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The European Mathematical Awakening: A Journey Through the History of Mathematics from 1000 to 1800

Frank J. Swetz
Dover Publications
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

This is a second slice from the book From Five Fingers to Infinity, edited by Frank Swetz and published by Open Court in 1994. Five Fingers was aimed at mathematics teachers and the (mythical?) general reader, as shown by the fact that I bought mine from the (now long defunct) Quality Paperback Book Club. It collected many articles about the history of mathematics organized in eight parts. The articles were a motley group: older and newer, from sources ranging from Scripta Mathematica to The Mathematics Teacher. The desire to remain accessible, however, clearly dictated that the majority of the articles should come from journals dealing in less technical materials, so there were many more articles from The Mathematics Teacher than from MAA journals.

A previous slice of Five Fingers, dealing with mathematics since 1800, was brought out by Dover with the title The Search for Certainty. That was also the title of part VIII of Five Fingers, from whence it came. There were a few bits of added material, but most of the articles were exactly the same.

The current selection has a more complex relation to the original: it collects (most of the) articles from parts IV (European Mathematics During the "Dark Ages"), VI (The Revitalization of European Mathematics), and VII (Mathematical Responses to a Mechanistic World Outlook)  of the original book and spans a much broader period, about 1000 to 1800. Apart from preface, notes, bibliography and other ancillary material, the new items here are a three-page introduction entitled "Perspective: The European Mathematical Awakening" and "Historical Exhibit 9", which shows multiplication algorithms of the 15th and 16th centuries. The majority of the articles were originally published by The Mathematics Teacher in the 1950s or 1960s, which must have been a golden era for historical material in that journal. It is good to have them gathered here, though of course one must be cautious about checking the accuracy of 50-year-old material. In many cases, Swetz has appended a brief editorial note pointing to more recent publications dealing with the material in the article, making this version more useful than the original Five Fingers version.

Three articles from MAA journals are included. From the American Mathematical Monthly in 1919, "Leonardo of Pisa and His Liber Quadratorum" is a useful survey of that work that gets extra points from me for not using "Fibonacci" in the title. From the College Mathematics Journal in 1987 come two well-known articles: Fred Rickey's "Newton: Man, Myth, and Mathematics" and Bill Dunham's "The Bernoullis and the Harmonic Series"; the latter eventually became a chapter in Journey through Genius.

I'll point out one missed opportunity: Swetz includes a couple of articles on the introduction of decimal fractions in Europe, but does not include two brilliant (and hard to find) articles by David Fowler: "400 Years of Decimal Fractions" and "400.25 Years of Decimal Fractions", both from Mathematics Teaching, which I think is the British equivalent of our The Mathematics Teacher

The European Mathematical Awakening is a very useful collection of historical articles that can, among other things, be used as side reading for a history of mathematics course focusing on this period. (I did that recently with The Search for Certainty.) Students will love the fact that it is affordable and easy to read, and some of the articles may be useful jumping-off points for further research.

Fernando Q. Gouvêa teaches History of Mathematics (and many other things) at Colby College, where he is Carter Professor of Mathematics.


Perspective: The European Mathematical Awakening
Historical Exhibit 1: The Growth of Mathematical Knowledge
1. Counters: Computing if You Can Count to Five
Historical Exhibit 2: Bede's Finger Mathematics
2. Gerbert's Letter to Adelbold
Historical Exhibit 3: The Geometry of Gothic Church Windows
3. The Arithmetic of Medieval Universities
4. The Craft of Nimbryng
Historical Exhibit 4: Algorist versus the Abacist
5. Leonardo Fibonacci
6. Leonardo of Pisa and his Liber quadratorum
7. Some Uses of Graphing before Decartes
8. Adam Riese
9. Tangible Arithmetic: Finger Reckoning and Other Devices
10. The Cardano-Tartaglia Dispute
Historical Exhibit 5: Cardano's Technique for the Solution of a Reduced Cube Equation
11. Complex Numbers: An Example of Recurring Themes in the Development of Mathematiocs.
12. Robert Recorde's Whetstone of Witte, 1557
13. The Teaching of Arithmetic in England from 1550 until1880 as Influenced by Social Change
14. Tangible Arithmetic: Napier's and Genaille's Rods
15. The Life and Times of Johann Kepler
Historical Exhibit 6: Multiplication Algorithm of the Fifteenth and Sixteenth Centuries
16. Simon Steven and Decimal Fractions
Historical Exhibit 7: Mathematical Considerations on the Trajectory of a Cannon Ball
17. Viete's Use of Decimal Fractions
18. John Napier and His Logarithims
Historical Exhibit 8: The Evolution of Algebraic Symbolism
19. Projective Geometry
20. Pisa, Galileo, Rome
Historical Exhibit 9: Torricelli's Wine Glass
21. Analytic Geometry: The Discovery of Fermat and Descartes
22. The Young Pascal
Historical Exhibit 10: Roberval's Quadrature of the Cycloid
23. Isaac Newton: Man, Myth, Mathematics
Historical Exhibit 11: Newton's Method of Fluxions
24. The Newton-Leibniz Controversy Concerning the Discovery of Calculus
Historical Exhibit 12: Mengoli's Proof for the Divergence of the Harmonic Series
25. The Bernoulli Family
26. The Bernoullis and the Harmonic Series
27. The First Calculus Textbooks
28. The Origin of L'Hopital's Rule
29. Euler, the Master Calculator
30. Gaspard Monge and the Descriptive Geometry
31. Mathematicians of the French Revolution
32. The Ladies Diary...Circa 1700
Historical Exhibit 13: Women in Mathematics
Epilogue: The Process Continues
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