From the seventeenth to the early nineteenth century, cyphering books were the main tools for conveying mathematical knowledge to students. During that period in America teachers didn’t stand at blackboards and lecture. Textbooks were rarely used and if so it was mainly to copy from. There were no written examinations. The cyphering tradition incorporated a form of pedagogy whereby students became independent problems solvers by learning the rules for solving various types of problems. After working out a problem (perhaps on slate), students would have their work checked by a teacher or tutor who might quiz them on their understanding of the problem. If done correctly, they would copy the problem and their solution into their cyphering book, often with exquisite calligraphic style. Sometimes an embellished front cover would be added and often sometimes drawings or watercolors would be added.

In this volume, exquisitely illustrated with many images of the actual pages, eleven cyphering books composed in North American or Great Britain between 1630 and 1836 are examined. Seven deal with arithmetic, three with navigation, and one with measuring and surveying. Six were prepared in Great Britain, four mainly in North America, and one partly in Great Britain and partly in North America.

We learn about the *abbaco* tradition employed by tutors and supervisors, which included operations on whole numbers, currency exchange, gauging (measuring the capacities of containers), and the rule of three, to deal with problems relating to trade, partnership, and the handling of merchandise. In some cases, we learn about the lives of the authors of the cyphering books, the schools they attended, and the books they copied from.

Besides having collected over 1500 mathematics textbooks from the period, the authors have examined approximately a thousand handwritten cyphering books. They have a personal collection of over 500 such books, 350 of them North American, 100 from Great Britain, and 50 from other countries. The authors aptly note that this book is more than just a complement to their *Rewriting the History of School Mathematics in North America 1607–1861: The Central Importance of the Cyphering Tradition,* Springer, 2012.

Each chapter begins with an abstract and a list of key words used. With each image of a page from a cyphering book, there is commentary on the text and a discussion of its historical significance. After an introductory chapter, there is a chapter devoted to the oldest extant North American cyphering book, a seven-page work by an unknown author in New England around 1667. The book covers basic arithmetic operations and the rule of three. The next chapter deals with a cyphering book began by Thomas Prust in England and completed in New England by his stepson James Collins. It is an abbaco-inspired cyphering book containing pages devoted to the four operations on whole numbers: addition, subtraction, multiplication, and division. These operations are followed by mediation (factoring a number into a product of two numbers), reduction (expressing a quantity in smaller units of measurement), the rule of three, and common fractions.

Chapter 4 features the cyphering book of two sisters from Bertie County, North Carolina: Martha and Elizabeth Ryan, who were teenagers around 1780. The authors of the present work point out that it is rare to have a southern cyphering book prepared by a woman. In this case, it might have been necessary for the sisters to have a feel for business accounts. Their cyphering book is festooned with drawings of ships and patriotic sayings and exhibits a high quality of penmanship, calligraphy, and detailed artwork. The book begins with numeration (where numbers are spelt out). Quite remarkably, the topics in the book go well past basic arithmetic operations (the zig-zag method for division is employed) to the rule of three and fellowship problems in which a quantity is divided up among people who have contributed different amounts of capital to a venture for different periods of time. The book continues with basic business topics including simple and compound interest, profit and loss, decimal currency, equation of payments, alligation (a method for solving arithmetic problems related to mixing ingredients), single and double false position, tare & tret (rules for calculating the net weight of goods), and concludes with vulgar factions. Both the introduction-rule-case-example-exercise and the problem-calculations-answer methods are employed in their book.

The next chapter focuses on Mary Walter’s Cyphering Book which was composed around 1820 in Baltimore. In her cyphering book, arithmetic is studied through poetry and song, exhibiting strong links with Irish tradition. Such books are very rare. Problem are posed in poems with a human interest situation added to fit the basic arithmetic sequence, reduction and the rule of three. It was a singular way to learn the *abbaco* curriculum. The authors suggest that she had an Irish-heritage school teacher and attended a school where students speak aloud, sing to themselves and others throughout the school day.

Most schools in agrarian communities operated in the winter and possibly in the evening at other times of the year. Abraham Lincoln attended grammar school for at least three winter terms. His cyphering book is the oldest surviving manuscript of his handwriting. It follows the *abbaco* curriculum and was composed between 1819 and 1826. It is not clear whether it was copied from a textbook or from dictation. It includes work with the rule of three but not with decimal fractions. He used casting out nines as a check on his arithmetic. Later in life, Lincoln studied geometry, logarithms, trigonometry, and surveying. Another leaf from his book was discovered recently in Harvard’s Houghton Library. The authors provide a new ordering for the extant pages. There is a discussion of what textbooks the problems may originate from, what schools he attended, and who his teachers were. What makes the book extraordinary is undoubtedly the conditions under which it must have been prepared.

The next chapter focuses on Thomas Dixson’s 302-page cyphering book. It is the oldest book in the selection, dating from around 1630. It follows the Sacrobosco curriculum, in which the galley method is used for division. The next chapter features the cyphering book of George Bickman, a writing master, dating from 1740. The book was commercially prepared with an aim to show fine calligraphy and penmanship and was intended for sale to others for use in schools. There is an extended discussion on the link between writing education and mathematics education. Chapter 9 features a 241-page navigation cyphering book by an unknown author. Even though the voyages depicted in the book appears to be fictitious, the book is richly illustrated with 80 illustrations and 51 water-color miniatures. Extra topics covered include mensuration, fortification, and navigation. Drawing was an important component form 1750 to 1900 in public schools in England, even at Oxford and Cambridge. Locke, Pepys, and Wren advocated that drawing be an integral part of the curriculum in all middle-class schools.

The 671-page cyphering book composed by Charles Page around 1825 is discussed in the next chapter. Page was a student at London’s Christ’s Hospital, the first English school in which mathematics became an important part of the curriculum and one of the world’s best training school for navigators. The method of education at Christ’s Hospital and its contribution to the development of cyphering system is described in some detail. Topics in Page’s book include, arithmetic (including logarithms), geography, Euclidean geometry (first six books of Euclid), plane trigonometry, chronology, algebra (which was unusual in the* abbaco* curriculum), definitions and principles of navigation, spherical trigonometry, astronomy, the use of globes, and practical methods for navigating.

Chapter 11 deals with two manuscripts composed by William Beattie. One is a 446-page text compiled in 1810, while he was a student at the Clarencefield Academy in Dumfriesshire, Scotland; it focuses on mensuration problems. The other is a 309-page text composed during the period from 1835 to 1836 and is devoted to navigation problems. The books from which he copied his cyphering books are identified and described. Beattie was the personal physician and private secretary for the Duke of Clarence, who would later become King William IV.

There are several recurrent themes in the volume: the intended curriculum, the implemented curriculum, and the notion of whether students actually learned the material found in their cyphering book. The authors point out that the intended mathematics curriculum that was present in an individual’s cyphering book was not always the same as the implemented curriculum. Cyphering books show what students studied but not what they learned. The authors conclude that it is almost impossible to estimate the quality of a student’s mathematical learning by merely examining their cyphering book, or for that matter the textbooks they used. With respect to many cyphering books, it appears that the goal of reaching the rule of three was a powerful motivator. Anyone who cyphered up to the rule of three was regarded as an efficient reckoner. Those who studied beyond could claim to be advanced reckoners. The remarkable value that many students placed upon their cyphering books is striking. It appears that next to the Bible, one’s cyphering book was the most valuable book owned. Undoubtedly, for some it was a guidebook for life.

The authors have put together a brilliant sample of cyphering books and in the process discuss a number of relevant issues in mathematics education. They present a cogent argument on why the early history of school mathematics should be rewritten. There is some confusion in Chapter 3 as to who John Collins is and some repetition throughout the volume, but neither distract from this very interesting and informative sourcebook.

James Tattersall is Professor of Mathematics at Providence College.