### Conclusion

Napier's Binary Chessboard Calculator, almost completely overshadowed in *Rabdologia* by his Bones and Promptuary, is perhaps the earliest comprehensive, practical use of the concept of binary numbers. Here we have a calculator that can be used to add, subtract, multiply, divide, and take square roots of whole numbers.

The precedents for the use of powers of two, and the processes of doubling and halving (duplation and mediation) to accomplish multiplication can be traced to the Rhind (or Ahmes) papyrus from ancient Egypt (1650 BCE). The process was commonplace in the Middle Ages, and an interesting instance from relatively modern times is found in the so-called "Russian Peasant Multiplication" algorithm. Napier's Binary Chessboard Calculator is thus a brilliant synthesis of mathematical concepts that long preceded him, realized however with the classic simplicity inherent in commonplace objects.

The Chessboard Calculator has proven to be an excellent vehicle for teaching the binary number system and binary arithmetic, the distributive property, and place value. The first author has introduced and demonstrated Napier's Binary Chessboard Calculator as an enrichment topic to elementary and secondary teachers (at universities and at an Eisenhower Institute) and to college students in a Liberal Arts Mathematics class with great success, and to enthusiastic response. All this activity required were bingo chips and cardboard checkerboards!

In retrospect, the Chessboard Calculator seems to have been quite properly treated by Napier's peers and by history; that is, it was regarded as a curiosity and overshadowed by his other inventions. In commercial practice, where both casting counters and Napier's "bones" were readily available, it is not surprising that the Chessboard Calculator failed to find acceptance. Moreover, while more advanced practitioners of mathematics might have found the device somewhat intriguing, Napier had already given them the much more powerful tool of logarithms and ultimately the slide rule. Thus, while many examples of the Bones (and one of the Promptuary) have survived, it seems unlikely that examples of Napier's Chessboard Calculator will be found. It most likely never found common use and, if used at all, reverted back to a game board when no longer needed.

### Acknowledgments

I thank my co-author, Erwin Tomash, who was a mentor and good friend to me. I spent many happy hours perusing his extensive collection of books related to the history of computer science, including many early works on computing by hand like those of Napier. I am delighted to be able to finally finish this article that we started so many years ago and I hope it honors his memory. I also thank my daughter Allison Kolpas, Associate Professor of Mathematics at West Chester University, for reviewing a late draft of the article for me.

### About the Authors

*By Janet Beery, Editor*

Sidney J. Kolpas, a frequent contributor to *Convergence,* has long been interested in the history of mathematics, especially as it relates to teaching, and has been a longtime collector of historical mathematical books and objects. Kolpas holds an MS in mathematics and an EdD in mathematics curriculum and instruction. His first career, which he pursued for 19 years, was as an award-winning junior high and high school mathematics teacher in Burbank, California. His second career was as professor of mathematics at Glendale College in California from 1990 to 2011. He received many awards for his outstanding teaching during his tenure at Glendale, including the college's Distinguished Faculty Award in 2004 and the State of California's Hayward Award for Excellence in Community College Education in 2010. In 2011, he moved to Pennsylvania, and became a mathematics professor at Delaware County Community College, where in 2016 he received the college-wide Gould Award for outstanding faculty performance. He retired in 2018.

Erwin Tomash (1921-2012) was a pioneering computer scientist, helping launch the U.S. computer industry from the 1940s onward. During the 1970s he became interested in the history of computer science, and founded the Charles Babbage Society, and its research arm, the Charles Babbage Institute. The Institute, an archive and research center, is housed at the University of Minnesota. Its Erwin Tomash Library on the History of Computing began with Tomash's 2009 donation to the Institute of much of his own collection of rare books from the history of mathematics and computing. (*Source:* Jeffrey R. Yost, Computer Industry Pioneer: Erwin Tomash (1921-2012), *IEEE Annals of the History of Computing,* April-June 2013, 4-7.)