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Publisher:

Dover Publications

Publication Date:

2013

Number of Pages:

250

Format:

Paperback

Price:

14.95

ISBN:

9780486498058

Category:

Anthology

[Reviewed by , on ]

Fernando Q. Gouvêa

08/7/2013

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.

Preface | |||||||

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 | |||||||

Suggested Readings | |||||||

Index |

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