At this time, of course, there was no provision in Britain for the higher education of women,^{1} so there would never have been any question of Lovelace attending De Morgan’s lectures. But quite apart from her gender and high social status, she would have differed from De Morgan’s middle-class male students in several other respects. Since school education was not yet compulsory and the school leaving age was, on average, around fourteen, the students at UCL in De Morgan’s day were substantially younger than they are today, ranging from around 15 to 18 years of age. At 25, therefore, Lovelace would have been noticeably older than the usual college undergraduate. She would also have been far more motivated and mathematically literate than the typical UCL student at this time.

How then did the course of study undertaken by Lovelace in the early 1840s compare with the contemporaneous program offered by De Morgan at UCL? As we have seen, by the time Lovelace started studying with De Morgan, she was well acquainted with both arithmetic and elementary geometry, having essentially taught herself the early part of Euclid’s *Elements*, although probably no further than the end of the fourth book. She had also acquired some knowledge of algebra and trigonometry although, as seen above, her level of proficiency in these subjects is debatable. Thus, while certainly knowing more than was required for a student entering De Morgan’s lower junior class at UCL, her mathematical attainments in mid-1840 seem to place her roughly midway through the syllabus for his higher junior division. Despite this, however, her letters to De Morgan leave no doubt as to the subject in which she was most interested: calculus. And as we will see, material from this area dominates the correspondence.

The course of study that Lovelace began with De Morgan in July 1840 would nowadays be called a ‘correspondence course’. It consisted of guided independent reading along with a variety of relevant problem exercises. This material came largely from De Morgan’s own published textbooks and articles, supplemented on occasion by other relevant works, such as *A Collection of Examples of the Application of the Differential and Integral Calculus* (1820), written by Charles Babbage’s undergraduate contemporary and De Morgan’s erstwhile Cambridge tutor, George Peacock. The study methods fostered by De Morgan were very different to those of Lovelace’s former tutor, Dr. King, who had encouraged rote learning when reading Euclid’s Elements: ‘The words of each proposition . . . must be fixed in your mind, with their number, so that you could repeat the Book through’ [LB 172, 15 March 1834, f. 129r]. In contrast, De Morgan offered Lovelace a far more progressive and modern approach to her mathematical studies:

*Festine lente*,^{2} and above all never estimate progress by the number of pages. You can hardly be a judge of the progress you make, and I should say that it is more likely you progress rapidly upon a point that makes you think for an hour, than upon an hour’s quick reading, even when you feel satisfied [LB 170, 15 Sept. 1840, f. 14r].

As far as De Morgan was concerned, the key study skills required for success in mathematics were not speed or memorization, but thinking and understanding.

By the time the correspondence came to an end after about eighteen months, Lovelace had reached a level of familiarity, if not expertise, with much of the material covered in De Morgan’s higher junior and lower senior classes: algebra, trigonometry, complex numbers, functions, limits, infinite series, differentiation, integration, and differential equations.^{3} But it wasn’t all plain sailing. Let’s look now at ten sources of difficulty she encountered during her course of study with De Morgan.

1. The earliest women’s colleges in Britain were founded later in the 1840s, with De Morgan giving lessons in elementary mathematics at one of them, Bedford College, in 1849 during its first year of operation.

3. Perhaps inevitably given her concentration on calculus-related material, she could not be said to have completed a course of study entirely equivalent to those classes. Spherical trigonometry, projective geometry and conic sections are all absent from the correspondence, for example, and there is similarly no reference to subjects from the higher senior class, such as probability or the calculus of variations..