# Pascal's Arithmetical Triangle: The Story of a Mathematical Idea

JHU Press, 23 lug 2002 - 202 pagine

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.

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### Indice

 The figurate numbers 1 Three combinatorial rules 20 The combinatorial numbers in India 27 The combinatorial numbers in the West 34 The binomial numbers 50 Pascals Treatise on the Arithmetical Triangle 57 Pascals Treatise Part II and associated tracts 70 The Arithmetical Triangle in analysis 87
 Bernoullis Ars conjectandi 121 Epilogue 138 Pascal and the Problem of Points 144 The Gamblers Ruin 157 Commentary on Ars conjectandi 167 Table of binomial coefficients 182 References 189 General index 198

 The binomial and multinomial distributions 112

### Riferimenti a questo libro

 The Mathematics of Great AmateursJulian Lowell CoolidgeAnteprima non disponibile - 1990
 Excursions in Calculus: An Interplay of the Continuous and the ..., Volume 13Robert M. YoungAnteprima limitata - 1992
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### Informazioni sull'autore (2002)

A.W.F. Edwards is a professor of biometry at the University of Cambridge and a fellow of Gonville and Caius College, University of Cambridge, England. He is the coauthor of Annotated Readings in the History of Statistics and author of Foundations of Mathematical Genetics and Likelihood, the last is available from Johns Hopkins.