#### Conclusion

The mathematical content of Ada Lovelace’s correspondence with Augustus De Morgan reveals a great deal to the modern reader. It shows that in a period of just eighteen months, her mathematical proficiency grew dramatically, as evidenced by the noticeable increase in the sophistication of her questions: from re-arranging simple algebraic expressions to the solution of second-order differential equations. But from a modern-day perspective, perhaps the most striking feature is the similarity between the deficiencies Lovelace displayed in her studies with De Morgan and the shortcomings exhibited by present-day students of mathematics. These include a lack of fluency in algebra, the misunderstanding of terminology and concepts, circular reasoning, and errors in applying rules or interpreting questions. The ten problems we have highlighted could all be given to today’s calculus students, not just as valuable exercises in mathematics, but as examples of pitfalls to avoid. Indeed, for both student and teacher, spotting Lovelace’s errors is arguably as instructive as solving the problems themselves. Fundamentally, they all illustrate the value of practice, the necessity of asking questions, and the essential role of a good teacher. In this respect, the difficulties faced by Ada Lovelace when learning calculus 180 years ago—and the remedies for them—differ very little from those facing our students today.

**References**

**Archival Sources**

Lovelace Byron Papers (Bodleian Library, Oxford), Boxes 41, 118, 170, 172, 174, 175. Reproductions and transcripts of Box 170 (the correspondence with De Morgan) may be viewed on the website of the Clay Mathematics Institute. In the citations, ‘Lovelace Byron Papers, Box *n*’ is abbreviated as ‘LB *n*’.

Somerville Papers, (Bodleian Library, Oxford), Dep. c.367.

**Published Sources**

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De Morgan, A. 1838. Induction (Mathematics). In *The Penny Cyclopædia of the Society for the Diffusion of Useful Knowledge*, vol. 12, 465–466. London: Charles Knight and Co.

De Morgan, A. 1840. Negative and Impossible Quantities. In *The Penny Cyclopædia of the Society for the Diffusion of Useful Knowledge*, vol. 16, 130–138. London: Charles Knight and Co.

Hollings, C., Martin, U., and Rice, A. 2017a. The early mathematical education of Ada Lovelace, *BSHM Bulletin: Journal of the British Society for the History of Mathematics* 32, 221–234.

Hollings, C., Martin, U., and Rice, A. 2017b. The Lovelace–De Morgan mathematical correspondence: A critical re-appraisal, *Historia Mathematica* 44, 202–231.

Hollings, C., Martin, U., and Rice, A. 2018. *Ada Lovelace: The Making of a Computer Scientist*. Oxford: Bodleian Library.

Lovelace, A. A. 1843. Sketch of the Analytical Engine invented by Charles Babbage Esq. By L. F. Menabrea, of Turin, officer of the Military Engineers, with notes upon the memoir by the translator. *Taylor’s Scientific Memoirs* 3, 666–731.

Peacock, G. 1820. *A Collection of Examples of the Applications of the Differential and Integral Calculus*. Cambridge: J. Deighton and Sons.

Rice, A. 1996. Augustus De Morgan: Historian of Science. *History of Science* 34, 201–240.

Rice, A. 1997. Inspiration or desperation? Augustus De Morgan’s appointment to the Chair of Mathematics at London University in 1828. *British Journal for the History of Science* 30, 257–274.

Rice, A. 1999. What Makes a Great Mathematics Teacher? The Case of Augustus De Morgan. *The American Mathematical Monthly* 106 (6), 534–552.