Pascal's Arithmetical Triangle: The Story of a Mathematical Idea (Johns Hopkins Paperback) [Paperback]

A. W. F. Edwards (Author)

Book Description

Series: Johns Hopkins Paperback | Publication Date: July 23, 2002

Imagine having some marbles, pebbles, or other objects that you want to lay out in a neat triangular pattern. How many do you need to end up with a complete triangle? Three will do; so will 6, 10, 15, and... These numbers are called triangular numbers. Ask the same question for a triangular pyramid. Four will do; so will 10, 20, 35, and... the so-called pyramidal numbers. This book looks at the discovery of the multiplicity of properties and uses triangular numbers and their many extensions possess.

Although often displayed in a triangular array named after the seventeenth-century French philosopher Blaise Pascal, triangular numbers were known many centuries earlier. In this book A.W.F. Edwards traces the Arithmetical Triangle back to its roots in Pythagorean arithmetic, Hindu combinatorics, and Arabic algebra, and gives an account of the progressive solution of combinatorial problems from the earliest recorded examples to the work of Renaissance and later mathematicians. He shows how Pascal's work -- so modern in style -- in establishing the properties of the numbers and their application in various fields led to Newton's discovery of the binomial theorem for fractional and negative indices and to Leibniz's discovery of calculus.

Editorial Reviews

Review

An impressive culmination of meticulous research into original sources, this definitive study constitutes the first full-length history of the Arithmetic Triangle.

(Mathematics of Computation 2004)

A fascinating book... giving new insights into the early history of probability theory and combinatorics, and incidentally providing much stimulating material for teachers of mathematics.

(G.A. Barnard International Statistical Institute Review )

Scrupulously researched... Carries the reader along in a rewarding manner. It is a scientific who-dun-it, and one must admire the author for the scholarly yet unpedantic manner in which he disperses some of the mists of antiquity.

(A.W. Kemp Biometrics )

Recommended not only to historians and mathematicians, but also to students seeking to put some life into the dry treatment of these topics to which they have doubtless been subjected.

(Ivor Grattan-Guinness Annals of Science )

A dependable, accessible resource for college mathematics majors to use in learning about specific historical topics. Professor Edwards has carefully researched and tightly organized his historical/mathematical account of Pascal's triangle.

(Richard M. Davitt Convergence )

From the Publisher

"An impressive culmination of meticulous research into original sources, this definitive study constitutes the first full-length history of the Arithmetic Triangle."—Mathematics of Computation

"A fascinating book . . . giving new insights into the early history of probability theory and combinatorics, and incidentally providing much stimulating material for teachers of mathematics."—G.A. Barnard, International Statistical Institute Review

"Scrupulously researched . . . Carries the reader along in a rewarding manner. It is a scientific who-dun-it, and one must admire the author for the scholarly yet unpedantic manner in which he disperses some of the mists of antiquity."—A.W. Kemp, Biometrics

"Recommended not only to historians and mathematicians, but also to students seeking to put some life into the dry treatment of these topics to which they have doubtless been subjected."—Ivor Grattan-Guinness, Annals of Science

See all Editorial Reviews

Product Details

• Paperback: 224 pages
• Publisher: The Johns Hopkins University Press (July 23, 2002)
• Language: English
• ISBN-10: 0801869463
• ISBN-13: 978-0801869464
• Product Dimensions: 9 x 6 x 0.5 inches
• Shipping Weight: 12.8 ounces
• Average Customer Review: 4.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #553,383 in Books (See Top 100 in Books)

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5 of 5 people found the following review helpful:

4.0 out of 5 stars A world history of the origin of Pascal's triangle, October 5, 2002

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
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This review is from: Pascal's Arithmetical Triangle: The Story of a Mathematical Idea (Johns Hopkins Paperback) (Paperback)

Pascal's famous triangle is much like the Fibonacci numbers, in that there always seems to be room for more properties and uses. With all these possibilities, it is easy to become overwhelmed by all the information about them. In this book, the emphasis is on the original discovery of the triangle rather than the mathematical properties that it has, even though the two are inseparable.
Of course, the idea for the triangle that bears his name did not spring fully formed from the mind of Blaise Pascal. It was centuries in the making and Edwards traces through many of the historical roots leading to the publication of Pascal's, "Treatise on the Arithmetical Triangle." The oldest of these thoughts dates back to the Pythagorean brotherhood that existed almost three thousand years ago. The explanation of these ideas, largely from original sources and covering the entire world, develops a foundation of idea followed by consequence followed by new idea and new consequence that is much like a mathematical proof. Once the concept of combinatorial numbers arose in usage, it became necessary to understand them mathematically, which led to new uses and new mathematics. These threads make this an excellent book for learning the history of mathematics.
In most cases math students are like those in other fields, they do the work but have no idea how and why the concept was developed. That deficiency is where books like this can be so valuable. Explanations as to the why and how are two alternate routes to the what that are often overlooked. This book will help you get to the what by taking you through the why and how avenues of the historical development of Pascal's amazing triangle.

Published in Journal of Recreational Mathematics, reprinted with permission.

4 of 4 people found the following review helpful:

4.0 out of 5 stars Plain but solid account of all things binomial, February 2, 2006

By 

Viktor Blasjo - See all my reviews
(REAL NAME)   

This review is from: Pascal's Arithmetical Triangle: The Story of a Mathematical Idea (Johns Hopkins Paperback) (Paperback)

Pascal's triangle flirts with the Greek theme of "figurate numbers": the third row are the triangular numbers, the forth row are the pyramidal numbers, and so on for the higher dimensional analogs (chapter 1). But the entries of the triangle, i.e. the binomial coefficients, turn up more naturally in "n choose k"-type combinatorics, ignored by the Greeks but picked up by everyone else: the Indians, the Arabs, the Chinese, and the Renaissance Italians (chapters 2-5). The binomial coefficients have an inherent desire to be tabulated triangularly, and we see several charming variants of this reproduced from original sources. Pascal then put the triangle in its definite form, and provided neat inductive proofs of its marvellous properties (chapters 6-7). The most exciting twist of the story is Newton's discovery of the general binomial expansion, i.e. the expansion of (a+b)^n also for non-integer exponent (chapter 8). Such infinite series are powerful tools in the calculus; indeed, Wallis had already used similar tricks to integrate (1-x^2)^(1/2), yielding his infinite product formula for pi. Meanwhile, the binomial coefficients kept up their combinatorial life (chapters 9-10), especially in the fields favoured by Pascal himself--probability theory and gambling--culminating with Bernoulli's famous Ars Conjectandi.