Ada Lovelace & The First Algorithm

“The Analytical Engine has no power of originating anything. It can only do what we know how to order it to perform.”

London, July 1843. A woman sits at a writing desk in her home on St. James’s Square, surrounded by papers covered in mathematical notation, cross-references, corrections, and marginal observations written in a hand that has grown increasingly hurried over the preceding weeks. The papers are the working drafts of a translation — a rendering into English of an Italian paper describing a remarkable machine — but the translation itself has long since been buried under an avalanche of additions that have taken on a life of their own.

The woman is Ada Lovelace, Countess of Lovelace, twenty-seven years old, mother of three, daughter of Byron, and the only person in the world who fully understands — perhaps more fully even than its inventor — what the Analytical Engine of Charles Babbage could become.

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What We Think We Know — and What We Don’t

Ada Lovelace is one of the most celebrated figures in the history of computing. Her profile in this series has told the broad story of her life — her unusual education, her partnership with Babbage, her rediscovery after a century of neglect. Her image is on the British fifty-pound note. There is a programming language named after her. Ada Lovelace Day is celebrated annually.

This article is an attempt to remedy that. We are going to go into the mathematics — not at the level of a technical paper, but deeply enough to understand what Ada Lovelace actually did, why it was remarkable, and what it means for our understanding of computation, programming, and the nature of intelligence.

The journey requires us to understand four things: what the Analytical Engine was and how it worked; what the Bernoulli numbers are and why computing them was a worthy challenge; what Ada’s algorithm actually said, step by step; and what the broader philosophical ideas in the Notes meant — and still mean.

Let us begin at the beginning.

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The Analytical Engine: A Computer Before Computers

To understand Ada Lovelace’s algorithm, you first need to understand the machine it was written for. And to understand the machine, you need to understand the problem it was designed to solve — and the gap between that problem and the far more extraordinary thing the machine turned out to be.

Charles Babbage spent his life in a state of productive fury at the errors in mathematical tables. In the early 19th century, mathematical tables — tables of logarithms, trigonometric functions, astronomical data — were the essential infrastructure of navigation, engineering, and science. A ship’s navigator used logarithm tables to calculate positions. An artillery officer used trigonometric tables to calculate trajectories. An insurance actuary used mortality tables to calculate premiums. The accuracy of these tables was, quite literally, a matter of life and death.

The tables were calculated by human computers — teams of men who worked through the calculations by hand, following prescribed methods, checking each other’s work. And they contained errors. Not because the computers were careless or unintelligent, but because calculation by hand, sustained over hours and days and weeks, inevitably produced mistakes.

The Difference Engine was never completed, for reasons we have already discussed. But as Babbage wrestled with its construction through the 1820s and 1830s, his thinking evolved. He began to see that the Difference Engine’s limitation — it could only compute by finite differences, only perform the specific sequence of operations built into its mechanism — was not a necessary feature of a calculating machine but an arbitrary constraint. What if the operations to be performed were not fixed in the machine’s construction but could be changed — specified separately, as a kind of instruction to the machine?

This thought led Babbage to the Analytical Engine.

The Analytical Engine also incorporated something that the Difference Engine had not: the ability to branch. Based on the result of a comparison — is this number greater than that one? Is this value zero? — the machine could select different sequences of operation cards to execute.

Babbage never built the Analytical Engine. The engineering challenges were immense, the funding inadequate, and Babbage’s own temperament — brilliant but restless, always moving to the next refinement before the current design was complete — made the sustained focus required for a project of this scale almost impossible for him to maintain. Attempts by his son Henry Babbage to build portions of it after his death were partial and unsatisfying.

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The Italian Paper and the Translation That Became Something Else

The chain of events that produced Ada Lovelace’s Notes began in 1840, when Babbage visited Turin and gave a series of lectures on the Analytical Engine to an audience that included several of Italy’s leading mathematicians and engineers.

Among the audience was Luigi Federico Menabrea, a young military engineer who would later become Prime Minister of Italy. Menabrea was impressed by Babbage’s lectures and wrote them up as a paper — “Notions sur la Machine Analytique de M. Charles Babbage” — published in a Swiss scientific journal in French in 1842.

The paper was a competent summary of what Babbage had presented, explaining the Analytical Engine’s basic architecture, the role of the operation and variable cards, and some illustrative examples of how it could be programmed to perform specific calculations. It was not, by any means, a complete or technically deep treatment. Menabrea was reporting what he had heard, not developing the ideas further.

In 1843, Charles Wheatstone — the physicist and inventor who had developed the electric telegraph — suggested to Ada Lovelace that she translate Menabrea’s paper into English. Ada was a natural choice: she was fluent in French, she had been corresponding with Babbage about the Analytical Engine for a decade, and she understood it more deeply than almost anyone outside Babbage’s immediate circle.

She undertook the translation. And somewhere in the process of rendering Menabrea’s French into English — a process that forced her to engage with every sentence, to understand what it was saying well enough to say it differently — she began to see the gaps. Things Menabrea had not explained. Implications he had not drawn out. Aspects of the Engine that he had not addressed, either because he did not fully understand them or because his paper was not intended to be comprehensive.

She decided to add notes. She wrote to Babbage explaining her plan, and he enthusiastically supported it — suggesting additional topics, providing technical information, reviewing her drafts.

The notes grew. By the time Ada was finished, they were nearly three times the length of Menabrea’s paper. There were seven of them, labeled A through G. Each addressed a different aspect of the Analytical Engine or its implications. Together, they constituted something quite different from an annotated translation. They were an original work.

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Notes A Through F: The Foundations

Before we reach Note G — the algorithm — it is worth surveying the earlier notes, because they establish the conceptual foundation on which the algorithm rests and contain ideas that are important in their own right.

Note B discusses the mechanical arrangement of the Engine in more detail, explaining the Store and the Mill and how numbers moved between them. This note is more straightforwardly technical than the others — it is explaining the hardware, in modern terms — but it contains careful work clarifying points that Menabrea had left ambiguous.

Note C introduces one of the most important concepts in the Notes: the distinction between different kinds of operations, and the way the Engine handles operations that involve different numbers of steps. Lovelace discusses what happens when an operation produces a result that is larger than the Store location allocated for it — the equivalent of an overflow condition in modern computing.

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The Bernoulli Numbers: What Note G Was Actually Trying to Compute

Now we arrive at Note G — the algorithm. To understand it, we first need to understand what it was computing: the Bernoulli numbers.

Computing Bernoulli numbers by hand is tedious and error-prone. The standard method, deriving from Bernoulli’s original work, requires solving a system of equations in which each Bernoulli number is expressed in terms of all the previous ones. To compute the eighth Bernoulli number, you need the first seven. To compute the tenth, you need the first nine. The calculation grows more laborious with each successive term.

The recurrence relation that defines the Bernoulli numbers — the formula expressing each one in terms of its predecessors — can be written in several equivalent ways. The form that Ada used in her algorithm was based on a formula that expressed each Bernoulli number as a sum involving all the preceding ones, weighted by binomial coefficients. The formula is:

This formula has a specific structure that is important for understanding Ada’s algorithm: to compute Bₙ, you need all of B₀, B₁, …, Bₙ₋₁. This means the computation has a natural sequential structure — you compute the Bernoulli numbers in order, using the ones you have already computed to produce the next one. The algorithm needs to manage this accumulation of results, keeping track of which ones have been computed and making them available for subsequent calculations.

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Note G: The Algorithm, Step by Step

Ada Lovelace’s Note G presents the algorithm for computing Bernoulli numbers in the form of a table — a “diagram of the development of the operations,” as she calls it. This table is the algorithm. Understanding it requires understanding how she is representing computation.

The algorithm proceeds in several phases.

Phase 1: Initialisation

The algorithm begins by loading the initial values needed for the computation into specific Store locations. These include the value of n — the index of the Bernoulli number to be computed — and the values of the previously computed Bernoulli numbers that will be needed.

Phase 2: Computing the binomial coefficients

For each term in the sum, the algorithm needs the appropriate binomial coefficient — the combinatorial number C(n+1, k). Rather than looking these up from a table, the algorithm computes them on the fly using the recurrence relation that connects consecutive binomial coefficients: C(n+1, k) = C(n+1, k−1) × (n+1−k) / k.

Phase 3: The main loop

The heart of the algorithm is a loop that iterates over the terms of the sum, accumulating the result.

Phase 4: Computing the final result

Once the loop has processed all the terms in the sum, the algorithm computes the final result from the accumulated total, applying the factor of −1/(n+1) that appears in the formula.

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What Made the Algorithm Remarkable

It is easy, reading Ada’s algorithm from the perspective of modern computing, to underestimate how remarkable it was. We live in a world saturated with algorithms. They are everywhere, running on every device, performing computations of unimaginable complexity billions of times per second. The concept of an algorithm — a precise, step-by-step procedure for solving a problem, written in a form that can be mechanically executed — is so familiar that it barely seems like a concept at all.

There had been algorithms before Ada Lovelace, of course. Euclid’s algorithm for computing greatest common divisors is more than two thousand years old. The long division algorithm taught in schools has been around for centuries. But these were algorithms in the informal sense — descriptions, in words or symbols, of mathematical procedures intended to guide a human calculator.

The algorithm also demonstrated, for the first time, the power of the loop structure — the ability to repeat a sequence of operations a variable number of times, depending on the data. This is not a trivial capability. An algorithm without loops can only perform a fixed number of operations, predetermined at the time of writing. An algorithm with loops can perform computations of any length, scaling to the complexity of the input.

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The Dispute Over Credit

No account of Ada Lovelace’s algorithm can avoid the question of credit — specifically, the question of how much of the algorithm was Ada’s and how much was Babbage’s.

The case for Babbage’s substantial contribution rests primarily on a passage in his autobiography, Passages from the Life of a Philosopher (1864), in which he writes that he provided Ada with “the indication of the method” for computing Bernoulli numbers, and that she “worked the details.” Some historians have interpreted this as meaning that Babbage supplied the key idea and Ada merely carried it out.

The case for Ada’s primary authorship rests on the surviving correspondence, which tells a more complex story. The letters show Ada working through the algorithm independently, identifying errors, correcting them, and challenging Babbage’s suggestions. In one crucial letter, Ada points out an error in Babbage’s version of the algorithm — an error in the arrangement of the operations that would have produced an incorrect result — and presents her corrected version. She does not defer to Babbage on the technical details. She argues with him, and she is often right.

What seems most accurate, and most intellectually honest, is to describe the algorithm as Ada’s while acknowledging that it was produced in close collaboration with Babbage and that the boundary between Ada’s contributions and Babbage’s contributions cannot always be precisely drawn from the surviving evidence. This is, in fact, how most collaborative intellectual work operates — through a dialogue in which ideas are proposed, challenged, refined, and eventually owned by the work itself rather than cleanly divisible between the contributors.

What is not in dispute is that the Notes as published — the translation, the seven annotations, the algorithm in Note G — were presented under Ada’s initials and represented her intellectual work. She was not a transcriptionist for Babbage. She was an independent thinker who used her collaboration with Babbage as raw material for her own analysis and synthesis.

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Lady Lovelace’s Objection: The Argument Against Herself

Among all the ideas in the Notes, none has been more debated than a passage in Note G in which Ada appears to argue against the very significance of what she is demonstrating.

This statement seems, at first reading, to undercut the significance of the algorithm she has just written. If the Engine can only do what it is told to do — if it has no capacity for genuine origination — then isn’t it, at bottom, just a sophisticated tool? A very fast, very accurate calculator, but not anything that could be described as intelligent or creative?

This reading of the passage has been influential. Turing used it in his 1950 paper as a formulation of the principal objection to machine intelligence — the objection that machines are fundamentally limited by their programming, that they cannot transcend their instructions, and that this limitation puts genuine intelligence permanently beyond their reach.

But the passage deserves more careful reading than this standard interpretation allows.

What she could not have known — because the technology and the theory had not yet been developed — was whether future machines, built on different principles, might be capable of something that would meaningfully be called origination. Machines that learned. Machines that modified their own behaviour based on experience. Machines that developed internal representations not specified by their designers.

The deep irony is that Ada herself, elsewhere in the Notes, anticipated something like this possibility. Her discussion in Note F of the Engine operating on musical notes — of computing as general symbolic manipulation rather than arithmetic — was gesturing toward a conception of computation that went beyond what the Analytical Engine could do. She was imagining, even as she accurately described the Engine’s limitations, a broader conception of what computation might become.

She was arguing against herself — and both sides of the argument were right, in different ways.

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The Algorithm as Art

There is something that the technical account of Note G — the operations, the Store locations, the loop structure — necessarily leaves out: the quality of the work as a piece of thinking.

Ada’s algorithm has this quality. It is not the simplest possible algorithm for computing Bernoulli numbers — you could write one that was more straightforward in some ways but would require more Store locations or more operations. Her version finds a path through the computation that is both correct and efficient, that manages the Engine’s resources thoughtfully, that handles the variable repetition of the loop cleanly.

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What Ada Got Wrong — and Why It Doesn’t Matter

In the spirit of intellectual honesty that Ada Lovelace herself exemplified, it is worth acknowledging that the algorithm in Note G contains an error.

Various explanations for the error have been proposed. Some historians believe it was a typographical error — a mistake in the transcription of the algorithm from working notes to the published version, rather than an error in Ada’s understanding of the computation. Others believe it was a genuine oversight in the algorithm’s design. The surviving correspondence does not definitively settle the question.

What is clear is that the error was in a specific implementation detail, not in the overall structure or approach of the algorithm. The fundamental ideas — the loop structure, the management of Store locations, the computation of binomial coefficients on the fly, the accumulation of the running sum — are all correct. The algorithm as a whole represents a valid approach to computing Bernoulli numbers on the Analytical Engine. The error is in one step of the execution, not in the conceptual architecture.

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The Hundred-Year Gap: Why the Notes Were Lost

Ada Lovelace published her Notes in 1843. For the next hundred years, they had almost no influence on the development of computing.

The reasons for this obscurity are multiple. The Notes were published in an obscure scientific journal. They were attached to a translation of a paper about a machine that was never built — a machine that became, increasingly, an eccentric historical footnote rather than a live engineering project. Babbage died in 1871, his great project unrealised, and interest in the Analytical Engine faded with him.

Ada herself died in 1852, at thirty-six, before she could write the larger work on the foundations of mathematics and computation that she had mentioned to Babbage as a future ambition. Had she lived another twenty years, she might have produced work that could not be ignored — work that would have kept her ideas alive and in circulation.

As it was, the Notes survived in libraries, available to anyone who looked for them but sought by almost nobody. The handful of historians who knew about them treated them as curiosities — impressive but disconnected from the mainstream of mathematical and scientific progress. The idea that they were foundational documents in the history of computing could not have occurred to anyone before computing existed.

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The Living Legacy: From the Notes to Modern AI

The direct line from Ada Lovelace’s Notes to modern AI runs through several generations of computing technology, each building on the one before.

The stored-program computers of the late 1940s — the Manchester Baby, EDVAC, UNIVAC — implemented, in electronic form, the logical architecture that Babbage had designed and Ada had programmed. The concept of a program stored in memory alongside the data it operated on was the enabling innovation. Without stored programs, computers were fixed calculators. With them, they became general-purpose machines that could run any algorithm their programmers could write.

The development of programming languages — from assembly language to FORTRAN to LISP to C to Python — was a progressive elaboration of the ideas that Ada had introduced in the Notes. Each new language made it easier to express complex algorithms clearly, to manage data structures, to organise the flow of execution. The discipline of programming that Ada pioneered became a mature engineering practice, then a creative medium, then the infrastructure of civilisation.

The concept of the algorithm — the precise, executable procedure for solving a problem — that Ada’s Note G demonstrated for the first time became the central concept of computer science.

And the broader vision of Note F — computation as general symbolic manipulation, applicable to any domain that could be formalised — became the foundation of AI. When Newell and Simon wrote the Logic Theorist in 1955, they were implementing Ada’s vision: a machine that operated on symbols representing logical propositions, manipulating them according to formal rules to produce new symbolic results. When modern language models process text, they are operating on symbolic representations of language, transforming them according to learned statistical rules to produce new symbolic outputs. The specific mechanisms are unrecognisably different from anything Ada imagined. The conceptual foundation is the one she identified.

That is what genius looks like. Not the convenient genius of comfortable circumstances and receptive audiences, but the inconvenient genius of a mind so far ahead of its time that the world it is addressing does not yet exist.

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Further Reading

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