# Great mathematics books by pre-modern authors

Last summer, I read Euclid's Elements, and it was an eye-opening experience; I had assumed that three thousand years' difference would make the notation incomprehensible and the reasoning alien, but his proofs were beautiful; I had never experienced synthetic geometry (at least since middle school), and it was very enjoyable, especially his 3-dimensional geometry and the classification of platonic solids.

The experience made me realize that older math books could be worthwhile to study; for instance, I've heard that Euler wrote some incredibly popular calculus books, and that others (like Maclaurin and L'Hopital) wrote popular textbooks.

What math books from before 1900 (or from the beginnings of newer areas like topology and category theory) have you read and enjoyed? Are there any you would recommend?

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Way too broad a question for my taste. It's good the OP realized that older math books could be worthwhile, but asking for a list is kind of like asking for "the greatest books of all time". There are just too many. But criticism aside, I found my horizons broadened by "The Mathematics of Egypt, Mesopotamia, China, India, and Islam," edited by V. Katz. That's where I first realized that I could read and enjoy much older math texts, especially those from outside the Eurocentric canon. You could start with Katz's book as a source for excerpts, and look up full books when interested. –  Marty Apr 19 '13 at 13:12
Related MO question: mathoverflow.net/questions/28268/do-you-read-the-masters –  Timothy Chow Apr 19 '13 at 20:15
I look forward to the complementary question, "Great mathematics books by post-modern authors". –  Gerry Myerson Jan 11 '14 at 20:05
@Gerry--the fully dual would be: Lousy books by post-modern authors. –  Włodzimierz Holsztyński Jan 12 '14 at 13:33
@WlodzimierzHolsztynski I've read some of those. –  Brian Rushton Jan 12 '14 at 23:45

Here is an incomplete list of pre-1900 books that I read, enjoyed and strongly recommend (I apologize for some repetitions):

1. Collected works of Archimedes.

2. Ptolemy, Almagest (yes, this is a math book:-)

3. Kepler, Stereometry of wine barrels.

4. Newton's Principia,

5. Complete works of Abel and Riemann, Laguerre and Stieltjes.

6. Gauss, General investigation of curved surfaces (available in English)

7. Fourier, Analytic theory of heat.

8. Fourier, Analyse des equations determinees (this is a rare book. Available on my web page).

9. Complete works of Chebyshev (available in Russian and French)

10. Maxwell, Treatease on Electricity and Magnetism. (There is a nice paper of F. Dyson, Missed opportunities, where he explains how much Mathematics would gain if mathematicians read this book. I completely agree with Dyson).

11. Painleve, Lecons, sur la theorie analytique des equations differentielles, professees a Stockholm, 1897.

12. Picard and Poincare, of course...

BTW, I disagree with designation "pre-modern" for the period before 1900. From my point of view, "modern period" begins with Abel. There is no substantial difference between Laguerre or Stiletjes and XX century mathematics.

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Many of these books can be found scanned on the internet archive. For example Maxwells: archive.org/details/electricandmagne01maxwrich –  Michael Bächtold Jan 11 '14 at 18:41
And even more are available on the Internet for free in excellent Russian translations, frequently with commentaries:-) –  Alexandre Eremenko Jan 11 '14 at 18:49

The question is kinda off topic but I will give an answer because I really, really like Disquisitiones Arithmeticae.

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I never tried to read it. It would be great if you tell us more about what of his themes and sections you find most interesting to read! –  Thomas Riepe Apr 19 '13 at 10:59
The second half (about quadratic forms) is tough going but the first half is just this elegant, concise, well-written introduction to elementary number theory. The proof of quadratic reciprocity is a bit harder than the rest of the first half, but is manageable. If you want to read it for fun, start on page one. –  Felipe Voloch Apr 19 '13 at 11:56
When I was an undergraduate in Mexico, an important rite of passage was attending the geometry and number theory courses of A. Barajas. He was a legend as one of the founders of Mathematics in Mexico, having worked with Einstein, and organizing the famous 1956 International Symposium on Algebraic Topology. He always started the number theory class by citing the first chapter of Disquisitiones from memory: "Wenn die zahl $a$..." –  Rodrigo A. Pérez Apr 19 '13 at 12:11
Si numerus $a$ numerorum $b$, $c$ differentiam metitur, $b$ et $c$ secundum $a$ congrui dicuntur, sin minus, incongrui... –  Chandan Singh Dalawat Apr 19 '13 at 12:36
"Wenn die Zahl a in der Differenz der Zahlen b, c aufgeht, so werden b und c nach a congruent, im andern Falle incongruent genannt. Die Zahl a nennen wir den Modul. Jede der beiden Zahlen b, c heißt im ersteren Falle Rest, im letzteren aber Nichtrest der anderen." Although, as Chandan points out, the original WAS in latin :) –  Rodrigo A. Pérez Apr 19 '13 at 17:05

F. Klein's "Development of mathematics in the 19th century". It is a history book; it is a Math book; it is a great read.

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Reading Bhaskara II's Lilavati (written in 1150) was an eye-opening experience and provided me with many gems with which to liven up a calculus course. It's quite readable, and its approach is playful and refreshing. I'm sure it was even better in Sanskrit.

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books.google.co.in/… –  Chandan Singh Dalawat Apr 19 '13 at 10:47

Well, it bears a copyright date of 1902, but I think Eduoard Goursat's Cours D'Analyse is such an excellent text that it should be mentioned. I have the English translation, Mathematical Analysis, translated by E.R. Hedrick, usually referred to as just "Goursat-Hedrick". (My 528 page volume 1 has a price of six dollars and a quarter pencilled in on the inside cover. It is currently available on eBay for \$85) http://www.ebay.com/itm/MATHEMATICAL-ANALYSIS-Vol-1-By-Goursat-Hedrick-1904-Ginn-Co-/261154396415

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Weil used to prefer Jordan's Cours d'Analyse (gabay-editeur.com/epages/300555.sf/fr_FR/?ObjectPath=/Shops/…) –  Chandan Singh Dalawat Apr 19 '13 at 9:48
It was recommended as a supplementary text when I was a (what might be considered an undergraduate) student in Jagiellonian University (Krakow, Poland) in mid-to-late 1980s. –  Margaret Friedland Apr 19 '13 at 14:18

Euler's Elements of Algebra, Newton's Principia, and Riemann's works, seem rather obvious suggestions.

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Heath (whose English translation of the Elements is what most people read when they read the elements) also has a translation of Treatise on Conic Sections by Appollonius of Perga.

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"Arithmetica" by Diophantus. English translation is contained in T. L. Heath, Diophantus of Alexandria. A Study in the History of Greek Algebra. For other translations see e.g. http://web.math.pmf.unizg.hr/~duje/refclas.html

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Heinrich Weber's 3vol Algebra (but the 3rd vol is not entirely correct). And "the short version": Miller, Blichfeld, Dickson "Theory and Appl. of finite Groups".

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Euler's two-volume Introductio in analysin infinitorum is an excellent read, from the preface all the way to Appendix on Surfaces at the end of Volume II. Here is the opening paragraph laying out the principles of the treatise, which I translated from the 1961 Russian translation:

I had noticed on many occasions that the majority of difficulties encountered by the mathematics students in the analysis of the infinite arise because having hardly digested elementary algebra, they direct their thoughts to this higher art, and consequently not only do they remain at the doorstep, but they even form perverse impressions about the kind of infinite which is used there. Although analysis of the infinite does not require perfect command of elementary algebra and all the techniques pertaining to it, yet there are many questions whose resolution is important for preparation of the students to the higher art, but which are either omitted altogether in elementary algebra or are considered cursorily. Therefore I have no doubt that the content of these books will help to abundantly compensate for the gap just mentioned. I have attempted not only to give a fuller and more precise treatment to everything that is required for the infinite analysis, but also to develop a good many questions that would allow the readers to come to grip with the idea of the infinite in an unobtrusive and natural way. Many questions that are commonly treated in the infinite analysis I have resolved here by means of the laws of elementary algebra, so that later the very essence of both this and the other method become clearer.

In spite of the modest reference to "elementary algebra", Euler's treatise gives the first modern exposition of analysis based on the notion of function, including a systematic treatment of exponential, logarithmic and trigonometric functions, and contains plethora of examples of working with infinite series. Among the highlights of the first volume: the method of generating functions is applied to the enumeration of partitions and the computational aspects of the series (including Euler's zeta function) are given a superb treatment not reached in the contemporary curriculum until a good course in numerical methods, if then. The second volume is devoted to coordinate geometry of curves and surfaces and lays down the foundations of differential geometry.

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I am surprised that it has not been mentioned yet : Klein's Lectures on the Icosahedron is a pure gem (not easy to read, though).

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William Burnside, Theory of groups of finite order (1897).

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Boole's Treatise on the Calculus of Finite Differences (~1860)

The book where modern discrete calculus is invented and applied to PDEs, with a very pleasant linear algebraic flavour.

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