An Elementary Treatise on the Differential and Integral Calculus. ADA LOVELACE, BABBAGE, CHARLES, AUGUSTUS DE MORGAN.
An Elementary Treatise on the Differential and Integral Calculus
An Elementary Treatise on the Differential and Integral Calculus
An Elementary Treatise on the Differential and Integral Calculus
An Elementary Treatise on the Differential and Integral Calculus
An Elementary Treatise on the Differential and Integral Calculus
An Elementary Treatise on the Differential and Integral Calculus
An Elementary Treatise on the Differential and Integral Calculus
An Elementary Treatise on the Differential and Integral Calculus
An Elementary Treatise on the Differential and Integral Calculus

An Elementary Treatise on the Differential and Integral Calculus

ADA LOVELACE’S PERSONAL ANNOTATED COPY OF CHARLES BABBAGE’S FIRST SIGNIFICANT PUBLICATION. A MAJOR MATHEMATICAL TEXT IMPORTANT FOR HER INTELLECTUAL DEVELOPMENT.



WITH AT LEAST 35 INK ANNOTATIONS IN LOVELACE’S HAND – THE MAJORITY CORRECTIONS TO MATHEMATICAL EQUATIONS – AND AT LEAST 18 PENCIL ANNOTATIONS IN THE HAND OF HER TUTOR, THE FAMED MATHEMATICIAN AUGUSTUS DE MORGAN. WITH LOVELACE’S INITIALS (“A.A.L.”) IN GILT ON THE SPINE AND WITH EAST HORSLEY TOWERS (THE LOVELACE ESTATE) BLINDSTAMPS ON ENDPAPER AND TITLE.

The importance of Lacroix’s Elementary Treatise:

The key text driving the advance of English science and logic during the first half of the 19th century, Lacroix’s 1802 treatise on the Calculus was collaboratively translated by the members of “The Analytical Society”: a triumvirate of students at Cambridge University comprised of Charles Babbage, John Herschel, and George Peacock. Lacroix was the leading advocate for algebraic analysis in Europe, and his 1802 text on Calculus epitomized the advanced state of Continental mathematics – a state which the Analytical Society sought to promote in England. The book’s impact in England was substantial; and our research suggests that the book was in fact a common factor interlinking many of the major players in English science of the period. Certainly Babbage, Somerville, De Morgan, and Boole were all familiar with and influenced by LaCroix’s work. In Babbage’s case, a direct line can be traced from this book through Babbage’s 1815-16 essay on the Calculus of Functions onward to his seminal 1826 essay “On a Method of Expressing by Signs the Action of Machinery” -- and thence forward to his subsequent development of the Analytical Engine, the first programmable computer. And De Morgan’s own thinking and writings were so highly influenced by LaCroix’s work, that it was thought De Morgan was himself plagiarizing the French scientist in his own writings. Ada Lovelace’s own copy of the book – only recently re-emerged to light -- affirms that her own intellectual development was also significantly indebted to LaCroix’s text.

On Ada Lovelace:

Famous in her own century for being Lord Byron’s daughter, Ada Lovelace is now recognized as one of the leading female scientists of the 19th century and specifically celebrated for publishing the first algorithm and being one of the first people to envision a machine that could perform tasks beyond mere calculation.

Lovelace moved in the upper circles of English society and was intimately connected with the intellectual echelon that was driving the transformation of English mathematics and science. Lovelace’s early education was focused on mathematics – her mother did not want Ada following the wayward poetic ways of her father Lord Byron – and as a grown woman she studied mathematics both with Mary Somerville (the leading female mathematician of England and the translator of Laplace) and later with the great mathematician Augustus De Morgan. Lovelace first met Babbage when she was 18 (in 1833) and their friendship and interaction strengthened over the years. (Indeed, Babbage might have married Lovelace, but Lady Byron insisted that Ada not only marry rich but also nobly.) In 1843, she translated and annotated Menabrea’s “Sketch of the Analytical Engine Invented by Charles Babbage, Esq”, the work for which she is now most remembered.

It is almost certain that Lovelace acquired and read this book during the 1840-43 period – the most intellectually intense and productive period of her life – when she both studied calculus with De Morgan and also worked together with Babbage to annotate the “Sketch of the Analytical Engine.” It is known that Lovelace both worked with De Morgan’s own treatises on calculus during this period and also with George Peacock’s 1820 A Collection of Examples of the Differential and Integral Calculus. An 1843 letter from Lovelace to Babbage, however, clearly evidences that she is seeking further guidance and instruction in matters of calculus, with Lovelace stating “I cannot understand the Examples” and specifically asking Babbage for a copy of “your Calculus of Functions”. But whether the request is for a copy of Babbage’s two-part 1815-16 article “An Essay Towards a Calculus of Functions” (or his 1817 article “Solutions of Some Problems by Means of the Calculus of Functions”) or for the present book remains intriguingly ambiguous. It is known that Lovelace herself actually purchased a copy of Peacock’s Examples for her studies with De Morgan, but it is an open question whether and when Lovelace bought this present book or had it gifted it to her (by Babbage, De Morgan, or someone else).

The Annotations:

The ink annotations in Lovelace’s hand suggest a close reading of the text with an eye to correcting errors both typographical and critical – similar in character to the keen editorial eye she applied to Babbage’s and De Morgan’s own writings. Babbage himself records in his 1864 autobiography (Passages from the Life of a Philosopher) that it was Lovelace who “detected a grave mistake which I had made in the [algorithmic] process” for computing Bernoulli numbers; and “this keen eye for mathematical detail” is similarly displayed in Lovelace’s correspondence with De Morgan, which contain “multiple [valid] claims by Lovelace to have spotted errors or misprints in the various textbooks she was reading” (cf. Hollings, Martin, and Rice,“The Lovelace-De Morgan Mathematical Correspondence”, 2017).

The pencil annotations in De Morgan’s hand are marks, symbols, or single words mostly for emphasis, except for two longer comments. At the end of the Advertisement leaf, in response to the ad leaf’s promise that a companion volume (George Peacock’s calculus book A Collection of Examples) will be ready “in a few months”, he writes: “How vain are the affectations of Man! Four years later this book was printed.” Near the end of the book, on page 635 he has added in pencil: “You must not trouble yourself – I shall not read the one or the other – and I’ll be – A” followed by very faint text that appears to be “D. Morgan”. The De Morgan annotations further highlight the historical significance of this copy – he certainly handled and read it, implying that he either gifted it to Lovelace or used it while teaching her calculus.

De Morgan’s assessment of Lovelace’s work:

De Morgan had a very high opinion of Ada Lovelace’s mathematical abilities. In a now-famous letter from January 21, 1844, he wrote to Lovelace’s mother, Lady Noel Byron, informing her of her 28-year old daughter’s progress and potential in mathematics:

“I feel bound to tell you that the power of thinking on these matters which Lady L[ovelace] has always shewn from the beginning of my correspondence with her, has been something so utterly out of the common way for any beginner, man or woman, that this power must be duly considered by her friends, with reference to the question whether they should urge or check her obvious determination to try not only to reach but to get beyond, the present bounds of knowledge... Had any young [male] beginner, about to go to Cambridge, shewn the same power[s], I should have prophesied ... that they would have certainly made him an original mathematical investigator, perhaps of first rate eminence.”

Although contemporaries like De Morgan and Babbage recognized Lovelace’s brilliance, memory her achievements faded after Babbage‘s death in 1871, and it was only in the 1950s that Alan Turing “rediscovered” her and brought her legacy back into public awareness. Though many books have been written about Lovelace and her life, even now her properly scientific work remains under-explored by academics.

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Lacroix’s text was central to the intellectual development of an important and tightly connected core of 19th-century English scientists – a coterie of luminaries whose work was in large measure responsible for the birth of our digital age – and this copy, owned and read carefully by Lovelace, places her firmly within this distinguished group. Read and used during the 1840-43 period of her greatest intellectual creativity – during the time she worked on the “Sketch of the Analytical Engine” – Lovelace’s copy of Lacroix attests to the fact that she thoroughly studied the text and confirms her expertise with a high level of mathematics.

Unknown to scholarship, the present book is a seminal addition to the historical record of Lovelace’s life. This book both illuminates the background and context in which Lovelace created the first algorithm and offers future researchers a significant new source of insight into the mind of “the first computer programmer.”

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An Elementary Treatise on the Differential and Integral Calculus, translated from the French [by, Charles Babbage, John F.W. Herschel, & George Peacock]. First edition in English. Cambridge, for J. Deighton and Sons, 1816. ADA LOVELACE’S COPY (with East Horsley Tower blind-stamps on front free endpaper and title and gold initials AAL at tail of spine) WITH AT LEAST 53 AUTOGRAPH ANNOTATIONS (by our count, 35 by Lovelace in ink and 18 by De Morgan in pencil). Contemporary half calf gilt; light rubbing, the text with some general toning and slight waterstain in a few upper margins.

Price: $135,000 .