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|