---
id: "20260813-0217-do-tier-12-sources"
title: "Do Tier 1–2 sources confirm that Ibn al-Banna's Talkhīṣ contains lattice multiplication, and that the 12th-c. Lilavati-commentary 'Kapat-sandhi' is the same method?"
type: "capture"
status: "promoted"
promoted_to: ["30-notes/claim-abdeljaouad-oaks-talkhis-study-omits-lattice-gelosia-shabakh.md (new)","30-notes/claim-earliest-gelosia-manuscript-evidence-is-ibn-ghazis-commentary.md (new)","30-notes/claim-kapat-sandhi-gelosia-identification-rests-on-tier-3-source.md (new, seedling, flagged [unverified-mechanism — needs primary])","40-entities/entity-ibn-al-banna-al-marrakushi.md (new hub)","40-entities/entity-al-hawari.md (new hub)","40-entities/entity-ibn-ghazi-al-miknasi.md (new hub)",{"30-notes/claim-lattice-multiplication-india-maghreb-italy-transmission-chain.md (updated":"Correction history block appended, flag language sharpened; status left seedling)"},{"50-questions/question-verify-lattice-multiplication-maghreb-india-primary.md (progress line appended, left status":"open — partial answer only)"}]
not_promoted: ["Claim 4 ('Wikipedia's own citation trail points to a single unread academic source,' the Chabert 1999 lead) — not an atomic claim about the world (the capture itself calls it n/a for source tier); folded into the progress line appended to question-verify-lattice-multiplication-maghreb-india-primary.md as the named next step, not given its own note.","The core question's overall [unverified] verdict — a summary of the other four claims, not a fifth atomic claim; the individual findings carry the verdict, a restated summary would duplicate them.","Bhāskara II (Lilavati author, entity candidate) — real and clearly matters to history of mathematics generally, but no source about him or his own text was directly read this session; he appears only via a Tier 3 secondary paraphrase (Rastogi & Nitin) and Tier 4 Wikipedia. Left as a mention inside claim-kapat-sandhi-gelosia-identification-rests-on-tier-3-source.md rather than promoted to a hub; revisit if a source about the Lilavati or its commentarial tradition is read directly.","Gaṇeśa Daivajña (entity candidate) — same reasoning as Bhāskara II: named only via the Tier 3 Rastogi & Nitin paraphrase, not directly engaged this session. Left as a mention, not a hub.","Jean-Luc Chabert (entity candidate) — editor of the unread A History of Algorithms (Springer, 1999); his significance here is entirely prospective (he is the trailhead, not yet a source this vault has read). Left as a mention and as the named target of the question's open progress line, not promoted to a hub. Revisit the moment his book is read.","David Eugene Smith (entity candidate) — classic historian of mathematics, cited only secondhand via Rastogi & Nitin; not read directly this session. Left as a mention.","Mahdi Abdeljaouad and Jeffrey A. Oaks (entity candidates) — authors of the Tier 1 source actually read and quoted this session (a stronger claim to a hub than the above), but not yet established elsewhere in the vault as recurring authorities the way de Blois was for the sibling Kalila wa-Dimna cluster; captured fully in claim-abdeljaouad-oaks-talkhis-study-omits-lattice-gelosia-shabakh.md's provenance instead. Revisit if their work recurs in a future capture.","Frank J. Swetz (entity candidate) — same reasoning as Abdeljaouad/Oaks: author of the Tier 2 source read this session, captured in provenance rather than promoted to a standalone hub.","Talkhīṣ aʿmāl al-ḥisāb (entity candidate, concept/term) — the vault's practice on this exact cluster (see entity-ibn-al-muqaffa.md / no separate Kalila wa-Dimna hub) is to carry a text's identity through the claim-notes and the author's entity page rather than a separate text hub; covered inside entity-ibn-al-banna-al-marrakushi.md instead.","Kapat-sandhi (entity candidate, concept/term) — same reasoning: carried by claim-kapat-sandhi-gelosia-identification-rests-on-tier-3-source.md rather than a separate term hub; it is a disputed identity claim, not yet an established recurring term independent of that one claim."]
origin: "batch"
writer_model: "claude-sonnet-5"
date_created: "2026-08-13T00:00:00.000Z"
provenance: "batch run, 2026-08-13, researching [[question-verify-lattice-multiplication-maghreb-india-primary]]"
derived_from: []
tags: ["history-of-mathematics","lattice-multiplication","islamic-golden-age","transmission-of-knowledge","verification","calculating-instruments"]
sources: [{"source_url":"https://mathshistory.st-andrews.ac.uk/Biographies/Al-Banna/","source_title":"al-Marrakushi ibn Al-Banna","source_author":"J J O'Connor and E F Robertson","source_date":"1999-11 (last update)","source_venue":"MacTutor History of Mathematics, University of St Andrews","source_tier":3,"source_sha":"1d49c41fd4a8c3b81442fb0f254cb4c9acb714b7241c667b1f56c2b1d6e291b4","source_quote":"Perhaps al-Banna's most famous work is Talkhis amal al-hisab (Summary of arithmetical operations) and the Raf al-Hijab which is al-Banna's own commentary on the Talkhis amal al-hisab. ... There are, however, many interesting mathematical ideas and results which appear in the Raf al-Hijab. For example it contains continued fractions..."},{"source_url":"https://al-furqan.com/al-hawaris-commentary-on-ibn-al-bannas-talkhi%E1%B9%A3-contents-and-influences/","source_title":"Al-Hawārī's commentary on Ibn al-Bannā's Talkhīṣ: Contents and influences","source_author":"Mahdi Abdeljaouad and Jeffrey A. Oaks","source_date":2013,"source_venue":"Suhayl: International Journal for the History of the Exact and Natural Sciences in Islamic Civilisation, Vol. 12, pp. 9-44 (article reproduced by the Al-Furqan Islamic Heritage Foundation)","source_tier":1,"source_sha":"e80daba090816d52d14a98c095c8959ef55067991cdea7a9cfff0d5b988bb247","source_quote":"Below is a list of the numbers of problems of different types taken by Ibn Ghāzī from al-Hawārī's book that are not in Rafʿ al-ḥijāb: Sum of numbers : 16 / Casting out nines or sevens : 3 / Types of multiplications : 4 / Computing the numerator of a fraction : 11 / Operations on fractions : 10 / Operations on roots : 26 / Double false position (scales) : 6 / Algebra : 27"},{"source_url":"https://web.archive.org/web/20260106010306/https://old.maa.org/press/periodicals/convergence/mathematical-treasure-ibn-al-bannas-operations-of-calculation","source_title":"Mathematical Treasure: Ibn al-Banna's Operations of Calculation","source_author":"Frank J. Swetz (The Pennsylvania State University)","source_date":"2014-01","source_venue":"Convergence, Mathematical Association of America (archived copy; original old.maa.org URL returned HTTP 521 repeatedly this session)","source_tier":2,"source_sha":"fa3f585eb2ac3b685c846b57f4590449e248b5a580e644f3ec1ab5313e910bcc","source_quote":"The Desire of the Students for an Explanation of the Calculator's Craving was compiled by Muhammad ibn Gāzī (ca. 1437-1513)... This book was a commentary on ibn Gāzī's own commentary, The Craving of the Calculators, on ibn al-Bannāʾ's The Abridgement of the Operations of Calculation. ... Figure 3. Examples of the gelosia method of multiplication using both right and left slanting diagonals within the grids in The Desire of the Students for an Explanation of the Calculator's Craving"},{"source_url":"https://instavm.org/wp-content/uploads/2021/05/H16.pdf","source_title":"History of Multiplication: Classical Period in Indian Mathematics","source_author":"Prabha S. Rastogi and Sandhya Nitin (JJT University, Rajasthan)","source_date":"undated (paper); c. 2020 per web metadata","source_venue":"hosted at instavm.org — no journal name or peer-review notice visible in the document itself","source_tier":3,"source_sha":"a91dd13c4d75b2a39e514c8cee7404c31e0c5be9b706a6b3f61cfe9b815f5b0d","source_quote":"Sridhara suggested another method called the 'Kapat-sandhi', which is another well-known method of multiplication... D.E. Smith in his History of Mathematics (1958) states that another method of multiplication called the Gelosia Method appears in the 'Ganita-manjari' (16th century) as the kapata-sandhi method. Ganesa in his commentary on the Lilavati also calls the gelosia method as the kapat-sandhi method."},{"source_url":"https://en.wikipedia.org/wiki/Lattice_multiplication","source_title":"Lattice multiplication","source_author":"Wikipedia contributors","source_date":"2026-01-07 (last edit per page footer)","source_venue":"Wikipedia","source_tier":4,"source_sha":"3135918f398b433b8452522b48861c95a08fb82dbef75874cc533df83e0ff076","source_quote":"Though lattice multiplication has been used historically in many cultures, a method called 'Kapat-sandhi' very similar to the lattice method is mentioned in the commentary on 12th century 'Lilavati'... [5] The earliest recorded use of lattice multiplication: [6] in Arab mathematics was by Ibn al-Banna' al-Marrakushi in his Talkhīṣ a'māl al-ḥisāb, in the Maghreb in the late 13th century.","source_note":"Both footnotes [5] and [6] on this page cite the same underlying academic source: Jean-Luc Chabert, ed., A History of Algorithms: From the Pebble to the Microchip (Berlin: Springer, 1999), p. 21 and pp. 21-26 respectively. That book could not be accessed this session (no open-access copy found; academia.edu and archive.org copies returned 403/401 to the vault's fetch tools) — see Further leads."}]
seek_code_commit: "17d9798"
---


This capture researches the open question at
[[question-verify-lattice-multiplication-maghreb-india-primary]], itself raised
against [[claim-lattice-multiplication-india-maghreb-italy-transmission-chain]].
That note's two load-bearing attributions — that **Ibn al-Banna's** *Talkhīṣ aʿmāl
al-ḥisāb* (late 13th c., Maghreb) contains the lattice/gelosia method, and that
**'Kapat-sandhi'** in a 12th-century *Lilavati* commentary is the same method —
rest entirely on Tier 4 Wikipedia. This capture went looking for Tier 1–2
confirmation of both. It found partial, complicating evidence rather than
confirmation.

## Claim: The most thorough modern scholarly study of the Talkhīṣ's actual contents does not mention lattice, gelosia, or shabakh multiplication anywhere

**Claim type**: historical / technical-mechanism (an inventory claim: what is and is not inside a specific named text). **Source tier required**: Tier 1–2.

Abdeljaouad and Oaks (2013), published in *Suhayl*, a peer-reviewed history-of-
science journal, is a close study of al-Hawārī's 1305 commentary on Ibn al-Banna's
*Talkhīṣ* — written by one of Ibn al-Banna's own students during the master's
lifetime, and the earliest commentary on the text that exists. The article
catalogues, operation-by-operation, everything al-Hawārī's commentary adds to the
*Talkhīṣ* and Ibn al-Banna's own *Rafʿ al-ḥijāb*, including a table of material a
15th-century successor (Ibn Ghāzī) later borrowed from al-Hawārī: "sum of numbers,"
"casting out nines or sevens," "**types of multiplications**," "computing the
numerator of a fraction," "operations on fractions," "operations on roots,"
"double false position," and "algebra." The 44-page article, read in full this
session, never uses the words "lattice," "gelosia," or "shabakh," and "types of
multiplications" is never itemised further than the bare count of 4. A study built
specifically to catalogue the *Talkhīṣ* tradition's contents in detail does not
independently confirm that lattice/gelosia multiplication is among them.

*Provenance*: Abdeljaouad, Mahdi; Oaks, Jeffrey A. "Al-Hawārī's commentary on Ibn
al-Bannā's Talkhīṣ: Contents and influences." *Suhayl* 12 (2013), 9-44.
https://al-furqan.com/al-hawaris-commentary-on-ibn-al-bannas-talkhi%E1%B9%A3-contents-and-influences/

## Claim: The earliest published manuscript evidence for gelosia-grid multiplication in the Ibn al-Banna tradition is a 15th/16th-century commentary, not Ibn al-Banna's own 13th-century Talkhīṣ

**Claim type**: technical-mechanism / historical. **Source tier required**: Tier 1–2.

Frank J. Swetz, writing in the Mathematical Association of America's *Convergence*
and captioning manuscript images sourced from the World Digital Library / Library
of Congress, shows gelosia-grid multiplication examples ("Examples of the gelosia
method of multiplication using both right and left slanting diagonals within the
grids") — but they appear in *The Desire of the Students for an Explanation of the
Calculator's Craving*, a work compiled by **Muhammad ibn Gāzī (Ibn Ghāzī
al-Miknāsī, ca. 1437–1513)**, two centuries after Ibn al-Banna. Ibn Ghāzī's book is
itself a commentary on his *own* earlier commentary (*The Craving of the
Calculators*) on Ibn al-Banna's *Talkhīṣ* — three removes from Ibn al-Banna's
original 13th-century text, not a description embedded in the *Talkhīṣ* itself.
This is real manuscript evidence that gelosia multiplication circulated *within
the Ibn al-Banna commentarial tradition*, but it dates the earliest documented
instance to the Ibn Ghāzī branch of that tradition, not to Ibn al-Banna's own hand
— a materially different claim from the one Wikipedia (and the seedling note it
feeds) currently makes.

*Provenance*: Swetz, Frank J. "Mathematical Treasure: Ibn al-Banna's Operations of
Calculation." *Convergence* (January 2014), Mathematical Association of America.
https://web.archive.org/web/20260106010306/https://old.maa.org/press/periodicals/convergence/mathematical-treasure-ibn-al-bannas-operations-of-calculation
(original old.maa.org URL returned HTTP 521 on every direct-fetch attempt this
session; cited via the Wayback Machine capture instead).

## Claim: The Kapat-sandhi / Ganesa's-Lilavati-commentary identification with gelosia rests, in this search, only on a Tier 3 source

**Claim type**: technical-mechanism (an identity claim between two named methods). **Source tier required**: Tier 1–2.
**Flag: [unverified-mechanism — needs primary]**

A named-author paper (Rastogi & Nitin, JJT University, Rajasthan) states plainly
that "Ganesa in his commentary on the Lilavati also calls the gelosia method as the
kapat-sandhi method," citing D.E. Smith's *History of Mathematics* Vol. II (1958),
p. 115, and Datta & Singh's *History of Hindu Mathematics* (1935), p. 138, as its
authorities. This is the single most direct statement of the Indian-leg identity
claim found this session, and it is more specific than Wikipedia's own "very
similar to" phrasing. But the paper's publication venue could not be established —
it is hosted at instavm.org with no journal name, volume, or peer-review notice
visible anywhere in the document — so it does not clear the Tier 1–2 floor this
claim type requires. The two underlying classic sources it cites (D.E. Smith 1958;
Datta & Singh 1935) were not independently locatable in full text this session
(see Further leads). The claim is recorded here, not promoted, and stays flagged.

*Provenance*: Rastogi, Prabha S.; Nitin, Sandhya. "History of Multiplication:
Classical Period in Indian Mathematics." Undated paper hosted at
https://instavm.org/wp-content/uploads/2021/05/H16.pdf

## Claim: Wikipedia's own citation trail for both halves of the original claim points to a single unread academic source

**Claim type**: historical (source-provenance claim about the trailhead itself). **Source tier required**: n/a — this documents what remains to be checked, not a claim about the world.

Both the Ibn al-Banna/Talkhīṣ attribution and the Kapat-sandhi/Ganesa mention on
Wikipedia's "Lattice multiplication" page cite the same footnote: Jean-Luc
Chabert (ed.), *A History of Algorithms: From the Pebble to the Microchip*
(Springer, 1999), p. 21 and pp. 21-26. This is a real, reputable academic-press
book (a standard reference in the history of computation, with named
contributing historians including work touching Arabic mathematics) and is very
plausibly the actual primary/secondary source that would resolve this question
outright. It could not be read this session: no open-access text was found, the
academia.edu mirror returned 403, and an archive.org copy returned 401
Unauthorized to the vault's extract tool. This is the single highest-value
unread lead this capture produced.

## Further leads

- Jean-Luc Chabert (ed.), *A History of Algorithms: From the Pebble to the Microchip* (Springer, 1999), pp. 21-26 — the actual source behind both Wikipedia claims; unread this session (academia.edu 403, archive.org 401). Chasing a library/institutional copy would likely settle this question directly.
- D.E. Smith, *History of Mathematics*, Vol. II, "Special Topics of Elementary Mathematics" (1925; Dover reprint 1958), p. 115 — cited by Rastogi & Nitin for the Gelosia/kapata-sandhi identification; a full-text copy exists on archive.org but the specific passage was not located this session (search-inside tooling did not surface it).
- Datta & Singh, *History of Hindu Mathematics* (1935), p. 138 — the other classic source Rastogi & Nitin cite for kapat-sandhi; not independently checked.
- muslimheritage.com's "Decimal Arithmetic" page attributes the popularisation of "sieve"/"lattice" multiplication to **al-Karaji** (successor to al-Khwarizmi, 11th c.), not to Ibn al-Banna — a competing attribution worth chasing against a Tier 1–2 source, since it would predate Ibn al-Banna by two centuries.
- manasganit.com, "Multiplication by Kapat Sandhi Method — Gelosia" — an Indian mathematics-education site with a page devoted to exactly this identification; unread this session, tier unassessed.
- Al-Hawārī's own 1305 commentary (*al-Lubāb*) — discussed at length in the Suhayl article above but not itself read directly; as the earliest and most numerically detailed commentary on the *Talkhīṣ*, written within Ibn al-Banna's lifetime, it is the single best candidate primary text to check directly for multiplication-method content.
- Grokipedia's "Lattice multiplication" page surfaced repeatedly across searches this session. It is an AI-generated encyclopedia (xAI), not a human-edited one; treat as below Wikipedia's own Tier 4 and do not cite as evidence — noted here so a future pass doesn't mistake its repeated appearance in search results for independent corroboration.

## Answer to the core question

**[unverified — could not confirm or deny after search]**, and the search
produced evidence that complicates the original claim rather than confirming it.
The one Tier 1 source that specifically catalogues the *Talkhīṣ*'s actual
contents (Abdeljaouad & Oaks 2013) does not mention lattice/gelosia/shabakh
anywhere. The one piece of real manuscript evidence located (Swetz/MAA,
Tier 2) places the earliest documented gelosia grids in the Ibn al-Banna
tradition two centuries after Ibn al-Banna himself, in Ibn Ghāzī's commentary.
The Kapat-sandhi/Ganesa identification is stated plainly by a named-author paper
but that paper does not clear the Tier 1–2 floor. The specific academic source
Wikipedia's own footnotes point to for both halves of the claim (Chabert 1999)
was not accessible this session and remains the most promising unresolved lead.

> [!note] Seek's commentary:
> Went in expecting to either confirm or fail-to-confirm a thin Wikipedia claim,
> and came out with something more interesting: the strongest source that
> actually examined the *Talkhīṣ*'s contents in detail is silent on lattice
> multiplication, and the strongest manuscript evidence for the method inside
> the Ibn al-Banna tradition belongs to a different author two centuries later.
> That's not "unverified," that's mildly *against* the specific claim as
> Wikipedia states it — though it's not strong enough evidence to assert the
> negative outright, since absence from one catalogue and one manuscript sample
> isn't proof of absence from the whole tradition. The parent note should stay
> at seedling, and the flag on it should probably sharpen from "unconfirmed" to
> "the confirmed evidence points at Ibn Ghāzī, not Ibn al-Banna" once this
> capture is promoted. The Chabert book is the loose thread worth pulling next.
> — Seek

## Entity candidates

- Bhāskara II (Bhāskarāchārya) — person — author of the 12th-c. *Lilavati*, the foundational Indian text every "Kapat-sandhi" claim is a commentary *on*; the whole Indian leg of the transmission chain rests on his original text, not just on its commentators.
- Gaṇeśa (Gaṇeśa Daivajña) — person — 16th-c. Indian astronomer/mathematician whose *Lilavati* commentary is the actual named source of the "kapat-sandhi = gelosia" identification per every source found this session; the foundational figure the Indian-leg claim measures itself against, and currently under-cited relative to Wikipedia's vaguer "a commentary."
- Ibn al-Banna al-Marrakushi — person — subject of the core question; see [[entity-ibn-al-muqaffa]] for an unrelated same-era Arabic-transmission figure already in the vault (different person, note only for domain-adjacency).
- Ibn Ghāzī al-Miknāsī (Muḥammad Ibn Ghāzī al-ʿUthmānī al-Miknāsī, 1437–1513) — person — author of the commentary tradition where gelosia-grid multiplication is actually attested in manuscript form per Swetz/MAA; may be the correct "first Arabic attestation" figure rather than Ibn al-Banna.
- Al-Hawārī (ʿAbd al-ʿAzīz ibn ʿAlī ibn Dāwud al-Hawārī al-Miṣrātī, fl. early 14th c.) — person — Ibn al-Banna's own student, wrote the earliest (1305) commentary on the *Talkhīṣ* during the author's lifetime; the closest primary-adjacent source to Ibn al-Banna himself.
- Jean-Luc Chabert — person — editor of *A History of Algorithms: From the Pebble to the Microchip* (Springer, 1999), the actual academic source both Wikipedia claims trace back to; unread, high-value target.
- David Eugene Smith — person — author of *History of Mathematics* Vol. II (1925/1958), the classic secondary source underlying the Kapat-sandhi/gelosia identification per Rastogi & Nitin.
- Mahdi Abdeljaouad and Jeffrey A. Oaks — people — historians of Arabic mathematics, authors of the 2013 *Suhayl* study of al-Hawārī's commentary used above.
- Frank J. Swetz — person — Penn State historian of mathematics, MAA *Convergence* author whose captioned manuscript images are the best evidence located this session.
- *Talkhīṣ aʿmāl al-ḥisāb* — concept/term — Ibn al-Banna's core arithmetic treatise, the specific document the whole question is about.
- Kapat-sandhi — concept/term — the Sanskrit multiplication-method name ("junction of doors") at the center of the Indian-leg identification claim.