The Educational Times Database: Building an Online Database of Mathematics Questions and Solutions Published in a 19th-Century Journal

Robert M. Manzo (University of North Carolina at Chapel Hill)

More to the point, however, is that nobody knows what these volumes contain.
… A bibliographer of great stamina is called for; and meanwhile a historian
will find many interesting surprises as he browses among its forgotten pages.

– Ivor Grattan-Guinness [1992]

The Educational Times and Journal of the College of Preceptors (ET) was an English academic journal first published in 1847. Aimed at educators across disciplines, it included a mathematical department to which any reader could submit a proposal or a solution for a mathematical problem. Demand for mathematical content from readers, in fact, was high enough to warrant the establishment in 1864 of Mathematical Questions with their Solutions taken from the “Educational Times” (MQ), a sister publication that reprinted and expanded on the ET’s mathematical sections. In 1918, the MQ ceased publication and the ET stopped printing mathematical problems. Over seventy years, the ET/MQ had published over 18,000 problems and 28,000 solutions. The resulting archive has remained of interest to mathematicians in the century since. Out of this interest have emerged various attempts to index the entire run of problems and solutions, so that interested scholars could look up the name of a mathematician to find out if and when she or he contributed to the journals. Most recently, Sloan E. Despeaux, professor of mathematics at Western Carolina University (WCU), North Carolina, has guided the creation of an online database that allows users to search for ET/MQ mathematical contributions by author name, question type, year, and/or volume number. The substantial index underlying the database is a creation of Dr. James Tattersall, Jr., professor of mathematics at Providence College, Rhode Island, who has been the “bibliographer of great stamina” that Grattan-Guinness called for in 1992. Now completed, the ET database aims to fulfill research and pedagogical needs in today’s global mathematics community, with particular relevance to historians of mathematics. The present article consists of five sections on various aspects of the ET database, starting with the two items of greatest practical relevance, followed by three items about the history of the ET/MQ and the database itself:


The Educational Times Database: How to Search the Database

Robert M. Manzo (University of North Carolina at Chapel Hill)

The ET database ( has three major features:

  • basic search,
  • advanced search, and
  • references look-up.

The basic search function enables users to look up the name of a mathematician and see all of the problems and solutions he or she contributed to the journal. The advanced search function enables users to limit results by date range or volume range, by question type, and/or by the author’s country of origin. The references feature links to tables that show a partial list of libraries possessing physical copies of ET/MQ volumes, as well as hyperlinks to digitized copies of selected volumes. The figures below illustrate how to use the three major database features.

Illustration of search field in Educational Times database of mathematical problems.

Figure 1. To do a basic search, type a name in the search box (circled in red here).
To search by categories other than name, select “Advanced Search” at the top of the home page.

Illustration of advanced search interface in Educational Times database of mathematical problems.

Figure 2. To do an advanced search, fill in any of the boxes under the three types of filter:
“Question,” “Solver,” and “General.” Fill in as many boxes as needed.

Illustration of references search in Educational Times database of mathematical problems.

Figure 3. To do a references look-up, select “References” at the top of the home page.
The references tables show a partial list of institutions where physical copies of ET and MQ
are held, as well as links to digitized copies of the journals.

The Educational Times Database: Using the Database in Mathematics Research and Pedagogy

Robert M. Manzo (University of North Carolina at Chapel Hill)

The ET database is intended to be helpful to researchers and teachers of the history of mathematics. There are many ways the database can aid teaching and research. To start with, researchers could leverage the database’s typology of questions in order to chart patterns in the timing and authorship of ET/MQ questions. For example, a search for all “Mechanics/Forces” questions published in 1876 yields eleven results, while a search for that year’s total “Conics” questions yields thirty-seven results. One could compare the relative popularity of specific question types among ET/MQ readers in specific years, and even compare what was discussed in the ET to topics addressed in peer journals. Such a comparison could reveal general historical trends in mathematics research, or, on the other hand, it could reveal how different mathematics journals appealed to different sub-disciplinary audiences.

Arithmetic (Mixture / Rate) 1 Triangle & Circle Geometry (Trilinear Coordinates) 41
Algebra 2 Projective Geometry / Vanishing Points 43
Trigonometry 3 Non-Conic Curves & Loci/Envelopes 44
Plane Geometry & Analytic Geometry 4 Conics (General) 45
Calculus 5 3-Dimensional Geometry / Surfaces / Tetrahedra / Spheres 46
Analysis (Real & Complex) 6 Geometric Constructions 47
Logic 7 Conic Sections (Parabolas, Ellipses, Hyperbolas) 48
Probability and Expectation 8 Differential Equations / Orthogonal Trajectories 51
Number Theory 9 Calculus (Maxima / Minima) 52
Combinatorics 10 Calculus (Integrals) 53
Applied Mathematics / Physics 11 Analysis (Sequences & Series) 61
Recreational Mathematics (Magic Squares, Chess) 12 Analysis (Inequalities) 62
Geometric Probability, Average value 13 Analysis (Approximation) 63
Average Value / Average Distance 14 Difference Equations 64
Optics 15 Knot Theory 71
Electricity & Magnetism 16 Statistics 81
Mechanics (Forces) / Statics / Equilibrium / Velocity / Acceleration 17 Philosophy 91
Hydrostatics / Hydrodynamics / Gas Dynamics 18 Astronomy 92
Thermodynamics 19 Glottochronology 93
Linear Algebra 21 Calendar Problems 94
Equations & Solving Equations 22 Partitions 95
Abstract Algebra (Group Theory) 23 Continued Fractions / Convergents 96
Polynomials / Invariants / Forms 24 Voting 97
Matrices & Determinants 25 Set Theory / Topology / Graph Theory 98
Quaternions 26 Music Theory 99
Spherical Trigonometry 31    

Table 1. The database’s typology of mathematical questions, created by Dr. James Tattersall.
All questions from the ET/MQ have been assigned to one of these categories.
Each category has its own numerical code, to ease processing of searches by the database.

The database also lends itself to analysis of historical patterns of authorship, revealing which individuals or classes of people contributed to the ET. In a 2004 article, for example, Tattersall used his then-unpublished data tables (now the backbone of the ET database) to create a profile of the ET’s female contributors. Several findings emerged, including the fact that women were most active in the ET from the 1870s to the 1890s. Women’s increased engagement with mathematics (as detailed in the “Women Contributors” section below) was precipitated by the 1860s education reforms that brought more women into secondary and higher education in Britain. The three most frequent female contributors to the ETChristine Ladd (1847–1930), Sarah Marks (1854–1923), and Belle Easton (dates unknown)—together submitted a total of 359 questions and solutions. Tattersall’s data tables allowed him to compile statistics that served as evidence to support broader historical claims. “The number of mathematical contributions made by women to pedagogical journals such as ET,” claimed Tattersall and co-author Shawnee McMurran, “increased dramatically in the late nineteenth century, indicating that women were taking advantage of educational opportunities, becoming more mathematically active, and establishing themselves as intelligent and competent analytical thinkers. In giving women credit for their mathematical contributions, ET helped promote an emancipated view of women” [Tattersall and McMurran 2004, 107]. Future researchers could use the database to explore patterns of participation by other categories of people and then connect those patterns to broader historical contexts, as Tattersall and McMurran have done. The potential for creative uses of the database in research thus is boundless.

Name Solutions Submitted Problems Posed Total Active ET Period
Christine Ladd 82 53 135 1872–1899
Sarah Marks 95 22 117 1881–1899
Belle Easton 81 26 107 1874–1893
Elizabeth Blackwood 23 76 99 1872–1897

Table 2. A table that compares the contributions of the most frequent female contributors to the Educational Times.
Excerpted from a pre-publication typescript of the article [Tattersall and McMurran 2004].

In the classroom, the ET database can be used as a tool to “integrate” or “permeate” the history of mathematics into conventional instruction [Siu and Tzanakis 2004, viii]. “Appropriate integration of mathematics history into the curriculum,” according to Tattersall and McMurran, “can lead students to make connections between various mathematical ideas; it can help students appreciate the integral role that mathematics has always played in society; and it can guide students to reach for a more meaningful understanding of major mathematical concepts” [Tattersall and McMurran 2004, 103]. For example, through exposure to mathematics history, students can gain an appreciation of the careers of famous mathematicians and learn the value of publishing in small venues such as the ET/MQ. Ivor Grattan-Guinness noted that the MQ contains “the first publication of Bertrand Russell” and that “at least twice Lord Kelvin put in an appearance, once on properties of determinants … and once on determining latitude” [Grattan-Guinness 1992, 77–78]. With the use of the ET database, other early publications by well-known mathematicians might be found, and qualifiers such as “at least” could be discarded and replaced by exact statistics on a particular author’s ET/MQ submissions. Besides famous names, numerous under-celebrated or otherwise unknown mathematicians contributed to the ET/MQ, and their presence is equally a part of mathematics history. Through contact with the work of people such as the lesser-known ET contributors Sarah Marks and S. Narayana Aiyar (whose biographies are sketched in the “Women Contributors” and “International Contributors” sections, respectively), students can gain a deeper appreciation of the appeal and vitality of mathematics beyond the lives of “great men” such as Bertrand Russell (1872–1970). Additionally, instructors can use the database to search for problems on specific topics, in order to extract example problems and have students compare the kinds of questions asked on those topics now versus then. This can shed light on the historical emergence, transformation, and/or disappearance of particular lines of mathematical inquiry. Beyond the few ideas mentioned here, use of the database for pedagogical purposes is open to myriad possibilities.

The Educational Times Database: History of the Educational Times and Mathematical Questions, 1847–1918

Robert M. Manzo (University of North Carolina at Chapel Hill)

The following three subsections provide an overview of the importance of the ET (and its offshoot MQ) and its contributors to the history of mathematics:

When the collective contents and widespread readership of the ET and MQ are appraised together, their historical significance is clear. The ET and MQ were read seriously by many of the nineteenth century’s leading mathematicians. It is hoped that this brief history illuminates the role of the ET/MQ in British and transnational mathematics communities, as well as shows how the journals validated women’s participation in a profession that typically excluded or marginalized women. No definitive history of the ET/MQ yet exists, but more details on the journals, their readers, editors, printers, and publisher can be found in excellent articles by Janet Delve [2003], Sloan Despeaux [2017], Ivor Grattan-Guinness [1992], and Shawnee McMurran and James Tattersall [2004]. These articles cannot be recommended highly enough for readers interested in the ET/MQ. They offer inspiration and encouragement for scholars to pursue further research on a remarkable pair of historic journals.

The Educational Times Database: Founding of the Journals and Early Contributors

Robert M. Manzo (University of North Carolina at Chapel Hill)

In 1846, a “group of committed private Schoolmasters” in England founded the College of Preceptors, a distinctly nonsectarian, secular institution dedicated to improving the quality of teachers in Britain [Delve 2003, 145]. In practical terms, the College provided training to aspiring teachers and administered exams which, if passed, resulted in a certificate of advanced academic ability and teaching skill. The College published the journal ET as a way to share news of the College itself, to circulate the newest teaching methods to its teacher-readers, to advance the professionalization of the teaching occupation, and to publish the latest research on a variety of scientific topics [Delve 2003, 151]. One of these topics was mathematics. Although the ET’s mathematics section started out as a didactic tool “to demonstrate solid methods to assist both teachers and students,” over time it also accommodated a spirit of competition among sporting mathematicians—among both women and men, professors and lay people [Despeaux 2017, 222]. Mathematics became such a popular topic that the College began issuing the MQ in 1864, which reprinted and expanded much of the ET’s mathematical content.

Covers of Educational Times and Mathematical Questions. 

Figure 4. Images of the May 1861 issue of the ET and one of two 1864 volumes of MQ.
Images collected via the “References” tab at

The ET emerged in the historical setting of Victorian England, an era which began in 1837 when eighteen-year-old Victoria, of the royal House of Hanover, assumed the throne. She succeeded her deceased uncle, the reformer King William IV (r. 1830–1837), and her grandfather King George III (r. 1760–1820). Over a long reign, King George III had presided over the creation of the United Kingdom of Great Britain and Ireland, witnessed industrial revolution, fought the creation of the United States, and won an epic war against Napoleonic France. Following an age of wars and political reforms, Victorian-era Britain was comparatively peaceful. Abroad, however, this was the height of the British empire, with its ruthlessly exploitative policies towards other countries. Yet at the same time, the empire facilitated cultural and intellectual exchange across continents; in mathematics, Britain had much to offer, as the “Victorian period coincided with a revival” of the discipline after “its mid-18th-century slump” [Rice 2011, 3].

The center of mathematical activity in Victorian Britain was Cambridge University, where the College of Preceptor’s Board of Examiners for Mathematics was based. The university had made mathematics a central part of its general curriculum in the mid-eighteenth century, instituting the Mathematical Tripos, a course of study that culminated in the Tripos final examination [Crilly 2011, 19–20]. Each year, the top scorers on the exam had the honor of being ranked as “wranglers,” further sub-ranked from “senior wrangler” to “second wrangler,” “third wrangler,” and so on. Among the early-nineteenth-century Cambridge mathematicians who went on to influence the mathematical renaissance of the Victorian era were John Herschel (1792–1871), George Peacock (1791–1858), and Charles Babbage (1791–1871) [MacTutor]. In the 1812 tripos exams, Herschel and Peacock were senior and second wranglers, respectively. In 1811 the trio had co-founded the Analytical Society at Cambridge, which famously promoted wider adoption of French and German approaches to calculus and algebra in Britain. The activities of Analytical Society members were diverse and impactful even though the group only met formally for about a year. Collectively, the Society translated several continental European mathematics papers into English. Individually, Peacock redefined a branch of mathematics in his 1830 Treatise on Algebra, the same year that Herschel published the well-received Discourse on Natural Philosophy, and around the time that Babbage began designing his “analytical engine, the forerunner of the modern electronic computer” [MacTutor].

Trinity College, Cambridge.

Figure 5. A view of Trinity College, Cambridge, where Charles Babbage, John Herschel,
and George Peacock matriculated in the early 1810s [Esteve 2016].

The students of Herschel, Peacock, and Babbage were the ones who, in the early Victorian era, would drive the rise of mathematical research and pedagogy and contribute to the ET’s mathematical section. For example, among the top half-dozen wranglers in the 1837 Tripos exam were J. J. Sylvester (1814–1897), George Green (1793–1842), Duncan Gregory (1813–1844), and A. J. Ellis (1814–1890) [Crilly 2011, 20]. Before his untimely death in 1844, at age thirty, Gregory wrote important papers that expanded on Peacock’s algebraic treatise; he also mentored and influenced the mid-Victorian algebraist George Boole; and he served as founding editor of The Cambridge Mathematical Journal in 1837 [Rice 2011, 4–5; MacTutor]. Although that journal floundered after Gregory’s death, it was revived in 1846 as The Cambridge and Dublin Mathematical Journal, which shared contributors with the ET [Crilly 2011, 24]. J. J. Sylvester, meanwhile, was not granted a degree from Cambridge because his Jewish faith prevented him pledging an oath to the Church of England, as was required of graduates at that time [MacTutor]. In 1841, finally, Sylvester earned degrees at Trinity College, Dublin. But when he began applying for faculty positions, his Cambridge professors Herschel and Babbage served as professional references. The Cambridge network remained strong. About twenty years older than Sylvester and Gregory—in fact, about the same age as the elder teachers Herschel, Peacock, and Babbage—was George Green, a self-taught and non-degreed mathematician (prior to attending Cambridge). Green became an undergraduate at Cambridge only at age forty, in 1833, after some of his published papers gained the attention of the Analytical Society founders [MacTutor].

Another early Victorian mathematician of note, who ranked as senior wrangler in the 1842 Tripos exam, was Arthur Cayley (1821–1895), whom Peacock had tutored [MacTutor]. So impressed with Cayley was Duncan Gregory, then the editor of The Cambridge Mathematical Journal, that he published several of Cayley’s undergraduate papers. In the 1850s, Cayley and J. J. Sylvester collaboratively developed invariant theory in algebra, describing a class of algebraic expressions that retain their basic form even when subjected to transformations [Rice 2011, 6]. Cayley and Sylvester’s theory provides just one example of the importance of this generation of British mathematicians to mathematical innovation. These pioneering Victorian mathematicians—Sylvester, Green, Ellis, and Cayley, although not the prematurely deceased Gregory—were among the ET’s earliest contributors. Thus, the ET quickly became an important forum for innovative British mathematicians. At this juncture, it bears repeating that apart from the great British men of the era, whose careers are already well-documented, women, non-Brits, and many anonymous individuals also found a welcoming forum for their mathematical passions in the ET.

Portraits of Arthur Cayley, Duncan F. Gregory, and J. J. Sylvester.

Figure 6. A few of the notable Cambridge mathematicians who matriculated in the early Victorian period.
From left to right: Arthur Cayley, Duncan Gregory, and J. J. Sylvester [MacTutor].


The Educational Times Database: Women Contributors

Robert M. Manzo (University of North Carolina at Chapel Hill)

In the early Victorian era, women were conspicuously absent from Cambridge mathematical circles and from the pages of the ET. According to one historian, women “did not feature” in the plans of Cambridge’s administrators, “coming onto the Cambridge scene only in the 1880s” [Crilly 2011, 19]. Actually, women began to be admitted as non-degree students at Cambridge in 1869, with the founding of Cambridge’s Girton College, “one of … Britain’s oldest residential colleges for women and the first to offer a university-level education for women” [McMurran and Tattersall 2017, 5]. Girton College was partly an outcome of Queen Victoria’s 1864 Royal Commission on Secondary Education, which had concluded that girls should have the same level of access to education as boys.

In 1876, Girton admitted two skilled mathematicians who would later become important to the ET: Charlotte A. Scott (1858–1931), a specialist in geometry, and Sarah Marks (1854–1923), a specialist in mechanics and engineering [Tattersall and McMurran 1995, 88–89; Tattersall 1999]. Scott and Marks became close colleagues, and they co-founded the Girton College Mathematical Club with classmates Margaret Ker (18571925) and Helen Sheldon (1859–1945), in part to prepare for the Tripos exam [McMurran and Tattersall 2017, 13; Tattersall and McMurran 1995, 90]. At that time, “mathematical textbooks as a rule did not contain pages of diverse exercises,” so the members of the Club—like numerous other British students of mathematics in the late nineteenth century—turned to the ET for practice problems [McMurran and Tattersall 2017, 13; Tattersall and McMurran 2004]. Unfortunately, despite the Royal Commission rulings and the founding of Girton College, women were not automatically allowed to sit for Cambridge exams. Girton student Sarah Woodhead (18511912) discovered this contingency in 1873, when she had to obtain special permission from the male examiners and became the first woman to pass the Tripos exam [McMurran and Tattersall 2017, 8]. In 1880, Scott and Marks were among the last cohort of women who had to obtain special permission to sit for the exam, and then “beginning in 1881 women were admitted to Cambridge examinations normally and not just by the courtesy of the male examiners” [Tattersall 1999]. Scott became the first woman to rank as a wrangler in the Tripos exam, while Marks passed with a third-class merit [Tattersall and McMurran 1995, 92].

Portraits of Charlotte Angas Scott and Sarah Marks.

Figure 7. Charlotte Angas Scott, left, and Sarah Marks (aka Hertha Ayrton), right, achieved a first- and third-class rank, respectively, on the 1880 Cambridge Tripos exam. While Cambridge allowed women to study at its Girton College, the university did not grant degrees to women at that time, so Scott and Marks ultimately obtained degrees elsewhere [MacTutor; Chaplin 1995; Bruton 2020].

In the 1880s, Scott and Marks contributed significantly to the ET. Scott solved a total of twenty-five problems and posed nine [Tattersall and McMurran 2004, 106]. Impressively, Marks was “responsible for submitting 3.5 percent of the solutions appearing between 1883 and 1889,” a total of ninety-five solutions, in addition to submitting twenty-two questions [McMurran and Tattersall 2017, 17]. Scott eventually completed her doctoral dissertation on algebraic geometry under Arthur Cayley’s supervision, although her doctorate came from the University of London, as Cambridge did not then grant degrees to women. She continued to lecture at Girton College until 1885, when she accepted a professorship in the United States [Tattersall 1999]. On a different path, Marks pursued further study in physics, outside of Cambridge, and in 1885 she married a physics professor, William E. Ayrton, F.R.S.(18471908), whom she joined as a teacher at Central Technical College in London [Tattersall and McMurran 1995, 96–97]. By the end of the nineteenth century, women were very much “on the Cambridge scene,” and the ET counted thirteen frequent women contributors, a significant if still disproportionate number [Tattersall and McMurran 2004, 106].

The Educational Times Database: International Contributors

Robert M. Manzo (University of North Carolina at Chapel Hill)

Apart from British mathematicians, a number of continental European and Indian mathematicians contributed to the ET. One of the most prolific French contributors to the ET was Joseph Neuberg (1840–1926) , who submitted 365 questions and ninety-six solutions between 1885 and 1915. Most of his contributions concerned geometry, the topic he taught most frequently. Neuberg had grown up in Luxembourg during the 1840s and later attended the University of Ghent in Belgium, graduating in 1862 [MacTutor]. While teaching at various colleges and universities in Belgium, Neuberg co-founded two French-language mathematics journals, Nouvelle Correspondance Mathématique (pub. 1874–1880) and Mathesis (pub. 1881–1915, 1922–1965). The ET’s distinguished reputation in the European mathematical community is evidenced by Neuberg’s decades-long correspondence with the British journal. A less frequent French contributor to the ET was Charles Hermite (1822–1901), whose work concerned algebra and number theory. Hermite worked with invariant theory in the 1850s, at the same time that the theory was being developed by Arthur Cayley and J. J. Sylvester. In related work on elliptic functions, Hermite famously showed that “although an algebraic equation of the fifth degree cannot be solved in radicals,” it “could be solved using elliptic functions” [MacTutor]. Between 1866 and 1896, Hermite contributed twenty-eight problems and six solutions to the ET, mainly on number theory and integral calculus.

Half a world away, one important Indian mathematician who submitted often to the ET was S. Narayana Aiyar (1874–1937). Employed as an accountant at the Madras Port Trust Office in India (from 1900 to 1934), Narayana Aiyar was interested in more than practical accountancy. At the Trust Office he worked with, befriended, and tutored Srinivasa Ramanujan (1887–1920), and in 1914 he “strongly advised Ramanujan to accept [G .H.] Hardy’s invitation to come to Cambridge” [Berndt 2011, 770]. Also, Narayana Aiyar was an early member of the Indian Mathematical Society, which served as a link between the professional mathematical communities in England and India. Several prominent Indian mathematicians contributed both to the Journal of the Indian Mathematical Society and the ET, including M. T. Naraniengar (18711940) , S. Narayanan, K. J. Sanjana, R. Ramachandra Rao  (18711936), and V. Ramaswami Aiyar (1871–1936) [Berndt 2011, 771].

Narayana Aiyar's solution of a mathematics problem.

Figure 8. A solution co-authored by S. Narayana Aiyar, in response to an anonymously posed polygon problem.
Narayana Aiyar was a prominent Indian mathematician of the early twentieth century.
Excerpted from Educational Times, April 1908, page 187.

The place of the ET in nineteenth- and early twentieth-century mathematical communities, both in Britain and abroad, can be appreciated not only in terms of the individuals who contributed to it but also in terms of contemporaneous publications. Some of the most puzzling mathematical problems of their day circulated beyond a single institution or publication. These problems might be unsolved, impartially solved, or open to alternate solutions. It is possible to trace some of the journals which reprinted the same or similar problems as those featured in the ET, and thereby identify a few peer publications. For example, in 1900 Benjamin F. Finkel, editor of the American Mathematical Monthly (AMM), noted that a particular geometric problem had been adequately solved in the ET after remaining unsolved in several other publications. The problem was: “If the two bisectors, trisectors, quadrasectors, etc., of a triangle are mutually equal, show that the triangle is isosceles” [Finkel 1900, 227–228]. Some form of this problem had appeared in the Nouvelles Annales de Mathématiques in 1842, The Lady's and Gentleman's Diary in 1856, and The London, Edinburqh, and Dublin Philosophical Magazine in 1874. Another puzzling problem that circulated for many decades concerned the “problem of determining the curve of pursuit in the case of the circle” or, in mathematical terms, integrating the “differential equation of the curve of pursuit for a circle” [Archibald and Manning 1921, 92]. The problem appeared in The Mathematical Monthly (AMM) in 1859, Nouvelle Correspondance Mathématique in May 1877, Mathesis in December 1883, the ET in February 1888, Revue de Mathématiques Spéciales in February 1894, and L'Intermédiaire des Mathématiciens in October 1894. The August 1894 issue of AMM declared the problem unsolvable, but it continued to be discussed in the AMM and Nouvelles Annales for decades thereafter. Finally, mathematician A. S. Hathaway offered a detailed solution in the AMM of February 1921. The publication history of these two example problems demonstrates the ET’s participation in a robust ecosystem of mathematical journals that encompassed England, Scotland, Ireland, India, the United States, and France.

Problem proposed by Arthur Cayley and solved by Joseph Wolsteholme.

Figure 9. A selected problem and solution from Mathematical Questions, volume 16, 1871, pages 65–66.
The problem was proposed by Arthur Cayley. It was solved by Joseph Wolstenholme (1829–1891), a Cambridge graduate and professor at the Royal Indian Engineering College, a civil service training school located
near London at Cooper’s Hill, now the site of a retirement village.

The influence of the ET in the United States is evident in the biographies of two of the journal’s most famous American readers, Benjamin F. Finkel (1865–1947) and the expatriate Charlotte A. Scott (1858–1931). Finkel’s career is well-known to historians of mathematics. In 1894, while employed as a professor of mathematics at Drury College in Kidder, Missouri, he co-founded the AMM with his colleague John M. Colaw (1860–1931) [Finkel 1931, 308]. By this time, Finkel was already a subscriber to the ET. The AMM was later adopted, in 1915, as the official publication of the Mathematical Association of America (MAA), of which Finkel was a leading member [Parshall 2016, 194]. Finkel’s subscription to the ET, along with his prominent roles as an AMM founder and MAA member, indicate the British journal’s influence across the Atlantic. English mathematician Charlotte A. Scott, a teacher at Girton College until 1885, afterward accepted a position at Bryn Mawr College, Pennsylvania, where she taught for almost four decades [Tattersall and McMurran 2004, 110]. She had achieved her wrangler rank on the Cambridge Tripos exam and established her early reputation, in part, because she had studied with and contributed to the ET. In the United States, her achievements included supervising seven doctoral students, serving as a founder and a chief examiner of the College Entrance Examination Board, co-editing The American Journal of Mathematics with Frank Morley (1860–1937), another notable ET contributor, and being elected as a council member and vice president of the American Mathematical Society. Finkel’s and Scott’s biographies highlight the ET’s connection to the evolution of a professional American—and transnational Anglo-American—mathematical community in the late nineteenth and early twentieth centuries.

The Educational Times Database: Indexing MQ and ET’s Mathematical Problems, 1940s–2000s

Robert M. Manzo (University of North Carolina at Chapel Hill)

The development of the ET database project encompasses the interlinked work of many individuals over many decades. The relevant germinal work dates to the 1940s and 1950s, when Dr. Raymond Clare Archibald (1875–1955) at Brown University, Rhode Island, led an effort to index the full run of questions and solutions published in MQ, from its first issue in 1864 to its last in 1918. Archibald’s project aimed at a specific kind of indexing, in which the “contents of a serial publication,” such as a newspaper or magazine, are cited in a list by reference to author and/or subject [Reitz 2020]. Archibald was predisposed to initiate such a project, since his twin interests were in mathematics and librarianship.

Two portraits of Raymond Clare Archibald. 

Figure 10. Portraits of Dr. Raymond Clare Archibald, a librarian and professor of mathematics
at Brown University from 1908 to 1955 [“Archibald” 2020; MacTutor].

Originally from Nova Scotia, Canada, Archibald graduated from Harvard University in 1895, completed his doctorate in mathematics at the University of Strasbourg in France at the turn of the century, and then worked several years as a college librarian in Canada [MacTutor]. In that capacity, he oversaw the acquisition of thousands of books, and he hand-wrote tens of thousands of library catalog cards. In 1908 he earned a position as instructor of mathematics at Brown University, working his way up to full professor by 1923, before retiring to an emeritus role after 1943. Sometime during his emeritus years, if not before, Archibald began indexing MQ content with the help of his students. His interest derived principally, it seems, from the fact that he had been a major contributor to the journal in his younger years [Tattersall and McMurran 2004]. Also, as a keen book collector who “frequently … went to Europe for the summer” to acquire mathematics books, journals, and dissertations for Brown University’s library, Archibald would have been aware of the historical significance of the ET for the Anglo-American mathematics community [MacTutor].

Archibald’s indexing project was left uncompleted upon his death in 1955, and it might have remained so if, decades later, another mathematician had not discovered the preserved remains of the project on the shelves of the Brown University Science Library. Dr. James Tattersall, Jr., a professor and historian of mathematics at Providence College, Rhode Island, spent considerable time in Brown’s Science Library as his own career was taking off in the late 1960s and early 1970s [Tattersall 2020]. During one of his library sojourns, Tattersall discovered five small boxes shelved next to a full set of MQs. In the boxes he found hundreds of handwritten file cards penned by Archibald and his students: the beginnings of the unfinished index. Recognizing the scholarly value of an MQ index, and sharing the late Archibald’s twin interests in librarianship and the history of mathematics, Tattersall had the idea to complete the project, and even improve on it by incorporating the ET. The task was immense, since MQ had run for 110 issues and the ET for 75 issues. In the final count, the total number of mathematical questions and solutions was in the tens of thousands. Nevertheless, Tattersall pursued the project. He built his index in the form of a series of tables, created with the aid of computer software. Thus the index transitioned from physical to digital media. Tattersall eventually shared his tables with Dr. Sloan Despeaux, another historian of mathematics, and they discussed the idea of publishing the tables as an online database, so that scholars around the world could make use of them. The indexing project was an ambitious idea, which has been carried out collaboratively across multiple generations of mathematicians.

Physical cards used as source material for the Educational Times database. 

Figure 11. Two boxes, with a sample card, of the file cards created by Archibald in the 1940s–50s and discovered
years later by Tattersall in the Brown University Science Library. Images courtesy of James Tattersall.

The Educational Times Database: Building a Database of ET/MQ Problems and Solutions, 2010s

Robert M. Manzo (University of North Carolina at Chapel Hill)

In the 2010s Despeaux, a professor of mathematics at Western Carolina University (WCU), North Carolina, took up the task of migrating Tattersall’s tables of data, stored across multiple computer files, into an integrated online digital database. Her main point of reference in designing the ET database was the popular database Auteurs des Nouvelles Annales de Mathématiques, developed by a team of researchers from two French institutions: the Archives Henri Poincaré (at Université de Lorraine) and the Groupe d’Histoire et Diffusion des Sciences d’Orsay (at Université Paris-Sud, now part of Université Paris-Saclay). In their database, the French team indexed 5,128 mathematical questions and answers contributed by 1,852 authors, printed in the journal Nouvelles Annales de Mathématiques between 1842 and 1927 [“Accueil” 2018]. Users can search the Auteurs database by author name, year published, author profession, and fourteen other categories. Similarly, the ET database is searchable by author name and other attributes.

Despeaux served as project manager, but realization of the ET database required several additional skill sets to come together. Most obviously, in order to build a database, the expertise of a computer scientist was needed. Dr. Mark Holliday, a colleague of Despeaux’s in the Department of Mathematics and Computer Science at WCU, outlined the steps required to turn Tattersall’s data files into a functioning database. First, the data had to be “cleaned up,” which meant transferring the data from Tattersall’s numerous computer files to a single computer file and imposing a standard format on all data points belonging to the same category (e.g., names, dates, volume numbers). Second, computer code had to be written that would refer to the single data file and make that file’s data into a searchable digital database. The first step of data clean-up was handled by Despeaux and Robert Manzo, a graduate student assistant with experience in librarianship, data management, and historical research. In the second step, Holliday and four computer science students wrote the computer code in two separate phases. In the first phase, completed in 2016–17, students Grant Brown and Logan Schmidt worked with Holliday to create the first iteration of the database, comprised of a limited set of about 1,500 mathematical questions. A research grant awarded to Despeaux by the WCU College of Arts and Sciences funded this first phase of coding. In the second phase, in 2019–20, students Kevin Filanowski and Caleb Tupone expanded the size and functions of the database. To fund this phase of coding, as well as the bulk of data clean-up, Despeaux applied for and won the annual Hunter Scholar Award, given by WCU’s Hunter Library to support data-focused projects.

To start the process of data clean-up, Holliday and Despeaux had to decide how to arrange Tattersall’s many computer files into a single file. Over the years, Tattersall had compiled his data into twenty-three separate files, four containing biographical information on authors and nineteen containing a list of about 18,000 proposed questions and about 1.5 times more solutions (since some problems elicited multiple solutions). The actual problems and solutions were not copied out, only information on who asked the question, who solved it, what type it was, and in which volume of the ET/MQ it appeared. In other words, Tattersall was dealing with metadata. Metadata indicates who communicated with whom, about what, and when. Per Holliday’s recommendation, the twenty-three separate files were integrated into a single unified file that could be referenced by the database’s code.

The unified file was created using Microsoft Excel. The file consisted of five large tables: authors, countries, question types, proposers, and solvers. The authors table was a master list of all contributors to the ET/MQ, with additional columns giving their gender, country of origin or primary residence, a biography, and the source for the biography. The countries table listed forty-four different countries and assigned each a numerical value, 1 to 44. (Each country number was used to fill in values in the country-of-origin column in the authors table.) The question types table listed and assigned a numerical value to about fifty different categories of mathematics questions (see Table 1). These values are used in the proposers table to indicate the type of each proposed question. The data entered into the countries and question types tables were straightforwardly copied from lists created by Tattersall. Originally, the numerical codes he created had acted as shorthand for Tattersall, so that he did not have to write out the full names of countries in every author’s biography, nor fully write out question types next to every problem listed. The codes were retained for the Excel file for the same reasons that Tattersall used them: as labor- and space-saving devices. Using the codes allowed the authors, proposers, and solvers tables to remain relatively compact and human-manageable. Lastly, the proposers and solvers tables are what their titles indicate: numbered lists of every question proposed and every solution offered in the pages of the ET/MQ, along with their respective authors, date published, volume number, and question type.

Excerpt from Excel spreadsheet showing Educational Times problem solvers.

Figure 12. A screenshot of the "solvers" table in the unified Excel workbook file. Note that female contributors are highlighted in red. The columns, from left to right, show the Excel row number, the ET problem number, the solver's first name, solver's unique ID, solver's display name, ET volume number and date, and problem type.

Unfortunately, in moving data from Tattersall’s files to the single Excel file, data could rarely be copied and pasted directly due to formatting issues. Author names had to be copied with Tattersall’s original formatting intact, since he had color-coded some names to indicate gender, language, or a pseudonym. But there was no easy way to copy data in bulk while also ensuring that color codes and other formatting features were retained. Another issue arose in later stages of computer coding, when Tupone and Filanowski dealt with the formatting of dates. The SQL code underlying the database required all publication dates to include a day, not just a month and a year. However, the ET was published monthly, without a day of the month stamped on the cover. The solution to this problem remains a work in progress. Currently, the database displays “01” for the day in most publication dates, until a satisfactory workaround can be implemented.

The most serious formatting issue involved names. Individual names needed to share a consistent spelling across all tables in which they appear, including solvers, proposers, and authors. The database team wanted to ensure that different spellings and/or forms of the same name were associated with the singular person who was known by those multiple forms. In the original data set, for example, one particular author was alternately referred to as “J.W. Sharpe” and “James William Sharpe.” Several authors had titles such as “General” or “Prince” printed before their names in some issues of ET but not in others. A few authors were credited as “Dr.” in one issue but “Prof.” in another. To avoid causing the database to dissociate data that should be linked, Despeaux directed that an additional column be added to the proposers, solvers, and authors tables. The additional column gave every author a unique identifier, typically the author’s last name followed by an underscore and a numeral. By this method, every author received both a “display name” and a “unique ID.” For example, one author who was variously credited as “A. Morel” or “Prof. Morel” was given the unique identifier “Morel.” The database would show the “display name” to the end user, while referencing the “unique ID” on the database’s back-end. The inclusion of a unique identifier also resolved confusion over different authors with the same last name. For example, author J.L. Jenkins has the identifier “Jenkins,” J.S. Jenkins has “Jenkins_2,” Morgan Jenkins has “Jenkins_4,” and so on. In the end, most formatting issues were resolved satisfactorily, and the team achieved the goal of a smoothly functioning ET database.

The Educational Times Database: Conclusion – Reflections on Opening Up the Contents of the ET/MQ

Robert M. Manzo (University of North Carolina at Chapel Hill)

For the researcher with sufficient time and finances, the contents of the ET/MQ have long been available to view at several of the world’s largest libraries, including the University of Toronto Library, Cornell University Library, the University of Göttingen Library, Oxford’s Bodleian Library, and the British Library, among others. Some of these institutions have even digitized select volumes of the ET and MQ, at times with the assistance of the Google Books digitization project. The unique contribution of the ET database is to offer a near-comprehensive index of the ET/MQ problems and solutions, as well as to collate in one central place (in the database’s “References” section) hyperlinks to digitized versions of most ET and MQ volumes. (As an aside, we wish to note here that the current ET database has not yet been updated to include a hyperlink to University College London’s (UCL) new digital archive of the full run of the ET, available at; the UCL archive is invaluable and we highly recommend that scholars make use of it.) Thus, while the ET database has its own distinct value as a tool for historians of mathematics, it also situates itself within a broader constellation of ET/MQ resources developed by a number of diligent scholars and institutions. Reflecting on the history of the ET/MQ and the database’s long development, it is fair to say that Raymond Clare Archibald was the crucial link between the journal’s life and afterlife. Archibald was an eminent contributor to the ET in the early twentieth century, and decades after the journal went out of print he initiated the indexing project. Later, fellow Rhode Island scholar Jim Tattersall’s passion for mathematics history and the ET led him to take up Archibald’s unfinished work, and finally he and Sloan Despeaux brought the project into the twenty-first century as a digital database. The creation of a database was never a foregone conclusion but was historically contingent, the result of Tattersall discovering a few boxes on a library shelf and of Despeaux developing a collegial relationship with a like-minded historian of mathematics. The team that worked on the ET database project, including Tattersall, Despeaux, Holliday, Brown, Schmidt, Filanowski, Tupone, and Manzo, hope that it is useful to mathematics scholars and look forward to seeing future research that utilizes the database.

The Educational Times Database: References

Robert M. Manzo (University of North Carolina at Chapel Hill)

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Archibald, Raymond Clare, and Henry Parker Manning. “Remarks and Historical Notes.” The American Mathematical Monthly 28, no. 2 (1921): 91–93.

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Chaplin, Stephanie. “Charlotte Angas Scott.” Biographies of Women Mathematicians. Agnes Scott College. April 1995.

Crilly, Tony. “Cambridge: The Rise and Fall of the Mathematical Tripos.” In Mathematics in Victorian Britain, edited by Raymond Flood, Adrian Rice, and Robin Wilson, 17–34. Oxford: Oxford University Press, 2011.

Delve, Janet. “The College of Preceptors and the Educational Times: Changes for British Mathematics Education in the Mid-Nineteenth Century.” Historia Mathematica 30 (2003): 140–172.

Despeaux, Sloan Evans. “Chapter 12: Constance Marks and the Educational Times.” In Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America, edited by J. L. Beery, S. Greenwald, J. Jensen-Vallin, and M.B. Mast, 219–229. Cham, Switzerland: Springer, 2017.

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The Educational Times Database: Acknowledgments and About the Author

Robert M. Manzo (University of North Carolina at Chapel Hill)

The author would like to thank project leader Sloan Despeaux for including him as part of the database development team. She graciously acted as an academic mentor and also referred him to many valuable resources in a discipline (history of mathematics) with which he was previously unfamiliar. Thanks also to Jim Tattersall and Mark Holliday for sharing, via e-mail, their experiences as contributors to this project. Last but not least, the author is grateful for his many meetings with Caleb Tupone and Kevin Filanowski, who took time to explain the requirements and inner workings of a well-formed database.

About the Author

Robert Manzo earned a degree in Library Science from the University of North Carolina at Greensboro in 2015 and an MA in History from Western Carolina University in 2020. Between degrees he worked in central North Carolina as a public librarian and a curriculum designer for a nonprofit adult literacy and GED program. Reflecting his interests in disability history and library history, his MA thesis explored the development of public libraries in North Carolina from 1896 to 1929, interpreting the state’s first public libraries as a product of the complex, contradictory Southern Progressive movement active at the turn of the twentieth century. Currently a PhD student in the School of Information and Library Science at UNC–Chapel Hill, Robert researches print and online publishing spaces used by autistic self-advocates who are re-shaping public and scientific discourses about what autism is and means. This research is based in respect for subaltern and historically silenced peoples whose capacity for self-definition and self-determination has been, and still is, typically denied by hegemonic discourses and institutions. At present, Robert lives in southeastern North Carolina with his fiancée, who is a public reference librarian specializing in local history and genealogy.