Disquitiones Arithmeticae

The Disquisitiones Arithmeticae was published in 1801, when Carl Friedrich Gauss was 24, but a great part of the material must have been conceived when Gauss was in his teens.  When I began reading books on elementary number theory, I found that, in spite of the eclectic nature of number theory, their contents were surprisingly uniform.  When I began to read Disquisitiones Arithmeticae I realised that it was because they were all, to a large degree, based on this book.  Disquisitiones Arithmeticae is not only a work of genius, opening up areas of research that are still being pursued, but also an exemplary textbook, a lucid exposition of number theory accessible to anyone with a high school education.  It contains numerous numerical examples and it is clear from reading it that Gauss’ insights came not only from abstract reasoning but also from  numerical experimentation.

Unfortunately translations into English of Disquisitiones Arithmeticae have to be paid for, but Hermann Maser’s 1889 translation into German can be found here, and anyone with some schoolboy (or schoolgirl) German can follow it easily with the aid of a good German to English dictionary (at least two very good ones are available on-line).  The original Latin edition forms volume 1 of Gauss’ collected works, a link to which is given on the Collected Works page.

Algebra.  An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges.  By G. Chrystal M.A., LL.D.  Vol 1. 5th Edition, Adam and Charles Black, 1904, Vol 2. 2nd Edition, 1900.

In spite of the colourless title, this book gives a treatment of continued fractions more comprehensive than that encountered in modern books.  Chapter 33 deals with Lagrange’s solution to the Diophantine Equation

x^2-Cy^2=\pm H when H>\sqrt{C}

and Chapter 34 deals with general continued fractions including Lambert’s appliction of them to show that e and \pi are irrational.  pdfs of V1 and V2 can be found here.

Higher Algebra.  By S Barnard and J M Child.  Macmillan and Co.  London. 1936, reprinted 1959.

It is described as a text book for high school students in an advanced maths stream, but I would describe it as a text at first year undergraduate level.  Many beautiful proofs and exercises.  Copies in various formats with some early pages missing can be found here.

A Course of Modern Analysis. By E T Whittaker and G N Watson. Third Edition (1920) published by Cambridge University Press.

Although the adjective ‘Modern’ has not applied for some time, this is still probably the most beautiful, rigorous, and concise text on Complex Analysis. The fourth edition (1927) is still being reprinted by Cambridge University Press. Free electronic versions in various formats are available through the Open Library project here.

Theory and Application of Infinite Series. By Konrad Knopp. Second English Edition Reprinted (1954). An published by Cambridge University Press.
A widely cited textbook. Free electronic versions in various formats are available through here.

Euclid’s Elements
Euclid’s Elements needs no introduction. Volume 3 only of Sir Thomas Heath’s translation, first published by Cambridge University Press in 1908 and still in print, can be found at here. A modern edition by Richard Fitzpatrick is here. A convenient web-based reference with commentary by David E Joyce is here.

A Course of Pure Mathematics. By G H Hardy. Third Edition (1921)
A mathematics text book for beginning undergraduate students first published in 1908 and still in print as the tenth edition. The third edition is available as a pdf and in other formats at the Project Gutenberg website here.

An Introduction to the Theory of Numbers. By G H Hardy and E M Wright. Fourth Edition (with corrections) (1975)
First published in 1938 by Oxford University Press, still in print and now in its sixth edition. The fourth edition is available in a number of formats at here.

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