# Charles Sanders Peirce turns the tables

#### Modern Books and Manuscripts 2013-08-13

[Thanks to Houghton patron Scott Guthery for contributing this guest post.]

In putting together a story of mathematics in post-colonial America – in particular mathematics as found outside of colleges and universities – I found Google’s digitization of Harvard’s copy of *Mathematical Tables* by Solomon P. Miles and Thomas Sherman, the third stereotype edition of 1842 published by Benjamin B. Mussey. Miles and Sherman were both Harvard graduates, Class of 1819 and Class of 1825 respectively, and both taught at Boston English.

On an early page summarizing the book, Miles and Sherwin wrote, “The Tables, comprised in this volume, have been very carefully compared with the best English and French Tables; and they will be found, it is believed, not inferior, in point of correctness, to any similar Tables in use.” At the bottom of that page there was a barely-legible handwritten note that begged to differ. The note seemed to be signed by two brothers: Harvard mathematics professor James Mills Peirce (1834-1906) and philosopher and mathematician Charles Sanders Peirce (1839-1914), but I couldn’t be sure.

Flipping back to the front cover, I found the book’s call number at the top – Math 838.42. Below this was a partially obscured stamp that read “Transferred to Cabot Sc”. Consulting HOLLIS, I found that there were five copies of the Miles/Sherwin tables in Harvard’s collection. Four of them – the 1830 first edition, the 1836 second edition, the 1842 third edition, and the 1859 fourth edition were listed as being on the shelf at Cabot Library. The fifth was another copy of the 1842 third edition at Widener Library. Being the proud holder of a reader’s card, I immediately boarded the 86 bus and headed for Cabot.

As you may guess, the 1842 third edition on the shelf at Cabot was indeed the copy that had been digitized by Google and contained the handwritten review signed by the Peirce brothers.

The note reads in full:

Having had occasion to make a slight examination of these tables the undersigned are able to testify that they are less accurate than most similar tables, in use. J.M. Peirce C.S. Peirce

We don’t know which of the seven tables in the Miles/Sherwin book the Peirces examined but we can be sure that neither of the brothers would confuse accuracy and precision. The three largest tables in the book – logarithms of natural numbers, logarithmic sines, etc., and natural sines – are all to seven-places. While there are certainly tables of these values to more decimal places, seven places is by far the most widely-published level of accuracy for such tables.

Thus, what the Peirces are saying is a bit of a puzzle. It certainly doesn’t take more that a “slight examination” to determine that the tables are to seven places. It is also certainly true that the tables are less accurate than tables to eight or twelve decimal places but such tables are similar to the Miles/Sherwin tables only to the extent that the same values are tabulated.

Nor do we know when the Peirces wrote their review. We do know that in 1871 J.M. Peirce published “Three and Four Place Tables of Logarithmic and Trigonometric Functions” and had either Miles or Sherwin still been alive they could have written the identical note in his book of tables.

*Editorial note: Thanks to Scott’s discovery in the Cabot stacks, this book has been transferred to Houghton Library, where it joins a large collection of books that belonged to and/or contain annotations by Charles Sanders Peirce. To view other books that belonged to Peirce, search HOLLIS for “Peirce, Charles S. (Charles Sanders), 1839-1914, former owner.”*