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v2.1 Reference

Cogwheels of the Mind

The Story of Venn Diagrams

Used today in spheres of life as diverse as business strategy, creative writing, medicine, computer science, and theoretical physics, Venn diagrams possess fascinating properties. The basic Venn diagram is both elegantly simple—three overlapping circles that intersect to create eight distinct areas—and conceptually innovative. Devised by English logician John Venn (1834–1923) to visually represent complex logical propositions and algebraic statements, the diagrams drew the excited interest of both scholars and the general public.

In Cogwheels of the Mind, statistician and geneticist A. W. F. Edwards provides an accessible and engaging history of the Venn diagram, its reception and evolution, and its presence in such objects and images as Christian iconography, tennis balls, and flags which provide a rich source of Venn diagrams for Edwards, including those of Switzerland, Poland, and Japan (all one-set Venn diagrams), Greenland (a two-set Venn diagram), and Maryland (a three-set device).

Edwards begins with a sketch of Venn's life, his discovery of the three-circle design while developing a series of lectures on symbolic logic at Cambridge University, and the publication of his find in an 1880 paper, and, more influentially, his 1881 book, Symbolic Logic. Edwards discusses the rival diagrammatic scheme invented by Charles Dodgson, better known as Lewis Carroll, who also developed a board game based on his design. The author also recreates famous Venn diagrams from history, including Winston Churchill's of 1948 depicting the mutual interests of the British Empire, a united Europe, and the English-speaking world, with the United Kingdom located at the intersection.

Edwards goes on to show how different shapes can be linked together to form artistically beautiful and mathematically important, multi-set Venn diagrams, including the author's own influential Adelaide variation. And he delineates the possibilities for expanding the analytic power of these diagrams far beyond those first appreciated by Venn. Edwards even tells readers how to draw complex Venn diagrams on a spherical surface to create "Vennis balls." For anyone interested in mathematics or its history, Cogwheels of the Mind is invaluable and compelling reading.

About the Authors

A. W. F. Edwards is, like John Venn before him, a fellow of Gonville and Caius College, Cambridge. He is the author of Likelihood and Pascal's Arithmetical Triangle, both available from Johns Hopkins.


"Edwards arrives with this pleasing little history about who John Venn was, why he conceived of the diagram, and the properties that lie secreted beneath such a seemingly simple mathematical object... A world-class authority, Edwards... proves himself wholly accessible to anyone interested in reading about mathematics."

- Booklist

"Deserves to become a minor classic and may well go on to make many friends for mathematics."

- Jeremy Gray - Nature


- Boston Sunday Globe

"Edwards is a charming if earnest guide, and the many illustrations of the beautiful cogwheels will fascinate and satisfy."

- Ben Longstaff - New Scientist

"The kind of book that I can imagine giving to a wide range of readers: any junior high student would be able to follow the mathematics, and most professors would find it interesting... I found it to be a nice—if light—read, and it is well worth a look."

- Darren Glass - MAA Online

"Many excellent and graphically exciting illustrations of Venn diagrams transform what might have been a simple math book into one that shows that mathematics can generate pictures that could be considered... in the forefront of modern art."

- School Library Journal

"Give this book to any youngster with an enquiring mind, and watch delight develop."

- John Haigh - Significance

"There is a nice balance between personal anecdotes, history, aesthetics and attention to detail... The overall result should become something of a classic."

- B. I. Henry - The Physicist

"Venn diagrams are familiar as pictorial representations of relationships among sets, and statistician Edwards discusses their development... He shows how to interlink shapes to form beautiful Venn diagrams."

- Choice

"I heartily recommend it for readers interested in knowing more about John Venn and the geometric properties of Venn diagrams. It will also be appreciated by those interested in the process of mathematical discovery."

- Frank Ruskey - American Scientist

"An insightful history of the diagrams."

- Scientific American

"Lovely little book... which should not be summarized. It should be bought and enjoyed first-hand. It is a book in which mathematics and its history are combined in a lovely autobiographical account of a voyage of discovery. The mathematics is interesting, the history is interesting, the personal account is interesting, and the book, with its elegant full-colour pictures and diagrams is beautifully produced. The author and his publisher are to be warmly thanked and congratulated."

- Peter M. Neumann - British Society for the History of Mathematics Bulletin

"A short book, with a fittingly large number of illustrations, it summarizes a wealth of logical and geometric ideas."

- Rob Hardy - The Times of Acadiana

"If you have an interest in the history of mathematical ideas and the creative process of mathematicians, then I recommend this book."

- John Wilkins - Mathematics Teacher

"Edwards' fascinating study relates this invention to the study of mathematics, scientific thought, graphic design, and modern art."

- School Library Journal

"A very short readable book on attempts to physically represent the intersection of any number of sets."

- Jim Kiernan - Convergence

"There is no better place to start than with Cogwheels of the Mind. "

- Frank Ruskey - American Scientist

"This is a wonderful book which should be taken simply for what it is, the story of the Venn-Edwards diagrams."

- Amirouche Moktefi - Review of Modern Logic

"An engaging, very readable, and profusely illustrated account."

- Historia Mathematica

"An easy, friendly, and enlightening book to read... Would be of particular interest to college professors, especially those involved in teaching a History of Mathematics class and/or a Graph theory class."

- Darlinda Cassel - School Science and Mathematics


"Cogwheels of the Mind is not only interesting but inviting. Edwards' penetrating insight into the contentious history of John Venn and his diagram is fascinating. Read it!"

- Jack Kuipers, Calvin College

"You simply can not be knowledgeable of Venn diagrams until you have read Professor Edwards' delightful book. It is a real treat for the logical mind."

- Paul J. Nahin, author of Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age
Johns Hopkins University Press
From 18

9780801874345 : cogwheels-of-the-mind-edwards-stewart
128 Pages
$31.00 USD

Other Titles by A. W. F. Edwards

Likelihood, expanded edition

A. W. F. Edwards
Oct 1992 - Johns Hopkins University Press
$31.00 USD - Paperback / softback

Other Titles in MATHEMATICS / Applied


Anna Weltman
Sep 2020 - The Johns Hopkins University Press
$24.95 USD - Hardback
$24.95 USD - Electronic book text

Republic of Numbers

David Lindsay Roberts
Sep 2019 - The Johns Hopkins University Press
$29.95 USD - Hardback
$29.95 USD - Electronic book text

Calculus in Context

Alexander J. Hahn
Apr 2017 - Johns Hopkins University Press
$95.00 USD - Hardback
$95.00 USD - Electronic book text

Other Titles in Applied mathematics

Republic of Numbers

David Lindsay Roberts
Sep 2019 - The Johns Hopkins University Press
$29.95 USD - Hardback
$29.95 USD - Electronic book text

Calculus in Context

Alexander J. Hahn
Apr 2017 - Johns Hopkins University Press
$95.00 USD - Hardback
$95.00 USD - Electronic book text

Emmy Noether's Wonderful Theorem, revised and updated edition

Dwight E. Neuenschwander
Apr 2017 - Johns Hopkins University Press
$30.00 USD - Paperback / softback
$30.00 USD - Electronic book text