1. A bishop asks for an explanation

In 1734 George Berkeley published a short book attacking the most successful new instrument in mathematics. That same year he became Bishop of Cloyne in the Church of Ireland. Philosophers remember him for his account of perception and material existence; a Californian city would later take his name. Here, however, his subject was neither the furniture of the world nor the government of a diocese. It was the authority of an argument written on a mathematician's page.

The book was The Analyst. Its long subtitle asked whether the objects and inferences of modern analysis were really clearer than religious mysteries and articles of faith. It addressed an unnamed unbelieving mathematician, commonly identified as Edmond Halley. That identification remains an inference, not a name printed on the dedication. Newton, who had died in 1727, is a poor candidate. Berkeley himself distinguished the religious commitments of the great mathematical author from the irreligion of people who admired his mathematics.

A bishop attacking calculus may sound like the opening of a familiar play in which religion tries to stop science. This is a different play. Berkeley did not demand that calculations cease. He did not deny their extraordinary success. He asked why people who insisted on clear concepts and demonstrative reasoning when judging other people's beliefs were prepared to accept obscurities in their own.

The accusation deserves to be taken seriously. Some widely taught explanations did make a questionable move: an increment had to be nonzero for an operation to be legitimate, then seemed to be treated as zero when the desired answer was extracted. Berkeley had identified a real problem in the explanation. That is considerably less than proving that calculus was contradictory, but considerably more than expressing a cleric's dislike of mathematics.

It is also important not to make his opponents helpless by definition. They had answers, and some of those answers distinguished mature Newtonian arguments from careless presentations of them. Their disagreement concerned what the methods required and what counted as an adequate justification. Berkeley was not a mathematical innocent stumbling upon a mistake that every expert had missed. He was an informed philosophical critic reading mathematical reasoning without the confidence that successful daily use could give its practitioners.

By then the new analysis had been at work for roughly half a century. It had transformed questions about trajectories, planetary motion, pendulums, and the motion of fluids. Its applications had moved far beyond the few examples with which its inventors first introduced it. An objection to one apparently small transition in a derivation therefore put pressure on an enormous and productive practice.

Over the following decades the replies came from British mathematicians, from a French encyclopedist, and from a professor who wrote more than seven hundred pages on fluxions. Their proposals did not all mean the same thing. Some defended geometrical limits, some clarified the rules governing infinitesimals, and some tried to replace the troublesome objects with algebra. Nineteenth-century accounts of limits, continuity, and convergence would reorganize the problem again.

Calling the nineteenth century the final answer would be too neat. Later mathematics also found rigorous ways to work with infinitesimals. But the standard limit-based reconstruction made a decisive change: it no longer needed an increment to survive and disappear in the same sense. It specified a relation among finite quantities and made the relevant approximation precise.

The episode is not simply about a defective tool awaiting repair. It concerns the interval between reliable practice and an explanation of why that practice is reliable. That interval can be intellectually uncomfortable without making every calculation performed within it invalid. Berkeley's challenge made the discomfort unusually difficult to ignore.

2. Quantities that flow, quantities that differ

Average speed presents no obvious mystery. Divide the distance traveled during a minute by the duration of that minute. Instantaneous speed is harder. If one tries to repeat exactly the same operation over an interval of no duration, both the elapsed time and the distance traveled are zero. The quotient 0/0 supplies no answer. This does not prove that instantaneous speed is impossible; it shows that the ordinary average-speed recipe cannot simply be evaluated at a zero interval.

Newton approached the difficulty through quantities generated by motion. A moving point produces a line, a moving line an area, and other continuous processes produce other magnitudes. The generated quantity is a fluent; its rate of generation is a fluxion. The rate need not be an indivisible piece of matter or a smallest possible stretch of time. Newton repeatedly emphasized finite rates and continuous generation, not a world assembled from tiny, ultimate fragments.

Leibniz developed a different organizing language. Two variables are related; a change in one corresponds to a change in the other. Their differentials make those relations available for symbolic calculation. The question is not only how quickly a quantity grows, but how its change is linked to another change and how a general notation can carry that link from problem to problem.

What, then, is an infinitesimal difference? Leibniz's writings do not yield a single uncontested philosophical formula. Sometimes the quantities can be understood as useful ideal entities or fictions whose consequences admit a formulation in terms of ordinary assignable quantities. Historians disagree about how literally to read particular statements and how the different interpretations fit together. We should not settle that dispute merely by translating every differential into the derivative of a modern textbook.

The word “fiction” recalls the previous essay's imaginary quantities. In both settings it can designate a permission to calculate, not an order to expel the offending object. Yet the comparison is interpretive: it does not establish that complex quantities and infinitesimals had the same logical status, or that every practitioner gave the same justification for either.

Modern analysis allows us to translate much of Newton's and Leibniz's mathematics into a common framework. Historical translation is not historical identity. Their starting questions, favored representations, and means of teaching differed. Newton's early work connected quadrature, infinite series, rates of change, and the inverse recovery of changing quantities. Leibniz's first printed presentation, in 1684, offered a compact algorithm for tangents and maxima and minima, with a notation intended to travel across complicated expressions.

The notation has its own chronology. Leibniz's private papers place the emergence of the elongated summation sign, the ancestor of ∫, in October 1675; his differential notation took shape in the work of that period, including November. Publication came later. The differential method appeared in 1684, whereas the integral sign entered his published work in 1686. A dated notebook and a public article answer different historical questions.

The celebrated 1684 paper occupied only about seven journal pages, conventionally cited as pages 467–473. It was not a comprehensive foundational treatise. It introduced rules, demonstrated uses, and let readers see how a new kind of calculation might be carried out. Its compression was part of both its power and its difficulty. What circulated was an operative language before there was a universally accepted account of every object appearing in it.

There is nothing automatic about the relation between these two forms of success. A notation can be learned, taught, extended, and applied by people who disagree about its interpretation. Conversely, a cautious account of its foundations may persuade readers while being awkward for routine calculation. The people who make an algorithm productive need not be the same people who explain its legitimacy.

The previous essay followed unfamiliar objects entering mathematics because rules for using them proved fruitful. Here the scale changes. An entire family of methods acquired authority through what it could do, while its practitioners continued to debate what its central operations meant. The difference between Newton and Leibniz matters precisely because the later common language can so easily hide that unfinished work.

3. Why the Principia speaks geometry

Open the Principia of 1687 and you do not encounter a modern introduction to differential equations. You encounter geometrical quantities, ratios, similar figures, conic sections, and constructions. Near the beginning, Newton develops arguments involving first and ultimate ratios. Quantities can diminish while their ratios approach a determinate value. The point is not to preserve a final nonzero infinitesimal triangle after every finite triangle has disappeared.

A familiar story explains the book's shape by saying that Newton first solved everything using calculus and then translated the solutions into classical geometry so that his readers could understand them. There is something behind that story, but it cannot serve as a description of the entire book's composition.

Newton later presented analysis as a means of discovery and synthesis as a means of establishing the results securely, in the manner he attributed to the ancients. But those retrospective claims were made in the atmosphere of the priority controversy. They are evidence of his methodological self-presentation, not a proposition-by-proposition diary of what he had done in the 1680s. Drafts and related papers reveal several kinds of work: series, quadrature, fluxional techniques, and genuinely geometrical reasoning.

At least four considerations help explain his published choices.

First, proof had a hierarchy. A method that enabled one to find a result was not necessarily the method in which one ought to demonstrate it publicly. In the classical tradition as Newton reconstructed it, analysis could belong to the workshop and synthesis to the finished exposition. This was not simply a concession to slow readers. It expressed a view of what completed mathematical knowledge should look like.

The third essay asked what could count as a principle. Here the question is one level further along: what could count as the publicly acceptable demonstration of a proposition? A calculation might be indispensable in discovering a relation and still not supply the form of proof its discoverer regarded as authoritative.

Second, the physical meaning of the quantities mattered. Newton resisted making natural philosophy depend on indivisibles to which he could assign no satisfactory geometrical or natural existence. Continuous generation offered another account. A fluxion was a finite rate, and finite lines could represent the ratios required by the argument. His repeated explanations show that this was a live issue for him, not a difficulty invented by later historians.

Third, geometry belonged to the research problem itself. Newton wanted to expose relations between forces, orbits, areas, and motions. In central-force problems, a construction can reveal an invariant or a proportional structure directly. It need not be a clumsy disguise for an equation that would obviously have been better. Later analysts would place greater weight on general equations and uniform computational procedures, but those priorities should not be silently imposed on every choice made in 1687.

Fourth, publication did not contain the whole mathematical practice. Manuscripts, correspondence, and exchanges with disciples preserved techniques that were not displayed in the same way in the finished book. The boundary between private calculation and public demonstration affected what readers could learn and reproduce. It also affects historians: a printed theorem and a working sheet must be compared rather than assumed to tell identical stories.

The productive question is therefore not whether Newton secretly used calculus. It is what he regarded as sufficient to complete an argument in natural philosophy. Once that question is asked, the Principia ceases to look like an encrypted modern textbook. Its form becomes an intellectual choice, made within a particular understanding of evidence.

The comparison with Archimedes in the fourth essay is illuminating but limited. Archimedes distinguished a heuristic method from demonstrations that met another standard. Newton likewise distinguished discovery and justification in some of his accounts. Yet we should not imagine that every geometrical proposition in the Principia had a matching analytical proof sitting intact in a drawer. The surviving workshop is mixed. Its very mixture is part of the history.

4. A discovery becomes a dispute

The public war was not there from the beginning. Through Henry Oldenburg, Newton and Leibniz exchanged information in the 1670s. Newton described results concerning series and concealed parts of his methods in anagrams. Meanwhile, Leibniz's Paris papers show his own developing differential practice. His 1684 publication and Newton's relatively friendly acknowledgment in the first edition of the Principia do not read like the opening declarations of an inevitable feud.

Suspicions hardened later. Nicolas Fatio de Duillier entertained doubts in the early 1690s and published an insinuation in 1699 that Leibniz owed the method to Newton. A further escalation followed the appearance of Newton's De quadratura curvarum as an appendix to the Opticks in 1704. An anonymous review in the Acta Eruditorum, generally attributed to Leibniz, used wording that Newton's supporters took to imply borrowing in the opposite direction. The attribution should not erase the fact that the review appeared anonymously.

John Keill then made the accusation more explicit: Newton had invented the fluxional arithmetic first, and Leibniz had published it under another name and notation. Priority was no longer merely a question about the dates of particular results. It had become an allegation about honesty. Honor can be shared, but these participants increasingly treated concession as personal injury.

Their institutional positions were also unlike. Newton moved from Cambridge into the Mint and became president of the Royal Society. Leibniz worked as a court counselor, librarian, and historian of a dynasty, with obligations that competed with his mathematical ambitions. Describing the dispute as a quarrel between two university professors erases much of the social setting.

Leibniz complained formally to the Royal Society in 1711. A committee was appointed in March 1712. Its members were not simply a uniform collection of Newton's private dependents; several had genuine mathematical competence, and the membership was enlarged, eventually including a Prussian diplomat. Nevertheless, the Society's president was himself the central interested party.

The investigation concentrated on letters and papers, especially those associated with John Collins. Newton supplied and arranged substantial material. Leibniz had no equivalent role in editing the dossier and no comparable opportunity to explain each point before the committee. The resulting collection, the Commercium Epistolicum, appeared under the Society's authority. Its conclusions defended Newton's priority and rejected the injury alleged in Keill's statement.

Newton's involvement is not inferred solely from his presidency. Surviving drafts in his hand establish his work on the controversy and on an account of the collection published anonymously. The Newton Project's manuscript catalogue identifies him as author and locates the relevant papers in Cambridge. President, litigant, supplier of evidence, and drafter of a public interpretation were overlapping roles. This procedure did not provide the independence that we would now demand.

That does not establish that the age possessed no concept of impartial judgment. It establishes a failure of separation in this particular case. Nor did one sensational discovery suddenly reveal everything. De Morgan argued in 1852 for Newton's authorship of the anonymous account using textual and bibliographical evidence. The transfer of scientific papers to Cambridge in 1872, followed by systematic cataloguing and the 1888 catalogue, made manuscript evidence more accessible.

The conclusion of independent development is not an equitable compromise invented to calm two nations. Newton's papers document work in the 1660s. Leibniz's papers around 1675 document a distinct sequence of experiments, revisions, and changes in notation. Independent development rests on comparing those sequences, not on allocating half the honor to each man.

It would be equally inaccurate to claim that Leibniz never saw Newtonian material. His copies show that he did. The crucial question is what he saw, when he saw it, and whether it supplied the method whose emergence can be traced in his own notes. Contact is established; possession of a complete Newtonian calculus before his own development is not. The surviving evidence does not substantiate the plagiarism charge.

The equally familiar verdict that the quarrel put British mathematics to sleep for a century also requires restraint. The dispute helped nationalize differences of method and notation, and aggravated separation from Continental developments. But British work on series, differences, fluxions, and analysis continued. Newton's dot notation was not itself a prohibition on calculation. Institutional and linguistic isolation is a more plausible contributing explanation than a century-long intellectual coma.

Finally, this history depends on paper. Newton's enormous surviving archive was dispersed and reorganized over generations. Leibniz's papers comprise tens of thousands of items; his correspondence alone includes roughly fifteen thousand letters with around eleven hundred correspondents, as the German UNESCO Commission records. What survived, when it was catalogued, and which sheets an editor could consult changed the history that could be written. The judgment of priority was not merely argued again. Its evidential basis became larger.

5. The increment that disappears

Return now to Berkeley and to the small increment. To determine a rate at an instant, an exposition begins with a short but nonzero interval. The changing variable receives an increment. A corresponding difference is formed, and that difference is divided by the increment. So far, the assumption that the increment is nonzero is essential.

After simplification, some terms still contain it. The calculation then announces that the increment has vanished, discards those terms, and retains the desired result. Berkeley's question is precise: what licenses the change? If the increment was zero from the beginning, the division was illegitimate. If it was not zero, why can the surviving terms simply be treated as nothing?

A modern elementary example makes the issue visible without pretending to reproduce every detail of an eighteenth-century text. For the function x², the difference quotient is

((x + h)² − x²)/h = 2x + h, provided h ≠ 0.

The condition belongs to the equality. Setting h equal to zero in the original quotient is not allowed. The derivative is nevertheless 2x, because the values 2x + h approach 2x as the nonzero increment approaches zero. Taking that limit is a different operation from pretending that division by zero has become permissible. The modern explanation distinguishes the two operations explicitly.

Berkeley objected to presentations in which the needed distinction was missing or obscure. His attack was not merely a metaphysical question about whether very small objects existed. It was an objection to the apparent substitution of one premise for another inside a derivation. The argument required the increment to be something and subsequently appeared to require it to be nothing.

His famous phrase, “ghosts of departed quantities,” condensed the objection into an image. The quantities had performed indispensable work, then departed, leaving their effects behind. The joke had a theological edge. A critic of religious mysteries who accepted invisible and inadequately explained entities in calculation could hardly dismiss another discipline merely for containing difficult objects.

The original Analyst makes that comparative purpose plain. But a successful phrase should not be allowed to enlarge the scope of the demonstration. Berkeley had not shown that every formulation of calculus required an increment simultaneously to equal and not equal zero. Nor had he abolished the difference between an ultimate ratio and the ratio of two ultimate, vanished quantities.

Newton's defenders could point to precisely that difference. A ratio can tend to a finite value while both the numerator and denominator diminish. The claim concerns the limiting relation, not a final pair of nonzero fragments lurking at the end of an infinite process. Whether Newton's formulations met every later requirement is a separate question from whether they committed the crude contradiction Berkeley described.

The most vulnerable target was therefore a familiar pedagogical performance: carry out the algebra, dispose of the inconvenient terms, and offer the success of the answer as reassurance. Familiarity makes such a transition easy to overlook. A skilled practitioner can supply qualifications that a printed explanation omits; a beginner cannot be assumed to possess them.

This suggests a broader lesson, but it is a lesson rather than a universal historical law. A discipline may expose its weakest explanations in introductory teaching, where difficult qualifications have been removed to make a procedure usable. Its strongest practitioners might know why a shortcut works while the shortcut travels without that knowledge. An external critic can then discover a genuine gap without having refuted the most careful version of the theory.

The ghost is not the derivative. It is the missing account of the passage from one kind of statement to another. Once that distinction is held steady, Berkeley's criticism can be respected without converting a dispute over justification into the claim that a century of mathematics consisted of meaningless arithmetic.

6. What Berkeley wanted

The subtitle was not ornamental. Berkeley had long opposed freethinkers who contrasted the certainty of mathematics and natural philosophy with the supposed obscurity of Christian doctrine. His strategy was to turn their criterion back upon them. If clarity and stepwise demonstration were the entrance requirements for rational belief, the mathematicians should explain fluxions, vanishing increments, and infinitesimals by the same standard.

His proposed symmetry did not prove Christianity true. Unexplained success in mathematics cannot establish a theological proposition. The narrower point was that conceptual difficulty, by itself, could not be invoked consistently as a sufficient reason to reject religion while comparable difficulty was tolerated in an admired science.

This explains his persistent use of the vocabulary of faith and mystery. He placed the analyst's obscure objects beside the religious concepts the analyst's allies mocked. To turn Berkeley into a nineteenth-century analyst born early would lose this motive. To turn him into an anti-scientific cleric with no mathematical point would lose the objection that made the polemic worth answering.

The mathematical criticism and the religious purpose are not rival accounts of the book. Both can be true. A purpose we do not share cannot invalidate a well-aimed objection; a well-aimed objection cannot validate every purpose for which its author uses it. Intellectual history becomes flatter when either half is removed.

His position outside the community of routine calculation may have helped him notice what successful practice encouraged others to pass over. That is a plausible interpretation of the episode, not evidence that outsiders are generally clearer thinkers. He read the inferential transitions as transitions requiring justification, rather than trusting them because the technique had repeatedly produced the right result.

How far was he right? Older histories often treated his criticism with considerable respect, especially where an increment was first used as a divisor and then discarded without explanation. That assessment remains important. But it is not a license to identify every Leibnizian infinitesimal with an ordinary nonzero real quantity and then convict the system of violating that identification.

Leibniz had principles concerning continuity and the treatment of higher-order terms. His transcendental law of homogeneity governed the comparison of terms of the relevant order. Whether those principles supplied a complete foundation, and how they should be interpreted, remain questions for scholarship. They at least show that dropping terms was not necessarily intended as an arbitrary declaration that a quantity had become exactly zero.

Berkeley also offered his own explanation of the correct answers in terms of compensating errors. Those arguments are not uniformly satisfactory. A critic can detect an inadequate proof without correctly explaining why the result survives. The fact that his diagnosis had force does not make every part of his alternative account sound.

The most defensible conclusion is therefore bounded. He exposed genuine foundational questions and inadequate contemporary presentations. He did not prove an unavoidable contradiction in calculus, did not establish that Newton's and Leibniz's methods shared every defect, and did not provide the later theory of limits.

His intervention did stimulate substantial and immediate replies. It would overstate the case to say that he alone made foundational work unavoidable forever after: the concern predated him, and the development of analysis had many causes. Yet the pamphlet gave a public and memorable form to a demand that successful calculation could not by itself satisfy.

Over the next century, mathematicians pursued several different answers. The variety matters. Their disagreement was not simply about finding a better word for one tiny quantity. It concerned whether the subject should rest on motion, geometry, algebraic expansion, limits, or a disciplined calculus of ideal entities. Before returning to those answers, another mathematical tradition makes the question of what counts as calculus less easy to take for granted.

7. Another tradition, another measure

Move back some three centuries from Berkeley and turn toward southern India. The network now called the Kerala school connected teachers, families, commentators, and shared astronomical problems. “School” is a historian's convenient label, not evidence of a modern academy with a building and a membership list. Madhava, usually dated approximately to the late fourteenth and early fifteenth centuries, stands near the beginning of a tradition that continued through several generations.

Among its achievements were infinite series recognizable in modern notation as expansions for sine, cosine, and arctangent. A special value of the arctangent yields the alternating series for π/4: 1 − 1/3 + 1/5 − 1/7 + …. The slow convergence of such a series is not a minor practical inconvenience. These mathematicians also developed correction terms to improve what a finite calculation could deliver.

The dates require care. Later texts attribute central results to Madhava; that is documentary evidence for an attribution. It is not the same thing as possessing his autograph manuscript or a dated record of the moment each formula was discovered. The origin of a result, the composition of a surviving text, and the age of a particular manuscript copy may be different dates.

Nor do the texts merely announce remarkable numerical coincidences. Their purposes include astronomical computation, trigonometric tables, and accurate values related to the circle. The Yuktibhasa, composed in Malayalam by Jyeshthadeva in the first half of the sixteenth century, is particularly important because it supplies rationales. Its title itself emphasizes explanation.

The arguments use the geometry of arcs, division into many small parts, comparisons among those parts, and relations involving sums of powers. By refining the divisions, the writer arrives at expressions that a modern reader can recast as series. There are real chains of reasoning here. One cannot dismiss them as fortunate guesses merely because they do not appear in the form of a later European textbook.

Equally, a rationale of that kind is not already a nineteenth-century theory of real-number completeness, limits, and convergence. Some justifications of correction terms require qualifications when examined by modern standards. The same care must be applied symmetrically: seventeenth- and eighteenth-century European analysis did not begin with those later foundations either.

Does this work count as calculus? An affirmative answer can emphasize infinite processes, local change, summation, area reasoning, and the control of finite approximations. A more restrictive answer reserves “a calculus” for a general system of differentiation and integration, their systematic relationship, and algorithms applicable across wide classes of problems. A third response questions the framing: why judge Kerala astronomy principally by how near it came to a later European arrangement of subjects?

The question is not settled simply by pointing again to the series, whose existence is not the main disagreement. The classification depends substantially on the criterion, although gaps and uncertainties in the surviving record still matter. The issue resembles earlier questions in this series: whether the Elements inaugurates “axiomatization,” and whether particular Chinese arguments qualify as “proof.” Labels and analysis of the materials do different jobs.

Possible transmission to Europe is another question and requires another kind of evidence. Portuguese maritime networks, missionary activity, and practical demand for calendars and astronomical information created opportunities for contact in the sixteenth and seventeenth centuries. Opportunity, motive, and chronological possibility are relevant. They are not a documented chain of transmission for a specific series.

The evidence presently cited does not establish a continuous route from a particular Kerala text through a translation, letter, notebook, or identifiable European reader to the European discoveries. That does not prove such transmission impossible. It means that a possible route should not be narrated as a demonstrated event. A categorical claim that nobody outside Kerala knew anything of the results would go further than this evidential conclusion warrants.

A secure later point of contact is Charles Matthew Whish. His paper was read on 15 December 1832 and published in 1834 in the Transactions of the Royal Asiatic Society. The original paper identifies the texts he used and the date of presentation. An East India Company official, he recognized an extensive mathematical achievement and described it through the European language of fluxions.

His intervention did not immediately transform standard histories. Sustained publication, translation, and reassessment developed much later, especially from the mid-twentieth century. But “ignored” does not automatically mean deliberately suppressed. Linguistic access, difficult manuscripts, the lack of critical editions, disciplinary networks, and established narratives are all relevant considerations; their relative weight cannot be reduced to one undocumented motive.

The physical record matters too. Palm-leaf manuscripts survive through care and repeated copying, particularly in climates where insects, moisture, and wear threaten them. A text composed centuries earlier may be known through much later witnesses. Survival is an activity, not simply an object's ability to remain untouched.

The point of this detour is not to nominate a replacement winner in an invention contest. It is to keep mathematical achievement, documentary evidence, and later classification distinct. The Kerala materials contain substantial reasoning. Understanding that reasoning on its own terms is more demanding, and more illuminating, than awarding or withholding one retrospective name.

8. Did rigor change its meaning?

Mathematicians did not answer Berkeley with a century of silence. In 1734 James Jurin, writing as Philalethes Cantabrigiensis, published Geometry No Friend to Infidelity. The pamphlet exchanges continued. Benjamin Robins's more mathematical response in 1735 emphasized that mature Newtonian methods should be understood through first and ultimate ratios, not through mysterious actual infinitesimals. Defending Newton did not require accepting the version of his method that Berkeley attacked.

Colin Maclaurin's two-volume Treatise of Fluxions appeared in 1742. Its main text ran to more than seven hundred pages. Geometry, exhaustion, proportion, and arguments about limiting relations served a comprehensive treatment of analysis, not merely a point-by-point reply to a pamphlet. Maclaurin sought a defensible account based on finite magnitudes and controlled relations. His own introduction places the preservation of demonstrative certainty alongside the expanding usefulness of mathematics.

To defend an algorithm developed roughly seven decades earlier, he returned extensively to a geometrical language with ancient credentials. The contrast is striking, but it must not become the claim that calculus was already two centuries old in 1742. Its novelty and the antiquity of its preferred justificatory resources belonged to the same intellectual moment.

By the middle of the eighteenth century, d'Alembert was explicitly advocating limits as the foundation of differential calculation. The relevant object was the value approached by the ratio of finite differences, not an unexplained little entity. His program anticipated an important direction of nineteenth-century analysis, though it did not yet contain all the later precision of that subject.

Lagrange tried another route. His Théorie des fonctions analytiques of 1797 proposed to ground differential calculus in algebraic expansions. Expand a function after an increment has been introduced; take the coefficient of the first power of that increment as the derived function; continue with the higher coefficients. In the announced program, infinitesimals, vanishing quantities, limits, and fluxions could all be dispensed with.

The difficulty had moved rather than disappeared. Not every differentiable function, or even every infinitely differentiable function, is represented near a point by its Taylor series. Where a series is available, convergence and the behavior of remainders still require justification. Formal algebra alone cannot guarantee the analytical properties the program needs. Its foundational ambition failed in that generality, although its mathematical contributions were substantial.

Did “rigor” therefore change its meaning, allowing later generations to condemn earlier work by newly invented standards? Partly. Newton valued classical synthetic demonstration. Leibniz emphasized general rules for symbolic operations. Many eighteenth-century analysts placed considerable weight on formal generality and the fertility of calculation. Later analysis increasingly required explicit hypotheses concerning functions, limits, continuity, and convergence.

But the qualification is essential. The foundational worries were not manufactured retrospectively in the nineteenth century. Seventeenth-century critics had raised objections; Berkeley sharpened some of them; Jurin and Robins thought replies necessary; Maclaurin constructed a substantial defense; d'Alembert and Lagrange proposed different foundations. Their own disagreements are evidence that there was something to explain.

Two statements must be retained together. Later technical standards cannot be treated as a timeless court before which every earlier proof fails. Earlier mathematicians nevertheless recognized gaps and difficulties by the standards and questions available to them. Neither the story of universally confused ancestors nor the story of purely anachronistic accusations fits the record.

Meanwhile, analysis changed physics. Euler's Mechanica of 1736 organized the motion of particles analytically. Lagrange's Mécanique analytique of 1788 brought constraints, virtual displacements, differential equations, and variational methods into a powerful general framework. Mechanics increasingly proceeded through reusable analytical procedures instead of requiring a new geometrical construction for each problem.

Celestial mechanics made the change particularly consequential. The ideal two-body problem is not the full solar system. Higher-order treatment of lunar apsidal motion in the eighteenth century became important to evaluating gravitational theory. Laplace's five-volume Mécanique céleste, published from the late eighteenth century into the nineteenth, organized planetary and lunar perturbations, tides, the Earth's figure, and questions of long-term stability on a vast scale. This was not a complete proof of every stability claim. It was a new analytical undertaking, not Newton rewritten with different symbols.

Fluids, elasticity, and vibration generated further problems. Fourier's Théorie analytique de la chaleur appeared in 1822, drawing on work developed during his years as prefect of Isère. An earlier memoir had been submitted in 1807; his work received the Academy's prize in 1812, accompanied by reservations about generality and rigor. The use of trigonometric series to represent temperature distributions made questions about functions, convergence, termwise operations, and integration unavoidable in new forms.

Applications did not merely wait for foundations. They changed what foundations had to accomplish. The assertion that an “unrigorous calculus” supported two centuries of physics is too blunt. A family of methods, with disputed foundations and changing local standards of proof, developed faster than a unified account of its legitimacy.

Berkeley asked why the mathematician claimed greater clarity than the theologian. The subsequent century did not answer solely by philosophical argument. It multiplied the phenomena that analysis could explain and the difficult mathematical operations those explanations required. A moon's changing orbit, a vibrating string, a bending beam, and heat spreading through metal made the demand for justification both more urgent and more specific.

One way to interpret this history is that foundations are rebuilt not only when someone points to a crack, but when the structures resting on them become heavier. That is the essay's interpretive image, not a single-cause law of intellectual progress. The calculations continued. What exactly entitled them to count as knowledge remained a question under construction.