds me of E.T. Bell's book, Men of Mathematics. It contains the history of mathematical discoveries as they are known to scholars. For instance, it shows that certain theorems were known to the oriental nations like China and India, and that a lot of things had to be rediscovered after the whole rigmarole with the fall of empires and nations and the destruction of ancient repositories of knowledge.

It starts with counting and goes on through the Egyptians, Babylonians, Greeks and Romans. After the Decline and Fall of the Roman Empire, we follow mathematical thought to India, China and Arabia. Throughout the book, it covers quadratics and how the ancients thought of them and goes on through the founding of Calculus and Analysis. The Giants are all covered, with Euler and Gauss each getting their own chapters. Basically, with every big name or thought in mathematics, the book is there, offering an opinion on stuff. Most of the stuff is priority of discovery, which is a huge thing to mathematicians.

This book is really interesting, but it takes a while for me to read the notation. I really wish I was better at that, but I am working on it.

[Thomas Ray · Rating 4 out of 5]

I can't figure this out: Third Edition page 33 (First Edition page 39): Babylonians knew that for any whole p and q, p > q, a right triangle with whole-number sides a, b, c is a = (p^2 - q^2); b = 2pq; c = (p^2 + q^2). So far, so good. Babylonians left a table of values of a, b, and (secant^2 of the smallest angle), at about 1-degree increments. (Tablet readable, inscribed sometime 1900-1600 BCE.) OK. Now Boyer says, for q < p < 60, and a < b, there are just 38 possible pairs of p and q. I have no clue where he gets this. There are (59×58)/2 = 1711 such (p, q) pairs: excluding duplicates they give 722 unique shapes of right triangle, smallest angle 0.98° to 44.96°. The largest jump in angle in the set is .59 degree from 36.28 to 36.87. There's only 1 other jump over .5 degree, and only 7 other jumps of more than .25 degree.

What is he talking about, "only 38 (p, q) pairs???"

Nicomachus (around 100 CE) noticed that if the odd integers are grouped in the pattern 1; 3 + 5; 7 + 9 + 11; 13 + 15 + 17 + 19; . . . , the successive sums are the cubes of the integers. This observation, coupled with the early Pythagorean recognition that the sum of the first n odd numbers is n^2, leads to the conclusion that the sum of the first n perfect cubes is equal to the square of the sum of the first n integers. (p. 160, 3d ed.)

No other city has been the center of mathematical activity for so long a period as was Alexandria from the days of Euclid (ca. 300 BCE) to the time of Hypatia (415 CE). (pp. 161, 171)

Zu Chongzhi (Tsu Ch'ung-chih, 430-501 CE) found 3.1415926 < pi < 3.1415927 (and, less accurate but more compact, pi is about 355/113. p. 181).

"Pascal's" Triangle of binomial coefficients was known in China by about 1100 CE. (p. 184)

Laws of exponents, x^m · x^n = x^(m + n) and (x^m)^n = x^(mn) are from 1360, Nicole Oresme.

Logarithms, 1614 John Napier, Scotland.

Slide rule, 1630s.

Euler, 1707-1783.

Then it gets complicated.

The book answers questions such as:
goodreads.com/trivia/work/316791-a-hi...

[Jason · Rating 2 out of 5]

A rather dull book. I know it is old, but I've read older (math) books that were far more interesting. It isn't the material, it is the way it is presented. There is no enthusiasm for the topic at hand. Nothing flows, it feels like list of facts in paragraph form.

I know a book like this can't go too in-depth, but an in-depth look into one proof or problem of the greatest minds wouldn't be too much. Euclid's proof of the Pythagorean theorem would have been nice. There are little to no proofs and this really disappointed me.

Journey Through Genius, The Great Theorems of Mathematics (Dunham), Number Theory and Its History (Ore) and Zero: The Biography of a Dangerous Idea (Seife) are all superior in how it presents its material.

[Vladimiro · Rating 5 out of 5]

La domanda preliminare ogni volta che ci si avvicini ad un libro scientifico ma divulgativo è: lo può leggere anche chi non ricorda nulla di matematica o non l'ha studiata all'università? Rispondo subito per il libro di Boyer: è un "ni" tendente al no (al contrario di quanto si dice nell'introduzione).

Il problema è la mole di risultati, teoremi ecc. esposti in 700 pagine che raccontano l'intera storia della matematica dall'antico Egitto fino ai primi del '900. In generale, ritengo impossibile che un "lettore non esperto di matematica ma intelligente e curioso" abbia la pazienza di sorbirsi pagine e pagine di "paragrafi elenchi" dallo stile piano e fin troppo noioso; tanto più che arrivato alla matematica '700-'800 i termini e i risultati si accumulano, quindi un neofita inevitabilmente si perderà.

Comunque, il libro ha il grande merito di raccontare TUTTA la matematica dagli inizi. E' importantissimo, perché il respiro dato alla narrazione dei fatti e dei risultati è amplissimo; è finalmente possibile avere sottomano, in uno stesso testo, i teoremi dei greci e le loro riscoperte in epoca rinascimentale; i numeri indiani, l'algebra araba e la sua ripresa nell'Italia del rinascimento; la ripresa della geometria ad inizio '800 e molto altro. Insomma, l'autore riesce ad individuare le "linee di sviluppo" della matematica, a giudicare le scuole sorte nei vari paesi, i loro propositi e i risultati ottenuti; è questo ad alzare il livello medio di una narrazione che altrimenti sarebbe stata molto piatta.

Probabilmente, un banale glossario con alcune definizioni di base (numero trascendente e algebrico, postulato euclide ecc.) avrebbe aiutato il lettore inesperto.

Quindi, in definitiva, io metto 5 perché da laureato scientifico ho seguito tutto "abbastanza" facilmente; ma appunto, sono uno del settore. Non lo consiglio dunque a chi non ha fatto almeno un paio di corsi di matematica all'università (analisi 1 e geometria).
Per tutti gli altri, se sono anche appassionati di storia o di riflessioni sui fondamenti della materia, è consigliato (anche se il libro è po' vecchiotto).

[Nita]

Purchased near the Penn campus at a lovely used bookstore apparently tended to by a lovely grey-haired couple (think: camel wide wale cords and maroon boiled wool slippers).

http://www.biblio.com/bookstore/house...

Unfortunately this book is difficult to read in bed as I keep wanting to fetch the quad rule paper and work on math problems as the Babylonians did. So it's slow going.

Oh, base 10; how status quo you have become! What better commentary on norm-shifting than the "fixed" (<-- two syllables, please) truth of math? Am I right? 2+2=5? R D R R ?

[Ali Al-ismail · Rating 4 out of 5]

الكتاب يعطي نظرة عامة إلى تاريخ الرياضيات بداية من الحضارات القديمة إلى العصر الحديث. مفيد جدًّا لمن لديه اهتمام في معرفة الأدوار التي احتلتها الشخصيات المشهورة في الرياضيات. أيضًا في أكثر من مناسبة يقدم للقارئ الطرق التي اتبعها الرياضيون في التوصل للنظريات الجديدة.

اكتفيت بأربع نجوم لأن الاثباتات التي قدمها كان بإمكانها أن تكون أكثر تنظيمًا، إضافة إلى أنه يحتاج المزيد من الرسوم البيانية في وجهة نظري، بدلًا من الاعتماد على مخيلة القارئ.

[Samuel Miller]

DNF

I got through the first hundred of pages or so (through Euclid), and while I wouldn't say this is a bad book, it's not the ideal medium to consume this information.

I like the mixing of history with concrete mathematics and specifically viewing things through the eyes of mathematicians in their time (e.g. how the Egyptians used unit fractions, the way Euclid used geometric concepts in proofs in place of our modern algebraic ones).

I didn't like the obtuse prose and sparse diagrams. Some of this is slight differences in how formal English has evolved over the decades, but frankly a lot of it is that English descriptions are no substitute for precise notation. There were a number of times where I spent 5 min or so trying to parse what the author was saying, just to go to wikipedia and completely grasp the concept in ~20 seconds. After doing this enough times I realized this book just wasn't worth the trouble.

Some people might parse the English descriptions of math faster than me and enjoy it, but personally I'd prefer either an actual math textbook or a history textbook over this awkward combo of the two.

[Luke Lambert · Rating 4 out of 5]

Read for my history of mathematics class. Covers topic of mathematics from the start to the early 2000s. Was very dense and definitely would want quite a bit of mathematical background, I found the more advanced geometric topics difficult to understand, but enjoyed the structure of chapters between mathematicians in different places and time periods. I wish the titles of treatises/papers/memiors/ect would have been more often translated into English. Overall, would recommend.

[Martin · Rating 4 out of 5]

Excellent. Very well written, and readable with the history not taking a back seat to the mathematics. The mathematics are used to support the history. Those without a math background can enjoy this quite well, but the more math background the better. Mathematical knowledge up to and including differential and integral calculus helps!

[Lucy Gould · Rating 4 out of 5]

This book was actually really interesting! The only downside is that it is so thick.

[Haydn · Rating 3 out of 5]

The first thing to say is that it's big. Unreasonably big for a book. Meaning you can't take it anywhere (reasonably). Want to read about Euclid in the park? Not an option. How about Cantor on the train? Not possible. Which is annoying, because I would quite like to read about Cantor on the train.

This perhaps becomes less unreasonable as you discover that this book is better enjoyed as reference material. More like a textbook that a book book. The reading function of any sane person surely cannot be linear, monotonic, or even differentiable. I just used some fancy maths speak (because this is maths book!) there to say the following: you should dip in and out, read specific sections of interest, and jump ahead frequently.

Because some areas of mathematical history just aren't all that interesting and, credit to Boyer, they all seem to be mostly covered here. One way this manifests itself, which led to skipping and minor frustration, was the long exploration of basic mathematical concepts developed in the distant past, compared to the speed run that occurred for the 19th and particularly 20th century. Whole chapters devoted to periods in which the author admits not a whole lot was going on (mathematically) but Kolmogorov axioms get one sentence? Boo.

Yeah ok this book was about history, not concepts so I can't complain too much I guess. And also the more recent topics are complicated and mostly fly at an altitude above my head. Fine. I had the nagging feeling that the distribution of devoted words was not equal to the distribution of interesting material to draw from.

[Mark Moon · Rating 4 out of 5]

This book was my first serious introduction to this subject. It seems like a good overview to start with!

[Metello · Rating 4 out of 5]

Un bel libro, secondo me solo per appassionati di matematica, in particolare da 3/4 in avanti adatto solo a chi possiede un livello universitario, perché inizia a riferirsi a aree e teoremi senza spiegarli più, rendendone quindi necessaria la conoscenza.

Ho trovato interessante notare la strana sequenza che ha seguito lo sviluppo della matematica, per certi versi l'inverso di quella spiegata oggi (interi, reali, limiti, derivate, integrali).
Piccola ma pure sempre rilevante la parte dedicata a matematici non occidentali (cinesi, arabi e indiani)

[Simone Scardapane · Rating 5 out of 5]

Uno di quei libri che merita l'articolo davanti: IL Boyer, il libro perfetto per chiunque si voglia avvicinare alla Storia della Matematica. C'è tutto quello che si potrebbe desiderare: chiarezza espositiva, perfetta divisione in capitoli, bibliografia eccellente per approfondire ogni argomento, esempi su esempi per illustrare ogni concetto... Consigliatissimo, peccato solo che chi denigra la matematica non arriverà mai a leggerlo ed a capire quale eccezionale impresa, collettiva ed individuale, essa sia.

[Silviu Popescu · Rating 3 out of 5]

This is a piece of collection with lots of interesting information on the subject. In many parts of it, mostly in the later chapters I felt the need for more description on the mathematical concepts involved. Of course the later the advance in mathematics the higher the abstractness and complexity of the theory, but I felt like more emphasis has been given to more basic concepts which are easier to grasp.

[Susan Mazur Stommen · Rating 3 out of 5]

Batter and deep fry any subject with a crispy coating of history and I am your gal. I know this was written in 1968, but the ancient and prehistory sections are not as deep or comprehensive as they could be. The upside is, makes me want to look into the archaeology of math, and if there isn't such a field already, one should invent it.

[Debi Emerson · Rating 3 out of 5]

Actually, I didn't finish this book (which is rare for me). It is a textbook that requires an understanding of math beyond my capabilities. Or at least what I remember. I'm sure it is a good textbook, but not what I was looking for to satisfy my curiosity about the development of math as we know it today. I will need to find a book that is directed more towards laymen than students.

[Michal Paszkiewicz · Rating 4 out of 5]

A pretty solid description of the history of mathematics with a fairly comprehensive list of persons and mathematical advances. It isn't a textbook on topics though, so when theorems are introduced, one can only understand them if one has studied them already, so this is not a book for the innumerate.

[Matthew Richards · Rating 5 out of 5]

An excellent history of mathematics from the beginning of counting to the 1960s, all in one volume!

Back in 2015, I read a book called Journey through Genius by William Dunham. This was a very good book which I still recommend to any lay person interested in a treatment of various mathematicians and important theorems throughout history. Dunham's book is written in a very accessible way that goes into some mathematical depth and makes even complicated theorems easy to understand. However, the book was relatively short, and it left me wanting more. It also had the annoying habit of treating the history of mathematics as if it were conducted by solitary geniuses pockmarked throughout history, rather than a collaborative effort of hard-working people.

Boyer's book, on the other hand, is a truly expansive work written by a proper historian! It reads more like a history book than a book of mathematics. Rather than focusing exclusively on individuals, he showed how different countries, mathematical schools, and cultural centers have influenced mathematics and interacted with each other. It provided a scaffolding to view the entire history of math without gaps, see how one age influenced the next, and more fully understand mathematicians within the culture contexts they were working in. He dispels many popular misconceptions that have built up through the ages and sets the record straight.

One of my favorite sections was on the mathematics of Ancient Mesopotamia. He explains how their mathematics weren't evaluated fairly by historians until much later because even though there are more existing records from Mesopotamian society than Egyptian society, the modern world had a head start interpreting hieroglyphics due to the Rosetta Stone. The Mesopotamians were actually surprisingly advanced in algebra before the time of the Greeks or al-Khwarizmi, including in the treatment of quadratic and even cubic equations! There's even evidence that they engaged in more recreational mathematics, or at least mathematics that wasn't expecting to be immediately applicable to a problem. For anyone interested in learning developments in this fascinating field of research since Boyer published his book, I recommend checking out the Cuneiform Digital Library Initiative's website, or the recent works of Swedish math historian Jöran Friberg, who discusses the fascinating early Bronze Age developments in Sumer's sexagesimal counting system(s).

Another one of my favorite periods to read about in this volume was the 19th and early 20th century, which was such a creative period in the history of mathematics! I appreciate that he's not afraid of diving into these mathematical topics, even though it's on an undergraduate or first-year graduate level. I enjoyed reading about the abstractification of mathematics and rapid development of many different algebras and non-Euclidean geometries. I also loved reading about the life and work of mathematicians like Poincaré, Boole, Riemann, Lebesgue, and especially Emmy Noether.

The downsides: this might be an entitled complaint about a work over 600 pages long, but I wish I had gotten more! Everyone else rightly praises Boyer for his sections on Hindu, Arabic, and Chinese mathematicians, but I actually think these could have been longer and more in depth! I've found a lot of interesting reading about Chinese mathematics in particular that weren't included in this volume. It's possible there wasn't a lot of good research in English at the time Boyer was writing. I also think the sections about mathematics from the turn of the 20th century to the age Boyer's writing, which Boyer agrees is the *true* Golden Age of mathematics, could have had several more chapters devoted to them! There really was an explosion in the amount of mathematics at that time that warrants it. With a few exceptions, his treatment of mathematicians of this age was frustratingly short. While I didn't expect a treatment of mathematicians like Grothendieck, who won his first Fields Medal while Boyer was writing this, he could have written in more depth about figures such as Gödel or Ramanujan (the latter of whom he doesn't mention at all). One thing that left a particularly bad taste in my mouth was the second-to-last chapter of the book, where he erases Alan Turing's homosexuality and glosses over the circumstances of his death. This was a striking omission in a work that includes lengthy biographical sections about the tragic life of Évariste Galois and Condorect committing suicide in prison during the French Revolutionary period.

Going back through primary and secondary sources, I also noticed a few errors in the book, such as Boyer giving more credit to al-Mamun than the early al-Mansur to the establishment of The House of Wisdom in Baghdad and commissioning a translation of Euclid's Element, or his mistranslation of Fibonacci's "Liber Abaci" to "Book of the Abacus" rather than "Book of Calculations". These are relatively minor quibbles, and the history in the text holds up better against primary sources than some of the works I've read today, including Dunham's work and a math history textbook for a course I'm in that was written in 2015.

Overall, I give this book 4 1/2 stars, which I'll round up to a generous 5!

[Nia · Rating 3 out of 5]

I am certain that I read the English version of this book as part of my History of Mathematics class while doing my MAT in secondary mathematics.
I loved doing the ancient mathematics!

Must re-read for a proper review.
Shira

[Roberto Zanasi · Rating 5 out of 5]

Un'opera colossale sui progressi della matematica. Dalle origini del concetto di numero fino a Bourbaki.

[Jaci · Rating 5 out of 5]

All the history of Mathematics is here.
This book is interesting and there’s more than you imagine.

[Ramon Mirabal · Rating 5 out of 5]

Excelent to mathematicians!

I highly recomend this book to math students,teachers or proffesionals, because it is very complete and almost encycloepedical.Enjoyed it much.

[K. R. B. Moum]

well ... started reading to have a grasp of the origination of ideas before getting into exercise and while reading, found that this piece was meant to be grasped after being able to exercise

[Demis Biscaro · Rating 5 out of 5]

Il Boyer, ossia il classico dei classici della storia della matematica. Acquistato e letto molti anni fa, quando volevo a tutti i costi farmi un'idea dell'origine delle formule e dei teoremi che per anni mi avevano fatto studiare alle scuole superiori e all'università. Non avevo nessuna idea di quali fossero i testi di riferimento nell'ambito della divulgazione matematica e in particolare in quello storico. Oltretutto era prima del 2000 per cui non c'era molta scelta: continuavo a vedere questo testo sugli scaffali in libreria e mi son detto: "Da qualche parte bisogna pur cominciare.", mi sono buttato e sono caduto bene. Alla grande, direi.

L'opera del Boyer è un caposaldo nel suo genere che copre l'evoluzione della regina delle scienze dalle origini egizio-mesopotamiche fino ai primi del '900. Grazie a uno stile piano ma preciso e ad una esposizione lineare, l'autore presenta personaggi, concetti, innovazioni, scarti di pensiero con naturalezza e rigore, soddisfacendo in pieno sia il lettore curioso che l'appassionato in cerca di approfondimenti. Naturalmente, la mole della materia non consente di scendere oltre un certo grado di dettaglio, ma l'aureo compromesso raggiunto da Boyer continua a fare di questo libro un riferimento imprescindibile per chiunque voglia saperne di piú sullo strano mondo dei matematici.
Forse il imite maggiore è proprio la mancanza di un excursus nel '900, che richiede una doverosa integrazione con altri testi, primo fra tutti La matematica del Novecento di Odifreddi.

Consigliato a tutti, nessuno escluso: leggetelo e vi renderete conto di quanta umanità ci possa stare in una singola formula.

---
Precedente: Tutti i racconti del mistero, dell'incubo e del terrore
Successivo: Il fuzzy-pensiero

[Lance Johnson · Rating 4 out of 5]

Sitting down to read this book is quite the undertaking. The authors explore the history of math with the rigor and thoroughness one would expect from mathematicians, but less of the narrative flair that one would expect from historians. The book could have benefited from a bit better formatting for some of the descriptions of mathematical methods. All in all, this book provides a wonderful insight into the history of mathematical thought and provides a thorough understanding of the wandering path history has taken in this regards.
I recommend this book to anyone who is interested in it, but caution the reader not to be afraid to skim or skip sections which seems to tedious.

[Sanane · Rating 2 out of 5]

Öncelikle genel matematik tarihinin güncel olmayan (hem geçmiş hem günümüz açısından) kronolojik bir sırası için ideal bir kitap. Yalnız, yazarının tarihçi olmaması ve tarih usullerini bilmemesi zaten yaptığı yorumlar ve çıkarımlardan belli.

En önemlisi de çevirmenin rezilliği. İnsan çevirdiği konu hakkında bir bilgi sahibi olur öyle 700 sayfalık kitabı çevirir. Çevirmen akademisyen olmasına rağmen ne matematikten, ne de tarihten anlıyor. Kullandığı dil de felaket. Koca kitaba yazık olmuş.

[Paige McLoughlin · Rating 4 out of 5]

I like that the history has chapters on India, China, and Islamic contributions to mathematics. It covers the major mathematicians from Egypt and Mesopotamia to Greek and Hellenistic mathematics middle ages where math developed outside the Latin west and its incorporation into western Europe during the Renaissance and of course Giants like Newton, Gauss, Euler, and other figures familiar to someone with a mathematics background.

[Steven Keays · Rating 5 out of 5]

Simply put: THE BEST book ever written on the subject. No need to understanding mathematics to understand it. Not even a need to like mathematics: the author manages to make this extraordinary topic, one that spans the entirety of the human experience, into a joyful journey of discovery and surprises from start to end.