An Enchantress of Number? Reassessing the Mathematical Reputation of Ada Lovelace

Introduction

Figure 1.

Ada Lovelace, pictured around 1843.

If any reader of the Notices were asked to name a female mathematician of the 20th or 21st century, they would have no difficulty whatsoever. Giving them the same task but changing the parameter to the 19th century would present more of a challenge, but the names of several notable figures would no doubt still emerge. Mathematicians such as Sophie Germain, Sofia Kovalevskaya, Christine Ladd-Franklin, Charlotte Scott, and Mary Somerville would probably all feature—along perhaps with Emmy Noether, who although born in 1882 did all her significant mathematics in the 20th century. But if this second task were given to the general public, far fewer names would appear, and the occurrence of one would probably outweigh all others by a considerable margin: Ada Lovelace.

This is not surprising as today Lovelace ranks as one of the most famous female scientists in history. Lauded as “the first computer programmer” and praised for her visionary ideas, she is held up regularly as an inspiration for women in science and mathematics. She has been immortalized in literature, in plays, and on the screen. Exhibits relating to her life and legacy crop up in numerous science museums around the world, while her biography and earnest appraisals of her work adorn countless blogs and websites. Her homeland has been similarly keen to commemorate her as one of its own: between 2015 and 2020 her picture featured inside every newly issued British passport and in 2023 the UK’s Royal Mint issued a commemorative £2 coin in her honor. And all of this despite the fact that she made no major mathematical or scientific discovery, proved no theorems, and her entire scholarly output amounts to a single paper published in 1843.

Augusta Ada King, Countess of Lovelace was born in London on December 10, 1815. The only legitimate child of the romantic poet Lord Byron, she lived a privileged but somewhat isolated life. After her parents’ separation in early 1816, she was raised largely by her mother, the redoubtable Lady Byron, but never managed to escape the all-embracing effects of her father’s immense fame. She was what we would today call a “celebrity,” famous in her lifetime simply for being Lord Byron’s daughter, with her every move watched and scrutinized by polite society. Even after her death on November 27, 1852, at the early age of 36, her obituaries still focused on her Byronic connection. A fitting modern analogy would be the life of Lisa Marie Presley. Both women were the only children of fathers who died young, having experienced unprecedented levels of fame, enormous wealth, and notoriety. Both women lived lives defined, whether they liked it or not, by their status as the sole offspring of a dead celebrity. And both women themselves sadly succumbed to untimely deaths.

But that is where the analogy ends. Lovelace lived the privileged life of an English Victorian aristocrat, moving in social circles that included royalty and influential nobility, whose servants tended to her every need, and whose leisure activities included riding to hounds, lavish soirées, and nights at the opera. She was unusual for a woman in the 19th century, not only for having interests in science and mathematics, but for having been encouraged to pursue them, first by her mother and then by her husband, the Earl of Lovelace, who also cultivated an amateur interest in scientific matters. Her mathematical proclivity culminated in her publication of the 66-page paper on which her present-day fame rests Men16. It contains a theoretical account of a machine called the analytical engine, designed by the Victorian mathematician, inventor, and polymath Charles Babbage, which, had it ever been built, would have been the world’s first general purpose computer—100 years before the beginning of the modern computer age.

Lovelace’s paper contained seven lengthy appendices, or “Notes,” with the last one, “Note G” Men16, pp. 94–105, being her chief claim to fame. In this, in addition to some thoughts on the possibility of artificial intelligence, she outlined an iterative process by which Babbage’s machine could compute Bernoulli numbers, coefficients of the $x^n/n!$ terms in the infinite series expansion

$$\begin{equation*} \frac{x}{e^x-1}=\sum _{n=0}^{\infty } B_n \frac{x^n}{n!}. \end{equation*}$$

Lovelace described how to use this definition to derive a useful identity that would generate each Bernoulli number in turn. Since the above equation is equivalent to

$$\begin{equation*} \frac{x}{x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots }=1-\frac{x}{2}+\sum _{n=1}^{\infty }\frac{B_{2n-1}}{(2n)!}x^{2n} \end{equation*}$$

or

$$\begin{equation*} \frac{1}{1+\frac{x}{2!}+\frac{x^2}{3!}+\cdots }=1-\frac{x}{2}+B_1 \frac{x^2}{2!}+B_3 \frac{x^4}{4!}+\cdots , \end{equation*}$$

multiplying through by $1+\frac{x}{2!}+\frac{x^2}{3!}+\cdots$ would give

$$\begin{equation*} 1=\left(1+\frac{x}{2!}+\frac{x^2}{3!}+\cdots \right)\left(1-\frac{x}{2}+B_1\frac{x^2}{2!}+\dots \right) \end{equation*}$$

or

$$\begin{align*} 1&=1+\left(\frac{B_1}{2}+\frac{1}{3!}+\frac{1}{2\cdot 2!}\right)x^2\\ &\quad +\left(\frac{B_1}{2}\cdot \frac{1}{2\cdot 3}+\frac{1}{5!}-\frac{1}{2\cdot 4!}+\frac{B_3}{4!}\right)x^4+\cdots . \end{align*}$$

From this expression, it followed that the general coefficient of the $x^{2n}$ term would be

$$\begin{equation*} \frac{1}{(2n+1)!}-\frac{1}{2}\cdot \frac{1}{(2n)!}+\frac{B_1}{2!}\cdot \frac{1}{(2n-1)!} + \frac{B_3}{4!}\cdot \frac{1}{(2n-3)!}+\cdots +\frac{B_{2n-1}}{(2n)!}=0 \end{equation*}$$

or, when multiplied by $(2n)!$

$$\begin{equation*} -\frac{1}{2}\cdot \frac{2n-1}{2n+1}+B_1\cdot \frac{2n}{2}+B_3\cdot \frac{2n\cdot (2n-1)\cdot (2n-2)}{4!} +\cdots +B_{2n-1}=0. \end{equation*}$$

Lovelace then described how, using a series of punch-cards, the analytical engine could be instructed to successively plug $n=1,2,3,\dots$ into this formula to produce the famous but nonintuitive sequence of Bernoulli numbers, $B_1=\frac{1}{6}$, $B_3=-\frac{1}{30}$, $B_5=\frac{1}{42}$, $B_7=-\frac{1}{30}$, etc. (The Bernoulli numbers $B_2,B_4,B_6,\dots$ are all zero. It should also be observed that Lovelace’s notation for the Bernoulli numbers differs somewhat from that commonly used today, in which they would be listed as $B_0=1$, $B_1=-\frac{1}{2}$, $B_2=\frac{1}{6}$, $B_3=0$, $B_4=-\frac{1}{30}$, $B_5=0$, $B_6=\frac{1}{42}$, $B_7=0$, $B_8=-\frac{1}{30}$, etc.) Although the algorithm she devised was never run and the computer for which it was intended was never built, it is the process based on this algebraic manipulation for which Ada Lovelace is chiefly remembered today. And this is what has gone down in history as the world’s “first computer program.”

As we will see, in the 180 years since the appearance of Lovelace’s paper, opinions on her mathematical proficiency have varied considerably, with some assessments verging on the hagiographic, giving her credit for more than is perhaps due, while other writers have cast considerable doubt on whether she had the ability to understand—let alone write—such a technically proficient account, implying that it was in fact due to Babbage. On the one hand, in order to explain the process of calculating Bernoulli numbers, Lovelace must have had an understanding of the mathematics underlying the procedure, which was by no means trivial. On the other, this would largely have been beyond the capability of anyone who had not had some kind of university-level training in mathematics, which in the mid-19th century was unavailable to women, since they were not allowed to attend college or university. So which view of Lovelace’s mathematical ability is correct? Did she have the mathematical competence to write and understand the mathematics contained in her 1843 paper? Or was she essentially a mathematical ignoramus, as some have argued?

In this article, we will attempt to answer this question. While others have focused their attention on assessing Lovelace’s theoretical understanding of the mechanical operations of the analytical engine, we limit our analysis purely to an assessment of her mathematical ability. We begin with a brief survey of the mathematical topics to which she would have been exposed in the ten-year period culminating with the publication of her 1843 paper. We then look at the changing perspectives on Lovelace’s intellectual caliber over the years, from the 19th century to the present day, before focusing on the three most fundamental (and compelling) arguments that have influenced almost all negative assessments of her mathematical ability. Finally, we close with a timely postscript on Ada Lovelace and artificial intelligence. Much of the analysis below is informed by research carried out in collaboration with my colleagues Christopher Hollings and Ursula Martin, comprising a detailed study of the mathematical papers in the Lovelace archive, now housed in the Bodleian Library in Oxford. This research resulted in the research papers HMR17a and HMR17b, as well as an illustrated expository book HMR18, all of which may be consulted for further details.

Lovelace’s Mathematical Education

Throughout her childhood, Ada’s keen interest in science and mathematics was actively fostered by her mother. By the age of 18, her mathematical education consisted of a thorough grounding in arithmetic and some basic geometry, already surpassing the level attained by most children at the time. Her introduction to London’s high society in 1833 led to her meeting two of the foremost British mathematical scientists of the day, Charles Babbage and Mary Somerville, both of whom encouraged the continuation of her mathematical studies.

On the advice of a family friend, Dr. William King, who had studied mathematics at Cambridge in the early 1800s, she began a focused course of reading under his direction. He advised:

Begin with Euclid, then Plane & Spherical Trigonometry, which is found at the end of Simson’s Euclid, then Vince’s Plane & Spherical Trigonometry, then Bridge’s Algebra HMR17a, p. 229.

But this guidance, while standard for a British university course circa 1800, was quite out of date by the 1830s. By this time, the old-fashioned synthetic, geometrical methods in vogue in England at the beginning of the century were quickly being replaced by newer and more progressive analytic and algebraic techniques imported from continental Europe, with which Lovelace’s friends like Somerville and Babbage were quite familiar. Somerville in particular was able to provide some useful assistance on various mathematical points, later recalling that Ada was “much attached” to her and “always wrote to me for an explanation when she met with any difficulty” Ste85, p. 50. Indeed, by 1835, shortly after her marriage, we find the 19-year-old Ada writing to Mrs. Somerville about her current mathematical studies Too92, p. 83:

I now read Mathematics every day & am occupied in Trigonometry & in preliminaries to Cubic & Biquadratic Equations. So you see that matrimony has by no means lessened my taste for those pursuits, nor my determination to carry them on….

Figure 2.

Augustus De Morgan.

Although her marriage caused no immediate interruption of her mathematical education, bearing three children in quick succession between 1836 and 1839 certainly did. But she was eager to resume her studies and, in the summer of 1840, she began a course of study under the tutelage of the mathematician and logician Augustus De Morgan. After about eighteen months, Lovelace had covered much of the material included in a typical undergraduate course of the time: basic algebra, trigonometry, logarithms, complex numbers, functions, limits, infinite series, differentiation, integration, and differential equations. It was during this period that she learned the skills that would be essential for her mathematical commentary on Babbage’s analytical engine a couple of years later. In particular, it was via her study with De Morgan that she first came across Bernoulli numbers.

In late 1842, Lovelace was invited by the scientist Charles Wheatstone to translate an account of Babbage’s analytical engine, recently published in French by the Italian politician and engineer Luigi Menabrea. On learning that she was working on an English translation, Charles Babbage asked her

why she had not herself written an original paper on a subject with which she was so intimately acquainted? To this Lady Lovelace replied that the thought had not occurred to her. I then suggested that she should add some notes to Menabrea’s memoir; an idea which was immediately adopted Bab94, p. 136.

This led to the composition of Lovelace’s seven lengthy “Notes” which, at 41 pages, when added to her 25-page translation, resulted in a paper more than double the length of the original.

Lovelace spent the first half of 1843 feverishly working on her “Notes,” with multiple revisions and corrections flying back and forth between her and Babbage. As she told him, “I am working very hard for you; like the Devil in fact” Too92, p. 198. It was in July that she came up with the idea for her final note:

I want to put in something about Bernoulli’s Numbers, in one of my Notes, as an example of how an implicit function may be worked out by the engine, without having been worked out by human head & hands first Too92, p. 198.

Figure 3.

Charles Babbage.

Babbage later recalled that he wrote out the necessary algebra for her, “to save Lady Lovelace the trouble,” but that “she sent [it] back to me for an amendment, having detected a grave mistake which I had made in the process” Bab94, p. 136. The result was the famous “Note G” containing her well-known “program.”

When the paper was published at the end of the summer, Lovelace made no attempt to hide the obvious pride she felt in her work, declaring herself “very much satisfied with this first child of mine” Too92, p. 211—although before long this satisfaction had morphed into considerable hubris. Using the word “analyst” as it was often used in those days, to mean “mathematician,” she wrote Too92, p. 215:

I do not believe that my father was (or ever could have been) such a Poet as I shall be an Analyst…. The more I study, the more irresistable do I feel my genius for it to be.

Yet despite this bold claim (and several others), her paper of 1843 remained her only publication and the only other mathematics she wrote—none of it original—is contained in her surviving letters to Babbage, Somerville, and De Morgan. It is perhaps for this reason that subsequent opinions on her mathematical merits have not always been favorable, as we shall see.

Evolving Perspectives on Lovelace

Given Lovelace’s high social rank and the scarcity of women in any branch of science in those days, it is perhaps unsurprising that contemporary evaluations of her mathematical talent were uniformly positive. But, even taking these factors into account, they are still overwhelmingly effusive. Shortly after her paper on his machine had appeared in 1843, Babbage was referring to her as “the Enchantress of Number,” going even further in a letter to Michael Faraday, in which he described her as

that Enchantress who has thrown her magical spell around the most abstract of Sciences and has grasped it with a force which few masculine intellects (in our own country at least) could have exerted over it. Jam96, p. 164

The following year, De Morgan, who as her tutor had witnessed her mathematical development firsthand, wrote that her “power of thinking on these matters …[was] utterly out of the common way for any beginner, man or woman.” He went on to say

Had any young [male] beginner, about to go to Cambridge, shown the same power[s], I should have prophesied …that they would have certainly made him an original mathematical investigator, perhaps of first-rate eminence Ste85, p. 82.

After her death, obituarists were at pains to emphasize her mathematical talents. The Athenæum reported Ano52a: “Like her father’s Donna Inez, in Don Juan—‘Her favourite science was the mathematical,’” while the Examiner observed Ano52b:

Her genius, for genius she possessed, was not poetic, but metaphysical and mathematical, her mind having been in the constant practice of investigation, and with rigour and exactness. With an understanding thoroughly masculine in solidity, grasp, and firmness, Lady Lovelace had all the delicacies of the most refined female character.

The years that followed saw similarly fulsome praise come from a variety of sources. In 1860, the English author John Timbs praised “the noble and all-accomplished” translator of Menabrea’s memoir, remarking that

The profound, luminous, and elegant notes forming the larger, and by far the most instructive, part of the work, and signed A. A. L., are all by that lamented lady. Tim60, p. x

In his 1864 autobiography, Babbage concurred with this assessment of Lovelace’s “Notes,” believing that “Their author has entered fully into almost all the very difficult and abstract questions connected with the subject” Bab94, p. 136.

A generation later, in an effort to collate and preserve documents relating to his father’s life’s work, Babbage’s son Henry published a substantial volume of writings in 1889, including the entirety of Lovelace’s 1843 paper. Sixty years later, as the modern computer age dawned, this book was rediscovered by the Cambridge physicist Douglas Hartree, who found himself drawn to Lovelace’s paper. Struck by the fact that “some of her comments sound remarkably modern” and impressed by her discussion of the numerical computations and iterative procedures, he observed that “she must have been a mathematician of some ability” Har49, p. 70.

Hartree’s book was read by Alan Turing, who in a famous paper of 1950, challenged Lovelace’s views on artificial intelligence Tur50 (see Postscript). Turing also contributed to a volume of essays on computing edited by the scientist and educationalist B. V. Bowden, who agreed with Hartree’s assessment of Lovelace’s abilities, describing her as “a mathematician of great competence” Bow53, p. xi. But we note here that, with the passage of time and increasing reliance on second- and third-hand information, assessments of Lovelace’s intellect, while still positive, had become a little more measured, with words like “some ability” and “competence” replacing the more indulgent uses of “first-rate” and “genius” by Lovelace’s contemporaries.

By now, along with Babbage, the name of Ada Lovelace was known and accepted among those in the burgeoning community of computer programmers and engineers as one of the pioneers in the prehistory of their field. It was in this spirit that the US Department of Defense chose Ada as the name of its newly developed programming language in 1979.

The 1970s saw the publication of the first full-scale biography of Lovelace, by experienced Byron scholar Doris Langley Moore. This study, like many which were to follow, benefitted substantially from access to the large collection of unpublished Lovelace-Byron manuscript papers, made available to scholars via their recent deposit in the Bodleian Library. Although a fine historian, Langley Moore was less authoritative when it came to mathematics, able to offer little more than a description of Lovelace’s “curious letters” to De Morgan as being full of “pages with equations, problems, solutions, algebraic formulae, like a mathematician’s cabalistic symbols.” LM77, p. 99

The first Lovelace biography to focus on the mathematics and science was Ada: A Life and a Legacy (1985) by Dorothy Stein. Also the first intellectual study to delve deeply into the archives, Stein’s book examined Lovelace’s mathematical and scientific papers in painstaking detail, and laid bare lesser-known aspects of her life, including extramarital liaisons, gambling and drug addictions, and physical and mental illnesses. Her key thesis was, essentially, that Lovelace was “a figure whose achievement turns out not to deserve the recognition accorded it” Ste85, p. xii. Perhaps inevitably, given Lovelace’s own grandiose claims about her mathematical ability, Stein’s conclusion is not flattering:

It is unusual to find an interest in mathematics and a taste for philosophical speculation accompanied by such difficulty in acquiring the basic concepts of science as she clearly displayed. We can only be touched and awed by the questing spirit that induced her to launch so slight a craft upon such deep waters Ste85, p. 280.

To this day, Stein’s text remains a standard work for anyone interested in Lovelace’s scientific endeavors, its authority bolstered by the sheer weight of documentary evidence employed to defend its thesis. For this reason, Stein’s conclusions have strongly influenced subsequent studies, particularly those with negative viewpoints. For example, in 1990 historian of computing Allan Bromley concluded: “Not only is there no evidence that Ada ever prepared a program for the Analytical Engine, but her correspondence with Babbage shows that she did not have the knowledge to do so” ABCK$^{+}$90, p. 89. More damning still was Bruce Collier who described Lovelace as “mad as a hatter …with the most amazing delusions about her own talents,” calling her “the most overrated figure in the history of computing” Col90, preface. Recent studies have been more measured, however, with Thomas Misa’s 2016 survey of the debate concluding that the 1843 “Notes” were “a product of an intense intellectual collaboration” between Lovelace and Babbage Mis16, p. 18.

It was against this background that Hollings, Martin, and I undertook our research project to provide what we hope is a more nuanced and historically accurate assessment of Lovelace’s mathematical proficiency. In particular, our work challenged—and, we argue, refuted—Stein’s judgement that

The evidence of the tenuousness with which she grasped the subject of mathematics would be difficult to credit about one who succeeded in gaining a contemporary and posthumous reputation as a mathematical talent, if there were not so much of it Ste85, p. 90.

Stein’s case against Lovelace’s mathematical competence boils down to three main arguments, which, at first sight, seem quite compelling. And these arguments have influenced the subsequent judgements of others for nearly four decades. But as we will now show, all three arguments fail when subjected to further contextual analysis.

Argument 1: A Trigonometric Identity

The first argument given by Stein is that “despite hard work, skill and ingenuity in the manipulation of symbols did not come easily to her” Ste85, p. 56. And certainly, as we found in our study, many of the letters from Lovelace to her tutor De Morgan, particularly early on in her course of study, contain either algebraic errors or evidence of difficulty in symbolic manipulation. However, the first example given by Stein to illustrate this algebraic incompetence predates these letters by five years. Dating from 1835, it is contained in two letters to Mary Somerville, concerning the algebraic manipulation of trigonometric identities.

Figure 4.

Mary Somerville.

At this point, Lovelace was studying trigonometry and had attempted unsucccessfully to derive certain identities from others algebraically. To give a flavor, we will just look at the first (and easiest) problem. Given the addition formulae

$$\begin{gather*} \sin (a+b)=\frac{\sin a\cos b + \sin b\cos a}{R}\\ \cos (a+b)=\frac{\cos a\cos b - \sin a\sin b}{R}, \end{gather*}$$

the question was to deduce the corresponding difference formulae

$$\begin{gather*} \sin (a-b)=\frac{\sin a\cos b - \sin b\cos a}{R}\\ \cos (a-b)=\frac{\cos a\cos b + \sin a\sin b}{R}. \end{gather*}$$

In her letter to Somerville, Lovelace quotes her textbook as saying: “The values for the sine & cosine of the differences $(a-b)$ may easily be deduced from these two formulas.” But she admits, “Now however easily deduced, I have not succeeded in doing it” Ste85, p. 55.

On seeing this, the modern reader will undoubtedly have two principal reactions. The first would be to ask “what is $R$?” And the second would be to entertain serious doubts, along with Stein and others, about Lovelace’s mathematical competence if she could not even make simple substitutions such as $\sin (-x)=-\sin x$ and $\cos (-x)=\cos x$.

The convention of teaching trigonometry with reference to a circle of radius 1 is a relatively recent development and had certainly not been uniformly adopted in Lovelace’s time. In her day, all trigonometric functions were defined with respect to a circle of arbitrary radius $R$. Rather than ratios, they were understood as simple chord sections, with respect to an arc, which of course would correspond to an angle, subtended from the center of the circle. So, for example, in Figure 5, the angle $\angle$ACM would correspond to the arc APM. The sine of that angle was defined as being “a line drawn from one extremity of the arc perpendicular to a diameter drawn through the other extremity” Vin00, p. 44. Thus $\sin \angle$ACM is the line segment MH, $\cos \angle$ACM is the line segment CH, and $\sin ^2(\angle$ACM) + $\cos ^2(\angle$ACM) = CM$^2$ = $R^2$.

Figure 5.

We referred earlier to Ada’s somewhat outmoded course of study in the 1830s, as recommended by her family friend, Dr. King. This included reading on algebra (largely comprising methods for solving polynomial equations), Euclid’s Elements for basic plane geometry, and A Treatise on Plane and Spherical Trigonometry (1800) by the Reverend Samuel Vince. This last work was a highly synthetic, geometrical treatment of the subject, in which there was no mention of negative angles, no use of algebraic substitutions, and in fact, little use of algebra at all.

Here, for example, is the rather verbose—but entirely rigorous—way that Ada would have learnt how to derive the addition formula for the sine, with reference to Figure 5 Vin00, pp. 55–56. The reader is invited to follow the reasoning.

Let ACM, MCN be the two given angles, of which ACM is the greater; make the angle MCP = MCN, and then $\angle$ACN is the sum of the two given angles, and $\angle$ACP their difference. Draw the chord PN, intersecting CM in D; then as the angle $\angle$PCD = $\angle$NCD, PC = CN, and CD is common to the triangles PCD, NCD, we have (by Euclid, Book I, Prop. 4) PD = DN, and the angle $\angle$PDC = $\angle$NDC, therefore each of them is a right angle. Draw NG, DB, MH, PK, perpendicular to AC, and therefore parallel to each other, and DF, P$x$E parallel to AC, and therefore parallel to each other (by Euclid, Book I, Prop. 30). Now as ND = DP, we have (by Euclid, Book VI, Prop. 2) NF = FE = D$x$, and P$x$ = E$x$ = DF; therefore BK = GB = DF. Also, the triangle NDF is similar to the triangle CBD, for the angle $\angle$NDF = $\angle$CDB, each being the complement of $\angle$FDC, and the angles at F and B are right angles; but $\Delta$CDB is similar to $\Delta$CMH, the angle at C being common, and the angles at B and H right angles; therefore $\Delta$NDF is also similar to $\Delta$CMH.

The book then uses the similarity of these triangles to deduce that

$$\begin{align*} & \mathrm{CM} : \mathrm{MH} \Colon \mathrm{CD} : \mathrm{DB},\\ & \mathrm{CM} : \mathrm{CH} \Colon \mathrm{ND} : \mathrm{NF}, \end{align*}$$

and therefore

$$\begin{align*} \frac{\mathrm{MH}\cdot \mathrm{CD}+\mathrm{CH}\cdot \mathrm{ND}}{\mathrm{CM}} &= \mathrm{DB} + \mathrm{NF} \\ &= \mathrm{FG} + \mathrm{NF} = \mathrm{NG}. \end{align*}$$

Thus, letting $\angle$ACM $=a$, $\angle$MCN $=b$, and the line length CM = $R$, it arrives at

$$\begin{equation*} \sin (a+b)=\frac{\sin a\cos b + \sin b\cos a}{R}. \end{equation*}$$

The corresponding difference formula would therefore follow, not via the algebraic substitution of negative-angle identities, but by further geometric reasoning:

$$\begin{align*} \frac{\mathrm{MH}\cdot \mathrm{CD}-\mathrm{CH}\cdot \mathrm{ND}}{\mathrm{CM}} &= \mathrm{DB }-\mathrm{ NF} \\ &=\mathrm{DB }-\mathrm{ D}x\mathrm{ = B}x\mathrm{ = PK}, \end{align*}$$

which gives the desired result.

Given Lovelace’s exposure to this style of reasoning in trigonometry, it is not difficult to see why the thought of algebraic substitutions either did not occur to her, or did not come easily at first. Although we might regard such operations as obvious today, if we had not been trained in the algebraic style of doing mathematics, the need for such procedures would be far from evident. Thus, to base an argument of algebraic ineptitude on this example fails to take into account precisely what kind of mathematics and what style of mathematical thinking Lovelace had been exposed to up to this point.

Argument 2: The Case of the Missing Case

Stein’s next argument concerned what she believed to be “perhaps the most telling and consequential” Ste85, p. 90 piece of evidence against Ada’s mathematical proficiency. It was contained in the 1843 paper, not in Lovelace’s “Notes” at the end of her translation of Menabrea’s article, but in the text of the translation itself. In his original article, Menabrea had discussed how the analytical engine might deal with computations that theoretically required an infinite number of steps. The example he gave was the following multiple of the Wallis product

$$\begin{equation*} 2\cdot \prod _{k=1}^{n}\left(\frac{2k}{2k-1}\cdot \frac{2k}{2k+1}\right), \end{equation*}$$

the value of which, as $n\rightarrow \infty$, is $\pi$.

Since plugging in a value of $n=\infty$ would obviously be impossible, Menabrea suggested that, in this case, a punch-card containing the numerical value of the output could simply be substituted in place of having the machine actually do the calculation. In Lovelace’s translation, the full passage reads as follows Men16, pp. 53–54:

Let us now examine the following expression:

$$\begin{equation*} 2\cdot \frac{2^2\cdot 4^2\cdot 6^2\cdot 8^2\cdot 10^2\cdots (2n)^2}{1^2\cdot 3^2\cdot 5^2\cdot 7^2\cdot 9^2\cdots (2n-1)^2\cdot (2n+1)^2}, \end{equation*}$$

which we know becomes equal to the ratio of the circumference to the diameter, when $n$ is infinite. We may require the machine not only to perform the calculaiton [sic] of this fractional expression, but further to give indication as soon as the value becomes identical with that of the ratio of the circumference to the diameter when $n$ is infinite, a case in which the computation would be impossible. Observe that we should thus require of the machine to interpret a result not of itself evident, and that this is not amongst its attributes, since it is no thinking being. Nevertheless, when the cos of $n=\infty$ has been foreseen, a card may immediately order the substitution of the value of $\pi$ ($\pi$ being the ratio of the circumference to the diameter), without going through the series of calculations indicated. This would merely require that the machine contain a special card, whose office it should be to place the number $\pi$ in a direct and independent manner on the column indicated to it.

Although a little long-winded, the majority of this extract is intelligible enough, but there is one notable exception: the sentence concerning “the cos of $n=\infty$.” Not only is this phrase obviously meaningless, but it also raises the question of what on earth the cosine function has to do with this example anyway. Referring back to Menabrea’s article in French, we find that sentence originally was rendered as follows:

Cependant lorsque le cos. de $n=\infty$ a été prévu, un carton peut ordonner immédiatement la substitution de la valeur de $\pi$ ($\pi$ étant le rapport de la circonférence au diamètre), sans passer par la série des calculs indiqués.

This is exactly the same as in Lovelace’s translation. Why then is there mention of $\cos \infty$? The answer is simple: Menabrea’s paper contained a typo. The relevant phrase should have read: “lorsque le cas de $n=\infty$ a été prévu,” or “when the case of $n=\infty$ has been foreseen,” which obviously makes much more sense. Thus, in Stein’s view, “Ada had translated a printer’s error” Ste85, p. 91, thereby revealing her limited understanding of this section of Menabrea’s paper.

Figure 6.

Portion of Menabrea’s article containing the original misprint.

One can certainly agree that, by giving a literal translation of this passage without noticing the printing error, Lovelace did not show her eye for detail and proofreading skills in the best possible light. But at the same time it’s hardly indicative of mathematical incompetence. Several other typographical errors also slipped past her, including the misspelling of the word calculation as calculaiton in the same passage and the rendering of her initials A. A. L. as “A. L. L.” at the end of the paper. It should also be remembered that the “cos of $n=\infty$” typo did not just slip past Lovelace and Menabrea; it went unnoticed by several scholars who we know read the article in detail, including Babbage and Wheatstone in the 19th century and Turing and Hartree in the 20th.

Stein’s case is further weakened by her subsequent frank admission that this mistake “has been reprinted several times” Ste85, p. 91. In addition to its first occurrence in Menabrea’s paper and subsequent inclusion in Lovelace’s translation, it also appeared in Henry Babbage’s collection of papers in 1889, where for some unknown reason, he gave the phrase as “the cos of $n=1/0$,” which makes even less sense than before. This rendering was also subsequently reproduced in Philip and Emily Morrison’s Charles Babbage and his Calculating Engines, published in 1961. In fact, the mistake was only spotted in 1953, when B. V. Bowden reprinted Lovelace’s paper in its entirety with the error silently corrected Bow53, p. 359. But the fact that it took 110 years before the mistake was noticed gives some indication that it is not quite as “telling and consequential” as Stein would have us believe.

Argument 3: Chronic Algebraic Inability?

The final piece of evidence presented by Stein is, without doubt, potentially the most damaging. Indeed, if correct, this example alone would be enough to cast serious doubt on Lovelace’s ability to comprehend anything but the most elementary mathematics. Like the first argument, it implies that she had a chronic inability to manipulate algebraic expressions. That first argument is based on evidence from 1835, when Lovelace was only 19 and had yet to start her intensive study of university-level mathematics with De Morgan. But by contrast, Stein’s final argument dates its evidence from the end of 1842—two and a half years after the commencement of those studies and, most worryingly of all, almost exactly contemporaneous with the beginning of her work on her famous paper.

Stein insists that Lovelace continued to display an inability “to assimilate the symbolic processes with which alone highly complex and abstract matters may be rigorously treated,” which rendered her incapable of grasping algebraic reasoning and “in the end limited her understanding of science” Ste85, p. 84. This insistence is, of course, essential to her overall thesis that by 1843 Lovelace was in no way sufficiently prepared to write on a mathematical subject.

As we have already indicated, there is no doubt that Lovelace found algebra challenging at first, and her letters to De Morgan contain many examples where she has made elementary algebraic errors or displayed a novice’s misunderstanding. Several examples are highlighted, for example, in HMR17b and Ric21. But it is equally clear from their correspondence that not only did her algebra improve over time, but that it was soon applied to far more demanding topics, such as, for example, second-order differential equations, which she was studying by November 1841. Thus, while still prone to the odd algebraic mistake or misunderstanding, these increasingly arose from far more sophisticated mathematics.

To begin our analysis of Stein’s third argument, we quote the relevant passage from her book in full, in order to convey the full weight of the alleged evidence Ste85, p. 90:

The last surviving letters in Lovelace’s mathematical correspondence with De Morgan are dated 16 and 27 November 1842 (hence shortly before she translated the Menabrea memoir). In them we find her wrestling with an elementary problem in functional equations. (The problem was: Show that $f(x+y)+f(x-y)=2f(x)f(y)$ is satisfied by $f(x)=(a^{x}+a^{-x})/2$.) She was still unable to take a mathematical expression and substitute it back into the given equation. It was the same “principle” that had plagued her in her correspondence with Mary Somerville and in earlier letters to De Morgan. On 27 November, after having struggled for at least eleven days, she sighed,

I do not know when I have been so tantalized by anything, & should be ashamed to say how much time I have spent upon it, in vain. These functional Equations are complete Will-o-the-Wisps to me. The moment I fancy I have really at last got hold of something tangible & substantial, it all recedes further & further & vanishes again in thin air…. I believe I have left no method untried.

The obvious inference is that if Lovelace was unable to handle algebraic manipulations like these in November 1842, after more than two years of studying mathematics at a much higher level than this, how are we to believe that she was capable of writing such a mathematically fluent account of Babbage’s analytic engine in the first half of 1843?

Figure 7.

Problem from page 206 of De Morgan’s Elements of Algebra.

Lovelace’s course of study with De Morgan began around July 1840 and, although dominated from fairly early on by calculus-related topics, periodic excursions into more elementary topics were necessary due to gaps in her prior studies. As De Morgan reminded her in September of that year: “You understand of course that your Differential Calculus must be delayed from time to time while you make up those points of Algebra and Trigonometry which you have left behind” HMR17b, p. 206.

As part of this remedial study, Lovelace was instructed to read De Morgan’s 1835 textbook The Elements of Algebra, subtitled, appropriately enough, Preliminary to the Differential Calculus. By November, she had reached the book’s short Chapter 10 entitled “On the notation of functions.” There, in just four pages, De Morgan showed how to express functions—writing for instance $\phi x$ where we today would usually write $f(x)$—and introduced the idea of solving functional equations. (A simple example would be something like $\phi (xy)=\phi x + \phi y$, of which one possible solution is $\phi x=\ln x$.) It was this topic that caused Lovelace problems.

In a letter dated November 10, but with no year given, she wrote to De Morgan: “I do not know why it is exactly, but I feel I only half understand that little Chapter X, and it has already cost me more trouble with less effect than most things have. I must study it a little more I suppose” HMR17b, p. 224.

A subsequent letter written on November 16 finds her going into more detail HMR17b, p. 225:

I can explain exactly what my difficulty is in Chapter X…. That I do not comprehend at all the means of deducing from a Functional Equation the form which will satisfy it, is I think clear from my being quite unable to solve the example at the end of the Chapter ‘Shew that the equation $\phi (x+y)+ \phi (x-y)=2\phi x \times \phi y$ is satisfied by $\phi x = \frac{1}{2}(a^{x}+a^{-x})$’. I have tried several times, substituting first 1 for $x$, then 1 for $y$ but I can make nothing whatever of it, and I think it is evident there is something that has preceded, which I have not understood. The 2nd example given for practice ‘Shew that $\phi (x+y)=\phi x + \phi y$ can have no other solution than $\phi x = ax$’, I have not attempted.

The problem to which Lovelace is referring appears on page 206, at the end of Chapter 10 of De Morgan’s algebra textbook (see Figure 7). Again, she has given the date but no year on this letter. However, immediately below her handwritten date, a 20th-century hand, possibly that of an archivist, has added a further detail: the year “1842” appears quite legibly on the letter in pencil. Lovelace’s final contribution to this particular topic is the letter dated November 27 (but again, with no year) quoted by Stein above. It is thus easy to see why Stein believed that this particular episode dates from November 1842, as all the evidence thus far presented points in that direction.

It looks pretty bad for Lovelace doesn’t it?

But wait! There is a flaw in Stein’s analysis; namely, she does not appear to look at De Morgan’s replies to these letters from Lovelace. And it turns out that one letter in particular carries a vital clue.

In a letter seemingly composed in reply to Lovelace’s initial letter of November 10, De Morgan wrote back, saying: “The notation of functions is very abstract. Can you put your finger upon the part of Chapt. X at which there is difficulty[?]” HMR17b, p. 224. The content of this letter not only makes it obvious that it is an integral part of this specific epistolary conversation, but crucially, reveals it to be the one and only letter from this sequence that actually contains a full date. At the end, just to the left of his signature, De Morgan has written in his distinctive handwriting: “Nov^r. 14/40.” In other words, this conversation began, not in 1842, but in 1840. We are now left with the possibility that the first two letters in the discussion (from November 10 and 14) were both written in 1840, with two subsequent letters following two years later, on November 16 and 27, respectively. This looks potentially even worse for Lovelace’s mathematical reputation since it would imply that even after two further years of higher mathematical study, she still couldn’t solve this relatively simple problem!

But one final possibility also exists, and this only becomes clear when you read all of the letters in this particular conversation. There are actually six letters in this back-and-forth exchange (three from Lovelace and three from De Morgan) and, when read in the correct order, they shed crucial light onto what was really happening and, most importantly, when it was happening.

On November 10, 1840, Lovelace first wrote to complain that she felt “I only half understand that little Chapter X.” Four days later, in a letter dated November 14, 1840, De Morgan replied, asking “Can you put your finger upon the part of Chapt. X at which there is difficulty[?].” Lovelace then responded on November 16, “I can explain exactly what my difficulty is in Chapter X,” which she then proceeded to do. De Morgan’s answer to this, merely dated “Friday,” gave her a short hint, to which she then replied on November 27 HMR17b, p. 225:

I have I believe made some little progress towards the comprehension of the Chapter on Notation of Functions, & I enclose you my Demonstration of one of the Exercises at the end of it : “Show that the equation $\phi (x+y)=\phi x+\phi y$ can be satisfied by no other solution than $\phi x=ax$.” At the same time I am by no means satisfied that I do understand these Functional Equations perfectly well, because I am completely baffled by the other Exercise: “Shew that the equation $\phi (x+y)+\phi (x-y)=2\phi x\times \phi y$ is satisfied by $\phi x=\frac{1}{2}(a^x+a^{-x})$ for every value of $a$.”…These Functional Equations are complete Will-o’-the-Wisps to me….

Lastly, in a letter dated “Monday,” De Morgan closed the episode with a final reply HMR17b, p. 225:

I can soon put you out of your misery about p. 206. You have shown correctly that $\phi (x+y)=\phi (x)+\phi (y)$ can have no other solution than $\phi x=ax$, but the preceding question is not of the same kind; it is not show that there can be no other solution except $\frac{1}{2}\left(a^x+a^{-x}\right)$ but show that $\frac{1}{2}\left(a^x+a^{-x}\right)$ is a solution: that is, try this solution. [He then demonstrates the solution.]…I think you have got all you were meant to get from the chapter on functions.

It is therefore clear that this entire discussion, rather than dating from 1842 as Stein claimed, actually took place two years earlier in 1840, close to the very beginning of Lovelace’s studies with De Morgan, rather than right at the end of them. And since Stein’s argument hinged on these letters dating from 1842, our redating of them leads to the inevitable conclusion that her negative assessment of the state of Lovelace’s mathematical abilities by late 1842 was in fact incorrect. (For the first in-depth analysis of the Lovelace–De Morgan correspondence by historians of mathematics, see HMR17b.)

On the face of it, and taken together, all three of Stein’s key arguments seemed at first to present a compelling picture of Lovelace as someone whose mathematical skills were considerably weaker than had previously been supposed. But as this article has shown, further contextual analysis reveals fundamental flaws in each of these arguments, throwing considerable doubt, not just on Stein’s negative conclusions, but on those of subsequent authors who accepted her findings. This results in a more nuanced assessment of Lovelace’s abilities. And while it does not prove that she was solely responsible for the 1843 “Notes,” nor does it in any way suggest that she was a genius, it does provide strong evidence that she did indeed have the mathematical competence to write and understand the mathematics contained in her famous paper of 1843.

Postscript

In recent months, the subject of artificial intelligence (AI) has featured prominently in news stories, due to rapid developments in the abilities of AI software applications such as ChatGPT, Google Bard, and Microsoft Bing. Academics and educators have exhibited a mixture of excitement and terror at the sheer volume of convincingly humanlike responses now capable of emanating from their computers. But of course, speculation about the possibility of AI is nothing new and, as is well known, also appeared in Lovelace’s 1843 paper. For this reason, it seems appropriate to conclude this article with a timely postscript on Ada Lovelace and artificial intelligence.

Lovelace’s views on the possibility of artificial intelligence can be summed up in her own words on the potential of Babbage’s theoretical machine: “The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform” Men16, p. 94. A century later, while agreeing with her that all the thinking had to be done beforehand by the human hardware and software designers, Douglas Hartree countered: “This does not imply that it may not be possible to construct electronic equipment which will ‘think for itself,’ or in which, in biological terms, one could set up a conditioned reflex, which would serve as a basis for ‘learning’” Har49, p. 70.

This view was famously expanded by Alan Turing in his seminal 1950 paper “Computing machinery and intelligence,” in which he posed the classic question: “Can machines think?” Tur50, p. 433. Describing himself as being “in thorough agreement with Hartree” on the question of whether machines might one day have the power to think for themselves, he speculated “that the evidence available to Lady Lovelace did not encourage her to believe that they had it” Tur50, p. 450. Turing believed that by the end of the 20th century, “one will be able to speak of machines thinking without expecting to be contradicted” Tur50, p. 442; he then gave—and attempted to refute—nine counter-arguments to this opinion, the sixth of which was “Lady Lovelace’s Objection” Tur50, p. 450.

Figure 8.

Alan Turing.

Turing chose to intepret Lovelace’s view as being that a machine can never “take us by surprise” and tried to counter it with a few rather lame examples of how humans can be surprised by computers, due perhaps to insufficient or erroneous assumptions on the part of the programmer. But he admitted that this wasn’t particularly convincing. Of greater importance, of course, was his “imitation game” or Turing test, as it is now known. This was a thought experiment via which, if successful, a computer could produce output that was indistinguishable from that of a human brain. But while most today would agree that the Turing test has been satisfied, Lovelace’s objection remains a harder criterion to fulfill.

For this reason, a 21st-century upgrade of the Turing test, known as the Lovelace test, was formulated. Instead of focusing on the mere replication of humanlike intelligence, this new benchmark concentrates on Lovelace’s use of the word “originate” to ask whether a machine could ever be able to produce an original, creative piece of work unforeseen by its programmer. Put simply, a computer would pass this test if it created something new and original (such as a poem or a piece of music) in such a way that no programmer could explain the process.

To this end, the author of this article conducted a casual experiment. On one occasion, he instructed ChatGPT to “write a limerick about Ada Lovelace.” A limerick was duly composed. He then made up a limerick of his own. Finally, on a separate occasion, he gave the same instruction again to ChatGPT, and the software produced another limerick. The three limericks are presented here, in a different order from their initial composition.

There once was a woman named Ada
Whose writings are said to have played a
Part in the formation
Of machine computation
That’s now used to analyze data.

Of computing and numbers a fan,
In mathematics as strong as a man;
With all of her might
She worked day and night
To invent the computer program.

But was she as smart as they say?
Some answer this question, “No way!”
While others maintain
She did all that they claim
And her legend persists to this day.

Three things should now be noted. Firstly, the limericks are all atrocious, for which the author can only apologize. Secondly, in this experiment there is no doubt that ChatGPT passed the Turing test, as the two AI-generated limericks are indistinguishable in quality from the human-composed verse—all three limericks being equally terrible. But thirdly and most significantly, the software failed the Lovelace test because, although it can be seen to have “originated” something, namely two brand-new limericks, it only did exactly what it was instructed to do. And if it had been told to write a good limerick, the chances are that it would have failed entirely. It is thus clear from this very unscientific experiment that this particular piece of AI software still has some way to go before it can be said to have satisfied the Lovelace test.

At the beginning of her famous “Note G,” while warning of “exaggerated ideas that might arise as to the powers of the Analytical Engine,” Lovelace noted

a tendency, first, to overrate what we find to be already interesting or remarkable; and, secondly, by a sort of natural reaction, to undervalue the true state of the case, when we do discover that our notions have surpassed those that were really tenable Men16, p. 94.

It is ironic that she was in fact warning against attributing too much (or too little) to the machine with respect to its abilities, when that is exactly the fate that subsequently befell perceptions of her own mathematical skill. Let us hope that this article goes some way toward setting the record straight.