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Pythagorean Triangles (Dover Books on Mathematics) Paperback – October 5, 2011

by Waclaw Sierpinski (Author), Mathematics (Author)

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The Pythagorean Theorem is one of the fundamental theorems of elementary geometry, and Pythagorean triangles — right triangles whose sides are natural numbers — have been studied by mathematicians since antiquity. In this classic text, a brilliant Polish mathematician explores the intriguing mathematical relationships in such triangles.
Starting with "primitive" Pythagorean triangles, the text examines triangles with sides less than 100, triangles with two sides that are successive numbers, divisibility of one of the sides by 3 or by 5, the values of the sides of triangles, triangles with the same arm or the same hypotenuse, triangles with the same perimeter, and triangles with the same area. Additional topics include the radii of circles inscribed in Pythagorean triangles, triangles in which one or more sides are squares, triangles with natural sides and natural areas, triangles in which the hypotenuse and the sum of the arms are squares, representation of triangles with the help of the points of a plane, right triangles whose sides are reciprocals of natural numbers, and cuboids with edges and diagonals expressed by natural numbers.

Pythagorean Triangles (Dover Books on Mathematics)

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Product Details

Series: Dover Books on Mathematics
Paperback: 128 pages
Publisher: Dover Publications (November 2, 2011)
Language: English
ISBN-10: 0486432785
ISBN-13: 978-0486432786
Product Dimensions: 8.5 x 4.9 x 0.3 inches
Shipping Weight: 3.2 ounces (View shipping rates and policies)
Average Customer Review: 4.0 out of 5 stars See all reviews (1 customer review)
Amazon Best Sellers Rank: #1,293,358 in Books (See Top 100 in Books)
- #373 in Books > Science & Math > Mathematics > Trigonometry
- #1299 in Books > Science & Math > Mathematics > Geometry & Topology

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9 of 11 people found the following review helpful

A Compendium of Facts about Pythagorean Triangles

By watzizname VINE VOICE on September 4, 2010

Format: Paperback Verified Purchase

A pythagorean triangle (PT) is a right triangle (i. e. a triangle which has a right angle) the lengths of the sides of which are all 'natural numbers' (i.e. positive integers) The smallest and best known PT is (4, 3, 5).

Chapter 1, section 1.2 begins: In a pythagorean triangle (as in any right triangle) the biggest side is obviously the hypoteneuse; the other two sides, called arms, contain the right angle. If these (i. e. their lengths) are x and y and the hypoteneuse is z, then by the theorem of Pythagoras,

. . x^2 + y^2 = z^2

[Sierpinski used superscripts, but Amazon's text box doesn't provide for them]

Many (I think most) Americans are used to calling the shorter sides of a right triangle legs, not arms, and we might be more comfortable with 'called' instead of 'obviously.' Also, we are used to naming the legs a and b and the hypoteneuse c, rather than x, y, and z. While differences such as these make this book require a bit more effort to read, the effort is worth it for the many interesting facts you will find here.

For example, on page 16 we learn that if the lengths of the two legs of a PT are consecutive numbers, b = a + 1, then (3a + 2c + 1, 3a + 2c + 2, 4a + 3c + 2) is another PT, On page 17 he lists the first six such triangles: (4, 3, 5), (20, 21, 29), (120, 119, 169), (696, 697, 985), (4060, 4059, 5741), and (23660, 23661, 33461). Perhaps because it is so obvious, Sierpinski doesn't mention that therefore PTs exist with acute angles arbitrarily close (but never =) to 45 degrees.

While extensive, the information about PTs in this slim volume (107 pages) is not exhaustive. In addition to the omission cited above, Sierpinski says nothing about infinite matrices of PTs, of which there are two that I know of.Read more ›

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