''Pythagorean Triangles'' is a book on

right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...

s, the Pythagorean theorem, and

Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...

s. It was originally written in the

Polish language Polish (Polish: ''język polski'', , ''polszczyzna'' or simply ''polski'', ) is a West Slavic language of the Lechitic group written in the Latin script. It is spoken primarily in Poland and serves as the native language of the Poles. In a ...

Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and t ...

(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 Yeshiva University is a private Orthodox Jewish university with four campuses in New York City."About YU
on the Yeshiva Universi ...

(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 ''The Pirates of Penzance; or, The Slave of Duty'' is a comic opera in two acts, with music by Arthur Sullivan and libretto by W. S. Gilbert. Its official premiere was at the Fifth Avenue Theatre in New York City on 31 December 187 ...

'': "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 In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

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 number In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the first few are: :0, 1, 36, , , , , , , Expl ...

s. 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 In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...

. After a chapter on

Heronian triangle In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula. Heron's formula implies ...

s, 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 In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

, Chapter 14 discusses right triangles whose sides are

unit fraction A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc ...

s rather than integers, and Chapter 15 is about the

Euler brick In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime. A perfect Euler brick ...

problem, a three-dimensional generalization of Pythagorean triangles, and related problems on integer-sided

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

. Sadly, in giving an example of a

Heronian tetrahedron A Heronian tetrahedron (also called a Heron tetrahedron or perfect pyramid) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles. Every Heronian tetrahedron can be arranged ...

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 The Boolean Pythagorean triples problem is a problem from Ramsey theory about whether the positive integers can be colored red and blue so that no Pythagorean triples consist of all red or all blue members. The Boolean Pythagorean triples problem w ...

. Nevertheless, he highly recommends it to mathematics teachers and to readers interested in "thorough and elegant proofs". Reviewer

Eric Stephen Barnes Eric Stephen Barnes (1924–2000), was an Australian pure mathematician. He was awarded the Thomas Ranken Lyle Medal in 1959, and was (Sir Thomas) Elder Professor of Mathematics at the University of Adelaide. He was elected a Fellow of the Austra ...

rates Sharma's translation as "very readable". The editors of ''

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastruct ...

'' 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 , 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 The ''Journal of Number Theory'' (''JNT'') is a bimonthly peer-reviewed scientific journal covering all aspects of number theory. The journal was established in 1969 by R.P. Bambah, P. Roquette, A. Ross, A. Woods, and H. Zassenhaus (Ohio State ...

, 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 =

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