1. When the Cords Rotted

In the winter of 1983–84, archaeologists excavating Han tombs at Zhangjiashan, near Jiangling in Hubei, recovered a group of bamboo manuscripts from tomb 247. Dated material places the burial around 186 BCE, although that should be understood as an archaeological inference rather than a surviving certificate recording the instant the tomb was sealed.

Among the manuscripts was a book of calculation. It had once been a physical volume: about two hundred narrow bamboo strips, roughly thirty centimeters long and only several millimeters wide. Brush-written columns ran down the strips. A procedure might continue across several pieces, held in sequence by binding cords.

The cords had decayed. The bamboo survived in varying states of completeness, but the book's continuous reading order no longer arrived automatically with it. The result introduces a different material problem from those encountered earlier in this series. The words may remain while their arrangement requires reconstruction.

Editors compare lengths, bamboo nodes, binding traces, handwriting, ink, broken edges, and the way sentences and technical instructions can continue. Terms designating operations or quantities help establish possible joins. Move one strip into the wrong place and a procedure can appear to prescribe something quite different.

The difficulty is not solved entirely by mathematics. A historically surprising or erroneous operation cannot simply be corrected out of existence because a modern reader dislikes it. Yet semantic coherence inevitably enters the reconstruction. Knowing how a procedure could work helps determine how its surviving instructions may have been ordered.

That is a controlled interpretive circle, not a license to arrange the evidence at will. Physical clues constrain the possibilities, and independent forms of evidence can confirm or challenge a proposed sequence. The final order is an argued conclusion, not just something seen when the tomb was opened.

More than a decade separated excavation from the publication of a full transcription in 2000 and an annotated edition in 2001. Conservation, photography, paleographical reading, identification of hands, repeated arrangement, and comparison with other finds all contributed. The delay reflects a complex task, not simply slow reading.

The manuscript's own title, Suan shu shu—here understood as Writings on Calculation—is written on the back of the strip numbered six in the modern edition. That wording matters. The numeral identifies an editorial arrangement; it need not tell us that an ancient owner deliberately put the title on “the sixth page” in the modern sense.

A title on a roll's exterior could help someone identify its contents without opening it. For modern editors, it provides important evidence that the material formed a named textual object. It does not by itself prove the membership or order of every disputed fragment.

Several strips also carry the surnames Wang and Yang and notices conventionally read as saying that one or the other had checked the text. Their exact roles—copyists, revisers, owners, or participants in another arrangement—remain debated. The marks nevertheless attest to intervention in a written mathematical collection, not merely to isolated answers.

The early Han witness thus establishes a material tradition of reusable calculation procedures and textual checking. It does not need to become a precisely dated autograph to be important.

Clay, wax, parchment, and bamboo offer different failures of survival. A tablet can remain after a surface is smoothed; wax can lose its marks; scraped parchment can retain recoverable ink; a bound bamboo book can lose its cords. None leaves a self-explanatory record. In the bamboo case, the problem is especially visible: recovering a text includes recovering a defensible order in which it can mean something.

2. Purchased and Excavated

Other early calculation manuscripts enlarge the picture, but their evidential status is not identical. Yuelu Academy acquired a large group of Qin bamboo strips through the Hong Kong antiquities market in late 2007, followed by additional donated material in 2008. Among them is a work called Shu, Numbers or Calculation, published in 2011.

Peking University received another group in 2010. Its institutional description lists 762 bamboo strips together with wooden writing objects, a die, and sixty-one counting rods. Many strips bear writing on both sides. The Tsinghua collection acquired in 2008 includes twenty-one strips forming an elaborate decimal multiplication table.

These are important materials, but they are not controlled archaeological excavations. Authenticity and provenance are separate questions. Bamboo, ink, script, language, internal relationships, and scientific examination may support judgments about age and genuineness. They cannot recreate a lost tomb number, stratigraphic position, or original association among objects.

An authentic text can therefore support close mathematical analysis while remaining weaker evidence for its owner's occupation or the composition of a burial. Philology cannot retrospectively supply an excavation record. The appropriate response is differentiated confidence, not either indiscriminate acceptance or refusal to study the text.

The Peking material preserves some unusual physical information because parts arrived still rolled or grouped. Such arrangements can illuminate how strips were bound, how writing continued across surfaces, or how different texts shared a roll. The counting rods are especially valuable because they are tools rather than verbal descriptions of tools.

Even here, observations made during conservation cannot fully replace observations in situ. A market-acquired assemblage may preserve relationships, but the route by which its objects were combined or separated remains a relevant uncertainty.

Controlled finds offer a different anchor. Tomb 11 at Shuihudi, excavated in 1975, contained over a thousand Qin strips and fragments associated with a man named Xi. A chronicle records both political events and details of his life: his birth in 262 BCE, registration for service in 246, later scribal appointments, legal work, and military service.

Before Xi administered official records, an official record had already claimed him. That small biographical sequence connects literacy and calculation with a state that registered, assigned, measured, and audited people as well as goods.

The statutes concerning verification address discrepancies in stores and measuring equipment. They assign penalties to responsible officers when balances or capacity measures fail specified tolerances. The source essay's examples include penalties assessed in armor or shields according to the scale of an error. These are not mathematical textbook exercises. They establish the administrative reality in which accurate measures and accounts had consequences.

Liye, excavated from an ancient well in 2002, supplies a complementary archive: tens of thousands of Qin documents associated with the county administration of Qianling. Multiplication material occurs alongside working accounts and official records. Its importance is not the recovery of another complete mathematical canon but the embedding of calculation within an administrative setting.

The excavated book and the excavated accounts do not become the same object. They come from different sites and need to be connected through careful comparison. Together they support a stronger account of relevant practices than a textbook's practical vocabulary alone could provide.

Preservation imposes another selection. Bamboo and wood decay readily in ordinary oxygenated environments. Waterlogged, sealed, oxygen-poor tombs and wells can preserve them; extreme dryness can preserve other wooden documents. Neither kind of site is a random sample of everyday writing.

We cannot infer the original distribution of mathematical activity directly from today's map of manuscript discoveries. Workshops, ordinary burials, surface buildings, and heavily used discarded books may have left almost nothing. Surviving concentrations partly describe environments suitable for organic preservation.

The same discipline applies to earlier essays. The Greek-speaking world was not devoid of mathematical papyri or ostraca merely because famous authors' originals are lost. Mesopotamian tablets, Chinese bamboo books, and medieval Greek copies are different kinds of samples produced by different histories of survival. They are not neutral measuring instruments for the amount of mathematics each society once performed.

3. The Procedure

A characteristic format links the excavated calculation books to the later Nine Chapters: a problem beginning with a situation, an answer, and a procedure. The first two concern a particular case. The third describes operations capable of application to a class of cases.

The Chinese term shu can designate such a procedure. It may instruct the reader where to place quantities, what to multiply, what becomes divisor or dividend, how to divide, reduce fractions, or continue an operation. Replacing the example's numbers should in principle leave a usable sequence.

That is the crucial difference between a result and a method. An answer settles this instance. A procedure provides a way to handle related instances. Entries in the early Han manuscript, including conversions between grain forms, already display this movement from a case to a reusable rule. It was not invented by Liu Hui in the third century CE.

The received Nine Chapters contains 246 problems. Its chapter names classify technical work in ways that overlap with, but do not duplicate, modern school subjects. Rectangular Fields has thirty-eight problems on areas and fractions. Millet and Rice has forty-six on conversions. Proportional Distribution has twenty.

Short Width has twenty-four, including extracting roots from given areas or volumes. Construction Consultations has twenty-eight problems concerning earthworks and solids. Fair Transport has twenty-eight concerning taxation, transport, and labor apportionment. Excess and Deficit has twenty, using the consequences of two trial assumptions. Fangcheng has eighteen, and Right-Angled Triangles has twenty-four.

Calling fangcheng “linear algebra” can reveal familiar structures but can also erase the original organization of work. It refers to arrangements of quantities and procedures carried out with rods, not simply to symbolic equations written on paper. Similarly, the final chapter covers more than the one theorem a modern reader might expect from its title.

Five chapters—fields, grain conversion, distribution, construction, and transport—contain 160 problems in explicitly practical settings. The remaining chapters also contain measuring, trading, digging, and building. The practical language is unmistakable.

But a story about moving grain is not necessarily a surviving record of grain actually moved. Numbers may be designed to make a method easy to display. Different situations can carry the same procedure. Some problems explore technically useful cases rather than report administrative incidents.

No reliable census separates the 246 into so many real cases and so many classroom inventions. The archive from Liye and the verification statutes show that officials performed related kinds of calculations. The Nine Chapters shows an edited body of methods suitable for such work. Functional connection is not documentary identity.

This is not a ranking of practical and theoretical mathematics. A deliberately constructed example can carry general knowledge more effectively than an unedited account. Its artificiality may be part of its mathematical function.

The book itself accumulated in layers. Institutions, place names, units, and vocabulary do not all belong to one moment. Liu Hui's preface associates the restoration and revision of older calculation material with Zhang Cang and Geng Shouchang after losses traditionally linked to the Qin destruction of books.

Other historical sources connect those men with technical and administrative activity. Their precise roles in producing the received Nine Chapters are less directly established. Liu Hui's account, written centuries later, is evidence for a tradition about earlier compilation, not a contemporary editorial ledger.

Qing scholars already noticed chronological difficulties. References associated with the Shanglin imperial park seemed too late for an attribution of the entire book to the early Han Zhang Cang. The objection does not exclude his participation in some earlier stage; it excludes treating one name as the explanation of every layer.

We should therefore distinguish the historical officials, Liu Hui's account of their relation to older material, and the complex received text. The book's authority does not require a single author or a single day of composition. Its procedures reached their canonical arrangement through a history that remains partly hidden.

4. Cutting Again

Liu Hui's commentary is conventionally dated to 263 CE, the fourth year of Jingyuan under the Cao Wei regime. His biographical record is much thinner than the significance of the work might lead a modern reader to expect. The commentary and the material later transmitted as the Sea Island Mathematical Classic are our principal sources for his mathematics.

Precise modern birth dates, family pedigrees, and detailed local biographies often exceed what the early evidence can sustain. An important work does not automatically produce an abundant personal archive.

Even the date of the commentary requires a textual distinction. The received material includes references to a bronze capacity vessel associated with the Jin arsenal, whereas 263 belongs to Wei. Qing editors noted the problem. Liu Hui may have revised material later, or another layer may have entered the transmitted commentary.

Thus “a commentary associated with Liu Hui in 263” is more secure than “every word now labeled Liu Hui was written in 263.” The date identifies an important historical anchor, not the completion date of every surviving sentence.

His preface explains his relation to an older book. He recalls studying it from childhood, returning to it, and annotating it according to what he could understand. A branching image presents apparently diverse problems as connected to common roots. Grasp a relation in one place and analogous cases can become intelligible.

He occupies the position of an interpreter. The procedures are already there; his task is to explain why they work and connect their underlying patterns. That role is neither passive repetition nor the wholesale replacement of a predecessor's book. The commentary can expose a procedure's reasons while preserving the inherited textual framework.

The circle-area discussion offers the best-known example. Begin with a regular hexagon inside a circle. Use right-triangle relations involving the radius and chord to double the sides: twelve, twenty-four, forty-eight, ninety-six, and 192. Finer polygons leave smaller uncovered regions.

The important work includes controlling the remainder. Liu Hui is not simply stopping when a drawing looks close enough. He relates the uncounted area to geometrical quantities so that approximation has an accountable error.

His famous language about cutting ever more finely, losing ever less, and eventually coinciding with the circle belongs inside that argument. Detached from its surroundings, it can sound like a philosophical aphorism about infinity. In place, it accompanies constructions, numerical work, and estimates.

Modern historians understandably speak of a limit idea. That phrase can illuminate the structure, provided it does not claim that Liu Hui wrote a nineteenth-century epsilon definition. Historical comparison should identify similarities without exchanging his words for a later formal theory.

The contrast with Archimedean demonstrations is instructive. The preceding essay emphasized finite stages that defeat any proposed positive difference. Liu Hui's language evokes the continuation of cutting to its ideal end. How that expression functions within the whole argument is a real interpretive problem, not something solved by assigning both texts the same modern label.

The numerical record also has layers. The familiar sequence reaches a 192-sided polygon and supports the useful approximation 157/50. More accurate values and computations involving further subdivisions occur in the transmitted commentary. Their attribution and dating intersect with the Jin-vessel passage and later editorial history.

It is therefore unsafe to state without qualification that in 263 Liu Hui personally carried out every high-sided calculation now printed under his name. The mathematical reconstruction may be coherent while the textual attribution remains uncertain.

What is clear is a repeated concern with explanation. Why multiply here? Why divide there? Why does cutting and recombining preserve an area? Those are not merely requests for a successful answer. They concern the grounds on which a general procedure deserves confidence.

5. Leaving the Doubt in Place

Circle cutting is not an isolated outbreak of reasoning in an otherwise mechanical commentary. In the construction chapter, Liu Hui asks why formulas for solids contain divisors such as two, three, or six. His explanations decompose bodies into related pieces and compare their volumes.

Traditional names such as qiandu, yangma, and bienao designate specific solids. Roughly, they include wedge-like triangular prisms, pyramid-like bodies on rectangular bases, and tetrahedral pieces. Translating the names into modern shapes is useful only if the actual configurations and conditions are retained.

The argument concerning yangma and bienao continues a subdivision into smaller similarly related pieces. A remaining discrepancy is transferred into a smaller configuration. As in the circle discussion, the diminishing remainder is part of the reason the desired relation can be secured.

The broader pattern combines dissection, rearrangement, corresponding sections, and control of what remains. Liu Hui is explaining why operations yield necessary relations, not merely displaying a few favorable numerical trials.

But the diagrams have not survived intact from his time. References to red, blue-green, and yellow solids point to a visual component that later readers no longer possessed. Qing editors explicitly acknowledged the absence of the old figures and reconstructed drawings from the commentary.

Many attractive modern illustrations are therefore scholarly reconstructions, not reproductions of a third-century original. An interpretation may be convincing, but its visual completion cannot silently become additional ancient evidence.

The sphere problem is especially revealing because Liu Hui does not defend the inherited rule at all costs. He recognizes a difficulty in the Nine Chapters' volume procedure and introduces the mouhe fanggai, a solid commonly reconstructed as the intersection of two perpendicular cylinders of equal radius.

At corresponding heights, its sections have a stable relation to those of the sphere. If the volume of that intersecting body can be established, the sphere's volume can be obtained. The route is promising, but he does not supply the missing determination.

The commentary leaves the doubt explicit. Rather than force an explanation from an appealing shape and risk departing from sound principles, he waits for someone able to give an adequate account. The often-cited phrase gan bu que yi conveys the obligation to preserve uncertainty rather than conceal it.

This is not merely a graceful apology for running out of time. It marks a lower threshold for acceptance. An answer, a plausible diagram, or a suggestive comparison is not enough when the reason remains unsatisfactory.

Archimedes also marks a boundary, but at a different point. He can find a result mechanically and then insists on a geometrical demonstration. Liu Hui leaves a stage unresolved because he lacks the adequate account. In both cases the boundary is stated by the author rather than imposed solely by a modern historian.

Later work associated with Zu Chongzhi and his son Zu Geng, also known as Zu Xuan in some naming traditions, supplied the sphere relation. Zu Chongzhi's life is documented much more substantially than Liu Hui's, conventionally placed in the fifth century, about 429–500. His Zhui shu became an official mathematical classic but is now lost.

Tang commentary preserves material from the Zu tradition. The crucial principle compares solids through sections at corresponding heights: if those areas agree throughout, their volumes cannot differ. Applied to appropriately related bodies, the sectional argument resolves the volume left open in Liu Hui's discussion.

This can be compared with later principles associated with Cavalieri without claiming direct transmission or identical conceptual foundations. Once again, a modern resemblance should not erase a historical procedure's own form.

The historical record also preserves Zu Chongzhi's celebrated bounds for the circle constant: 3.1415926 < π < 3.1415927, an interval of width 0.0000001, together with 355/113 and 22/7. The results are better attested than the sequence of operations used to obtain them.

Claims that he necessarily reached a polygon of one particular enormous side count are reconstructions. Continued polygon refinement is plausible in the surrounding tradition, but the lost book prevents the result from identifying one unique algorithm. We may possess a numerical achievement without possessing its workshop.

6. Thirty Students

The transmitted Nine Chapters contains more than an early canonical text and Liu Hui's third-century explanations. Under the Tang, Li Chunfeng and colleagues produced further commentary under imperial authority. Reading a modern page can mean hearing several periods at once: base text, Liu's annotations, and Tang subcommentary.

Li Chunfeng's work joined mathematics to astronomy, calendrical calculation, official historical writing, and court technical service. Calling him a university professor would replace that institutional setting with a much later one.

The group later known as the Ten Mathematical Classics entered official teaching and examination. It included the Zhoubi, Nine Chapters, Sea Island, Sunzi, Zhang Qiujian, Five Departments, Xiahou Yang, Mathematical Procedures of the Five Classics, Continuation of Ancient Mathematics, and Zu Chongzhi's now-lost Zhui shu.

Institutional documents give the mathematical school a quota of thirty students. They also specify substantial study periods: the Nine Chapters and Sea Island together could occupy three years; other groups had shorter allocations; difficult works such as the Continuation and Zhui shu required several years.

The number thirty is a capacity or establishment figure, not proof that exactly thirty candidates passed the mathematical examination every year. School places, entrants, examination candidates, and successful appointees are different quantities. Administrative changes across periods further complicate any simple annual total.

Mathematics was one technical category within the examination system, but formal inclusion did not make its status equal to the prestigious literary routes. Its professional destinations lay largely within technical, calendrical, and other bureaucratic service. It did not create an independent modern class of mathematics professors.

The name Five Departments itself keeps calculation visibly connected with official functions. A canonical curriculum and practical administrative needs were not mutually exclusive descriptions of the same body of knowledge.

The book's later survival required additional institutions. An official Song printing project around 1084 established an important edition of the mathematical classics. Further copying and printing followed. Southern Song records describe Bao Huanzhi obtaining a Nine Chapters copy from the family of Yang Zhongfu and transmitting it to the imperial collection.

The text did not then remain available everywhere as an unchanged complete book. By the Ming, older complete witnesses were difficult to find. The Yongle Encyclopedia preserved extensive material by excerpting it into a classified compilation.

Qing scholars, prominently Dai Zhen in the work surrounding the Siku quanshu, gathered and collated mathematical texts from that encyclopedia and other witnesses. Modern editions depend substantially on this reconstruction while also drawing on the surviving branches of the tradition. “Lost and recovered” is a useful outline, but not a claim that every single line came through one exclusive route.

The present text is consequently a layered object: early procedures, third-century commentary, Tang annotations, Song editions, Ming excerpts, Qing scholarship, and modern collation. Its history differs from the palimpsest's, but neither is a transparent tube carrying an untouched original.

Counting rods bring us closer to execution. Slender sticks placed on a surface represent units, tens, hundreds, and other positions. Alternating orientations can distinguish neighboring places. Fangcheng procedures arrange quantities in parallel columns and eliminate them through prescribed transformations.

Modern matrix notation can reproduce much of the mathematical structure. It does not preserve every hand movement, placement convention, or temporary configuration. A written procedure is not a film of the calculation.

When a text instructs the reader to set down quantities of grain, the practitioner may make many changes to a rod arrangement before reaching the next written instruction. The book records a rule and representative configurations, not every intermediate bodily action.

Commentarial traditions distinguish positive and negative rods by colors such as red and black. That textual prescription is evidence, but it does not prove that every Han calculation used two standardized colored sets. The surviving physical objects and the written convention answer different questions.

There is always a distance between a description of how to calculate and a complete record of someone calculating. Making that distance explicit preserves both the generality of the procedure and the material specificity of its execution.

7. Does It Count as Proof?

The question cannot be settled by a single unexamined definition. Liu Hui plainly supplies reasons for procedures. The dispute concerns what kinds of reasoning the historian should call proof, how they were organized, and how modern reconstruction should relate to the ancient text.

One approach broadens the history of proof beyond the theorem-and-deduction form familiar from the Elements. It asks how mathematical communities offered reasons sufficient to establish the correctness of a general claim or algorithm. Under this description, many of Liu Hui's explanations are straightforwardly part of proof history.

A second approach emphasizes structural rigor together with formal difference. It recognizes genuine arguments about areas, volumes, and right triangles while insisting that the organization of the Nine Chapters and its commentary is not simply Euclid with different notation. Comparison should reveal how authority is established in each practice, not erase the differences by translation.

A third approach places philological caution first. Modern historians may render an instruction as an algebraic formula, prove that formula elegantly, and then attribute the newly supplied proof to the ancient author. The result can be mathematically correct and historically unearned.

This warning does not imply that Chinese authors gave no demonstrations. It distinguishes their surviving reasoning from ours. A reconstruction can explain what would make a procedure valid without showing exactly which justification the author or reader possessed.

A narrower use of proof reserves the word for deductions from explicitly stated definitions, axioms, postulates, and prior propositions within a systematically organized chain. Much of Liu Hui's commentary does not take that literary form. Even under this definition, however, “not this type of proof” does not entail “no argument for correctness.”

These are analytical positions, not four sealed camps into which every scholar fits. Their value lies in the cautions they articulate. A broad definition can include too much if it ignores differences. A narrow definition can prejudge the history by making one tradition the admission standard for all others.

The strongest comparative question is not simply who first obtained a listed result. It is what someone had to do to make a mathematical claim acceptable, to whom the explanation was addressed, and what counted as an adequate reason.

That question does not require imagining Greece and China as contestants on a single track toward modern European science. It requires close attention to texts, diagrams, procedures, readers, and institutions.

Excavated manuscripts change the inquiry further. The Nine Chapters can no longer stand as the only source from which every early procedure supposedly descended. The Zhangjiashan material demonstrates other arrangements and earlier forms of calculation writing. Canonization is a historical achievement, not evidence that the canon was the origin of all its contents.

Karine Chemla and Guo Shuchun's critical work brings the base text, commentaries, terminology, and reconstructed figures into one sustained investigation. An important concern is how algorithms themselves become objects whose correctness can be demonstrated. Proof and procedure are therefore not rival kinds of mathematics; one may be directed at the other.

Definitions still matter. If proof requires a complete explicit axiomatic architecture, many canonical procedures and annotations will not qualify in that sense. If proof concerns publicly articulated reasons for the necessary correctness of a general operation, dissection, root-extraction explanations, and recursive volume arguments become central evidence.

But a dictionary cannot settle the remaining questions. What did the missing original diagrams show? What role did the language of cutting until no further division remains play? What unspoken assumptions supported a recursive reduction of the remainder? Which reasons would the intended readers regard as sufficient?

These are substantive historical issues even after everyone agrees to use a broad word. Some cannot receive an unqualified answer because the relevant material is gone. A modern diagram can make the reasoning intelligible without restoring the exact ancient visual object.

Terminology decides whether an activity enters the historian's category of proof. Analysis of the evidence determines what the activity actually consisted of. Confusing those two decisions makes both the mathematics and the history easier than they really are.

8. Turning the Question Around

The title Another Criterion needs to withstand its own examination. Does it imply that one civilization possessed algorithms while another possessed deduction? If so, the title would reproduce the contrast the evidence has just undermined.

Liu Hui gives arguments as well as procedures. Greek mathematics includes numerical methods, astronomical calculation, algorithms, and practical measurement as well as axiomatic exposition. Selecting one famous book from each archive can manufacture two pure traditions that never existed in that form.

The local claim is stronger because it is narrower. The Nine Chapters and its commentaries organize much of their work around procedures and their explanation. That describes a textual tradition. It does not reveal an eternal Chinese mode of mind.

The supposed opposition between algorithm and proof also fails on its own terms. An algorithm can be the object of a proof: why do these operations succeed for every case of the intended kind? Liu Hui's commentary repeatedly occupies precisely that relation.

A third objection concerns mathematics itself. The modern discipline is an analytical category historians use across periods. Early Chinese writers spoke of numbers, calculation, procedures, and particular classified arts. They did not necessarily pose the modern philosophical question of what counts as mathematics in our sense.

The series' concern with criteria is therefore retrospective. It should be identified as the author's way of organizing a history, not placed as a speech in Liu Hui's mouth. He need not have contemplated a universal definition of mathematics in order to distinguish an adequate explanation from one he distrusted.

Excavated texts provide a fourth challenge. Knowledge may have circulated through short procedures, tables, excerpts, working collections, administrative documents, and oral instruction. The later authority of the Nine Chapters does not mean every earlier calculation was progressing toward that canonical book.

The new evidence turns an apparent starting point into one stage in a longer process. We see components, alternative arrangements, and partly shared techniques. We still do not possess a complete construction diary explaining how every element entered the finished structure.

A fifth objection concerns the boundaries of the comparison. Intellectual communities are not naturally sealed inside civilization-sized containers. Specific periods, texts, networks, and institutions make better units of investigation than a presumed national or civilizational temperament.

The defensible conclusion is consequently modest but substantial. Materials from the Warring States through the Wei–Jin period disclose practices centered on compiling, executing, and explaining procedures. Their grounds of credibility can be described in their own terms and compared with particular Greek practices. Neither “algorithmic” nor “deductive” is the permanent essence of a people.

Several false comparisons then become easier to recognize. Define mathematics in advance as an Elements-style text and the Nine Chapters will appear mainly as an absence of axioms. Define it as a collection of executable general procedures and the Elements can be made to appear deficient in calculation. In each case the chosen ruler disguises itself as a neutral standard.

Compare the bare Chinese base text with Euclid while excluding Liu Hui's commentary, and the outcome has been rigged by the selection of layers. Translate ancient categories into algebra, geometry, proof, and algorithm without marking the translation, and modern boundaries return disguised as ancient facts.

Return to the scattered bamboo strips. The loss of cords did not erase every trace of order; binding marks, material features, and local relationships can survive. But sequence must be reconstructed. Mathematical intelligibility helps constrain that work, while physical evidence prevents intelligibility from becoming the only judge.

The circle is not fatal. Historical interpretation often moves between parts and wholes, testing each against independent evidence. A proposed order can be revised when it makes a join physically implausible or conflicts with a handwriting pattern. The goal is accountable inference, not an impossible view without assumptions.

A related discipline applies to Liu Hui. Before asking whether his argument is sufficient, we must identify what claim it addresses, what resources it uses, and what its readers could supply. Our categories guide the question, but they must remain open to correction by the text.

His decision to leave the sphere problem unresolved gives the essay its firmest endpoint. He did not define the boundaries of a modern discipline. He identified one place where he lacked the reason he required. That negative judgment tells us something concrete about mathematical trust.

The broader suggestion—that criteria become visible through the success, criticism, and comparison of practices—is this series' interpretation. It is not a universal law that ancient people never discussed their own standards. In this instance, one author's refusal to smooth over uncertainty is enough. Calculation continued, and the conditions under which a result deserved acceptance remained part of the work.