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

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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

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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.

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