Helping Ada Lovelace with her Homework: Classroom Exercises from a Victorian Calculus Course – Augustus De Morgan and Mathematics at UCL

Augustus De Morgan was the founding professor of mathematics at University College London (UCL), where he had been appointed in 1828 at the age of only 21 [Rice 1997]. An intriguing character of high intellect, De Morgan was a prolific writer on all areas of mathematics, publishing numerous research papers, largely on algebra and logic, as well as on mathematical pedagogy [Rice 1999] and the history of mathematics [Rice 1996]. Unlike professorships at Oxford or Cambridge at this time, De Morgan’s post at UCL demanded a substantial amount of undergraduate teaching, but being deeply committed to education, he did not limit his teaching to the classroom. He also established a reputation as a clear and lucid mathematical expositor, authoring countless popular articles, book reviews, and textbooks, several published by the contemporaneous Society for the Diffusion of Useful Knowledge (S.D.U.K.) of which he was a prominent member. He wrote over 700 entries for the Society’s Penny Cyclopaedia (27 vols, 1833–1843), including one in which he introduced the term, although not the method, of ‘induction’ into mathematics [De Morgan 1838].

Figure 6. Augustus De Morgan, as sketched by his wife, around 1840.

At UCL, De Morgan’s mathematics course formed a core component of the undergraduate curriculum, which also included subjects such as Latin, Greek, and natural philosophy (i.e. physics). His students were divided into classes corresponding to the first and second years of undergraduate study, with each class being further divided into lower and higher divisions. For the lower junior class the mathematical requirements for entry were merely ‘that they shall be acquainted with Arithmetic to the extent of a knowledge of vulgar and decimal fractions’ [Anon 1827, 14]. From here, they would be introduced to further arithmetic, including the arithmetical theory of proportion, the first four books of Euclid, as well as some basic algebra up to linear equations. To provide extra material, examples and problems, De Morgan relied in part on his own textbooks, such as Elements of Arithmetic (1830) and his translation of the first three chapters of Louis Bourdon's Élémens d'Algèbre, which had been his first publication on joining UCL in 1828. Prior to the study of Euclid, De Morgan also employed material from his First Notions of Logic (1839), in keeping with his strongly held belief that, in order to foster the students' understanding of geometrical arguments, it was first necessary to introduce them to logical notions and processes.

De Morgan’s higher junior class began with Euclid Books V and VI, followed by an introduction to solid geometry, facilitated by the use of three-dimensional models. The complex theory of ratio and proportion, contained in Euclid Book V, was broached by means of the far more user-friendly approach contained in De Morgan's The Connexion of Number and Magnitude (1836), which also served as a basis for his initial treatment of trigonometry, subsequently developed more analytically in the Elements of Trigonometry and Trigonometrical Analysis (1837). After a quick review of the rules and procedures of arithmetic, students would then return to where they had left off in algebra in the lower junior course, progressing to higher order equations, logarithms and the binomial theorem. This led immediately to the study of series and the related notions of convergence and divergence, all centered on the recently formalized concept of the limit. Indeed De Morgan's Elements of Algebra (1835), used alongside this part of the course, was the first English work to contain a limit-based definition of continuity.

By the time they entered the lower division of his senior class, De Morgan’s students would have been poised to begin the study of calculus. But this was not the only topic featured. The lower senior class started with an introduction to spherical trigonometry, accompanied by a small textbook De Morgan had written for the S.D.U.K. in 1834. Projective geometry and conic sections then led into analytic geometry, which in turn motivated the study of higher algebra. According to the official college announcements, De Morgan’s senior classes were ‘confined principally to those parts of the subject which are necessary for the study of Natural Philosophy’ [Anon 1828, 42]—in other words, they were dominated by calculus-related topics. It was for this purpose that he wrote his treatise on The Differential and Integral Calculus, an 800-page compendium published in 25 installments by the S.D.U.K. between 1836 and 1842, and which De Morgan was still in the process of writing while tutoring Lovelace. Like the algebra textbook which preceded it, this book was entirely based on the concept of limits, and De Morgan used it to introduce his lower senior students to a host of differentiation and integration techniques, up to a preliminary survey of first-order linear differential equations. But the book went much further than this. In addition to its exhaustive coverage of pure calculus techniques, it also contained chapters concerning applications to algebra, geometry and mechanics; it would remain the most comprehensive English work on the subject for over a generation.

The vast majority of students at UCL proceeded no further than De Morgan's lower senior class, since that was all that was needed to pass the B.A. examination at the University of London, as well as to move on to the study of natural philosophy in the college. However, those wishing to take their studies further entered the higher senior class, in which they were exposed to ‘Subjects which all must learn who wish to become analysts, whether for Engineering or any other pursuit’ [Anon 1839, 6–7]. Here again, calculus dominated the material and De Morgan’s treatise was indispensable. Higher-order differential equations were studied at great length, particularly in relation to curves and surfaces, as well as the calculus of variations. Other subjects such as the theory of equations, three-dimensional geometry, probability theory and dynamics were also covered.

Having completed the higher senior class and been given a thorough grounding in most areas of contemporary mathematical science, De Morgan’s students would have had several options, including careers in the actuarial world, business, education and law. Although the concept of a graduate research student did not exist in Britain at this time, the higher senior class would almost certainly have served as a good starting point for those aspiring for an academic career in mathematics, since it provided guidance for those aiming for a University of London M.A. or preparing to embark on a course of higher study at Cambridge.