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

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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.

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.