# References — Chapter ᚛(Prologue): 0ur starting point *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-2, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch02.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 2.1 {#r-2.1} The current Oxford dating of the Bakhshali manuscript places the sampled folios in the late eighth to early eleventh centuries, with the manuscript modeled at about 799–1102 AD; the earlier Folio 16 result was inaccurate and Folio 17 was excluded as an outlier. *Status: confirmed* - [Chivall, Lladó-Buisán, Howell and Evison, Radiocarbon dating of the Bakhshālī manuscript, University of Oxford (2024)](https://ora.ox.ac.uk/objects/uuid:5a6d1dd7-f20c-4209-adb6-33849f5b08f4) — and the report PDF downloaded and read: Table 1 gives the five folios at 95.4% confidence as 773–986 (15), 901–1032 (16), 890–1014 (23), 885–995 (33), and 774–991 (55) cal AD; the OxCal phase model gives 799–892 cal AD for the start of production and 900–1102 cal AD for the end; the second determination on Folio 16 “demonstrated that the initial determination was inaccurate” and Folio 17 was excluded as an outlier. - [Bakhshali manuscript, Wikipedia](https://en.wikipedia.org/wiki/Bakhshali_manuscript) — summarizes the 14 October 2024 revision. Its table prints Folio 16 as 931–1032; the Oxford report itself gives 901–1032, and the report is the primary source, so the chapter follows the report. It also gives the superseded 2017 dates: “224–383 CE for folio 16 (Indo-Scythian), 680–779 CE for folio 17 (Turk Shahis), and 885–993 CE for folio 33 (Saffarid dynasty).” ### 2.2 {#r-2.2} The Bakhshali manuscript is a 70-leaf birch-bark mathematical text found near Peshawar in 1881 and held by the Bodleian Library. *Status: confirmed* - [Bakhshali manuscript, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Bakhshali_manuscript/) — confirms the 1881 discovery, birch bark, approximately 70 surviving leaves, and presentation to the Bodleian in 1902. - [Bakhshali manuscript, Wikipedia](https://en.wikipedia.org/wiki/Bakhshali_manuscript) — confirms the 70-leaf description, discovery near Peshawar, and Bodleian location. ### 2.3 {#r-2.3} Babylonian sexagesimal notation used blank places and later two wedge marks as a placeholder; Ptolemy used a barred circle as a sexagesimal placeholder around 130–150 AD. The derivation of that sign from the first letter of οὐδέν is commonly supposed and has been doubted (Neugebauer dismissed it, since omicron already stood for 70); in ninth-century Byzantine manuscripts the sign is still a full-size uncial omicron with a macron, and the form alters markedly only in some late medieval manuscripts. The placeholder signs were never used alone. The Sumerians counted in base sixty in the third millennium BC, and the Babylonians inherited it. In the Seleucid period, from 300 BC, Babylonian astronomical texts used a zero sign, never in final position. *Status: confirmed* - [Zero, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/) — describes Babylonian base 60, the two-wedge placeholder around 400 BC, and Ptolemy’s sexagesimal placeholder around 130 AD; on the omicron derivation: “Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number - it represented 70.” - [Earliest Uses of Symbols for Constants, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/constants/) — Burton: “there is no evidence that Ptolemy regarded the symbol as a number by itself that could enter into computations with other numbers”; Berggren: “Papyri from the period when Ptolemy lived show a small ‘o’ with a bar over it as the symbol for 0”; and “Babylonians in the Seleucid period (300 BC onward) used a symbol for zero, mainly in astronomical texts, and never in final positions.” - [Raymond Mercier, Ancient Greek Numbers: A Greek Zero Sign, Unicode Technical Committee document L2/04-054](https://www.unicode.org/L2/L2004/04054-greek-zero.pdf) — read as extracted PDF text. Under ORIGIN OF THE SYMBOL: “It is commonly supposed that these forms arise from an abbreviation of οὐδέν, but given the rather arbitrary and opaque character of the majority of the symbols, one might well doubt this rather simple explanation.” Under Ninth century: “In ninth century Byzantine manuscripts the character for zero differs little from the papyri … the zero consists of a full size uncial omicron with a macron.” Under Fourteenth century: “In the some late medieval manuscripts the form of the symbol has altered considerably.” - [Wikipedia, Greek numerals, § Zero](https://en.wikipedia.org/wiki/Greek_numerals) — read as raw HTML: Hellenistic astronomers limited each sexagesimal position to 59, “which uses only the letters up through nu (ν) and included otherwise unused omicron (ο) as a special symbol for zero,” a system “probably adapted from Babylonian numerals by Hipparchus c. 140 BCE” and then used by Ptolemy. - [0, Wikipedia](https://en.wikipedia.org/wiki/0) — the Babylonian placeholder never stood alone: “Significantly, however, these placeholder signs were not considered a numerical value, they were never used alone” - [Sexagesimal, Wikipedia](https://en.wikipedia.org/wiki/Sexagesimal) — carries the Sumerian origin and the Babylonian inheritance: “It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates.” Editorial note: Extended 17 September 2026 for the #greekzero card, which carried the origin dispute and a Byzantine transmission claim on no citation at all. The card’s sentence that the sign “had turned into a plain omicron” in tenth-century Byzantine copies did not survive the check and was replaced by what Mercier’s survey shows. The card names Otto Neugebauer as the scholar who rejected the οὐδέν derivation; that is the standard attribution and Wikipedia cites Neugebauer, The Exact Sciences in Antiquity, 2nd ed., Dover 1969, pp. 13–14 and plate 2, for the Greek zero — but those pages were not read in this pass, so the doubt is cited here to Mercier, who states it in terms, and the Neugebauer attribution is flagged for a later pass to confirm. ### 2.4 {#r-2.4} The Maya used base-twenty notation and a shell zero; positional use is attested by the seventh century and earlier Maya zero artifacts by the fourth century. The Long Count calendar in which the earliest dates are written is positional, and its two oldest surviving dates, Stela 2 at Chiapa de Corzo (36 BC) and Stela C at Tres Zapotes (32 BC), are Epi-Olmec rather than Maya. Maya numerals write one as a dot and five as a bar; Tres Zapotes Stela C carries the second-oldest Long Count date, 32 BC, in Epi-Olmec style. Several Mesoamerican cultures counted in base twenty, almost certainly from fingers and toes; Maya numerals use a dot for one, a bar for five, and a shell for zero, and the third place counts 360s rather than 400s. *Status: confirmed* - [Zero, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/) — confirms Maya base 20, a zero symbol by 665, and use of zero before the place-valued system. - [Earliest Uses of Symbols for Constants, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/constants/) — dates the oldest listed Maya zero artifact without positional notation to AD 357 and the oldest listed positional zero to AD 665. - [0, Wikipedia](https://en.wikipedia.org/wiki/0) — gives Stela 2 at Chiapa de Corzo with a Long Count date of 36 BC as the oldest, and says “it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs.” - [Maya numerals, Wikipedia](https://en.wikipedia.org/wiki/Maya_numerals) — the dot-and-bar notation: “The numerals are made up of three symbols: zero (a shell), one (a dot) and five (a bar).” - [Mesoamerican Long Count calendar, Wikipedia](https://en.wikipedia.org/wiki/Mesoamerican_Long_Count_calendar) — the Epi-Olmec earliest dates and Stela C: “La Mojarra Stela 1, the Tuxtla Statuette, Tres Zapotes Stela C and Chiapa Stela 2 are all inscribed in an Epi-Olmec, not Maya, style.” and “This is the second oldest Long Count date yet discovered. The numerals 7.16.6.16.18 translate to September 1, 32 BCE (Gregorian).” - [Mayan mathematics, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/) — “Almost certainly the reason for base 20 arose from ancient people who counted on both their fingers and their toes.”; “it changes in the third place and this denotes the number of 360’s up to 19 instead of the number of 400’s.” ### 2.5 {#r-2.5} Brahmagupta’s Brāhmasphuṭasiddhānta was composed in 628 and gave rules treating zero as a number, including the incorrect rule 0/0 = 0. Brahmagupta gave rules for negative numbers as debts; the Sanskrit name for zero, śūnya, is the source of cipher and zero. *Status: confirmed* - [Brahmagupta, Wikipedia](https://en.wikipedia.org/wiki/Brahmagupta) — confirms composition in 628 and identifies the work as the first book giving arithmetic rules applying to zero and negative numbers. - [Zero, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/) — describes Brahmagupta’s rules and his treatment of division by zero. - [Brahmagupta, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Brahmagupta/) — negatives as debts: “He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers)” - [0, Wikipedia](https://en.wikipedia.org/wiki/0) — the Sanskrit name: “Sifr evolved to mean zero when it was used to translate śūnya (Sanskrit: शून्य) from India.” - [Earliest Known Uses of Some of the Words of Mathematics (Z), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Miller/mathword/z/) — the descent of the names: “ZERO, like its synonym CIPHER, derives from the Sanskrit sunya.” ### 2.6 {#r-2.6} The Khmer inscription K-127 contains a positional zero and is dated to 683 AD; an Old Malay inscription at Kedukan Bukit in Sumatra carries a zero in the same Śaka year, and the two cannot be ranked against each other. Sources differ on the Kedukan Bukit Śaka year: Cœdès read 605 in 1930, and the MAA article reads 604, which is why no Śaka year is printed. Corrections welcome. The Gwalior inscription is dated to 876 AD, records zeros in 270 and 50, and is the oldest Indian zero whose date and genuineness are not disputed; older Indian candidates exist, among them inscriptions of AD 672, 683, and 684, but their dates or their authenticity are questioned. The Gwalior inscription is at the Chaturbhuj Temple. The inscription is in the Chaturbhuj temple in Gwalior Fort. Cœdès’s 1930 paper is his reading of the Sumatran stone; his publication of the Sambor inscription K-127 is the 1931 paper of 2.14, so the card’s 1931 and the MAA article’s 1930 are two publications, not a dispute. *Status: confirmed* - [Zero, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/) — confirms the 876 Gwalior inscription (“The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876”), the garden dimensions and the garlands; the 683 Khmer zero is on the Earliest Uses line. - [Earliest Uses of Symbols for Constants, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/constants/) — confirms the 683 Khmer point and discusses the Gwalior date and its priority claims. - [Frank J. Swetz, Shaharir bin Mohamad Zain and Iwan Pranoto, Mathematical Treasure: Indonesian Zeros, MAA Convergence](https://web.archive.org/web/2023/https://www.maa.org/press/periodicals/convergence/mathematical-treasure-indonesian-zeros) — through the Internet Archive, the live MAA address now returning HTTP 403. Three Sriwijayan stones, “known as Kedukan Bukit, Talang Tuo, and Kota Kapur, bear the numerals 604, 606, and 608 respectively, denoting years as reckoned in the Hindu Saka era calendar, which translate into our Common Era calendar as the dates 683, 684, and 686,” and “the stone’s zero symbol thus corroborates the evidence of an early zero.” On the reading of the date: in his 1930 paper Coеdes “hesitantly interpreted the numerical symbol as 605” with a footnote saying that the units digit “ne peut être que 4 ou 5, mais 5 est plus probable,” and he read the Sambor inscription as 605 as well. - [0, Wikipedia](https://en.wikipedia.org/wiki/0) — the temple: “a small circle, appears on a stone inscription found at the Chaturbhuj Temple, Gwalior, in India, dated AD 876.” - [Chaturbhuj Temple, Gwalior, Wikipedia](https://en.wikipedia.org/wiki/Chaturbhuj_Temple,_Gwalior) — the temple “is a Hindu temple excavated in a rock face in the Gwalior Fort”; “One of the temple’s inscriptions contains the earliest known inscription of the circular symbol “O,” to represent zero.” Editorial note: Gwalior is often called the oldest Indian zero outright. MacTutor is more careful: it is “the first record of the Indian use of zero which is dated and agreed by all to be genuine”. Older Indian candidates exist — Michael Closs lists inscriptions of AD 672, 683, and 684, and there are copper plates that have been read as sixth-century — but their dates or their authenticity are questioned. So the chapter says undisputed rather than oldest. The Khmer stone is not uniquely oldest either: Miller lists a small circle for zero in an Old Malay inscription at Kedukan Bukit, Sumatra, in the same year, 683, and the two cannot be ranked, because K-127 records only the Śaka year and the fifth day of the waning moon, with no month. The chapter therefore calls Sambor one of the two oldest rather than the oldest. ### 2.7 {#r-2.7} Qin Jiushao’s 1247 Mathematical Treatise in Nine Sections uses a circle for zero, and Fibonacci used zephirum in Liber Abaci (1202); ṣifr is the source of cipher and zero. Al-Khwarizmi’s treatise on Indian reckoning, written about 825, is what carried the digit westward. 〇 also writes zero in the Suzhou numerals, the only surviving form of the rod numerals, and is still used for zero in Chinese and in Japanese kanji notation; one hypothesis derives it from □, the mark for a missing character. *Status: confirmed* - [Zero, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/) — confirms Qin Jiushao’s 1247 circle and Fibonacci’s use of zero in Liber Abaci around 1200. - [Earliest Known Uses of Some of the Words of Mathematics (Z), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Miller/mathword/z/) — traces Sanskrit śūnya through Arabic ṣifr to Latin zephirum in Fibonacci’s 1202 Liber Abaci and later cipher/zero forms. - [0, Wikipedia](https://en.wikipedia.org/wiki/0) — “Qín Jiǔsháo’s 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol ‘〇’ for zero”, and “the origin of this symbol is unknown; it may have been produced by modifying a square symbol”. - [On the First Printed Edition of Mathematical Book in Nine Chapters (1842), Springer](https://link.springer.com/chapter/10.1007/978-3-031-49617-2_13) — the 1842 Yijiatang Congshu edition is the first printed edition of Qin Jiushao’s treatise, which until then circulated in manuscript. - [Suzhou numerals, Wikipedia](https://en.wikipedia.org/wiki/Suzhou_numerals) — “The Suzhou numeral system is the only surviving variation of the rod numeral system.” and “Zero is represented by the character for zero (〇).” - [〇, Wiktionary](https://en.wiktionary.org/wiki/%E3%80%87) — Chinese: “It is considered a placeholder for a missing Chinese character. Used to express a zero digit in a number since at least the (c.) 12th century, perhaps even earlier than the concept “zero” appeared in the Chinese language. Also written as □ in early literature.” Japanese: “Used to write zero in kanji positional notation.” - [Chinese numerals, Wikipedia](https://en.wikipedia.org/wiki/Chinese_numerals) — present-day use: “Usually 零 is preferred, but in some areas, 〇 may be a more common informal way to represent zero.” ### 2.8 {#r-2.8} Dionysius Exiguus constructed an Easter table in 525, which introduced the Anno Domini numbering of years, and Bede extended that tradition in 725; Bede’s N represents nulla or nihil. Roman numerals had no zero; Bede’s N stood for 0 in a table of epacts, in De temporum ratione, written in Northumbria in 725. *Status: confirmed* - [Dionysius Exiguus’ Easter table, Wikipedia](https://en.wikipedia.org/wiki/Dionysius_Exiguus%27_Easter_table) — confirms the table was constructed in 525 and that Bede’s Easter cycle developed from it; “This table introduced the first use of the modern system of numbering years,” its first year being “Anno Domini 532,” and “With his Easter table Dionysius introduced the Christian era.” - [Bede’s Easter cycle, Wikipedia](https://en.wikipedia.org/wiki/Bede%27s_Easter_cycle) — confirms Bede’s 725 extension of Dionysius’s Easter table. - [0, Wikipedia](https://en.wikipedia.org/wiki/0) — “The initial ‘N’ was used as a zero symbol in a table of Roman numerals by Bede—or his colleagues—around AD 725”, and gives Dionysius Exiguus’s 525 table with the Latin nulla (“none”); it does not say which word the N stands for (the Roman numerals line: “the initial of nulla or of nihil”). - [Roman numerals, Wikipedia, § Zero](https://en.wikipedia.org/wiki/Roman_numerals) — the Roman system “lacked a numeral for the number zero itself”; “Dionysius Exiguus used nulla alongside Roman numerals in a manuscript from 525 AD. About 725, Bede or one of his colleagues used the letter N, the initial of nulla or of nihil (the Latin word for “nothing”) for 0, in a table of epacts, all written in Roman numerals.” - [The Reckoning of Time, Wikipedia](https://en.wikipedia.org/wiki/The_Reckoning_of_Time) — “The Reckoning of Time (Latin: De temporum ratione, CPL 2320) is a Medieval Latin treatise written by the Northumbrian monk Bede in 725.” Editorial note: The N is attributed to Bede here, as the chapter attributes it; the fullest source available says “by Bede—or his colleagues—”, and no source seen names the hand that wrote it. ### 2.9 {#r-2.9} Zero arose more than once: the Maya development was independent of the Old World, while historians differ on how far the Indian zero owed something to Babylonian practice through Greek astronomy; and the chapter’s classification of placeholder, digit, and number is a historical synthesis. *Status: judgement* - [Zero, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/) — emphasizes that zero has distinct placeholder and number uses and describes Babylonian, Maya, Indian, and Arabic developments without endorsing a fixed count of independent inventions. - [Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol ‘zero’, Bodleian Libraries](https://www.glam.ox.ac.uk/article/carbon-dating-finds-bakhshali-manuscript-contains-oldest-recorded-origins-symbol-zero) — “the use of zero as a placeholder was seen in several different ancient cultures, such as among the ancient Mayans and Babylonians,” but “it was only in India that this zero developed into a number in its own right.” - [Indian numerals, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/) — “Some historians believe that the Babylonian base 60 place-value system was transmitted to the Indians via the Greeks.” Editorial note: The chapter’s three-part framing (placeholder, digit, number) is an interpretive synthesis. MacTutor: “many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers”; of the Maya zero, “This is a remarkable achievement but sadly did not influence other peoples.” ### 2.10 {#r-2.10} The Bodleian’s 2017 radiocarbon result for the Bakhshali manuscript, which dated one leaf to the third or fourth century AD, has been withdrawn by the laboratory that produced it: a second sample from the same leaf, taken in 2018 and published on 14 October 2024, showed the first determination to be inaccurate. *Status: contradicted* - [Chivall, Lladó-Buisán, Howell and Evison, Radiocarbon dating of the Bakhshālī manuscript, University of Oxford (2024)](https://ora.ox.ac.uk/objects/uuid:5a6d1dd7-f20c-4209-adb6-33849f5b08f4) — current Oxford record for the revised radiocarbon study. - [Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol ‘zero’, Bodleian Libraries (2017)](https://www.glam.ox.ac.uk/article/carbon-dating-finds-bakhshali-manuscript-contains-oldest-recorded-origins-symbol-zero) — records the superseded 2017 announcement: the manuscript “dates from as early as the 3rd or 4th century” and “is in fact composed of material from at least three different periods”; the dating is withdrawn, but not its description of the text as “a seminal mathematical text which contains hundreds of zeroes,” the dot that “can be seen throughout the Bakhshali manuscript.” - [Plofker, Keller, Hayashi, Montelle and Wujastyk, The Bakhshālī Manuscript: A Response to the Bodleian Library’s Radiocarbon Dating, History of Science in South Asia 5.1 (2017), 134–150](https://journals.library.ualberta.ca/hssa/index.php/hssa/article/view/22) — the five scholars argue that a manuscript is dated by its latest datable folio rather than its earliest, and that the manuscript’s zeros work as numbers in their own right and not only as placeholders. Oxford’s 2024 re-dating bears out the first point. Editorial note: Both datings are on the page. The 2024 report is stated as the fact, because it is a re-measurement by the same laboratory rather than a restatement; the 2017 result is kept as the history it corrects, since without it the scholarly objection and the correction make no sense. ### 2.11 {#r-2.11} Each displayed glyph denotes zero in its own notation, so multiplying the thirteen factors gives zero. *Status: confirmed* - **Own computation** (the author's own work) — The displayed product contains thirteen zero-valued factors; 0 × 0 × … × 0 = 0, with no denominator present. ### 2.12 {#r-2.12} The character 零 is read rei in Japanese and líng in Chinese and is used as a word for zero in both languages. In Japanese the preferred spoken reading of 0 is the English loanword zero (ゼロ). *Status: confirmed* - [零, Kanji Data Base (Kanjipedia)](https://www.kanjipedia.jp/kanji/0007256600) — confirms the character and its Japanese readings/meanings. - [Zero, Treccani](https://www.treccani.it/enciclopedia/zero_%28Enciclopedia-Italiana%29/) — the history of the word, not the Chinese or Japanese one: “Leonardo latinizzò in zephirum il vocabolo ṣifr usato dagli Arabi in corrispondenza al sanscrito śūnya (vuoto)”; it has nothing on 零. - [零, Wiktionary](https://en.wiktionary.org/wiki/%E9%9B%B6) — a phono-semantic compound of 雨 (rain) with 令; the original sense is a light rain or drizzle, with the extended senses of a remnant or fraction, and thence zero. - [Japanese numerals, Wikipedia](https://en.wikipedia.org/wiki/Japanese_numerals) — its table gives for 0 (零 / 〇) the on reading rei and, as preferred reading, “zero (ゼロ) (loanword from English, gairaigo)”; and “as a number, it is only written as 0 or rei (零).” ### 2.13 {#r-2.13} The feather mark ᚛ used in this Pictish Ogham inscription is punctuation rather than a letter; it can mark the beginning and reading direction. The shared alphabet has twenty-six Unicode letter characters, of which this book reserves Ruis for the left-hand side, leaving twenty-five for its twenty-five lettered chapters. *Status: confirmed* - [The Unicode Standard 16.0, Chapter 8: Europe-II — Ogham](https://www.unicode.org/versions/Unicode16.0.0/core-spec/chapter-8/) — “U+169B OGHAM FEATHER MARK and U+169C OGHAM REVERSED FEATHER MARK are used at the beginning and the end of Ogham text, particularly in manuscript Ogham,” and “In some cases, only the Ogham feather mark is used, which can indicate the direction of the text.” - [UnicodeData.txt, Unicode Character Database](https://www.unicode.org/Public/UCD/latest/ucd/UnicodeData.txt) — downloaded and read: 169B OGHAM FEATHER MARK carries General_Category Ps, open punctuation, and 169C OGHAM REVERSED FEATHER MARK carries Pe, while the twenty-six Ogham letters from 1681 OGHAM LETTER BEITH to 169A OGHAM LETTER PEITH all carry Lo. The feather mark is therefore not one of the letters and spends none. - **Own enumeration** (the author's own work) — the twenty-five-letters claim was enumerated against the book itself and holds. The twenty-six Ogham letters are U+1681 to U+169A; the glossary’s letters section reserves Ruis for the left-hand side, and the chapters carrying the remaining twenty-five run from Chapter ᚁ(B) to Chapter ᚚ(P), which is exactly twenty-five, with the prologue and the epilogue taking the two non-letter feather marks U+169B and U+169C. Editorial note: The prologue’s first two sentences describe the feather mark and what it does, and its closing paragraph says the mark spends no letter and that the piece has twenty-five letters for twenty-five chapters. None of the three carried a footnote, and the V10 pass recorded the letter count as unchecked. The Unicode Standard supplies the first two; the third is an enumeration of the book’s own chapters, and it is right. ### 2.14 {#r-2.14} The Sambor inscription K-127 was published by George Cœdès in 1931, disappeared during the Khmer Rouge period, and was found again in a shed at Angkor Conservation in Siem Reap in January 2013. Aczel told the story in his book Finding Zero (2015). *Status: confirmed* - [G. Cœdès, À propos de l’origine des chiffres arabes, Bulletin of the School of Oriental Studies 6/2 (1931), 323–328](https://www.cambridge.org/core/journals/bulletin-of-the-school-of-oriental-and-african-studies/article/abs/a-propos-de-lorigine-des-chiffres-arabes/E6EE00E7C7811888DF9E074114A0EBDD) — the journal record gives author G. Cœdès, volume 6, issue 2, year 1931, pages 323–328, and DOI 10.1017/S0041977X00092806, and its abstract describes the paper as answering G. R. Kaye’s objections to an Indian origin for positional notation. The full text was not read in this pass, so this source is cited for the 1931 publication itself and not for what the paper argues about K-127. - [Amir D. Aczel, Searching for the World’s First Zero, HuffPost (15 May 2013)](https://www.huffpost.com/amir-aczel/worlds-first-zero_b_3276709.html) — the finder’s own first-person account: Georges Cœdès published the K-127 inscription in 1931; it was lost when the Khmer Rouge stole or destroyed close to 10,000 artifacts; and on 2 January 2013 Aczel found K-127 in the Angkor Conservation compound near Siem Reap. (First cited here: Aczel, The Origin of the Number Zero, Smithsonian Magazine, December 2014, which refuses automated readers). - [Amir Aczel, Wikipedia](https://en.wikipedia.org/wiki/Amir_Aczel) — the book: “was awarded a Sloan Foundation grant to research his book Finding Zero (2015)” Editorial note: The prose sentence naming Cœdès, 1931, the Khmer Rouge years, and the 2013 rediscovery carried no footnote of its own, and neither of r-2.6’s two sources mentions Cœdès, 1931, the Khmer Rouge, or 2013; both were fetched and read in full during the V14 prose pass. Aczel’s article is first-hand for the 2013 find and secondary for the 1931 publication, which the Cambridge record confirms independently. ### 2.15 {#r-2.15} At most eight of the twenty primary ogham letter names are the names of trees. Beith is ‘birch’, fern ‘alder’, sail ‘willow’ (cognate with Latin salix), dair ‘oak’, coll ‘hazel’, and onn ‘ash’; idad is probably an artificial form of ‘yew’, a sense McManus finds the kennings support, and ailm may mean ‘pine’. The rest mean other things: luis ‘blaze’ or ‘herb’, nin ‘fork’ or ‘loft’, úath ‘horror, fear’, tinne ‘bar of metal, ingot’ (probably cognate with Old Irish tend, ‘strong’, or tind, ‘brilliant’), cert ‘bush’, muin ‘neck’, ‘ruse’ or ‘love’, gort ‘field’, gétal ‘killing’, straif ‘sulphur’, ruis ‘red’, and úr ‘earth, clay, soil’, while edad is of unknown meaning; edad and idad are a pair, idad perhaps altered to match. Of the supplementary letters, ór is ‘gold’, uilleand ‘elbow’, emancholl ‘twin of coll’, pín probably from Latin spina, and ébad unknown. On attestation, the distinction this book has to make because its own script is Pictish: úath is unattested in the orthodox Irish inscriptions, which is the corpus McManus is describing, but the letter is common in the Scottish inscriptions of the Pictish area — it occurs in a dozen or so words across eight or nine of them, singly at the start or end of a word and doubled as HH in the middle, the Bressay cross slab in Shetland among them. Fern is from Primitive Irish *wernā, so the letter’s original value was [w]. *Status: confirmed* - [Ogham, section Letter names, Wikipedia](https://en.wikipedia.org/wiki/Ogham) — gives the twenty primary names above letter by letter, and in its letters table óir ‘gold’, uilleann ‘elbow’, pín ‘spine; thorn’, eamhancholl ‘twin of coll’, and éabhadh ‘unknown’; it attributes the etymologies to McManus, naming A Guide to Ogam §3.15 for the etymologies of all twenty names, McManus 1988 p. 164 for idad as an artificial form of iubhar and for the eighth-century poem behind ailm, and McManus 1986 for the suggested value [y] of úath. It states flatly that “out of twenty letter names, only eight at most are the names of trees”; for Beith: “Old Irish Beithe means ‘birch-tree’, cognate to Middle Welsh bedw. Latin betula is considered a borrowing from the Gaulish cognate.”; for Sail: “Old Irish Sail means ‘willow-tree’, cognate to Latin salix”; for Fearn: “Old Irish Fern means ‘alder-tree’, Primitive Irish *wernā, so that the original value of the letter was [w]”; for Dair: “Old Irish Dair means ‘oak’ (Proto-Indo-European: *doru-)”; for Coll: “Old Irish Coll meant ‘hazel-tree’, cognate with Welsh collen,” and “Latin corulus or corylus is cognate”; for Uilleann: “Old Irish Uilleand ‘elbow’; the arboreal tradition has edleand ‘honeysuckle’”; for nGéadal: “Old Irish Gétal from the kennings has a meaning of ‘killing’, maybe cognate to gonid ‘slays’,” and “The value of the letter in Primitive Irish, then, was a voiced labiovelar”; for Straif: “Straif, Old Irish Straiph means ‘sulphur’. The Primitive Irish letter value is uncertain, it may have been a sibilant different from s, which is taken by sail, maybe a reflex of /st/ or /sw/”; and in its notes: “With regard to ailm, in the “King and Hermit” poem the hermit Marban says “caine ailmi ardom-peitet” – “beautiful are the pines that make music for me.” This is a reference to the idea that pine makes a pleasing, soothing sound as the wind passes through its needles.”; for Ceirt: “Old Irish Cert is cognate with Welsh perth ‘bush’, Latin quercus ‘oak’ (Proto-Indo-European: *perkwos). It was confused with Old Irish ceirt ‘rag’, reflected in the kennings.”; for idad: “McManus (1988, p. 164) also believes that the name idad is probably an artificial form of iubhar ‘yew’, as the kennings support that meaning.” - [Bríatharogam, Wikipedia](https://en.wikipedia.org/wiki/Br%C3%ADatharogam) — “edad and idhad are paired names of unknown meaning, although idhad may be a form of ‘yew-tree’, altered to make a pairing.”; “gort means “field” (cognate to garden).”; for tinne, “the word is probably cognate to Old Irish tend “strong” or tind “brilliant.”” - [Forfeda, Wikipedia](https://en.wikipedia.org/wiki/Forfeda) — for pín: “McManus states that the name Pín was probably influenced by Latin pinus (‘pine’), but a more likely explanation is that it derives from Latin spina (‘thorn’), as the kennings indicate a tree or shrub with sweet tasting fruit (therefore not a pine)”; for Éabhadh: “The name appears modelled after Eadhadh and Iodhadh.” - [BabelStone Blog, The Ogham Stones of Scotland (8 June 2013)](https://www.babelstone.co.uk/Blog/2013/06/ogham-stones-of-scotland.html) — “The letter uath ᚆ, which corresponds to H in the manuscript tradition, is not normally found in Ogham inscriptions outside Scotland, as it appears not to have been needed for writing Primitive Irish”; “However, this letter is very common in Scottish inscriptions from the Pictish area, occuring in 12 or 13 words in eight or nine inscriptions, usually written singly at the start or end of a word, and doubled as HH ᚆᚆ in the middle of words,” its list including “BREAY/1 : NAHHTVVDDADDS,” the “decorated cross slab from Bressay in Shetland.” Editorial note: The letters table used to give the book’s own meaning for each letter in its Represents column, which duplicated the Chapter column and had gone stale in places. The column now gives the letter name’s own meaning, so it needed an entry behind it. The primary scholarship is McManus; what was read here is the encyclopedia article that reproduces him, and this entry says so rather than implying the books were opened. ### 2.16 {#r-2.16} Three lists of Bríatharogaim, or word-oghams, survive from the Old Irish period, ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn. Each gives a two-word kenning for every letter name. The first two are in all three copies of the Ogam Tract; the Cú Chulainn list is absent from the Book of Ballymote and is known only from sixteenth- and seventeenth-century manuscripts, and it has no entry for four of the five supplementary letters. *Status: confirmed* - [Bríatharogam, Wikipedia](https://en.wikipedia.org/wiki/Br%C3%ADatharogam) — prints all three lists in full “as edited (in normalized Old Irish) and translated by McManus (1988),” which is the text quoted on the letter cards. Beithe: féochos foltchain, glaisem cnis, maise malach. Luis: lí súla, carae cethrae, lúth cethrae. Fern: airenach fían, comét lachta, dín cridi. Sail: lí ambi, lúth bech, tosach mela. Nin: costud síde, bág ban, bág maise. Úath: condál cúan, bánad gnúise, ansam aidche. Dair: ardam dosae, grés soír, slechtam soíre. Tinne: trian roith, smiur gúaile, trian n-airm. Coll: caíniu fedaib, carae blóesc, milsem fedo. Cert: clithar baiscill, bríg anduini, dígu fethail. Muin: tressam fedmae, árusc n-airlig, conar gotha. Gort: milsiu féraib, ined erc, sásad ile. Gétal: lúth lego, étiud midach, tosach n-échto. Straif: tressam rúamnai, mórad rún, saigid nél. Ruis: tindem rucci, rúamnae drech, bruth fergae. Ailm: ardam íachta, tosach frecrai, tosach garmae. Onn: congnaid ech, féthem soíre, lúth fían. Úr: úaraib adbaib, sílad cland, forbbaid ambí. Edad: érgnaid fid, commaín carat, bráthair bethi (?). Idad: sinem fedo, caínem sen, lúth lobair (?). The English renderings on the cards are this page’s, and the question marks are its own; the Auraicept line for Dair: “arirde dossaib duir “higher than bushes is an oak””; for Tinne, “the word is probably cognate to Old Irish tend “strong” or tind “brilliant””; for Iphín, the table gives “milsem fedo “sweetest tree”” and “amram mlais “most wonderful taste”” (milsem fedo is also one of Coll’s). For Muin, “the kennings connect this name to three different words, muin “neck, upper part of the back,” muin “wile, ruse,” and muin “love, esteem.”” For Ébad the table gives “snámchaín feda “fair-swimming letter,”” “cosc lobair “[admonishing?] of an infirm person”” and “caínem éco “fairest fish.”” - [Forfeda, Wikipedia](https://en.wikipedia.org/wiki/Forfeda) — prints the same three columns for the five supplementary letters and shows the Cú Chulainn column empty for all but Ébad (caínem éco). Editorial note: Twenty-eight letter cards quote these kennings, so the quotation needed one entry of its own rather than a footnote each. The translations are McManus’s as the encyclopedia reproduces them; where that page marks a reading uncertain, the cards mark it uncertain too. ### 2.17 {#r-2.17} Reading the ogham letter names as names of trees belongs to the medieval manuscript tradition, the Auraicept na nÉces and In Lebor Ogaim, and postdates the naming of the letters. The Auraicept itself finds only five of the twenty names transparent without glossing — beith, fearn, saille, duir, and coll — and concedes that some of its trees are not known. McManus dates the tree-alphabet idea to the Old Irish period and traces it to the letters themselves being called feda, trees, and nin, forking branches, from their shape. McManus counts onn, ‘ash’, as a further definite tree name and says the Auraicept wrongly glosses it ‘furze’. Modern Irish giolcach means reeds, and also broom. Ó Dónaill’s dictionary gives peith as ‘dwarf elder’ and as the letter P of the later ogham alphabet. *Status: confirmed* - [Ogham, section Letter names, Wikipedia](https://en.wikipedia.org/wiki/Ogham) — quotes the Auraicept conceding “Now all these are wood names such as are found in the Ogham Book of Woods, and are not derived from men” while admitting “some of these trees are not known today”; names the five transparent tree names; reports McManus that the idea “dates to the Old Irish period (say, 10th century), but it postdates the Primitive Irish period” and arose because the letters were called feda or nin for their shape; and names ruis, úath, and gort as letter names the glossators reinterpreted “more or less forcefully” as epithets of trees. It also reports that McManus “adds one other definite tree name: onn ‘ash’ (the Auraicept wrongly has ‘furze’).” - [Bríatharogam, Wikipedia](https://en.wikipedia.org/wiki/Br%C3%ADatharogam) — prints the Auraicept gloss for each name with the Irish plant word — caertheand rowan, uinnius ash, sce whitethorn, cuileand holly, aball apple, finemhain vine, edind ivy, cilcach broom or fern, draighin blackthorn, trom elder, crand giuis .i. ochtach fir or pine, aiten furze, fraech heath, crand fir no crithach test-tree or aspen, ibhar yew — together with the short medieval sentences quoted on the cards; for cert: “The Auraicept glosses aball “apple,” clithar boaiscille .i. elit gelt quert (.i. aball) “shelter of a boiscill, i.e. a wild hind is queirt, i.e. an apple tree””; and of the forfeda, “oir with feorus no edind “spindle-tree or ivy.”” - [Forfeda, Wikipedia](https://en.wikipedia.org/wiki/Forfeda) — “According to Kelly (1976) the name spín … appears in the Old Irish tree lists as meaning either gooseberry or thorn” (Fergus Kelly, The Old Irish Tree-list, Celtica 11 (1976) 107–124; no free copy of the article). - [giolcach, Foclóir Gaeilge–Béarla (Ó Dónaill), teanglann.ie](https://www.teanglann.ie/en/fgb/giolcach) — “giolcach … 1. Coll: Reeds. Páirc na giolcaí, the reed field … 2. Reed; cane. 3. ~ (shléibhe), broom.” - [peith, Foclóir Gaeilge–Béarla (Ó Dónaill), teanglann.ie](https://www.teanglann.ie/en/fgb/peith) — “Lit: 1. Dwarf elder. 2. Letter P in later ogham alphabet.” - [úathmar, eDIL (Electronic Dictionary of the Irish Language)](https://dil.ie/42831) — “úathmar … adj o-ā (úath) dreadful; terrifying; ‘terrible, horrible, awful, direful’.” Editorial note: The glossary intro already said the plant column records “later medieval and modern arboreal associations” and is not the secure original meaning of every letter name. That sentence had no entry behind it, and twenty-eight cards now rest on it, so it has one. The three layers the cards keep apart are exactly the layers this entry separates: the name, the medieval plant gloss, and anything later. ### 2.18 {#r-2.18} The five aicme forfeda were invented in the Old Irish period, centuries after the peak of ogham use, and the sounds they stood for are uncertain: the Forfeda page says they “appear to have represented sounds felt to be missing from the original alphabet, maybe” é(o), ó(i), ú(i), p and ch. Apart from Ébad they barely appear in the orthodox inscriptions: Ór twice (once in Ireland, once on the Formaston stone), Uilleann once, Pín in two or three, Emancholl not at all, though some later scholastic inscriptions use all five. Later oghamists reassigned all five to diphthongs, which changed the values of Pín, renamed Ifín, and of Emancholl, and left the alphabet with no letter for p, so Peith — beithe bog, soft beithe, also peithbog — was invented, and appears in manuscripts only. Sources differ on the name Pín: Latin spina (the Forfeda page, reporting McManus) or Latin pinus (Hayden and Stifter, OG(H)AM, 2022); corrections welcome. *Status: confirmed* - [Forfeda, Wikipedia](https://en.wikipedia.org/wiki/Forfeda) — states that the five aicme forfeda “were invented in the Old Irish period, several centuries after the peak of Ogham usage”; that four are glossed with plants in the Auraicept and Emancholl is not; that Emancholl is named for its shape, ᚙ resembling two ᚉ; that Ébad is frequent in inscriptions inside the formula word KOI, meaning roughly ‘here’ and answering to Latin hic iacet; that Ór appears at Killogrone in County Kerry and on the Formaston cross slab in Aberdeenshire, Uilleann once at Teeromoyle in County Kerry, and Pín at Cool East, Crickowel, and probably Margam; and, under the heading Peith, that [p] “had again to be expressed as a modification of [b], called peithe, after beithe, also called beithe bog,” (the Ogham line carries “appears in the manuscripts only”). It also reports McManus on pín and Latin spina, and Kelly on spín in the tree lists. Of Ébad it reads: “Four of these names are glossed in the Auraicept with tree names, ebad as crithach “aspen””; “The kennings for Éabhadh point to the sound éo or é, which is also the word for “salmon””; and “Éabhadh was in fact frequently used as part of the formula word KOI ᚕᚑᚔ, but with the value /k/ or /x/”; its table of “The forfeda letter names and their kennings, as edited (in normalized Old Irish) and translated by McManus (1988)” has rows for Éabhadh, Óir, Uilleann, Ifín, and Eamhancholl and none for Peith, and the four Auraicept tree glosses it gives are for ebad, oir, uilleand, and iphin. - [Ogham, Wikipedia](https://en.wikipedia.org/wiki/Ogham) — gives the same reassignment and the invention of peithboc, ‘soft B’, which “appears in the manuscripts only”; and “Unlike orthodox ogham, some medieval inscriptions feature all five Forfeda.” - [Deborah Hayden and David Stifter, Ogam and Trees, OG(H)AM project, University of Glasgow (20 December 2022)](https://ogham.glasgow.ac.uk/index.php/2022/12/20/ogam-and-trees/) — “Iphin, Pín is probably called from Lat. pīnus ‘pine, fir’, after the model of the other tree names in the core ogam set”; and of Gétal, “Medieval scholars explain it as gilcach ‘reed’”; and “Ór means ‘gold’, a loan from Latin aurum.”; of the forfedae: “Ébad, allegedly ‘aspen’ (we have seen that one before) or ‘elecampane’ (real name: elenn). Ébad is probably just a meaningless variation on edad.” - [Formaston Stone, Elphinstone Institute, University of Aberdeen](https://www.abdn.ac.uk/elphinstone/resources/bbaf/category/3/180/) — “A fragment of a class II Pictish Symbol Stone,” with “two vertical Ogam inscriptions”; Aboyne, Aberdeenshire, dated 800–1000 AD. Editorial note: Six of the book’s twenty-eight terms are supplementary letters or later inventions — Ébad, Ór, Uilleann, Ifín, Eamhancholl, and Peith — and their cards have to say that the ground under them is softer than under Beith or Dair. Ruling D21 applies: the uncertainty goes on the card. ### 2.19 {#r-2.19} The tree calendar and the divinatory meanings attached to the ogham letters today are twentieth-century. Robert Graves, in The White Goddess of 1948, read the letter order as an ancient seasonal calendar of tree magic and reduced the alphabet to thirteen consonant months, in which Ruis, the elder, is the thirteenth and last, 25 November to 22 December. Celtic scholarship rejects the reconstruction, and the letter order Graves followed with it. It remains the main source of modern ogham tree lore and divination. *Status: confirmed* - [Ogham, section Neopaganism, Wikipedia](https://en.wikipedia.org/wiki/Ogham#Neopaganism) — states that modern New Age and neopagan approaches to ogham “largely derive from the now-discredited theories of Robert Graves in his book The White Goddess”; that Graves explored the tree folklore of the letter names and proposed that their order formed an ancient “seasonal calendar of tree magic” (citing Graves 1961 p. 165); that his theories have been “discredited and discarded by modern scholars,” Macalister included; that he followed Macalister’s BLNFS letter order, which scholars reject; and that the meanings used in modern divination “are usually based on the tree ogham,” with Graves still a major influence. (Read again on 23 September 2026: the article does not cite Hutton’s The Triumph of the Moon, which an earlier note said it did.) - [Celtic Tree Calendar, Mary Jones, Jones’s Celtic Encyclopedia](https://www.maryjones.us/jce/celtictreecalendar.html) — “there is no evidence that the Celts ever named their months after trees”; lists the thirteen months and gives “Ruis (Elder) November 25 to December 22,” with 23 December ruled by no tree; describes the scheme as Davies and Graves “picking and choosing the facts to suit their purpose,” and records that Graves ignored Macalister’s advice not to take the tree alphabet too seriously. - [Brian Walsh, Robert Graves’ fabrication of the Celtic tree calendar, Spiral Nature, 21 August 2013](https://www.spiralnature.com/spirituality/graves-frabrication-of-the-celtic-tree-calendar/) — calls the calendar “an invention unprecedented in the original lore” and concludes that “the result is a calendar that bears no resemblance to any traditional Celtic time keeping system we have on record,” setting out how Graves rearranged the Song of Amergin and cut the alphabet to thirteen consonants to do it. - [The White Goddess, Wikipedia](https://en.wikipedia.org/wiki/The_White_Goddess) — identifies Robert Graves’s book-length essay The White Goddess: a Historical Grammar of Poetic Myth, “First published in 1948,” with revised editions in 1952 and 1961. Editorial note: The guardrail on this column is that nothing modern may be presented as ancient. Every letter card that mentions present-day tree lore carries this marker, and the wording is always that the association is modern. Chapter ᚏ(R) says Ruis is associated with the elder and with transitions: the elder is medieval and well sourced, the transition is not — it comes from Graves putting Ruis last in the year. That split is stated on the Ruis letter card. ### 2.20 {#r-2.20} Elder folklore, which is where the modern reading of Ruis as a letter of endings and thresholds comes from, is late and regional: the tree is a guardian to be asked before it is cut, on pain of the Elder Mother’s revenge; it was auspicious growing near a dwelling; there was a taboo on burning its wood; Midsummer’s Eve under an elder was the time to meet the fairy court; and legend made it the tree Judas hanged himself from. Scottish Gaelic ruis, one of its names, occurs only rarely in Scottish place names. *Status: confirmed* - [Elder, Trees for Life](https://treesforlife.org.uk/into-the-forest/trees-plants-animals/trees/elder/) — “The best time to encounter faeries was under an elder on Midsummer’s Eve, when the Faery King and Queen and their train would pass by”; “Like rowan, people saw elder as being a protective tree. It was auspicious if it was growing near one’s dwelling, especially if it had seeded itself there”; “Legend has it that it was the tree from which Judas Iscariot hanged himself”; the belief that burning it released the Devil “reinforced the taboo against burning the wood”; and the note that the Gaelic names ruis or droman occur only rarely in Scottish place names. - [Sambucus, section In culture, Wikipedia](https://en.wikipedia.org/wiki/Sambucus#In_culture) — “If an elder tree was cut down, a spirit known as the Elder Mother would be released and take her revenge,” and the tree “could only safely be cut while chanting a rhyme to the Elder Mother”; records that some traditions have the elder warding off witches and others have witches congregating under it, and that Romani people hold burning elder wood to bring bad luck. Cites Royal Botanic Gardens, Kew, and Howard, Traditional Folk Remedies (1987), 134–5. - [Elder Mother, Wikipedia](https://en.wikipedia.org/wiki/Elder_Mother) — read as raw wikitext: “to take wood from the elder tree one would have to ask the Elder Mother first, or else ill luck would befall the woodsman,” and “The spirit is said to haunt or torment people who build from elder wood unless they ask permission first.” Editorial note: This is folklore, cited as folklore. It is the layer the Ruis letter card labels modern, and it is kept apart from the Auraicept gloss trom, elder, which is medieval and sits under r-2.17. Neither of these sources connects the folklore to the ogham letter; the Trees for Life page mentions ruis only as a Gaelic plant name. That is why the card says what modern lore says about the tree, and does not claim a medieval letter of transition. ### 2.21 {#r-2.21} The chapter epigraph is Whitehead’s, from An Introduction to Mathematics of 1911: nobody goes out to buy zero fish, and zero is the most civilized of the cardinals. *Status: confirmed* - [A. N. Whitehead, An Introduction to Mathematics (1911), full text at the Internet Archive](https://archive.org/stream/anintroductionto00whituoft/anintroductionto00whituoft_djvu.txt) — read as extracted page text on the chapter on zero: “The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought..” Editorial note: Added 17 September 2026 to source the chapter epigraph, which carried no marker. **The epigraph drops three words.** Whitehead wrote “It is in a way the most civilized of all the cardinals” and the page prints “It is the most civilized of all the cardinals.” The sense is unchanged and the hedge is Whitehead’s; restoring it is a prose question and is recorded as such. ### 2.22 {#r-2.22} In positional notation a digit’s value depends on the column it stands in, so a fixed set of signs writes every number. The Babylonian base-60 system is credited as the first positional system, and it lacked a true zero: a space, and later a punctuation sign, marked an empty column. Two comparable sources disagree on the date of the sign and are recorded here rather than silently reconciled: Wikipedia’s Positional notation puts a space or punctuation symbol “such as two slanted wedges” at about 700 BC, while MacTutor’s history of Zero distinguishes them, dating a tablet using three hooks to about 700 BC and the two wedges to about 400 BC, while Wikipedia’s 0 has two slanted wedges repurposed as a placeholder by 300 BC. The Babylonian placeholder card and the chapter write ~300 BC, the latest of the three dates, by which all three sources have the wedges in use, and that card says that the sources differ. Corrections welcome. Either way the placeholder was never written alone or at the end of a number, so it could not distinguish a trailing empty place. The Indians were the first to develop a base-10 positional system, and our place-value system descends from it. *Status: confirmed* - [Positional notation, Wikipedia](https://en.wikipedia.org/wiki/Positional_notation) — read as raw HTML: “the Babylonian numeral system, credited as the first positional numeral system, was base-60. However, it lacked a real zero. Initially inferred only from context, later, by about 700 BC, zero came to be indicated by a ‘space’ or a ‘punctuation symbol’ (such as two slanted wedges) between numerals. It was a placeholder…,” and contrasts positional with sign-value systems such as Roman numerals. - [Indian numerals, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/) — read as raw HTML: “The Babylonians had a place-value system as early as the 19th century BC but the Babylonian systems were to base 60. The Indians were the first to develop a base 10 positional system,” and “our place-value system is a direct descendant of the Indian system.” - [0, Wikipedia, § Ancient Near East](https://en.wikipedia.org/wiki/0) — “By 300 BC, a punctuation symbol (two slanted wedges) was repurposed as a placeholder.” - [Zero, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/) — “It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero.” - [Babylonian numerals, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/) — “The numbers sexagesimal numbers 1 and 1,0, namely 1 and 60 in decimals, had exactly the same representation.” ### 2.23 {#r-2.23} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: A tablet from Kish of perhaps 700 BC marks the empty sexagesimal place with three hooks. *Status: confirmed* - [Zero, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/) — read raw: “This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation..” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 2.24 {#r-2.24} The Long Count calendar is a positional, mostly base-twenty numbering system for dates, attested in Epi-Olmec and Maya inscriptions from the first centuries BC forward, and it needed a placeholder zero to work as a place-value count. Its earliest date yet discovered, Stela 2 at Chiapa de Corzo, and the second oldest, Stela C at Tres Zapotes, are in Epi-Olmec style, and a shell glyph stood for zero. *Status: confirmed* - [Mesoamerican Long Count calendar, Wikipedia](https://en.wikipedia.org/wiki/Mesoamerican_Long_Count_calendar) — describes the positional base-20 (with one base-18 place) system and its oldest surviving dates: “The earliest contemporaneous Long Count inscription yet discovered is on Stela 2 at Chiapa de Corzo”; of Tres Zapotes Stela C, “This is the second oldest Long Count date yet discovered”; “La Mojarra Stela 1, the Tuxtla Statuette, Tres Zapotes Stela C and Chiapa Stela 2 are all inscribed in an Epi-Olmec, not Maya, style.”; and “The shell glyph was used to represent the zero concept.”; on the base-18 place: “the second digit from the right (and only that digit) rolls over to zero when it reaches 18.” and “Thus 0.0.1.0.0 does not represent 400 days, but rather only 360 days,” and its table gives 1 tun = 360 days, approximately 1 solar year.