''Pythagorean Triangles'' is a book on right triangles, the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

, and Pythagorean triples. It was originally written in the Polish language by Wacław Sierpiński (titled ''Trójkąty pitagorejskie''), and published in Warsaw in 1954. Indian mathematician Ambikeshwar Sharma translated it into English, with some added material from Sierpiński, and published it in the ''Scripta Mathematica'' Studies series of Yeshiva University (volume 9 of the series) in 1962. Dover Books republished the translation in a paperback edition in 2003. There is also a Russian translation of the 1954 edition.

Topics

As a brief summary of the book's contents, reviewer Brian Hopkins quotes '' The Pirates of Penzance'': "With many cheerful facts about the square of the hypotenuse." The book is divided into 15 chapters (or 16, if one counts the added material as a separate chapter). The first three of these define the primitive Pythagorean triples (the ones in which the two sides and hypotenuse have no common factor), derive the standard formula for generating all primitive Pythagorean triples, compute the inradius of Pythagorean triangles, and construct all triangles with sides of length at most 100. Chapter 4 considers special classes of Pythagorean triangles, including those with sides in arithmetic progression, nearly-isosceles triangles, and the relation between nearly-isosceles triangles and square triangular numbers. The next two chapters characterize the numbers that can appear in Pythagorean triples, and chapters 7–9 find sets of many Pythagorean triangles with the same side, the same hypotenuse, the same perimeter, the same area, or the same inradius. Chapter 10 describes Pythagorean triangles with a side or area that is a square or cube, connecting this problem to Fermat's Last Theorem. After a chapter on Heronian triangles, Chapter 12 returns to this theme, discussing triangles whose hypotenuse and sum of sides are squares. Chapter 13 relates Pythagorean triangles to rational points on a unit circle, Chapter 14 discusses right triangles whose sides are unit fractions rather than integers, and Chapter 15 is about the Euler brick problem, a three-dimensional generalization of Pythagorean triangles, and related problems on integer-sided tetrahedra. Sadly, in giving an example of a Heronian tetrahedron found by E. P. Starke, the book repeats a mistake of Starke in calculating its volume.

Audience and reception

The book is aimed at mathematics teachers, in order to inspire their interest in this subject, but (despite complaining that some of its proofs are overly complicated) reviewer Donald Vestal also suggests this as a "fun book for a mostly general audience". Reviewer Brian Hopkins suggests that some of the book's material could be simplified using modular notation and linear algebra, and that the book could benefit by updating it to include a bibliography, index, more than its one illustration, and pointers to recent research in this area such as the Boolean Pythagorean triples problem. Nevertheless, he highly recommends it to mathematics teachers and to readers interested in "thorough and elegant proofs". Reviewer Eric Stephen Barnes rates Sharma's translation as "very readable". The editors of '' zbMATH'' write of the Dover edition that "It is a pleasure to have this classic text available again".

References

{{reflist, refs= {{citation , last = Barnes , first = E. S. , author-link = Eric Stephen Barnes , journal =

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

, mr = 0191870 , title = review of ''Pythagorean Triangles'' {{citation , last1 = Chisholm , first1 = C. , last2 = MacDougall , first2 = J. A. , doi = 10.1016/j.jnt.2006.02.009 , issue = 1 , journal = Journal of Number Theory , mr = 2268761 , pages = 153–185 , title = Rational and Heron tetrahedra , volume = 121 , year = 2006, hdl = 1959.13/26739 , hdl-access = free {{citation , last = Holzer , first = L. , journal = zbMATH , title = Pythagoreische Dreiecke (review of ''Trójkąty pitagorejskie'') , zbl = 0059.03701 {{citation , last = Hopkins , first = Brian , date = January 2019 , doi = 10.1080/07468342.2019.1547955 , issue = 1 , journal = The College Mathematics Journal , pages = 68–72 , title = review of ''Pythagorean Triangles'' , volume = 50 {{citation , last = Lehmer , first = D. H. , author-link = Derrick Henry Lehmer , journal =

, mr = 0065574 , title = Review of ''Trójkąty pitagorejskie'' {{citation , last = Vestal , first = Donald L. , date = August 2004 , journal = MAA Reviews , publisher =

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

, title = review of ''Pythagorean Triangles'' , url = https://www.maa.org/press/maa-reviews/pythagorean-triangles {{zbl, 1054.11019 Pythagorean theorem Mathematics books 1954 non-fiction books 1962 non-fiction books 2003 non-fiction books