Book Description

John Wallis was appointed Savilian Professor of Geometry at Oxford University in 1649. He was then a relative newcomer to mathematics, and largely self-taught, but in his first few years at Oxford he produced his two most significant works: De sectionibus conicis and Arithmetica infinitorum. In both books, Wallis drew on ideas originally developed in France, Italy, and the Netherlands: analytic geometry and the method of indivisibles. He handled them in his own way, and the resulting method of quadrature, based on the summation of indivisible or infinitesimal quantities, was a crucial step towards the development of a fully fledged integral calculus some ten years later.

Customer Reviews (2)

Infinite series and Wallis's product formula
From Stedall's introduction: "In De sectionibus conicis, Wallis found algebraic formulae for the parabola, ellipse and hyperbola, thus liberating them, as he so aptly expressed it, from 'the embranglings of the cone'. His purpose in doing so was ultimately to find a general method of quadrature (or cubature) of curved spaces, a promise held out in De sectionibus conicis and taken up at length in the Arithmetica infinitorum ... Wallis's major contribution to the development of seventeenth-century mathematics was perhaps, as he himself recognized, the transformation of geometric problems to the summation of arithmetic sequences. Many of the results demonstrated by Wallis were already well known but, as he repeatedly pointed out, his aim was to establish a method by which those results, and others, could be systematically obtained ... From his startingpoint of simple powers, he could easily handle sums (or differences) of sequences, and hence eventually quadratures of any curve of the form y=(1-x^(1/p))^q provided p and q were integers. But because his ultimate aim was the quadrature of the circle, the curve he was really interested in was y=(1-x^2)^(1/2) ... In what was perhaps the one real stroke of genius in Walli's long mathematical career, he saw how to complete his [solution] by a method now set out in Proposition 191, and so arrived at his infinite fraction for 4/pi", namely 3*3*5*5*7*7*.../2*4*4*6*6*8*... So what was this "one real stroke of genius"? Well, to find pi we need to integrate (1-x^2)^(1/2). Today we would feel like using the binomial series expansion, but this is not available to Wallis so he more or less has to invent it. The binomial series expansion for an integer exponent is of course given by Pascal's triangle, but to expand things like (1-x^2)^(1/2) we need a corresponding "fractional entry" sitting between two lines in Pascal's triangle. We don't know what that entry should be, but remember that we can integrate (1-x^(1/p))^q, which can be interpreted as a fractional entry in Pascal's triangle. So we do this for a bunch of integers p and q to get an enlarged Pascal's triangle with fractional entries and then use ordinary properties of Pascal's triangle extended to the fractional setting to deduce the value for the entry we want.

A historical juwel
If you want to study mathematics, in particular rational numbers, concepts of infinity and relation with geometry, this book is not very effective, be it only for the tedious style of formulation.
But when you look at the book from its historical importance and understanding of the mathematical reasoning in the 17th century, this book is to be considered as a masterpiece, or (why not) a collectors item.
The book is a translation from latin and is further complemented with an introduction by Dr Stedall, helping a lot to put the work in the right frame of its importance, that a Geometrical problem is reduced to a pure arithmetic problem.
The text also illustrates, some of the shortcuts taken in that period of history, such as the concept of proof by induction.
I do not intend to read the book from start to end, but I am regularly studying one chapter, to immerse myself in the reasoning at that time. I can not stop to be amused and amazed by the way John Wallis contempories are corresponding with each other.
What to think about "To the most Distinguished and Worthy gentleman and most Skilled Mathematician Dr William Oughtred..."
This book gets four stars, and I would give six stars if there would be a mapping for each proposition into modern language.
The book is expensive, as all these type of books, but its worth the money.
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Book Description
Wallis was one of the most original mathematicians of theseventeenth century and he left his mark on mathematics in manyways. He introduced arithmetical limits into mathematics (hisfamous infinite-product expression for $\pi$ is an example). Hisresearches (for example, the means whereby he obtained theaforementioned product) led directly to Newton's work on thebinomial theorem and quadratures. He was the first to see thesignificance of fractional and negative exponents, and he isresponsible for the introduction of such symbols as $\infty$ andsuch terms as hypergeometric series. He was very influentialpolitically, very quarrelsome, and at the center of thescientific life of his time (for instance, it was owing to hisadvice that the Gregorian calendar was not introduced earlierinto England, as he did not like the Pope). This second editionincludes Foreword by E. N. da C. Andrade, a Bibliography and anIndex. ... Read more

Book Description
This is the first volume of a six volume compendium on the correspondences of John Wallis (1616-1703). Wallis was Savilian Professor of Geometry at Oxford from 1649 until his death, and was a founding member of the Royal Society and a central figure in the scientific and intellectual history of England. Along with his role as decipherer on the Parlimentary side during the Civil War, he prepared the ground for the discovery of infinitesimal calculus by Newton and Leibniz and played a decisive role in modernization of English mathematics. This volume provides fascinating insight into the life of Wallis through his correspondences with intellectual and political figures of the latter part of the 17th century. ... Read more

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And The Mayor And Police Commissioners Of Baltimore, In 1861. ... Read more

Book Description
This is the second volume of a six volume compendium on the correspondences of John Wallis (1616-1703). Wallis was Savilian Professor of Geometry at Oxford from 1649 until his death, and was a founding member of the Royal Society and a central figure in the scientific and intellectual history of England. Along with his role as decipherer on the Parlimentary side during the Civil War, he prepared the ground for the discovery of infinitesimal calculus by Newton and Leibniz and played a decisive role in modernization of English mathematics. This volume provides fascinating insight into the life of Wallis through his correspondences with intellectual and political figures of the latter part of the 17th century. ... Read more

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This volume is produced from digital images from the Cornell University Library Samuel J. May Anti-Slavery Collection ... Read more

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This digital document is an article from Southern Economic Journal, published by Southern Economic Association on April 1, 2000. The length of the article is 898 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase. You can view it with any web browser.

Citation Details
Title: The Economics of the Great Depression.(Review) (book review)
Author: John Wallis
Publication: Southern Economic Journal (Refereed)
Date: April 1, 2000
Publisher: Southern Economic Association
Volume: 66Issue: 4Page: 1019

Article Type: Book Review

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This digital document is an article from Southern Economic Journal, published by Southern Economic Association on April 1, 1994. The length of the article is 740 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase. You can view it with any web browser.

Citation Details
Title: Strategic Factors in Nineteenth Century Economic History: A Volume to Honor Robert W. Fogel. (book reviews)
Author: John Joseph Wallis
Publication: Southern Economic Journal (Refereed)
Date: April 1, 1994
Publisher: Southern Economic Association
Volume: v60Issue: n4Page: p1094(2)

Article Type: Book Review

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Product Description
It seemed to be the dawning of just another summer day in the year 1680. The Spanish settlement of Santa Fe sat in the middle of the fields of corn. The town was too still and the children were hanging onto their mothers in corners. In a field of dry corn outside the town squatted a tall, gray-haired, copper skinned man. His name was Pope'. With him were more than two thousand Pueblo Indians ready to follow him into battle. The story is a legend with the Pueblo's. Interesting. ... Read more

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Book Description
This digital document is an article from NBER Reporter, published by National Bureau of Economic Research, Inc. on March 22, 1995. The length of the article is 2502 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase. You can view it with any web browser.

From the supplier: The public has revived its interest in the federal system's functions and changes that have transpired through the years. The interest extends to those years prior to the 20th century, with questions on the role of the government on the economy, the existence of centralizing trends, and the interactions involved in federal, state, local government and private sector activities. A comprehensive, quantitative record of US fiscal federalism covering the years 1790-1995 is discussed.

Citation Details
Title: Historical economics: U.S. state and local government.(Research Summaries)
Author: Richard Sylla
Publication: NBER Reporter (Magazine/Journal)
Date: March 22, 1995
Publisher: National Bureau of Economic Research, Inc.
Page: p14(3)

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