Reciprocal tables made many division problems into multiplication in a base-sixty system.
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NYU’s exhibition pamphlet transcribes YBC 10529 as a reciprocal table: whole numbers 56 through 71 appear with values around the tablet’s edge. It also notes that only some reciprocals in the displayed range are exact and records an erroneous approximate reciprocal. The evidence is a useful reminder that a mathematical table can be both a computational aid and a surviving record with mistakes.
The table connects sexagesimal place value with reciprocal-based calculation. It does not show that every Babylonian division followed one universal procedure. Decimal fractions, punctuation, and the choice of a units place belong to a modern reading of the inscription rather than to the tablet itself.
Plimpton 322 is an Old Babylonian numerical table whose purpose and construction remain debated.
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MAA Convergence describes Plimpton 322 as a small Old Babylonian tablet with sexagesimal entries, usually dated c. 1900–1600 BCE. Its left side is broken, so some entries must be reconstructed; the object was purchased from an antiquities dealer and its original provenance is unknown. These are material limits, not minor footnotes to the mathematics.
The entries invite comparison with right-triangle relationships and reciprocal-pair procedures, but the tablet’s purpose and original construction remain debated. It is therefore best read as a numerical pattern with several scholarly interpretations, rather than as a settled trigonometric table, school exercise, or proof of one known generating method.
The tablet is incomplete, lacks a known excavation provenance, and its intended use is debated.
The Rhind Mathematical Papyrus preserves a long sequence of worked arithmetic and geometry problems.
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The British Museum identifies EA10058 as part of the Rhind Mathematical Papyrus, a hieratic text dated about 1550 BCE and written by the scribe Ahmose. Its eighty-four problems include division and multiplication tables, fractions, areas, and volumes. The object record calls it probably a mathematics textbook used by scribes learning to solve particular problems through examples.
The papyrus reveals an organized practice of procedures, units, and worked cases. It is not a summary of all Egyptian mathematics, and the presence of a problem does not identify one practical use for it. Modern symbols can clarify a selected problem, while the papyrus itself uses hieratic writing and its own layout.
Archimedes used inscribed and circumscribed polygons to place numerical bounds on a circle’s circumference.
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MacTutor’s account of Archimedes’ Measurement of the Circle reports bounds for the ratio of circumference to diameter: greater than 3 10/71 and less than 3 1/7. It explains that the inequalities came from estimating the perimeters of regular polygons inscribed in and circumscribed around a circle, reaching ninety-six sides.
The argument illustrates proof by bounds: two computable perimeters enclose a quantity that is not measured directly. A modern polygon diagram makes the idea visible, but it is not Archimedes’ original diagram. The surviving account establishes the bounding result without settling every intermediate numerical technique used in the calculation.
Liu Hui’s 263 CE commentary explains and extends an older, anonymously compiled mathematical text.
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The Nine Chapters is an anonymous compilation with a disputed development history; MacTutor dates Liu Hui’s commentary to 263 CE. Its discussion describes Liu Hui giving explanations for procedures and using a method of exhaustion in geometric work. The date therefore belongs to the commentary, not to a single fixed date for the underlying Nine Chapters.
The commentary separates an algorithm from an argument about why it works. This historical record does not support a contrast in which Chinese mathematics is merely practical and Greek mathematics is uniquely deductive. A present-day counting-board diagram is an explanatory reconstruction, not direct visual evidence for the layout of an ancient board.
The Sunzi remainder problem gives a concrete procedure for a special system of congruences.
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MacTutor reports that the date and authorial identity of the Sunzi suanjing are uncertain, with scholarly proposals ranging from the third century to approximately the fifth. Its famous problem asks for a number leaving specified remainders when grouped by 3, 5, and 7; the least result is 23. The text also gives a procedure using coefficients 70, 21, and 15.
Modern congruence notation and the later name “Chinese remainder theorem” help describe the problem, but they are not the text’s own vocabulary. The surviving procedure handles a special case; the author’s biography, the work’s date, and a fully general theorem remain less certain than the numerical example.
The work’s date and its attribution to a historically identifiable Sun Zi remain uncertain.
The Aryabhatiya, completed in 499 CE, records a compact programme of astronomical and mathematical calculation.
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MacTutor dates the completion of Aryabhata’s Aryabhatiya to 499 CE from the author’s own age statement. It records a sine table and a circle rule equivalent to 62832/20000 = 3.1416. The text gives the result as an approximation, while the exact method by which that value was obtained is not explained in the source.
The stated numerical rule makes the approximation available for comparison with π, while the route by which Aryabhata obtained it remains open. Proposed polygon derivations are interpretations rather than established fact. The result shows a dated mathematical calculation without settling a direct borrowing or transmission claim from another tradition.
The approximation is documented; its derivation is not settled by the cited source.
Brahmagupta’s 628 treatise includes rules for zero, positive quantities, and debts.
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MacTutor dates Brahmagupta’s Brahmasphutasiddhanta to 628 CE and records its rules for zero and signed quantities. It states that subtracting a number from itself gives zero, and gives correct rules such as adding zero leaves a number unchanged and multiplication by zero yields zero. The work is a dated textual anchor, not a single origin story for zero.
The same source reports attempts to define division by zero that do not agree with modern arithmetic. They show an effort to extend a number system, rather than current definitions of division. This dated text is not a date for the Bakhshali manuscript, nor evidence that every feature of decimal notation began with one person or work.
Arabic translations made Euclid’s Elements available within new scholarly settings and textual traditions.
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MacTutor’s account of Hunayn ibn Ishaq states that the first Arabic translation of Euclid’s Elements was made by al-Hajjaj during the reign of Harun al-Rashid, and that Ishaq ibn Hunayn became known for an Arabic translation of the Elements. This supports a timeline node about named acts of translation rather than an arrow labeled with a civilization alone.
Translation is not neutral copying: texts acquire new languages, terms, readers, and sometimes revisions. These named translations document a history of Arabic Euclid texts without reducing every later Arabic mathematical work to a single route from Greek geometry. Specific manuscript relationships remain questions for manuscript-level scholarship.
Ibn al-Haytham joined geometric reasoning and investigation of vision in a major work on optics.
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MacTutor describes Ibn al-Haytham’s Kitab al-Manazir as a seven-volume work on optics, with geometry and number theory among his mathematical interests. It reports a programme of criticizing premises and exercising caution in conclusions. In Book V, geometrical arguments address reflection from spherical, cylindrical, and conical surfaces.
A. I. Sabra distinguishes the medieval Latin translation, known in manuscripts as Perspectiva or De aspectibus, from Friedrich Risner’s Basel edition of 1572. Opticae thesaurus is the collective title of that printed volume, not the medieval Latin title of Ibn al-Haytham’s seven books. The translator and exact date of the medieval translation remain unascertained.
David E. Joyce · Clark University. Translated by David E. Joyce.
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Linked source; no third-party media reproduced. Atlas diagrams are original explanatory reconstructions. Consult the publisher or holding institution before reusing source images.
Linked source; no third-party media reproduced. Atlas diagrams are original explanatory reconstructions. Consult the publisher or holding institution before reusing source images.
J. J. O’Connor and E. F. Robertson · University of St Andrews. 2003.
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Linked source; no third-party media reproduced. Atlas diagrams are original explanatory reconstructions. Consult the publisher or holding institution before reusing source images.
J. J. O’Connor and E. F. Robertson · University of St Andrews.
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Linked source; no third-party media reproduced. Atlas diagrams are original explanatory reconstructions. Consult the publisher or holding institution before reusing source images.
University of Oxford Gardens, Libraries and Museums. 2017.
Bodleian Libraries
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Linked source; no third-party media reproduced. Atlas diagrams are original explanatory reconstructions. Consult the publisher or holding institution before reusing source images.
David Chivall, Virginia Lladó-Buisán, David Howell and Gillian Evison. 2024. Oxford University Research Archive.
Bodleian Libraries
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Linked source; no third-party media reproduced. Atlas diagrams are original explanatory reconstructions. Consult the publisher or holding institution before reusing source images.
Linked source; no third-party media reproduced. Atlas diagrams are original explanatory reconstructions. Consult the publisher or holding institution before reusing source images.
Linked source; no third-party media reproduced. Atlas diagrams are original explanatory reconstructions. Consult the publisher or holding institution before reusing source images.
A. I. Sabra. Archive for History of Exact Sciences.
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Linked source; no third-party media reproduced. Use original Atlas diagrams only; consult the holding institution before reusing images or scans.
Accessed 2026-09-20.
Diagrams and teaching explanations are original Atlas reconstructions unless otherwise identified. Source checking and automated tests do not imply a recorded human historical review.