Nerida Ellerton and Ken Clements’ most recent work, published in March 2017, is an excellent history of London’s Royal Mathematical School. It follows in the footsteps of three recent Ellerton/Clements books, namely

*Rewriting the History of School Mathematics in North America 1607–1861: The Central Role of Cyphering Books* (2012)
*Abraham Lincoln’s Cyphering Book and Ten other Extraordinary Cyphering Books* (2014)
*Thomas Jefferson and his Decimals 1775–1810: Neglected Years in the History of U.S. School Mathematics* (2015).

All four of their books have been published by Springer, all are first-rate, and all make substantial contributions to the history of mathematics, even as each of them arguably contains flaws and limitations.

Christ’s Hospital is a London school, which was founded in 1552 and still operates to this day, albeit in a different location. The Royal Mathematical School (RMS) was established within Christ’s Hospital in 1673 with the goal of educating poor children in mathematics and navigation to furnish the Royal Navy with competent sailors. The authors argue that this school taught the first secondary school mathematics curriculum. Students learned about logarithms, plane and spherical trigonometry, and other mathematical topics that were considered highly advanced in that era, and which correspond roughly with the level of contemporary secondary school math. Apparently there had been no previous attempts to teach the same mathematical topics at that level to an entire school population.

Ellerton and Clements do a remarkably thorough job of describing the history of this school from “the perspective of the international history of school mathematics”. Chapter 1 describes mathematics in Christ’s Hospital before 1673, that is, before the founding of the Royal Mathematical School. At the time, according to the authors, “it is almost certain that elementary arithmetic was the only mathematics taught”. The school emphasized the study of English grammar and writing, and taught some Latin and Greek.

Chapter 2 tells the story of how the Anglo-Dutch wars and other historical factors inspired King Charles II and his many advisors, including Samuel Pepys, to found the RMS. Pepys served as Clerk of the Acts to the Navy Board, and later, Secretary to the Admiralty Commission. He is known for the very detailed diary he kept from 1660 to 1669, an important historical primary source.

Chapters 3 describes ten leading figures of the RMS, namely Samuel Pepys, Jonas Moore, Christopher Wren, Robert Hooke, Isaac Newton, John Flamsteed, Edmund Halley, James Hodgson, John Robertson, and William Wales. Some of these figures were masters of the school for periods of time and others contributed to discussions of the curriculum. Chapters 4–8 give details on various periods of the school’s history and the evolution of the mathematical curriculum.

In Chapter 9, the authors examine three mathematicians’ attempts to introduce more sophisticated topics into the curriculum, namely, mechanics, rigorous algebra, and fluxions. They examine the time lag between the invention of a theory and when it is introduced as a topic in schools. Finally, in Chapter 10 they systematically answer six research questions. For example, the first of these six questions is “Why was RMS established in 1673?”

Ellerton and Clements have undergone painstakingly careful research in preparing this book, and they are highly knowledgeable about the history of school mathematics. They’ve published hundreds of scholarly articles, many of them as a husband-and-wife team. They own 2500 old mathematics textbooks and what they describe as the world’s largest collection of cyphering books. Without a doubt they are the world’s most prominent historians of mathematics in their area.

Ellerton and Clements examine the history of the RMS from several perspectives. For example, they distinguish between the “intended curriculum”, the “implemented curriculum”, and the “received curriculum”. A theme throughout this book is that consulting mathematicians frequently prescribed challenging mathematical topics to be taught in the RMS. The school’s administrators (masters) were more realistic about what the boys could and could not learn in the allotted time; students, however, were often unable to absorb the mathematics of the implemented curriculum. Ellerton and Clements studied students’ personal cyphering books (personal notebooks) to get a better sense of the implemented curriculum (what the teachers taught) and the received curriculum (what the students learned).

This book is of great value, but from this reviewer’s perspective it has two notable drawbacks. The first is the lack of page numbers in most citations. Probably a majority of the references cited are books, and few of these citations give page numbers. It isn’t reasonable for Ellerton and Clements to expect their readers to read an entire book each time they want to investigate the source of a reference.

Second, the authors are overly contentious. An entire nine-page appendix (Appendix E, pp. 281–289) is devoted to arguing with other scholars. Clifford Jones wrote an official history of the Royal Mathematical School, and the page 281 picks apart an alleged error in that book. The rest of the appendix takes issue with reviews of previous Ellerton and Clements books, all of these reviews having appeared in the now defunct *International Journal for the History of Mathematics Education*. There are many other instances in this book where the authors seem to be fighting battles against other scholars, in addition to telling their own compelling story. While there is some scholarly rationale for being a stickler for the truth as one sees it, the sideshow of intellectual squabbling distracts a bit from the main narrative.

Despite the drawbacks cited above, this book represents an outstanding scholarly work. There are many figures with pages from cyphering books, and various other primary source documents are interspersed throughout the book. These include, for example, official critiques of the RMS curriculum composed by Samuel Pepys in 1677, and by Isaac Newton in 1694. The authors’ analysis of issues is extremely well informed and comprehensive. Any reader who is interested in the origins of secondary school mathematics in general, or the Royal Mathematical School in particular, will find this book to be exceptionally valuable.

Andrew Perry is Professor of Mathematics at Springfield College in Massachusetts. His interests include Sports Analytics and the History of Mathematics, especially the history of pedagogy and textbooks.