HomeTimeline of the etymology of the term ‘algorithm’
Timeline of the etymology of the term ‘algorithm’
The following table lists over 100 sources from antiquity to modern times. Summaries and links to the originals are provided for almost every source. The table presents a coherent overall picture that comprehensively supports the findings of the van Helsing study on the etymology of the term ‘algorithm’.
The table provides detailed evidence as to why the previous etymological derivation of the term algorithm should be regarded as a speculative retrojection from the 19th century. The main problems lie in the fact that, for a thesis to be reliable, the general legal principle of evidence applies: according to this principle, assumptions and speculations must be identified as such. If no evidence is provided, it cannot be considered reliable proof.
First, here is a small selection of the inconsistencies listed in the timeline with links to all original sources:
The narrative originated in 1849 from a conjecture by Reinaud. He referred to an astrological book by Hellert from 1548. As early as 1877, Wüstenfeld pointed out that the last page of the book to which Reinaud referred did not mention any Arab or Indian scholars at all.
Not even a word similar to Algorismus or Alchoarizam is mentioned there. Wüstenfeld’s discovery is demonstrably correct, which is why Reinaud’s assumption could hardly refer to al-Hwarizmi. Even the ‘zero hour’ of the eponym thus raises considerable questions.
Furthermore, there is still no evidence that al-Hwarizmi wrote a book on arithmetic. His algebra also does not match Dixit Algorizmi, as the principle of the root (radix) does not correspond to his teachings.
Finally, there is no evidence that the Oxford manuscript of Dixit Algorizmi (verifiably 13th century or younger) has an older Latin original. The age of the book is only presumed to be the 12th century because its creators are believed to have been active during the time of the Toledo School of Translators. Therefore, the Oxford manuscript must be a copy.
Conversely, there is Ibn al Yasimin, a West Arabian scholar who is proven to have written the two books mentioned in Dixit Algorizmi, who describes the unit (root/radix) exactly as mentioned in Dixit Algorizmi and who also conclusively explains the use of the dust board.
All other contradictions of the eponym thesis, such as Fibonacci’s negative assessment of the algorithm and Jacopo Firenze’s mention of the art in 1307, are thus almost completely resolved, as shown chronologically below.
The al-ghubar thesis favoured by the Real Academia Española (RAE), according to which the term algorithm is derived from the West Arabic dust calculation (Alguaritmo/Alchoarismus), can be supported by a large amount of evidence and does not require the assertion that the entire Middle Ages ‘forgot’ the origin of the term as it relates to al-Hwarizmi.
Most importantly, the al-ghubar thesis was not established by the RAE at all: it was Enrico Narducci, Baldessare Boncompagni’s secretary, who first published it – after years earlier classifying the derivation with al-Hwarizmi as theoretically possible but not factually provable.
Overall, the timeline proves that the etymology of the term was not a ‘big bang’ but an evolutionary process with many influences. The overview shows that Arabic influences continued to be very significant. However, there were also influences from India and Europe, so that the concept of the algorithm refers to a unique intercultural origin in which many actors proactively collaborated.
Timeline of the etymology of the term ‘algorithm’
No
!
Time
Year
Actor
Link
Work
Context
Region
Link
1
a
Antiquity
unknown
HSB - Arithmetic Papyrus Rhind
H-S-B as the linguistic origin of Arabic hisab (al ghubar)? Egyptian antiquity is particularly relevant in the Maghreb context, as it is partly an ancient arithmetic tradition. Number of strokes as numbers 1-9. Found in Dixit Algorizmi: "Number is therefore nothing else but a collection of ones," translated: 1 = I, 5 = IIIII or seven = IIIIIII; according to Crossley/Henry 1990, other contents of Dixit also point to Egyptian roots; A particularly revealing example from the Rhind Papyrus (c. 1650 BC) is the so-called "Seven Houses Problem", the solution of which is structured as7 + 7^2 + 7^3 + 7^4 + 7^5. The problem shows a clearly defined, step-by-step calculation procedure that works independently of the external numerical form – a precise precursor to what the Middle Ages later referred to by Roger Bacon as viae algorismi ("arithmetic paths"). Crosby 1997 explains that it is numbers and digits.
Egypt
2
d
Antiquity
unknown
Papyrus Ebers
makes it clear that control stamps are already used in medical processes in the sense of "first this, then that". Also striking with regard to numbers are the page numbering, which are interesting structural markers with dashes, dots and individual symbols.
Egypt
3
d
Antiquity
Pythagoras
Mathematics
Establishes mathematics as a science with the four disciplines of arithmetic, geometry, astrology and music; it illustrates, among other things, the connection between numbers and geometry; whether and to what extent the abacus later named after him goes back to his person can be left open here.
Greek
4
d
Antiquity
Archimedes
Salamino Table
Early abacus, which already reveals the basic features of the later line algorithm (Middle Ages)
Greek
5
d
Antiquity
Various
Cyclical Computational Models
astronomy, accrual calculus; Partial foundation for later astronomical algorithms
Greek
6
d
Antiquity
unknown
Antikythera Mechanism
highly developed "primordial astrolabe"; proves the early possibility of sophisticated mechanical-deterministic calculations
Greco-Roman
7
d
Antiquity
Nikomachus of Gerasa
Introductio arithmetica
Number theory: unit ≠ number; The principle of unity is the basis of numbers, but not a number itself (which only arises through the union of several units). For him, numbers are the sum of units. A primal idea for algebra, so to speak; "al-ǧabr" "the joining of broken parts"; Similarity to the Egyptian approach of the "I" as an element of "IIII" and "IIIIIIII" etc.
Greek
8
c
Antiquity
Boethius
De institutione arithmetica
Logical writings (syllogisms, ratio, arithmetic); theoretical statements on unity, number and root in algorism texts also go back to Nicomachus/Boethius and are to be distinguished from operational arithmetic practice (ġubār / algorism); Logic of Music; Fundamentals of the Quadrivium -> refers to the four advanced mathematical subjects of the seven liberal arts: arithmetic, music, geometry and astronomy; Quadrivium, for its part, can be found as a reference in many medieval texts on algorism (e.g. Salem Codex)
Roman
9
c
8th century
774
Kanka
Siddhānta tradition (astronomical-arithmetic)
Indian scholar, as he is later mentioned by Al Andalusi as a master of hisab al ghubar (= dust calculation); important, since in early algorism poems (Carmen & Tractatus) a similar person is portrayed and dust calculation is practiced. Kanka himself is not a philosopher-king, but the account described by Al-Qifti in the Chronology of the Scholars is younger than that of Al-Andalusi, but very similar in content
India
10
d
8th century
776
Muḥammad al-Manṣūr
Sindhind al-Kabīr (Translation Project)
Indian numerals come to Eastern Arabia; important because Indian and Arabic numerals and calculation methods are not identical
Baghdad
11
d
9th century
Albumanzar
Astrological texts
Since his works were demonstrably translated into Latin earlier than al-Ḫwārizmī's algebraic works were received, it is plausible that the Latin term "algorism" first arose in the context of astronomical-astrological arithmetic practice of Johannes Hispalensis and Hermann of Carinthia. In this context, Albumanzar did not describe authorship, but an operational method of dealing with numbers, tables and positions. See for more details under Johannes Hispalensis and Hermann von Kärnten.
Badgad
12
d
9th century
al-Kindī
Kitāb fī isti'māl al-'adād al-hindī
essential philosopher for the foundation of hisab al hindi, the Arabic term for arithmetic with Indian numerals; Development of this teaching (al-Hindi) even before al-Hwarizmi
Baghdad
13
b
9th century
al-Khwarizmi
Kitāb al-ǧabr wa-l-muqābala (algebra)
The book is the foundation of classical algebra, the science of solving equations. For centuries, it shaped the character of algebra as a practical science without an axiomatic foundation; however, Latin translation did not take place until the end of the 12th century. However, this does not ensure whether and to what extent he was really known in the Latin world before; Al-Andalusi later mentions his name, but as an astronomer; his algebra is not mentioned by authoritative Western Arab scholars (including al-Andalusi, Yasamin, Hassan). Not even from Ezra. Al-Hwarizmi did not know Radix (unlike Yasamin). Nor did he write didactic poems, as was initially common for doctrines of algorism
Baghdad
14
a
9th century
al-Khwarizmi
arithmetic book mentioned in Dixit; fictitious, merely backward-projected fictitious titles such as "Kitāb fī ḥisāb al-Hind" or "De numero Indorum"
Such a book is often cited as evidence of Dixit authorship or lost original, but is not verifiable; the book is not lost, but not verifiable; even the Arab world says to this day: "It is universally known that the Arabic original has not been found so far" (see link); also no entry in Fihrist or Hadschi Calfa, despite the enumeration of his works; no citation evidence in the entire Arab world; lost book would hardly be identical in content with the later scholastic algorism or the content of Dixit Algorizmi; Crossley/Henry show many differences to algebra, which can be solved much better with Yasamin (e.g. radix and arithmetic board/Ghubar arithmetic);regarding hisab al hind: tradition as a whole uncertain; can hardly be traced back to al-Hwarizmi, therefore fictitious titles in this regard are hardly accurate in terms of content
Baghdad
15
d
9th century
Dunash ibn Tamim
Hisab Al Ghubar
early evidence of Hisab al Ghubar in North Africa/Western Arabia; Computing continues to develop independently in the Western Arabian world
Maghreb
16
c
10th century
Ahmad al Biruni
Astronomy, Mathematics, Chronology
important astronomer; is examined with regard to origin from Khorezm (al-Hwarizmi) by both Reinaud and Cantor et al as a potential namesake for Algorizmi; makes it clear that a total of more than five different scholars from the region were known as "al-Hwarizmi" - and Ben Musa could therefore hardly be the only person in the Middle Ages known only under the Nisba.
Badgad
17
b
10th century
Gerbert of Aurillac
Letters and Geometria
later Pope Silvester II; rediscovery of the abacus, knowledge of Western Arabic numerals, operational arithmetic practice; has a considerable influence on scholars of the time; propagates the added value of Arabic-Latin hybrid terms in the field of mathematics, which also formally legitimizes the interpretation of "algorism" as a hybrid abbreviation of "al-ghubar-sillogism"; Prefix al-ghubar (Arabic) & suffix -ism (Latin-Greek). Such hybrid terms were later questioned by Cantor as "linguistic peculiarities" in order to interpret the overall term "algorizmi" as a uniform word or name.
France/Spain/Italy
18
b
10th century
Gerbert of Aurillac
Abacus Calculation
Gerbert did not use the corresponding number of arithmetic stones (see Calculus) to represent a number, but rather horn plates inscribed with the corresponding Indian number (see Apices). He was able to perform unknown arithmetic operations at high speed with his abacus, which is why he came under suspicion of witchcraft. Comparable plates are mentioned in 1975 by Mazaheri from Persia: Quote: "We have carefully translated al-takht as "planchette" (plate / board) – not as "dust tablet"". In this respect, Gerbert already had the possibility of a symbolic-rule-based arithmetic practice, which is structurally closer to written procedures such as the hisāb al-ghubār than to older abacus traditions. He also mentions wiping/erasing, as is typical for Ghubar; Particularly important: In his time, there was a hardware problem, because you couldn't calculate Indian with the classic abacus and paper was in short supply. Therefore, a new type of arithmetic board was required in order to calculate effectively and efficiently (also) with Indian numerals. The hisab al ghubar would make this possible.
France/Spain/Italy
19
d
10th century
Richer of Reims
Historiae
prepares reports on Gerbert's arithmetic theory; the abacus rediscovered by Gebert, precisely circumscribed by Richer, was a calculating board in the sense of a "dust board" with which Arabic numbers could also be used, which would explain the high speed with which Gerbert could calculate (an aspect that later led to the accusation of witchcraft, see above)
France
20
a
10th century
Ibn al-Nadim
Fihrist
in the leading encyclopedia of the Arab world in the 7th volume several works by al-Hwarizmi are listed, but none on al-Ḫwārizmī's algorism or arithmetic book; the thesis that this book was already "forgotten" in Baghdad at that time can hardly be factually justified, even if other Islamic encyclopaedias such as Haji Calfa or al-Andalusi do not know it either; the same applies to the thesis of editorial errors in the Fihrist (e.g. accidental attribution of the arithmetic book to other scholars). Entries in the Fihrist are also made under the full name "Mohammed Ben Musa"; the nisba "al-Hwarizmi" is used by several scholars, not only by him, so it is neither exclusive nor "identity-forming" in the sense of an associated individual who would have been known as "algorizmi"; a heading, as Dodge introduced in translations in 1970 for lexical reasons, does not exist in the Arabic original
Badgad
21
d
10th century
unknown
Codex Albeldensis
documents knowledge of Indo-Arabic numerals in Europe and Spain; important, because in the region of northern Spain and Provence commercial abacus methods are later developed and taught (e.g. Alguarismo & reports by Jacopo)
Spain
22
d
10th century
Kushyar ibn Labban
Principles of Hindu Reckoning
The Ḥisāb al-Hind is significantly influenced by him; this proves that two Indian calculation methods existed in parallel in the Arab world in the Middle Ages: Indian arithmetic and dust arithmetic; In this respect, Western and Eastern Arabia continue to develop partly independently of each other; medieval algorithms, however, are more influenced by Western Arabic, less by Eastern Arabic
Badgad
23
d
10th century
Maslama al-Majriti
Muʿamalat
According to al-Andalusi, only revises the astrological tablets of al-Hwarizmi; criticizes errors of al-Hwarizmi astro tablets; is also the author of a commercial arithmetic book (Thimár 'Im al-'Adad/Muʿamalat). On the one hand, confirms that al-Hwarizmi was known, but not with reference to arithmetic, but in Western Arabia primarily with regard to astrology; not primarily in relation to the name "al-Hwarizmi", but "Mohammed Ben Musa"
Al-Andalus
24
a
11th century
Garlandus Composita
Regulae super abacum
For the first time, continuous use of Ghubar numerals and numerals in Latin abacus text; procedural logic: words/numbers (positional/relation-based). Parallel to singular syllogisms for specific topics (comparison to Dialectica v. Abelardus): operations valid by singularity, without 'omnis' (superfluitas omnis = everything superfluous falls away = maximum reduction). In addition, bridge: ġubār practice / algorism.Potential syllogism in Garlandus acts as a binary operator: an operation is either structurally valid (1) or invalid (0), regardless of the semantic meaning of the number words. This enables maximum networking of character positions with minimum use of resources (elimination of the 'superfluitas omnis'); Conversion of numbers into Greek letters and use in multiplication, abacus and rithmimachia corresponds to what Roger Bacon similarly compares as a practice
France/Switzerland
25
a
11th century
Garlandus Composita
Tabulae
Singular, rule-guided operations with signs (words = numbers); Validity from structural effectiveness, not semantic. Functionally identical to ġubār computational practice: Factual application of algorism as an "al-ġubār syllogism" (procedural, singular). Precursor to the Dialectica Logic of Abelardus
France/Switzerland
26
d
11th century
Abbo de Fleury
De syllogismis hypotheticis
positional algorithms for letters, but also transferable to digits (Gerbert transfer); potential precursor of Latinization of the ġubār method; relevant in that Ghubar was used for both characters and numbers, and numbers were also represented as words; Indian numerals have always had the challenge of writing and speaking to function, therefore sillogismic logic relevant for the positioning of both aspects, in this respect also well compatible with the thesis of the "al-ġubār syllogism": numbers/digits and letters/words for numbers are equivalent. Cf. similar approach later ibn Ezra
France
27
b
11th century
1068
Said al-Andalusi
Ṭabaqāt al-umam
The Ṭabaqāt al-umam (category of nations) was one of the best-known encyclopedias in the Western Arab world. It lists the biographies of the most important scientists and scientific achievements of eight nations.
The work mentions Indian "Phliolsophian king" as the mythical master of hisab al ghubar; mentions history that is reminiscent of Kanka (see above), but Kanka is not the king himself). A similar "Philosope king" is later found in Sacrobosco and Villa Die as the mythical founder of algorism;
Al-Andalusi explicitly mentions "Mohammed ben Musa al-Ḫwārizmī . In the Ṭabaqāt al-umam, however, he is primarily mentioned as an astronomer;
Important: In the translation of Said's work by Cheikho (1912), an "Abu Jafar Muhammad ibn Musa al-Khwarizmi" is also mentioned as the founder of al-Ghubar arithmetic. Quote: "It is the most concise form of calculation, the fastest available, the easiest to handle, the easiest to learn and the most wonderful in its structure. For India, it testifies to a sharp mind, a gift for new developments and ingenious inventiveness." Wüstenfeld gets to the heart of the problem in 1877: "Abu Jafar" was not al-Hwarizmi's first name. So it must be another "Abu Jafar" who brought this method of calculation to Al-Andalus - which is also proven by the works of Hassar etc. It shows once again how translations from Arabic are "supplemented" with the knowledge of modern times and thus cause distortions.
Conclusion: al-Hwarizmi is known in al-Andalus, but not as an artithmetician, but rather as a mediator of Sindhind; he is also known as Ben Musa - not under the Nisa, as this name is evidently used by several scholars.
Al-Andalus
28
d
11th century
Jacob ben Nissim
Sefer Yetsirah
potential author of Hebrew writings on Indian numerals and reference to ḥisāb al-ġubār; shows that this form of calculation was interculturally widespread in the Mediterranean region
Maghreb
29
d
11th century
Omar al-Khayyām
Algebraic work
is later compared by Woepcke to al-Hwarizmi and thus becomes a multiplier of al-Hwarizmi's popularity in modern times, since he was almost unknown in Europe before 1831
Persia
30
b
12th century
Abraham ibn Ezra
Book of Unity
originates from the Jewish-Andalusian or Maghreb chain (ġubār tradition via Toledo/Maghreb). Names "Galgal" (wheel) for the zero. For him, algebra is not related to al-Hwarizmi, but much more closely based on al-Yasamin's algebra; philosophical as divine art; Particularly important: Ezra combines the logic of letters and words with the logic of numbers to form a superordinate positional logic, which is reflected in the syllogism (words) and the later algorism (numbers). Implementation and positioning of Ghubar both as a font and as a sign. cf. Abbo v. Fleury.Ibn Ezra's positional logic is the direct precursor of hybrid value extraction. Since he understands letters and numbers as relational operators (ġubār tradition), the individual language becomes a mere shell. The 'divine art' here is the ability of the system to liberate the universal value(s) from the multiplicity of concepts (spoken/written in different languages as a word or as a number)
Spain
31
a
12th century
Petrus Abelardus
Dialectica
Abelard provides the philosophical operating system for the transition from semantics to procedural logic. His theory of singular syllogisms establishes the validity of an operation solely by the singularity of the term (= sol(a): a term represents a quality/substance without the ballast of general categories). He declares the universal quantifier (omnis / "all") to be superfluous (superfluitas 'omnis'), which massively increases the computational speed in the head and on the board. Procedural logic: As with ġubār arithmetic, it is not what a sign "is" that counts, but where it is placed and how it is procedurally linked. Defines signs as relational operators (similar to "AI tokens"): A word or number loses its "morality" (meaning weight) and becomes a functional building block in a network.The system extracts the value (quality) and ignores the language envelope (superfluitas). Exclusion of semantics in favor of structural inference velocity. The algorithm as "liberated logic".
France
32
d
12th century
Ibn al-Qiftī
Ta'rikh al-hukama
He describes how Indian arithmetic came to Badgad in the 8th century (Kanka); in terms of content, it is very similar to the description of Al-Andalusi, but without mentioning the philosopher king.
France
33
d
12th century
1154
Giovanni Scriba
Il Cartolare
Italy's oldest notary book with many records of trade relations and relationships between Genoa and other countries in the Mediterranean region (including Arabia, Tunis, Maghreb, Sardinia, Sicily). Uses Roman numerals only. However, it clarifies the importance of the operative rake with fractions, among other things ("6.5 denarii" or "half of .XVIII."). Mathematics was (as in Fibonacci/Jacopo/Villani) an operational task, e.g. in the sharing of values among people ("who gets how much?").
Italy
34
d
12th century
Abū Bakr al-Ḥaṣṣār
Amal al-ġubār (Hebrew translation)
He explicitly calls the arithmetic he teaches "the art of dust calculation", which would mean exactly "al-ghuar-ism"/alchoarism" in the sense of a Latin-Arabic hybrid term. High phonetic similarity of the terms and very high correspondence in terms of content with medieval algoric texts. He also does not mention al-Hwarizmi, which proves that he did not have the same significance in the western Arab region as in the eastern Arab region; The use of al-ghubar corresponds exactly to the doctrine of Florence in 1307 (quote "in Arabic it is called algho") and to the commercial understanding in Catalonia in the 14th century (alguarismo)
Egypt
35
d
12th century
Abū Bakr al-Ḥaṣṣār
Amal al-ġubār (original)
The original edition shows: Absolutely identical content to algoristic texts in the Latin world.
Egypt
36
a
12th century
Ibn al-Yasamin
al-Urǧūza fī l-ǧabr wa-l-muqābala
Important: The formulation of Dixit Algorizmi, which is attributed to al-Hwarizmi, also applies to ibn Yasamin: "Et iam manifestavi in libro algebrae et almucabala". In al-Hwarizmi, however, only algebra is attested. Yasamin has demonstrably written both books (see follow-up entry). Yasamin's teaching also fits the Dixit much better in terms of content than al-Hwarimi's algebra. Yasamin finally also wrote an instructional poem, as did Sacrobosco and Villa Die > the same style with the logic of "unity". He defined the "root" (jidhr) and the "number" in a way that was similarly taught in the Latin West. He saw unity as something that stands outside the series of numbers, but establishes everything – a philosophical-mathematical depth that went beyond mere arithmetic. Unity as root -> identical mathematical syntax as in Dixit. Radix, Census. He also does not mention al-Hwarizmi in his works, which raises the question of why the algebra works of the long-deceased al-Hwarizmi should have been better known in Al-Andalus and the Latin world than those of the still living ibn Yasamin.
Maghreb
37
a
12th century
Ibn al-Yasamin
Talqīḥ al-afkār fī ʿamal rušūm al-ġubār
His concept of "dust drawings" (Rushum) shows that it is about operative graphics – the drawing of symbols in the medium of dust. This is the exact precursor to what Narducci identified in 1883 as "Apices without Abacus". Core problem: The entire 19th century could not have known Yasimin's texts, as they were not translated from Arabic until the 20th century. There was no edition, no translation, no mention in the relevant European reference works of the 19th century. This explains why initially only al-Hwarizmi was suspected as a potential author of the Dixit: simply because there was still a lack of known alternatives at the time. The al-Hwarizmi eponym was consequently able to prevail and spread into the 20th century precisely in the absence of alternative explanations of the content of the Dixit, since it was unknown for a long time that Yasamin created two texts that fit exactly to the Dixit Algorizmi in combination, style and content
Maghreb
38
a
12th century
Hermann of Carinthia
Introductorium in astronomiam (translation of Albumazar)
The 1506 edition linked here is a printed version of a Latin translation, and the self-reference to "Herman(n)" and the translation difficulties clearly suggest that the text goes back to the translation of Hermann von Kärnten (Hermannus Dalmata / Hermann von Kärnten). Important: In the prologue, science is described as dealing with sand by the sea: not as meaning, but as an infinite number of discrete, movable units. This is exactly where the proximity to algorism in the sense of al-ghubar lies. Possibility: The ysagogarum Alchorizmi is the arithmetic counterpart to Abū Maʿšar's astronomical-astrological introduction.
Austria
39
a
12th century
possibly. Johannis Hispalensis
Flores astrologiae (translation of Albumazar)
A little earlier in time than the translation by Hermann of Carinthia.
Maghreb
40
a
12th century
possibly. Johannis Hispalensis
Introductorius liber qui et pulver is dicitur in mathematicam disciplinant
Vatican: Reg. Lat 1285: This work is one of the central documents on the early history of algorism. Narducci (1883) cites it as proof that algorism and dust calculation (powder = dust) were understood identically in the Latin Middle Ages; the dust calculation corresponds to the Arabic ġubār method. Eneström (1906) also shows that the text is essentially identical in content to the Liber algorismi de practica arismetrice edited by Boncompagni (1857).
Maghreb
41
a
12th century
unknown
Introductorius liber qui et pulver is dicitur in mathematicam disciplinant
This book has the almost identical title as the previous one, but is not explicitly attributed to any author. The content does not seem to be 1:1 identical to that of the Vatican: Reg. Lat 1285, but it also explains the place value calculation.Particularly significant, however, is the context of the anthology (Bodleian Lyell 52): The pulveris text is here immediately before the Liber ysagogarum Alchorismi. This is evidenced by an early systematic division of algorism: – as an operative practice (dust, line and figure calculation: practica) – and as a theoretical-propaedeutic introduction (ysagoge), in which the underlying principles of figures, positions and relations are explained. In the 12th century, the term algorism thus did not primarily refer to a method of calculation in the modern sense, but to a figure-based system of order that could be taught both operationally (by positioning, moving, deleting figures) and theoretically (by introducing the logic of figure, locus and positio). The coexistence of pulveris and ysagoge in the same codex probably marks the decisive phase of the "marriage" of ġubār practice and the Latin logic tradition, from which medieval algorism emerged. The age is probably younger, if this interpretation for the ysagogare Alchoarizmi should also apply to the Liber pulveris in the same book: https://manuscripta.at/_scripts/php/cat2pdf.php?cat=wichner&ms_code=AT1000-612
unknown
42
a
12th century
possibly. Johannis Hispalensis
Liber ysagogarum Alchoarismi (Bodleian Lyell 52)
Most famous edition: The work is often referred to as the oldest (astrological) algorism scripture that explicitly mentions the word "Alchoarismi" (= algorism). Speculative backdating to the time of Adelard of Bath (1150). Purely functional, not a didactic text like Carmen o. Tractatus.However, the age of the Bodleian edition is only estimated at the end of the 13th century on the basis of the writing, see: https://manuscripta.at/_scripts/php/cat2pdf.php?cat=wichner&ms_code=AT1000-612Die positional-structural identity in Middle Ages. Tables (Ysagogarum/Dixit) can be interpreted as a combination of "al-ghubar" and "sillogismi" as a superordinate practice (cf. e.g. Ezra/Yasamin); Links al-Ghubar syllogisms (positional-logical order of numbers/letters) with Christian theological concepts, primarily about cosmic time order as a mirror of divine creation and redemption. Mentions, among other things, Arabū/hebraus as a letter-number hybrid; as a quadrivial practice → reductio ad theologiam (Bonaventure). Gerbert: Scholastici model; Strikingly, the term "Alchoarismi" is phonetically very close to the word combination of "al-ghubar-ism" or the Spanish "alguarism" = "al-guar-ism" with "al" as a prefix -> see Catalan dictionaries. Since the text is considered particularly old, the phonetic proximity to al-ghubar would be all the more plausible. The addition of the word ysagogarum is also an indication that it is the "theory to a practice". If algorism had originally been theory, one would not have written a "Liber Ysagogarum Alchorismi" – but a "Liber Algorismi". A "theory to theory" is not very convincing.
Al-Andalus
43
a
12th century
possibly. Johannis Hispalensis
Incipit prologus in libro alghoarismi de practica arithmetice. Qui editus est a magistro iohanne yspalensi (France, lat. 16208)
Second copy of the book with the same intro, but without "Ysagogarum", but with "alghoarismi" and reference to in the title; many similarities with Bodleian. Also in the text: "alghoarismi"; Spelling with "alghoa" is phonetically closer to "al-ghubar" than to "al-hwarizmi"
Al-Andalus
44
a
12th century
unknown
Untitled: Liber ysagogarum Alchoarismi (BSB Clm 13021)
Third copy without title and without mention of algorism. But: With parts of the content of the Liber ysagogarum Alchoarismi. Same beginning, but only partially identical in content to Lyell 52.; According to estimates, the Munich manuscript is even older. Since the title "Liber ysagogarum Alchoarismi" is missing, the question arises as to whether it existed at all in what is probably the oldest text.
unknown
45
b
12th century
1145
Adelard of Bath
Astrological texts
was a Mozarab, came from Seville, knew Arabic and helped with translations; is therefore also considered a translator of Dixit; however, there is no evidence of this, but he certainly knew the astrological tables of al-Hwarizmi; important: The thesis that Adelard is said to have translated the Dixit Algorizmi is used as an argument for the backdated age of the text to the 12th century. However, the Oxford Manunscript itself is only documented in one version from the 13th/14th century. It is therefore only shown there in this table, as there is no reliable evidence for earlier translations. Important: The fact that, according to Crossley/Henry in 1990, the authors of the Dixit did not understand the algorism of al-Hwarizmi speaks against a translation by Adelard; this would hardly be justifiable in Adelard's case, since he knew his tables and was mathematically adept; Conclusion: If neither the Dixit Algorizmi nor the Salem Codex were created before Carmen, Liber abaci and Tractatus, but only afterwards, then the derivation with al-Hwarizmi is also questionable.
Al-Andalus/England
46
c
12th century
1150
Robert of Chester
Latin Algebra Translation
has demonstrably written the first Latin translation of al-Hwarizmi's algebra. However, Latin Manuskritpt was not found until the end of the 19th century, i.e. only after al-Hwarizmi eponym was already widespread; Steinschneider is the first to mention his translation train. At the time, the late find was repeatedly described as "retrospective proof" (e.g. Sterner 891) that the al-Hwarizmi conjecture had finally been confirmed as correct. Conclusion: If this "late proof" is emphasized so strongly several times, then it is clear that until the end of the 19th century there were unspoken doubts as to whether the al-Hwarizmi eponym could be true at all. Only the discovery of the translation could prove that it was at least theoretically possible that his algebra could have been known in the 12th century in Al-Andalus - the time to which the Dixit and the Salem Codex were dated (both are works from the 13th/14th century according to the current status). Similar to Adelard, Crossley/Henry's criticisms of the ignorance of the creators of al-Hwarizmi's algebra in Dixit Algorizmi can hardly be reconciled with his person, since Chester knew al-Hwarizmi's teachings for sure
Al-Andalus/England
47
d
12th century
Gerard of Cremona
Latin Algebra Translation
Second translation of the book by al-Hwarizmis. Algebra into Latin. As with Adelard and Chester, the critiques of Crossley/Henry can hardly be reconciled with him as a person, since he was also well acquainted with algebra and al-Hwarizmi's teachings
Al-Andalus/England
48
d
12th century
1200
John of Seville
Liber algorismi de pratica arismetrice
attribution by Boncompagni uncertain; often serves to give the anonymous text of the 13th century (Dixit) an artificial authority of the 12th century. Eneström's critique of early dating; here, too, it is questionable to what extent the text agrees with the previously mentioned Introductorius liber qui et pulver is dicitur in mathematicam disciplinant or is a work in its own right; if it should be at least partially identical, it is clear: Algorism in this work means dust calculation = al-ghubar
Al-Andalus/Italy
49
b
12th century
1200
Alexander de Villa Dei
Carmen de Algorismo
the most important and most widespread didactic poem of the Middle Ages on algorism; content similar to Yasamin; provides four alternative etymological derivations: "This contemporary art is called algorism, after King Algore, its inventor, or one says, after algos, which is art, and rodos, which is number; this is the art of counting or counting, for which art it is good to know that five (i.e. ten) numbers were found twice among the Indians"; Not a word reminiscent of al-Hwarizmi; This is important because, according to current knowledge, the Carmen is older than the Dixit and the Salem Codex (both of which can only be documented for the end of the 13th or the beginning of the 14th century). The only personal reference is "King Algor" > as mentioned in al-Andalus; or: "Algos" > exactly as in Florence in 1307; Relation to the quadrivium: Describes "VII especes" as operations of algorism (addition, subtraction, duplation, mediation, multiplication, division, radix extraction), linked to "Indorum characteros" (Ghubar figures/letters) and "levitas discentibus" (easy pos. logic); Phonetically important: Clear separation of the syllables algor & ism -> therefore not an overall term that would have phonetic similarity to al-Hwarizmi; Algor as king is not the same as al-Hwarizmi; neither in terms of content nor language; For Carmen, algorism is not only mathematics, but also a world view. Conversely, the second variant of interpretation "algos" is functionally compatible with "al-ghubar sillogism". Thesis by Cantor, Friedlein, Treutlein, etc., that the Middle Ages "forgot" the "true" origin of al-Hwarizmi, is not verifiable, but insinuated, if a total of four derivations are examined by Villa Die, but a fifth is said to have existed in addition. Villa Die should have known such a fifth alternative derivation due to the temporal proximity, even if a potential reference to Indian origin is discussed. So there will not even have been "rumors" that anyone believed that algorism came from al-Hwarizmi.
France
50
b
13th century
1202
Alexander de Villa Dei
Carmen de Algorismo
Important regarding the content and relation to syllogism/language logic: The Arabic numerals in Carmen are only written briefly at the beginning, after which they are only displayed as words. In the Carmen de algorismo, medieval algorism thus appears completely linguistic, but is not a mere word syllogism, because the contents are not general concepts, but markers of operations. Language functions here as the executing medium of an al-ġubār-based procedure, not as a logical justification.It is important to understand algorism from a medieval point of view as something that at the beginning of the 13th century was not only conveyed in writing, but above all orally with words instead of just numbers - > the Dixit Algorizmi (who speak algorisms), which according to the current state of knowledge originated after Carmen, could mean exactly these by using the word "dixit": The necessary "pronunciation" of the contents, because with Arabic numbers it was possible to calculate faster, but the pronunciation of a number: In fact, the Dixit later gives a detailed instruction on how to "pronounce", i.e. the oral passing on of very large numbers. This in turn fits exactly with the contents of Carmen.
France
51
b
13th century
1230
Leonardo Fibonacci
Liber Abaci
In the introduction he writes: "hoc totum etiam et algorismum ... errorem computavi"; Clearly: He calls algorismum flawed; For him, word has purely functional meaning; no reference to al-Hwarizmi, although all calculation methods and scholars of the time and of all countries are known through travels and many years of residence in the Maghred; The evident contradiction is particularly important: on the one hand, it wants to propagate Indian arithmetic, but at the same time criticizes algorism as outdated; this is hardly explainable if he had meant al-Hwarizmi doctrine by algorism; on the other hand, the evaluation fits all the better with the al-ghubar thesis, since this method of calculation was prone to error, among other things, later in Spain with warnings to authorities regarding "guarsismo/alguarismo" and was even banned in Florence. Commercial view of algorism: He does not explain the "exotic" word. He assumes it to be generally common and known to the target group. Fits exactly with the perative understanding of algorism in the Mediterranean region, as in Jacopo and Florence's Abacus-Scuola 1307; therefore, the concept could be assumed everywhere in trade - not only by a few scholars; Striking: Fibonacci hardly has any theological reference to algorism/quadrivium; hence algorism for him not a world view as e.g. with "Super Algorizmum" (Salem Codex); The thesis spread by Friedlein and Treutlein in the 19th century that Fibonacci had "forgotten" the true origin of al-Hwarizmi is not only verifiable, but also seems like speculative insinuation in order to be able to resolve the otherwise existing contradiction with al-Hwarizmi eponym
Italy
52
c
13th century
1230
Johannes de Sacrobosco
Algorism Vulgaris
Reason of Numbers (sillogismi/algorimsi), like Carmen, takes up mythical as well as functional interpretations of etymology in order to contrast them. In the case of the Algus/Algo variant, it refers to Al-Andalusi's Indian philogoph king.Algorism clearly refers to an ars numerandi. The entire text explains the four basic operations – addition, subtraction, multiplication and division – as well as how to deal with fractions. Says: gogos, quod est ductio -> lead or draw lines (in the dust); The repeated use of the term radix in the Tractatus de algorismo (15x) proves its functional-arithmetic meaning. The term is used exclusively to describe a calculation method (rooting) and is hardly embedded in al-Hwarizmi's algebraic theory; Use the Indian numerals only in two places - otherwise only text is written (syllogism): Digits appear only in two places: first at the very beginning (inventory / initialization) and then again when series have to be explicitly shown as examples (1·2·3·4·5·6 or 1·3·5·7·9 etc.); Sacrobosco does not want to explain signs, but to make a practice verbally feasible, which is why no Roman numerals are mentioned in the entire text; in the commentary Commentar of Petrus de Dacia he uses a total of 34 times "deinde dicit"; corresponds exactly to the "Dixit Algorizmi" in the sense of speaking numbers.Again, algorism is not read, but thought or conveyed "spoken", which in the non-Arab world requires not only linguistic logic (syllogism) but also additional linguistic-mathematical logic (= algorism).
France
53
b
13th century
1266
Johannes de Sacrobosco
Algorism Vulgaris
The logic that we also find in the Dixit Algorizmi and also in the Erfurt Codex in 1301 should be emphasized separately here: Dots mark what belongs together, as in the Dixit, i.e. 1 2 = 12 = twelve and 1. 2. = one and two separated. Dixit and Salem Codex extend this point use with regard to large numbers (see below). Compared to Erfurt Code 1301: Point serves as a superordinate switch and control element
France
54
a
13th century
1266
Roger Bacon
Opus Majus
Bacon studied algorism. He mentions Albumasar several times in the work, who lived and worked at the same time as al-Hwarizmi. Although al-Hwarizmis has already been translated into Latin, it is used in the context of algebra genannt.er clarifies the relationship between mathematics and logic: Without mathematical (i.e. h+C47:J47gicae a mathematica propter suum finem, sed propter medium et cor ejus, quod est liber posteriorum, nam ille liber docet artem demonstrandi". Quote: "Major vero pars praedicamenti qualitatis continet passiones et proprietates quantitatum"/"demonstratio est syllogismus faciens scire": logic as a demonstrable, demonstrable, comprehensible operation corresponds in fact to al-ġubār syllogism: operative, positional
England
55
a
13th century
1266
Roger Bacon
Opus Majus
Connection of Rithmimachia and Algorism as an operative practice: "Rithmimachia, id est, numerorum pugna, nam ad l rithmo ' Grece ' numerus ' est Latine, et machia\ media producta, 'pugna' dicitur in Latino, et hec traduntur in libris propriis per singulas practicas nominatis, ut in libro Rithmimachie et in Algorismo completo in integris et fraccionibus, et in Algebra que est ' negociacio'"; Bacon's comparison of Rithmomachia and algorism is only meaningful if algorism is understood as a relational and position-oriented computational practice. A value- and operation-centered place-value algorism in the sense of a purely numerical-arithmetic calculation method (as identified with al-Ḫwārizmī's algebra) demonstrably destabilizes the game structure of Rithmimachia and is therefore not a benchmark. In conclusion, algorism works universally without any reference to value: with Indian numerals, dots with and without a line or Greek letters for numbers (superfluitas 'omnis'). Example: "uno chien + quattro hund + 1 dog = six cane" or "dog = x x and then x + x + x = 3 dog", here "x" is a variable; at the same time, "3 x dog = dog, dog, dog" - here the "x" is a special character; also inversion of language and writing: 23 is pronounced the other way around: twenty-three, i.e. not "20" "3". The examples are not a computational process, but a relational bundling of signs, as is typical of gematric thinking: various linguistic signs are transferred into a new symbolic unit via meaning and number. In the case of algorism, it is not calculated, but ordered (e.g. by positioning, deleting and repositioning)
England
56
a
13th century
1266
Roger Bacon
Opera hactenus inedita
Describes algorism as "viae algorismi" and thus as an operational computational practice, and interprets algebra functionally (trade/balance), which is consistent with ġubār algebra (Yāsamīn type). For him, stamen, abacus, algorism are different media of the same operative logic. With this also a high degree of agreement with Fibonacci, Jacopo and the Florence School, 1307.
England
57
a
13th century
1275
Abdalla Filius Ali
Geomantia
The o/oo pattern used in the Geomancy manuscript of 1266 acts as an upstream control layer within a ghubār-based data matrix. The combination of the characters "o" / "oo" has no numerical value here, but only encodes an operative state (set / not set) that determines the permissible progress of the operation. The o/oo sequence is invariant (therefore not the result of a calculation), non-accumulative, explicitly instructive (instruction manual), and logically superior to the numerical data fields.This reveals a strict separation between control logic and computational data:Geomancy does not operate with numbers in the narrower sense, but with discrete states that regulate the timing, selection and continuation of the drawing operations. This structure comes as a translation by Abdalle from Arabia, but logically it also corresponds exactly to what Garlandus Compotista had already analysed ex vi terminorum in the 11th century as singular final operations: the validity of a step does not result from a calculation, but from the effectiveness of a set sign. The o/oo characters take on the function of singular information that carries implicit conditions without being numeric themselves. Universal or accumulative operators are superfluous; the decision value lies in the singular state of the sign itself. The Geomancy manuscript of 1266 thus documents the continuous application of an operational practice that had already been logically reflected on beforehand in a material, instructive form.Bacon does not call for the domination of mathematics, but for a discipline of the sciences: knowledge must be demonstrable – or it is didactically empty.
Maghreb/Erfurt
58
b
13th century
Ramon Llull
Ars Magna
Explicit yes/no decisions on the level of concepts and truth (affirmation / negation). Llull shows that yes/no binaries alone do not justify an algorithm. An algorithm needs at least states, transitions, rules that are capable of transitioning, independence from semantic understanding.True and False are not truth values, but markers for whether a common "figure in space" is still (jointly) viable for several observing actors. Mathematics is therefore not true because it is provable. It is provable because it is repeatedly figuratively stable from a communal point of view. Conclusion in the context of figural algorism: Truth corresponds to stability of a figure under repeated, independent execution in a common space.
England
59
a
13th century
unknown (Oxford tradition)
Dixit algorizmi (original from Camebridge)
the Dixit Algorizmi is often interpreted as the earliest evidence of the al-Hwarizmi eponym on the 12th century, although only a singular copy from the middle of the 13th century exists (which is calligraphically confirmed, since font = northern textualis libraria). Earlier dating is based on the thesis that al-Hwarizmi was the author and could have been translated by Adelard of Bath or Gerhard of Cremona; In terms of content, however, the reference of the "author without a name" mentioned in the text to his two books on algebra and arithmetic fits all the better with ibn Yasamin: He not only demonstrably wrote both of the books mentioned, he also taught a more modern variant of algebra than al-Hwarizmi (including Root/Radix). In addition, the Dixit also had to be implemented with Dustboard. According to Crossley/Henry 1990, however, the creators of the Dixit demonstrably did not know the "algorithm of Al-Hwarizmi". The script used as well as the school structure of the text suggest that the surviving content was created in the 13th/14th century in the Oxford environment. An attribution to a scholar known by name, such as Robert Bacon, cannot be substantiated from tradition. According to Crossley/Henry, the work has hardly been distributed even in England. Contradictions with the expertise of possible translators such as Adelard of Bath, Gerhard of Cremona or Robert Chester make them appear very questionable as copyists, see above.The translation with slight changes by Boncompagni as Algoritmi in numero Indorum 1857 is of great importance: this title was only introduced later by Boncompagni and never existed. This is important because it is precisely this subsequent title that seems to document a lost original text by al-Hwarizmi, the existence of which cannot be proven to this day (see above). Wording: "opus algoristicum". This is the calculation itself while it is happening. Bacon speaks of "modos algoristicos" as an ability to deal with quantity.
England
60
a
13th century
unknown (Oxford tradition)
Dixit algoritmi (Boncompagni/Comparison)
The Dixit Algorizmi describes the Indo-Arabic numeral script not as a system of symbols, but as a logic of place and state (like Carmen and Algorismum Vulgaris). The "zero" (circle, o) functions not only as a number, but also as an explicit marker of an inactive place.This operative reading is conceptually and practically comparable to the dot/empty system in late medieval tables, as materialized in the Erfurt Codex 1301. The sentence "so that you may write it in a book or talk about it" explicitly formulates the basic axiom of the al-ghubār syllogism: A number is valid if and only if it can be performed as a regulated action in both the written and speech channels. For example, Crossley Henry has the number: 1,180,703,051,492,863; comparable figures can be found. in the Salem Codex - but there with dots. This reveals a systematic "blind spot" of modern editions: the Dixit of the Oxford text, translated by Boncompagni, contains the points missing in Crossley/Henry, e.g. . . x. .xx. . xxx. .c. . m. etc. They symbolize self-contained units that enable the structuring and verbalization of large numbers. This can be explained by the fact that Boncompagni did not try to explain the book with al-Hwarizmi's algebra, but simply translated the text of the original. The translation into English by Crossley/Henry is different: If you explain algorism primarily via al-Khwarizmi, then this is done via his algebra – and there the point does not play the same central role as in ḥisāb al-ghubār (stamen-board arithmetic). The result: it is not only omitted, but replaced by blank drawings, as in the case of the number 1 180 703 051 492 863, which is not to be found in Boncompagni. Boncompagni edits absolutely precisely, quasi "point by point" - and without the illustration of a single Arab-Indian numeral. The text operates exclusively with linguistic operations, Roman letter signs and positional logical markers (dots, circles). The absence of Indian numerals in the Cambidger manuscript is therefore not an editorial deficit, but an expression of a computational practice that does not require graphic numerals. Conclusion: If the points are highlighted in Boncompagni's text, then al-Hwarizmi's algebra is also not the correct frame of reference to explain the point-based positional logic contained in the Dixit. The textual form of Dixit algorizmi, as handed down by Boncompagni, can be explained by an oral mediation situation. The text explains procedures and positions precisely, but does not include the graphical representation of the Indian numerals, which are only announced but not shown in the text. This suggests that the Latin scribe transcribed the lecture without having the figures themselves in mind. The consequence: The text is not a book about algorism – the text is algorism in the mode of speaking. Algorism speaks – it does not write! And that's exactly why "Dixit Algorizmi" is used at the beginning and in the text "dicere numerum", "pronuntiare numerum", "non de voce diximus ... sed de re"; Therefore, with the understanding of algorism at the time, the hybrid equation oriented towards "AI tokens" also works: "3 + trois - one = five" or "2fast4you" or "Fülrnihg. Die Vegöl zschierwtn". Even the algorism of the Middle Ages does not understand languages – it understands operations.
England/Italy
61
a
13th century
unknown (Oxford tradition)
Dixit algorizmi (original from Camebridge)
According to current knowledge, Dixit Algorizmi is not older, but younger than the Carmen de algorismo, which explains the Indian numbers "spoken" or "speak"; thus, the Dixit can be plausibly understood as a later sequel, which transforms or expands the already established spoken algorism. In doing so, the Latin oral teaching tradition and operational concepts of Maghreb algebra are intertwined, as they are assumed by al-Yāsamīn, for example. In addition, the two blank spaces, which Boncompagni marks with four points, are not a lack of tradition, but mark the didactic distribution of roles. The first blank space is designed as an insertion field for graphical digits that the user should already know; the second confirms that the text is not aimed at visual representation, but at linguistically executable operations. Thus, the Dixit Algorizmi is not a numerical textbook or introductory text, but a rule text for an already established, orally taught arithmetic practice. In this respect, it is difficult to explain the gaps with the assumption of a lost Arabic original, since the graphic introduction of the digits would have been one of its central concerns. The fact that the blanks were never filled in retrospectively, although the anthology contains many glosses and annotations, also leads to the conclusion that the Indian numerals were not missing at all. Finally, the position of Dixit Algorizmi within the anthology itself must also be taken into account. The text is not at the beginning, but follows a series of tables, prognostic and astronomical texts that already operate with punctual markings, blank spaces and state-based ordering principles. In this context, the dots and blank spaces of the Dixit do not appear as an isolated phenomenon, but as a continuation of a drawing practice previously established in the anthology. The Dixit Algorizmi thus does not function as an introduction to algorism, but as a downstream rule and reflection text that explicates and linguistically stabilizes an already practiced operational logic. The embedding in the anthology suggests that the text served less to convey new signs than to specify their correct use within an existing computational and didactic context.From this point of view, the Erfurt Codex of 1301 and the Dixit Algorizmi could even be interpreted as different manifestations of the same operative sign logic, within which points can be seen as explicit control instruments for structuring procedural arithmetic acts. All in all, this makes the already dubious attribution to al-Ḫwārizmī even more questionable, since the Dixit Algorizmi, as part of an astrological anthology, apparently did not aim to introduce a new calculation method, but to reflect on a point and control logic that had already been described in detail for tables.
England
62
d
13th century
Ibn al-Bannā' al-Marrākušī;
Talkhis amal al-hisab
documents the ġubār methods in an extended operative form at a time when algorism was spreading more widely in the Latin world. While the West Islamic world continues to develop the practice of material signs, in the Latin North the same arithmetic logic is still verbalized and handed down as a spoken instruction. It is not a matter of mutilation, but of a media transformation.While in Latin Europe work is still being done on an oral-performative algorism (1.0), which structurally follows Yāsamīn and Ḥassān, a further writing and operationalization of the same logic is already underway in the Western Islamic world (direction of an algorism 2.0). Attention: It is not easy to differentiate between al-Banna and al-Hawari (the line after next), since al-Hawari comments on al-Banna's work
Maghreb
63
b
13th century
1300
unknown
Salem Codex
Cited as "proof" of Cantor's eponym, as it uses "nominative". His thesis: The word "algorism" describes a real person whose identity has been forgotten, although it can also be recognizably an allegorical content with seven principles of the Holy Spirit (Quadrivium). In addition, Salem Codex is incorrectly dated by Cantor in 1865, as it is over 100 years younger than assumed in 1865. The second part of the text begins with "VII species sunt algorizmi, quia VII sunt dona spiritus sancti", which sacralizes Algo (Ghubar pos.): Sequential numbers = divine mysteries; this is Christian-heavenly "super-logic", not algebra according to al-Hwarizmi. Clear reference to Quadrivium.The algorism text handed down here explicitly teaches a point-controlled linguistic logic of large numbers.The dots function as meta-signs that structure groups of three and trigger the repetition of the numeral mille. Numbers thus do not appear as words, but as operational tokens that are executed identically in the written and speech channels; Example similar to Dixit in the original by Boncompagni: Exemplum. 495.827.361.052.951. explains exactly the meaning of points.
Germany
64
c
13th century
1301
al-Hawārī
Talkhiṣ aþmāl al-hisāb
Attention: It is not easy to differentiate between al-Hawari and al-Banna (before-line), since al-Hawari comments on al-Banna's work; al-Hawari creates the concept of the "Ghubār" (dust/sand) teaching, which refers to the Arabic numerals (number signs), especially the zero. He described how these "Ghubār" numbers (0-9) were used to perform calculations, as opposed to the earlier Indian numerals or the abacus.Again, no mention of al-Hwarizmi; Uniform understanding of what is referred to as Algorithm 1.0 in Europe and is under development as Algorithm 2.0 across regions.
Maghreb
65
a
14th century
1304
unknown
Amplonia Manuscript 12° 18
The Amlonia manuscript of 1301 documents a fully integrated algorithmic organization in which numerical data, linguistic representations, and operational status markers are coordinated within a unified computational process. Central to this is a binary control column (0/1), which has no numerical eigenvalue, but only shows the operational state of a cycle (carryover active / inactive). This status marker is not accumulated, but set situationally and consumed in the next step. A value of "2" does not occur systematically. The table thus explicitly does not work additively, but state-based: The linear sequence of the first column (months 1–12) acts as an index. The hours/minutes/seconds of the values next to the control column contain residual values. The binary column thus acts as a procedural flag that decides whether a cyclic carryover is triggered (validated algorithmically with AI). This makes an early separation of computational data and control logic visible, which goes beyond pure astronomy. The Amlonia manuscript thus shows the practical implementation of the logic that Garlandus explicated linguistically and logically and which appears materially stabilized in geomancy. The reader's interaction implicitly confirms the principle of vis terminorum described by Garlandus Compotista: effectiveness does not arise from the meaning of individual words or numbers, but from their operational position within a regulated process. Word, number and dot function equally as signs with action relevance.The selection carried out by the reader, which is preserved by means of notes, thus corresponds to a singular final operation in Garlandus's sense: a certain state is recognized, marked and transferred into a concrete action.1301 thus documents the complete chain of algorithmic organization: sign structure + state logic + user interaction.
66
c
14th century
1307
unknown
Incipit liber restauracionis
Explicit character control (virgula / punctum) to display state, diminution, negation and direction of operation within a combined word-number system. Evidence of an integrated al-ghubār practice in which syllogism and dust calculation are merged into a common algorithmic logic. The Liber restauracionis therefore does not document the adoption of Arabic algebra (as is often assumed), but the consolidation of a Latin al-ghubār syllogism.Algorism as an al-ghubār syllogism is thus no longer only etymologically plausible, but structurally verifiable with another document; this approach is confirmed in 1307 by Florence Abacus schools and Jacopo in exactly the same way
Al-Andalus/Italy
67
a
14th century
1307
School of Florence
Il primo abaco in Volgare Italiano
Important aspect: Abacus and algorism are taught in Italian in abacus schools in various cities; it is a rather easy level of difficulty that does not require any Latin education. Little theory, a lot of surgical application. The origin of the word algorism is also explicitly explained to the students: The word comes from an unnamed Arab scholar, who invented the method; at the same time, "algo" or "algho" is the word used in Arabia to describe the concept of art; phonetic derivation, for example: al-ġu-bār ≈ al-ghu-bar≈ al-gho-bar ≈ al-ghoQuote: "as already mentioned above, our treatise in Arabic is called alghorism". Spelling Alghorism is phonetically close to al-gubar or al-gobar (ghobar and ghubar are both common to this day); In addition, note: "doveremo scrivere ad ritroso et leggere a diritto" (we must write backwards and read forwards); this does not merely describe a technical direction of writing, but reflects the separation of notation and execution. Algorism is encoded backwards, but executed forwards – a characteristic of algorithmic systems of speech and action: one "writes" the number – but one "reads" the algorithm.Also striking: al-Hwarizmi and his algebra play no role in the Florence schools in Abaco & Algorism; Content would also be too demanding and abstract. The schools had the purpose of educating merchants who traded with North Africa, among other places, so it was primarily important for them to know the arithmetic prevailing in this region as a term (al-ghubar) and to be able to apply it. The reference to "algho" and the reference to "as the doctrine is called by the Arabs" is therefore important practical knowledge. It is also etymologically unambiguous, so it is not speculation or theory! The interpretation of the Florence School is still conveyed in the first Italian dictionary of 1623
Italy
68
a
14th century
jacopo di Firence
Tractatus algorismi,
teaches algorism in Italian; refers to independent understanding and tradition of algorism in Provence: home of Gerbert d'Aurillac; also here spelling "Algho -> Alghorismus" Quote: "Et l'arte è dicta in lingua arabia algho, el numero è dicto rismus, et perciò è dicto alghorismus"; in terms of content 1:1 as statements on Florence 1307; al-ghubar (algho) was the predominantly orally used term of merchants of Arabic origin -> 100% conform to Fibonacci and his critical view of algorism
Italy
69
d
14th century
1410
Haukr Erlendsson
Hauksbók (algorism material)
contains demonstrable reference to Carmen de Algorismo and Algorismus Vulgaris, but not certain to Dixit Algorizmi; this shows that Dixit in particular (as explained by Crossley/Henry in 1990) was comparatively little known and widespread. This also suggests that Dixit did not have an early original, but possibly. could even have been written after the Hauksbók. The easy translation of algoristic texts into the vernacular of the Hauksbók shows that algorism was not tied to a specific notation or language. It was not a system of signs that was translated, but a sequence of actions. This is precisely why algorism could be effortlessly transferred to Icelandic.
Iceland
70
c
14th century
Prosdocimus de Beldemandis
Algorismus de integris
This is followed by "Bohectius" (Boethius) in the definition of numbers and in the consideration of the unit as no number in itself; speaks of the necessity that a computer needs a blackboard from which he can easily delete what he has written. Deletion creates profit for the art of arithmetic (corresponds to al-ghubar, since "slate" is not possible with abacus or paper)
Italy
71
a
14th century
Giovanni Villani
Book 11 and 12
Provides important details about education: Abbacho was a type of commercial mathematics and Algorismo was a calculation using Arabic numerals, both training subjects for commerce, while grammar and logic were advanced subjects of study that included Latin; Here, too, algorism is clearly knowledge of action, not textual knowledge; practical arithmetic: rule-based approach; Demonstration on the device; Imitation and practice; complete agreement with Fibonacci.Important information about users: 8,000–10,000 children learned to read afterwards; 1,000–1,200 learned abbaco e algorismo, but only 550–600 learned gramatica e logica; Villani's "beans" are also not a folkloric detail, but the empirical proof that algorism was based on everyday, non-textual, above all operational arithmetic practice; Shifting of beans even potentially goes back to Gerbert's "plates".First Italian dictionary from 1623 from Florence quotes Villani and says: Abacco = Algorismo
Italy
72
a
14th century
unknown
Libro de arismética que es dicho alguarismo
The text is considered the earliest evidence of the use of alguarismo among merchants in Catalonia. In terms of content, algorism is described exactly as by the Italian schools or by Villani. The content of Alguarismo also corresponds 1:1 to Yasamin's al-ghubar arithmetic; al-guarismo is explained in Spanish lexicons in the 18th century by arithmetic with Arabic numerals; so that in Spain, France (cf. Jacopo/Provence), Italy, Al-Andalus and Maghreb exactly the same understanding of methods and terms as the basis for intercultural trade in the Mediterranean region: Algorism = al-ghubar = factual, "token-based" standard of action with universal adaptability
Spain
73
b
14th century
unknown
Algorism per manus / figura
Finger arithmetic is ancient. However, it is described here as "algorism" in the sense of a rule-based computational practice that operates with figurae (visible signs) and their positions (loci). The value of the number does not arise from the figure itself, but from its shape and the sequence (temporal position). The combination of finger calculation (per manus) and drawn figura shows algorism as a physical-operative art (ars numerandi), independent of algebraic theory or author attribution. As in the ḥisāb al-ġubār, they arise in the execution of the calculation and disappear immediately. Importance therefore does not lie in the figure itself, but in its position and in the course of the operation. This points to an operative logic of imposition and deletion, not to symbolic persistence.Since about 1850, "algorism" and place value calculation have been de facto equated, but actually wrongly, because in the Middle Ages the terms were not identical:- In algorism as "finger arithmetic", number is not created by an abstract symbol system, but by execution.- In contrast, place value calculation is based on a completely different logic. Here, numbers are created by form and spatial position. Digits are meaningless figures, whose value is only determined by their position in space (left/right, higher/lower). All positions are present at the same time; Number here is something that exists as a structure.
Germany
74
b
14th century
unknown
Oxford Bodleian Tanner 192
"The method of drawing these figures is the following: A line is drawn from left to right like this which in itself means nothing, yet four place values are defined on it like this EA whereby a is the first place, b the second, e the third and d the fourth. And you should be aware that at these four positions nine different signs are attached"Central sentence: "But note that I call the first position of the second numeral as in the algorism."
Al-Andalus
75
c
14th century
unknown
Aenigmata algoristica
Term for didactic puzzle texts whose solution is based on algoristic calculation methods; the term algoristica functions as a functional classification of computational practice, not as a reference to author, theory or etymology. Numerous algoristic puzzle and exercise texts (Aenigmata, Cautelae algorismi) prove algorism as a widespread, practice-oriented art of arithmetic: rule-based, procedural, didactic and independent of algebraic theory or author attribution. "Aenigmata Algoristica" was a widespread conceptual standard - in the Middle Ages, it did not refer to arithmetic "as theory", but to the art of rule-based calculation – the practical "how" of number processing.
Germany
76
c
14th century
unknown
Algorismus/Art Algorismus
The text is written in Anglo-Norman French. It is clearly aimed at practical calculators / merchants. The majority of the text consists of precise, step-by-step instructions for action ("Si vous volez faire...", "Prenez garde...", "Escrivez..."), not definitions or justifications.Algorism is presented as the art of calculating with numbers; central topics are place, digitus / articulus, as well as the four basic operations (addition, subtraction, multiplication, division). The presentation is didactic, operational and procedural, with clear reference to oral instruction ("entendre", "retenir", "prendre garde"). The text is closely related to the Italian Abbaco traditions in terms of content and has the same practice-oriented structure. Algorism as a standardized calculation practice, adapted to everyday commercial life in the northwestern European trading area.
France/England
77
c
15./16, Jhrd
unknown
Traité d'algorisme ou d'arithmétique, en provençal.
Language is not Latin; Occitan/Provence; "This art that follows here was taken from algorism." present libre de la sciencia de arismatica, vulgarment dit algorisme": Algorism as application-related calculation for merchants, exactly as in Fibonacci, Jacopo di Firence, Villani or Catalan alguarismo; concrete invoice example: "A merchant buys 30 cubits (canas) cloth"; speaks for the importance of algorism as merchant arithmetic: uniform, purely functional understanding throughout the Mediterranean region; no relation to scholars
France
78
b
15./16, Jhrd
1411
John of Gmunden
Algorismus de minutiis
The algorism de minutiis is an example of how medieval algorism was primarily conceived as an operational positional logic for temporal and astronomical units. Minutiae do not function as numerical values, but as relational units of position within cyclic systems (calendars, astrology), the treatment of which makes algorism indispensable. Astrology requires position-based procedures, fast, correctable operations, semantics-free character manipulation, as well as the use of the dust board (al-Ghubar at its core). In the calculation, the numbers are denoted by figures, and these figures are divided, put together and transformed according to degree (gradus) and order (ordo). Corresponds to the principles of line algorism, dust calculation (powder, minutia calculation, rithmomachy. The meaning (numerical value, name, unit) is secondary. Primary are position, relation, transformation; The final page shows square and cube roots as purely figural transformations in positional grids. The calculation is not symbolic, but through the operative rearrangement of figures. This confirms that medieval algorism is essentially not numerical value arithmetic, but a position-based figure practice. Here, too, place-value calculation is nothing more than positional logic: the value does not arise from the sign, but from its position in relational space. Conclusion: In algorism, there is no calculation, but universal placement.
Austria
79
a
15./16, Jhrd
unknown
Art of Nombryng
Oldest arithmetic text in English. Partly translation of Sacrobosco's Tractatus incl. the derivation with Algus and Indian king. Words such as Algorym or Augrym are preceded as synonymous. Here, too, algorism is clearly an "ars" numerandi in the sense of an executive art: operative, positional, temporary, erasable, rule-guided; Particularly relevant is the "digit" calculation on the calculation board: "the lynes do stande for the order of places". Arithmetic is the proper movement of characters in space without digits, only with dots. Reading/writing is not necessary; correct result follows from position rules, once with lines and once without lines (only points). The latter is the dealer mode (e.g. for coins).
England
80
d
15./16, Jhrd
Cornelius von Zierickzee
Algorism novus
The word "novus" makes it clear: In the 15th century, there was an "old" and a "new" algorism from the point of view of the time; the new algorithm is increasingly written with th; Algorithm describes transition to paper & ink with Indian numbers; Now also in the Latin world transition from "Algorithm 1.0" (see above al-Banna/al-Hawari) to "Algorithm 2.0". The algorism novus contains important markings: The new algorithm now also works "without deletion" (unlike ghubar / Rechenbrett): Wording "Algorismus novus de integris compendiose sine figurarum deletione compilatus"; Formal linguistic separation of old/new shows that only the "new" abstracted algorithm really goes back to the al-Hwarizmi theory of algebra, while the "old" and earlier operational algorithm refers to the "token"-based approach. Chronologically, etymological roots are basically based on old understanding.
Germany
81
d
15./16, Jhrd
1508
Balthasar Licht
Algorithm linealis
Algorithm on lines/table abacus; practice-oriented book in Latin also on the new algorithm with Indian numbers; shows again the transition from the old understanding of algorism (based on operational dust calculation) to the new paper-based, abstract-rule-based algorithm; now also widely distributed by book printing; Bridging the gap between the operational understanding of commercial practitioners and the increasing number of Latin-taught merchants in German-speaking countries
Germany
82
d
15./16, Jhrd
1525
Adam Ries
Calculation on the lines and springs
Writes the rule-based teaching in German for the first time. Thus introduces a place value system in Germany across the board; begins with the older line arithmetic and only then leads to the more sophisticated numerical arithmetic; here, too, transition from old algorithmism to new algorithm, but without using the term as such; Cover picture shows the different variants of arithmetic on an arithmetic board with coins, numbers and lines, among other things
Germany
83
a
15./16, Jhrd
1548
Joachim Heller
Isagogue in astrologiam
Very important book, because it is the direct reference to Reinaud's eponym thesis of 1845: The al-Hwarizmi eponym was based as an explicit conjecture on the reference to the last page of this book by Heller on astrology from 1548; The book is said to have contained the phrase "Alchoarisam Magistri Indorum" - this vague description served as the central basis for the fact that "Alchoarisam" meant a person, especially a scholar from the Khorezm region.Problem: See follow-up entry -> There is no passage in this book to which Reinaud refers, and this has been known since 1877 (Wüstenfeld). Conclusion -> one has always relied on Reinaud, but apparently no one has really read Heller's book except Wüstenfeld or this aspect has been deliberately ignored.
Germany
84
a
15./16, Jhrd
1548
Joachim Heller
Johannes, Hispanus: Epitome totius astrologiae
Hellert's book of 1548, linked here in the right-hand column, is exactly the work on which Reinaud's conjecture of 1845 is based: its content was the basis for the entire eponym. The obvious problem: There is not a single passage in this book like the one Reinaud mentions. Neither on the last page, nor before. If Reinaud possessed a work with a corresponding annotation, it was at best a handwritten addition by some unknown reader and not a text published by Heller himself. This allows for the following interpretation: The early historians of mathematics read Reinaud's conjecture, they take a liking to the story of the "Alchoarizam", they consider the reference on the last page to be plausible and quote Reinaud – without checking or being able to check the "conjecture" in the original source. Heller's book was then as now a rare antiquarian object; one therefore had to and wanted to trust, especially in the 19th century, the statements of Reinaud as a recognized orientalist. Despite the lack of a passage in Heller's work, it is always important to note that Reinaud was honest and always spoke only of a conjecture. Wüstenfeld points out in 1877 that there are only references to European, i.e. non-Arab or Indian scholars in this book by Heller. This means that the "zero hour" of the al-Hwarizmi eponym is based on a thesis that cannot even prove the reason for the first conjecture and thus leads to a citogenesis of "trust" because no one reads this book. Narducci (1858) is the first to critically question the eponym, to doubt it, but ultimately to consider it possible; Woepcke (1863) is probably the first to adopt the eponym in its entirety, and Littre (1873) is the first to manifest the story in an encyclopedia in such a way that it is hardly reversible. In 1877, Wüstenfeld simply came "too late" ...
Germany
85
a
15./16, Jhrd
1623
Accademia della Crusca
Vocabolario degli accademici della Crusca
the first Italian dictionary from Florence describes exactly the understanding of Abacco and Algorithm of 1300: the interpretation taught by the school of Florence and Villani; thus a consistency that is over 300 years old with regard to the understanding of content: abacus and algoritmo are almost equated; "Algorismo et lo stesso che arithmetica et abacco"; Abacco: "The Latins called abaculi those small stones that were expected." An aspect that can be traced back to Gerbert
Italy
86
c
17./18, Jhrd
1652
Katib Çelebi
Haji Chalfa (Kashf al-Zunun)
Next to the Fihrist is the second comprehensive encyclopaedia on Islamic scholars from Turkey. But as with Fihrist, there is no reference to an arithmetic book by al-Hwarizmi - despite the listing of 14,500 texts of Turkish and Arabic origin; this makes it difficult to prove the thesis that there was an original on the arithmetic of al-Hwarizmi, although two of the most important Islamic encyclopaedias know al-Hwarizmi as a person and several of his works, but none on arithmetic. Alternatively, both in Europe and in the entire Islamic world, this book must have been "forgotten". To insinuate or claim this without evidence is very questionable, since even al-Andalusi knows al-Hwarizmi (see above). In total, three of the leading Islamic encyclopedias must have "forgotten" the supposedly lost book.
Turkey
87
b
17./18, Jhrd
1697
Herbelot
Bibliothèque orientale
Herbelot has translated entries from the Hajji Chalfa into French, including entries on Ben Mūsā. He is mentioned in the comprehensive standard work by different names: sometimes as Ebn Al Hareth Al Khovarezmi, sometimes as Ebn Hareth natif de Khovarezm. He is only one of many scholars from the Khorezm region; at that time, he is not known in either the Islamic or European world as "the scholar" from the Khorezm region, so that only he would have been called "alchoarizmi"; rather, "al-Biruni" was the scholar who would have been associated (in the Islamic world) with the Khorezm region. The thesis that Ben Musa, of all people, as "the" scholar of the Khorezm region, could have transferred the name therefore requires considerably more reliable evidence than is the case, for example, with Friedlein or Treutlein
France
88
b
17./18, Jhrd
1703
Leibniz
Explication de l'Arithmétique Binaire
Theoretical formalization of arithmetic operations (binary arithmetic) and their transferability to mechanical calculating machines. Prepares the later reinterpretation of algorithm as an abstract, machine-capable procedure. Leibniz influences the modern meaning of "algorithm", not its origin. With him, however, begins the loss of his original concept of practice, which refers to al-ghubar and operative dust calculation. The concept of the algorithm is already of declining relevance, especially in northern Europe.
Germany
89
c
17./18, Jhrd
1726
RAE
Diccionario de Autoridades
Algorithm = arithmetic. The entry warns authorities against errors of algorism; fits exactly with Fibonacci's critique and the original interpretation as dust calculation; can also be related to the fact that the concept of algorithm is becoming less relevant overall.
Spain
90
c
17./18, Jhrd
1770
RAE
Diccionario de la lengua castellana
the entry refers to the Arabic article "al" and the combination with "guarismo"; the Arabic language and conceptual separation, as well as the description of the content, fit very well with a shortened form of "al-ghubar-sillogism" or - as Corriente suspected in 1996: "al-ghubar-ism"
Spain
91
d
17./18, Jhrd
1829
v. Humboldt
refers to al Biruni and the Gobar numerals in a seminal article; he also provides a highly interesting interpretation of the philosophy of numbers, which have similar structures across all continents worldwide (e.g. comparison of South American knotted script and Asian calculation models)
Germany
92
c
19th century
1831
Roses
Algebra
translates al-Hwarizmi's algebra from the Arabic original for the first time.Important: Here, al-Hwarizmi is still consistently referred to as "Mohammed Ben Musa". The nickname Nisba hardly plays a role even at this time. Before Rosen's translation, al-Hwarizmi was hardly known in Europe. Hence e.g. hints from Steinschneider 30 years later (1865) that Ben Musa is now "mentioned more often" - > conclusion: Before that, he was rarely mentioned in European literature and almost not at all before Rosen's translation.
England
93
d
19th century
1835
Dictionnaire de l'Académie française
Lexicon
Strikingly, there is no entry at all in edition 6 on algorism in Reinaud's time, only on algebra; this is important for Reinaud's conjecture, which cannot be based on contemporary dictionaries with regard to etymology; The 4th edition of 1762 still mentions the word and gives a functional interpretation. Conclusion: At the beginning of the 19th century, the word algorithm was a term that had disappeared from some dictionaries due to its lack of relevance before it was rediscovered - regardless of the etymology, it did not play a major role at times, presumably due to the post-abacus reference and pre-industrial low relevance of trade
France
94
c
19th century
1836
Chaslet
La Geometrie des Hindous
His conception of algebra as a procedural and transformative practice is fully compatible with transmission via practical computational media (such as arithmetic boards), even if he himself did not explicitly address such material channels. The fact that Chasles does not mention al-Ḫwārizmī is neither a coincidence nor a mistake – it is a historical marker for the state of knowledge before the al-Ḫwārizmī canonization: he was apparently just as little known or relevant as a person as the word algorithm
France
95
c
19th century
1836
Raynouard
Lexicon
The entry still traces the term back to arithmetic and alguarismo / algarismo; no reference to al-Hwarizmi, important for Reinaud, who mentions this entry, but does not adopt traditional functional derivation and instead refers to Hellert 1548, although the site is not verifiable there.
France
96
c
19th century
1839
Halliwell
Rara Mathematica
first of all makes it clear that al-Hwarizmi is not mentioned at this time - not even in the context of algebra; assessment that the Dixit is from the 14th century; Illustration of Account Table for merchants from the 14th century (corresponds to the line algorithm; merchants/such as Fibonacci); shows that the same understanding of calculation methods was also available in the trade in England
England
97
a
19th century
1845
Reinaud
Mémoire
For the first time, he suspects the principle of the eponym with al-Hwarizmi. Important: It clearly indicates that it is only a guess. He also checks whether al Biruni or al Hwarizmi could be meant, since both come from Khorasm. Cantor 1880 credits him with the rediscovery, as do Narducci in 1858, Woepcke in 1863 and Littre in 1873. In this respect, it is still undisputed in scholarly scholarship that Reinaud is the one who first mentioned the al-Hwarizmi eponym. Core problem: As shown under Hellert 1548 and Wüstenfeld 1877, there is no original reference to Alchorazem (no matter in which spelling) in Hellert's book. So there is no evidence that there could have been more than one handwritten entry by any reader. This circumstance also indirectly proves that hardly anyone except Wüstenfeld seems to have read the book, which is why Reinaud's reference to Hellert is quoted again and again without checking the sources (in some cases to this day). This is exactly the basic pattern of citogenesis in the early stages.
France
98
d
19th century
1857
Woepcke
Algebra
compares several algebra methods and the scholars behind them. Woepcke thus effectively makes al-Hwarizmi better known in historian and mathematician circles; but he always calls him under the name "Ben Musa" -> his epithet/Nisba is therefore still not an essential identifier of his person, but subordinate or partly even meaningless from von Woepcke's point of view.
France/Germany
99
a
19th century
1857
Boncompagni
Numero Indorum
This book is the first printed publication of the Dixit Algorizmi. The title does not exist in the original, but is fabricated by Boncompagni. Although Boncompagni's text is still cited as an important source for eponyms, it itself provides no reference to al-Hwarizmi; nevertheless, such a reference is claimed in many publications. Problem: The 1857 editions contain no reference to al-Hwarizmi by Boncompagni; there are hardly any texts on al-Hwarizmi in Boncompagni's estate either.Conversely, it is striking that Narducci, of all people, is the secretary of Boncompagni in 1883 who tries to enforce the Ghubar method as a derivation of the term; Eneström, who takes over Boncompagni's Bulletino, also criticizes Cantor's working method several times and supports Narducci's Ghubar thesis in 1906.This book is of great importance here from a completely different point of view: On the basis of Boncompagni's precise work, it becomes clear that the points used by the unknown author in the text (e.g. . . xxx.) have a much more important function than generally assumed.
Italy
100
a
19th century
1857
Boncompagni
Liber algorismi de pratica arismetrice (Liber algorismi de pratica arismetrice)
The second important algorism publication by Boncompagni, which is attributed to Ioannis Hispalensis. Problem: the original is given as "Parigi Codice contrassegnato Ancien Fonds, n.° 7359". It can be viewed under https://gallica.bnf.fr/ark:/12148/btv1b9068289w/f35.item.zoom. -> see above.
Italy
101
b
19th century
1858
Narducci
Saggio di voci italiane derivate dall'arabo
Narducci is relevant because he worked for Boncompagni. He critically examines the eponym, but considers it conceivable and wrongly dates the Dixit to the 12th century, which only later proves to be an error. He points out that the Nisba as an origin is not a reliable criterion for the connection to Algorizmi. However, he assumes that only al-Hwarizmi can be considered as the author of the book on algebra mentioned in the Dixit. The text of Yasamin with the same title could not have been known to him. For him, therefore, it is not the name that is the important indication (phonetic similarity), but the conviction that there is only one author of a book that refers to algebra in the way mentioned in Dixit. But the fact is: Yasamin (see above) has written (see above) both a text of the same name on algebra and an additional text on arithmetic; Yasamin's works, however, had only become known in Europe in the 20th century; Objectively, al-Hwarizmi was never without alternatives - but it was unknown at the time that ibn Yasimin's works existed
Italy
102
d
19th century
1863
Woepcke
Mémoire sur la propagation des chiffres indiens
traces algorism to names with multiple references to Reinaud in 1845 to al-Hwarizmi; although previously only Ben Musa was mentioned in his work, the Nisba suddenly comes to the fore: this is the beginning of the modern depiction, which Ben Musa now only mentions under the Nisba.
France/Germany
103
d
19th century
1864
Stone cutter
About the lunar stations
In this text, he warns several times of the danger of pseudepigraphies in the history of mathematics (= writings that were wrongly attributed to a famous person in order to give them greater authority); Criticism in this regard is made, among other things, with regard to Reinaud; Steinschneider is conspicuously critical of the thesis that the eponym can be traced back to Ben Musa, since he knows no negative figures; he uncovers many inconsistencies; considers, with reference to Boncompagni's translation, that algorism could come from al-Hwarizmi - nevertheless, clear doubts of the thesis are (still) recognizable. This is important, because Steinschneider will equate it with Cantor in 1865: a) the word algorism is an eponym b) it means al-Hwarizmi
Germany
104
d
19th century
1865
Cantor
Salem Codex
Cantor refers to "nominative" here as "proof" of eponym. In doing so, he ignores all other clues in the same text that allow for different interpretations. Nor does he give any references to Reinaud, Narducci and Woepcke, who discuss the principle of the eponym much earlier. Among other things, it is difficult to use terms such as "proof", although there are only minimal phonetic indications; It is also striking that the same volume X of the Journal of Mathematics and Physics contains articles by Steinschneider, who first fills in the "empty" eponym that Cantor proves grammatically (questionably) with al-Hwarizmi. Friedlein also publishes articles in it.
Germany
105
d
19th century
1865
Stone cutter
Middle Books of the Arabs
In the same issue as Cantor, he equates "Epyonm = al-Hwarizmi"; like Cantor, he gives no reference to the previous publications by Reinaud, Narducci and Woepcke, which have already addressed the eponym thesis and the person of v. al-Hwarizmi earlier; this gives readers of volume X the impression in the German-speaking world that it was Cantor and Steinschneider who first rediscovered the eponym u. al-Hwarizmi. At the same time, the comparatively high reach of the journal and its reputation ensured for the first time that the eponym's thesis has reached larger circles of science.
Germany
106
b
19th century
1866
Woepcke (Chaslet); Boncompagni
Au Calcul gobari
the content makes it clear that the Arabic treatise analysed in the text fully corresponds to algorism in terms of content. However, it is avoided to name the contents of the Arabic ġubār as algorism. Based on terminology, system maturity and comparison with datable Arabic authors, the original text was very likely written in the 10th or early 11th century, i.e. well before the Latin algorism texts of the 12th and 13th centuries.
France
107
d
19th century
1869
Friedlein
Number sign
The philosopher named "Algus" in the Carmen or Tractatus, who is not described in terms of his origin, suddenly becomes an "Arab" scholar through Friedlein's editorial additions to the originals. The fact that the person in the text is simply called "Algus" and not "Algorizmi" is also ignored so that the transformation of the name "Alkharizmi" used by him via "Algorizmi" and "Algoarismi" fits the algorism. Accuses Fibonacci of a wrong understanding of algorithms. This assumption is important, because otherwise the term algorism could not have fit the al-Hwarizmi eponym
Germany
108
d
19th century
1869
RAE
Diccionario de la lengua castellana
up to this edition, the RAE dictionary still contains the old derivation and reference to alguarismo/numerical arithmetic; an indication that the al-Hwarizmi thesis was first disseminated after Reinaud, but also after Cantor & Co
Spain
109
d
19th century
1871
Treutlein
History of numbers
postulates the forgetting thesis of the "true" origin in the Middle Ages. Mohammed Ben Musa is now only called al-Hwarizmi, as if it were certain that he would have been known only by his epithet in the Middle Ages. The eponym is finally solidified in the German-speaking world as "secured"
Germany
110
c
19th century
1872
Wings
Fihrist Translation
Flügel is relevant because he has translated many texts. Among others the Fihrist. However, he can only know fragments of the Fihrist. A mention of the "lost" original of al-Hwarizmi's arithmetic cannot be based on wings, since chapter VII, which mentions al-Hwarizmi with regard to algebra, was not yet available or found at that time; only later will all of Dogde's chapters be translated - but with editorial additions that are not contained in the Arabic originalProblem: To this day, there are links on the Internet to the Fihrist that say that the (never found) arithmetic work of al-Hwarizmi is mentioned in Flügel.
Germany
111
b
19th century
1873
Littre
Dictionnaire de la langue française
It is probably the first encyclopedia entry on the eponym ever; he refers to al-Hwarizmi exclusively with reference to Reinaud's thesis of which, despite explicit conjecture, is quoted: "according to M. Reinaud by al-Khwārizmī, the Khoresmic, famous Arab mathematician, who lived under the caliph al-Maʾmūn, in the first third of the 9th century; this etymology is very preferable, as it explains the meaning well"; thus, although there is a weighing of the traditional functional with the new eponymic derivation, there is no indication that it is a mere conjecture based on the phonetic similarity of a single book on astronomy and that several persons (even according to Reinaud) who could be al-Hwarizmi are also possible. Here, too, Heller's book of 1548 has obviously never been read. Interestingly, although al-Hwarizmi was hardly known in Europe until the middle of the 19th century, he now became a "famous Arab mathematician"; this is the pattern that is also mentioned by Steinschneider (1864), Wüstenfeld (1877) and even Cantor as an indication of pseudepigraphon - i.e. the need to trace a text back to well-known personalities of history
France
112
a
19th century
1877
Wüstenfeld
The translations of Arabic works into Latin since the XI century.
In 1877, Wüstenfeld documented the obvious discrepancy between the doctrine (eponym = al-Hwārizmī) and the primary source: he reveals that Hellert's text, on which the eponym theory since Reinaud has been based, does not in fact name any Arab or Indian scholars at all - so an entry like "Alchoarizam" could hardly refer to it. Wording: "I find no Arabs in it, however, but only Magistri astrologiae in general, and especially Ptolemy, Dorothius and Hermes." In addition, Wüstenfeld points out in the same text that in the 17th century there was a book "Algorism Magistri Gerardi in integris et minutiis." that does not name al-Hwarizmi as the namesake. He thus provides all the information that speaks against the al-Hwarizmi thesis. As with several others, the criticism of the eponym is presumably in a rather subtle form, since an open contradiction to the mainstream would have been risky for the reputation (similar to Narducci, Curtze and others, who mainly try to argue in terms of content and not confront).
Germany
113
a
19th century
1880
Cantor
Lectures
The lectures finally manifest the eponym thesis; Cantor derives the eponym with detailed justification; mocks other interpretations; At the same time, and surprisingly, Cantor himself provides indications of the danger of pseudepigraphon or citogenesis and gives examples of this; In addition, there is mockery and irony of derivations other than those of the eponym; Cantor points out in passing in the footnote that Reinaud was the first to discover the al-Hwarizmi eponym. Wording: "To have recognized the name Alchwarizmi in the algorithm is the great merit of Reinaud {Memoir lnde pag. 303 sq.), who expressed this idea as early as 1845, i.e. long before the discovery of the Cambridge Codex turned the conjecture into certainty"; the last sentence is explosive, as it reveals the doubts that existed about this thesis. It proves that the eponym thesis itself presented itself for almost 40 years as merely a suspicion for Cantor, which became the supposed "truth" with a pseudo-find; Inglorious climax: Cantor justifies al-Hwarizmi eponym with the term, which was only introduced by Boncompagni in 1857 (algoritmi with t instead of z). Wording: "Such aberrations cannot be surprising when one takes into account that, by playful coincidence, all other forms of the name of our Arab scholar that have become known do not sound nearly as related to the algorithm as the most recently published Algoritmi." Cantor's lectures are still cited in many sources today as evidence for the proof of the eponym. Great caution is required here, as Eneström's many warnings prove.
Germany
114
a
19th century
1883
Narducci
Ghubar thesis
Narducci in 1858 was one of the first authors to critically examine the eponymic explanation and at the same time he is the first to seriously elaborate the content of the Ghubār thesis. Although he does not explicitly claim that the word algorism is directly derived from al-ġubār; but he is the first to systematically establish the identity of algorism and dust calculation on the level of practice.The starting point is the text Introductorius liber qui et pulveris dicitur in mathematicam disciplinam (12th century, Vatican manuscript), which Narducci compares with Boncompagni's Algoritmi practicae in terms of content. It shows that both texts largely agree in structure, procedure and didactic structure. Boncompagni describes computing per pulverem in exactly the sense in which it has previously been reconstructed by Woepcke and Chasles for ḥisāb al-ġubār: as an operative arithmetic practice based on writing, wiping, and rewriting. In this sense, the titles liber pulveris and liber algorismi are functionally interchangeable. By identifying the content of algorism with dust calculation (ḥisāb al-ġubār), Narducci also creates the technical foundation for later linguistic interpretations. He shows that liber algorismi and liber pulveris refer to the same operational practice and that algorism is not primarily to be understood as a theory, but as a concrete computational technique. On this basis, it becomes plausible for the first time not to regard the conspicuous phonetic proximity between algor- and ġubār as a coincidence. Narducci himself does not formulate an etymological derivation; but his analysis prepares the ground for later theses such as that of Corriente, who explicitly interprets the word alguarism as al-ġubār-ism.
Italy
115
d
19th century
1884
Narducci
Ghubar thesis
Narducci's new Ghubar thesis is presented at the Royal Academy of Sciences of Turin; the action is supported by Prof. Siacci. He was a proven friend of Boncompagni. Narducci his secretary.However, the result and result are unknown; It is striking that both Narducci and Boncompagni must have been aware of the implications of the equation of pulveris and algorism, when Siacci, who was already highly respected at the time, actively supported the Ghubar thesis; at the same time, a strong indication that even Boncompagni was not only aware of the Ghubar thesis, but also supported it in terms of content; he himself had also edited and commented on the work of Woepcke/Chaslet on dust calculation.
Italy
116
d
19th century
1884
Diccionario de la lengua castellana; Various encyclopaedias;
Apply eponym.
first dissemination of the al-Hwarizmi thesis without evidence in Spanish encyclopaedias of the RAE; this illustrates the sudden departure from functional entries of all previous editions; thereafter, Europe-wide distribution took place in encyclopaedias without source references with short entries without references, including Oxford, Meyers, Brockhaus, etc.; the eponym is beginning to be institutionalized worldwide
Spain
117
c
19th century
1885
Narducci
Divisions. After the abacus in the 12th century
Reference to Joannes Olibaso (?). confirms Gerbert's divisions (De numerorum divisione, Caps. VI/VII/XIV/XV) as proto-abacus in Italy before 1200 – iterative reductions (diminutio), displacement, differentiae integrae. This is relevant insofar as Narducci obviously creates a new overall picture that focuses more and more on Europe and reassesses the possible roots of mathematical knowledge as a whole
Italy
118
d
19th century
1887
Hunrath
Understanding the word algorism
He finds a Parisian text from the year 1503 in which a philosopher is named by name "algorism" and sees this as proof that it was still known until the 16th century that al-Hwarizmi gave the name. Hunrath also says that Boncompagni 1857 is the proof of the correctness of the eponym adopted by Reinaud and Cantor (which is precisely wrong). This note in turn led to research by French readers of the Bibliotheca Mathematika, who in 1891 described the book as a copy of the Tractatus of Sacrobosco and therefore the philosopher "Algorism" is actually called "Algus" in the original. Thus, it would be a fantasy name that only says that it could be a person with this name - but not which one.
Germany
119
d
19th century
1889
Nagl
Algorism Script 12th Century
Doubts translation by Adelard v. Bath, as there are only vague clues. Quote: "while the two writings immediately following Alkharismi, which are to be mentioned immediately, testify only to the first acquaintance of individual men with this subject." Columnas are interesting in terms of content: Explicitly mentions "columnam" (column shift); Boethius-like. Full-Edition Roots: √625=25; √2401=49th iteration: "Subtrahe quadratum... remainder ad proximam columnam". Gerbert-Link: "Multiplicatio instrumenti"
Germany
120
c
19th century
1897
Curtze
Algorimsus script of the 12th century
mentions various references to the attribution of algorism to Gerbert; uses column-based multiplication (grid for fractions, position pairing for roots). Not an "algorism" label, but "multiplicatio" with two columns (41 lines/page) and figures – direct predecessor of Dixit/ġubār in Latin Europe
Germany
121
d
19th century
1897
Curtze
Algorsimus Vulgarem
comments on Sacrobosco's Algorithums Vulgaris; Dust calculation / arithmetic board is implicitly central to its algorism; it requires deletion, conversion, restarting, which depends on the hardware; Paper calculation (sine deletione) is a later development; Curtze, contrary to the now universally established custom of the time, does not mention al-Hwarizmi at all as the originator of algorism; Curtze explicitly says that Sacrobosco's algorism is "anonymous" and that the source text is translated from Greek into Latin by an Arab. He emphasizes dust calculation, according to which every step must be deleted again (al-ghubar).
Germany
122
d
19th century
1899
Bubnov
Gerberti Opera mathematica
Bubnov publishes Gerbert's "In De numerorum divisione"; in it, Gerbert describes the movement of the discs on the board; term "Sipos" in connection with the Apices (discs); Important source that proves a lot: Starting with similarities in content with algorism to the recommendation of the use of hybrid technical terms from different languages (Arabic, Greek, Latin)
Russia
123
b
19th century
1904
Eneström
Bibliotheka Mathematika
In this volume, several important aspects are highlighted that give an overall picture: Curtze 1897 and Narducci 1883 are dealt with in detail; Illustrates vividly that algorism can already be derived at this time completely differently from al-Hwarizmi; also massive criticism of Cantor. Original sound: "In fact, young authors have sometimes sent me articles for the Bibliotheca Mathematica, in which it is agreed without further ado that the Cantor lectures have duly taken into account everything that has been published so far on a certain question, so that it is too unnecessary to study the other relevant mathematical-historical literature. Under such circumstances, I have considered it right to point out the true state of affairs in my reviews of the lectures, although I knew that this would not be pleasant to some admirers of the highly deserving author." This makes it clear that even justified criticism was quickly perceived as an "insult to lèse-majesté" and required courage. Eneström also questions the existence of al-Hwarizmi's book on arithmetic. Original sound: "(otherwise completely unknown) work on the later so-called speculative arithmetic";
Germany/Sweden
124
b
20th century
1905
Eneström
Hypotheses for historiography
confirmed by Narducci in 1883; liber pulveris is part of Liber algorismi practicae (= Boncompagni 1857); Eneström is now also the editor of the journal BIBLIOTHECA MATHEMATICA; later integrates the journal once founded by Boncompagni (as Bulletino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche); Original sound: "In an earlier remark (BM 53, 1904, pp. 410-411) I have shown that the excerpts from the Introductorius liber qui et pulveris dicitur in mathematicamdisciplinam published by N a r d u c c i agree so essentially with the corresponding passages of the treatise: I o a n n is H i s p a l e n s i s liber algorismi de pratica arismetrice, that it can be asserted that it is not a question of two different treatises, but only of two manuscripts of the same treatise"; here, too, it is quite clear: dust calculation and algorism are two sides of the same coin
Germany/Sweden
125
b
20th century
1906
Eneström
Bibliotheka Mathematika to
methodological fundamental critique of exactly the kind of historiography on which the al-Ḫwārizmī eponym is based; Consensus does not replace source work. Eneström's methodological critique of the mathematical historiography of his time power makes it clear that the eponymic interpretation of the term algorism established in the 19th century is based on a hypothetical construction that is not supported by contemporary sources or by a consistent tradition. Conclusion: Overall, in addition to Eneströn, several actors (including Wüstenfeld and even Steinschneider and Cantor) emphasize the problems of the time: Hardly any reliable sources, few sources that are difficult to decipher, but a high willingness to heroize people, danger of citogenesis and pseudepigraphs. In addition, there is the national rivalry, in which history (no matter what kind) becomes the intellectual "surrogate theater of war". Boncompagni, for example, has stated that he did not bow to the Turin government's requirement to interpret historical themes in the sense of the still young unity of Italy (piegare la scienza a fini nazionali).
Germany/Sweden
126
d
20th century
1908
Smith
Rara Arithmetica
Comprehensive work with many algorithm examples from different times, including many examples that describe the deletion, as is also required by al-Ghubar: "The computer also always needed a stone (a tablet) or a similar object on which he could write and easily erase the digits with which he worked in his calculation" Mention of "galley form" (= shuttle form) of the division, which only makes sense when overwriting
England
126
d
20th century
1908
Smith
Rara Arithmetica
Comprehensive work with many algorithm examples from different times, including many examples that describe the deletion, as is also required by al-Ghubar: "The computer also always needed a stone (a tablet) or a similar object on which he could write and easily erase the digits with which he worked in his calculation" Mention of "galley form" (= shuttle form) of the division, which only makes sense when overwriting
England
127
b
20th century
1912
Eneström
Heidelberg Texts on the History of Mathematics
Again urgently warns against Cantor's scientifically questionable way of working; is important, since Cantor himself describes the danger of citogenesis or pseudepigraphs in the context of algorism in his lectures, but speculates himself without evidence. Although Eneström has questioned Cantor's eponym and working methods with facts in detail for years, an institutionalized narrative such as the al-Hwarizmi eponym was already untouchable at that time and survived any kind of criticism - thanks in part to worldwide lexicalization and worldwide citogenesis, which was able to spread particularly effectively due to a lack of space in books without source references
Germany/Sweden
128
d
20th century
1912
Decourdemanche
Ancestry of modern European numbers
provides systematic paleographic evidence that the European numerals originate from the Ghubār tradition and not from the forms of modern Arabic numerals. It is also another indication of the evolutionary development of the etymology: There is no "al-Hwarizmi Big-Bang", but a continuous, intercultural development.
France
129
d
20th century
1914
Benedict
The Hindu Art of Reckoning
Recognizes many errors in the content of Dixit Algorizmi; assumes earlier and therefore incorrect translation; even recognizes similarities in the content of the Dixit Algorizmi with Gerbert's "Characters" and the sand calculation of the Arabs
United States
130
d
20th century
1917
Ruska
On the oldest Arabic algebra and arithmetic
implicit criticism of Cantor's eponym: shows terminological instability of the personal name, pleads for the precedence of the historically evolved generic concept of algorism over modern personalization.
Germany
131
d
20th century
1921
Smith
Numismatic Notes and Monographs
it shows how limited the knowledge of the sources from the 14th/15th century from Italy & Spain. Quote: "It was therefore not appropriate to speak of a "line algorism", since algorithm was the exact opposite of calculating with counting stones"; that this is not true, but that algorism and abacus were sometimes used synonymously, is shown by the early encyclopedia entries in Italy, among other places; only the algorithm novus did justice to this description (16th century)
United States
132
a
20th century
1922
Steele
The earlies Arithmetics in English
Emphasizes the clear separation of "theory" and "implementation". Interpretation of "Ars numerandi" as implementation, not as theory.Even though he considers the al-Hwarizmi thesis to be possible as an origin, he also points to Latin texts that have exactly the al-ghubar meaning that is also described by Narducci in 1883: "Et modus suus erat in computando per quasdam figuras scribendo in pulvere. . . ." "Si voluerimus depingere in pulvere predictos digitos secundum consuetudinem algorismi . . ." "Et sciendum est quod in nullo loco minutorum sive secundorum . . . In pulvere debent scribi plusquam sexaginta." This shows how "torn" the statements regarding etymological consensus and content-related interpretation of algorism have been.
England
133
b
20th century
1931
Gandz
Arabic numerals
proves that hisāb al-ghubār and kitāb al-takht refer to abacus arithmetic terminologically and objectively; refutes Cantor's thesis that the abacus was unknown in Arabic literature; rehabilitates the Arab continuity between Indian place-value arithmetic and European algorism. At this point, at the latest, it is hardly scientifically tenable to explain algorism primarily as an eponym of al-Ḫwārizmī; but similar to Eneström, facts are not sufficient to critically question the eponym thesis
Austria
134
b
20th century
1968
Alfred E. Dear
Eastern Business Practices and Medieval European Commerce
Relevant remarks on trade in the Middle Ages: At the time of the 11th to 15th centuries, there was an intensive trade tradition between Italy and North Africa, among others. A common mathematical language was de facto necessary for Mediterranean trade. Especially the Ghubar method, which was standard in North Africa, had to be known and used by merchants: "each foreign merchant ... had to clear his goods through the diwan ... elaborate registers". Conclusion: For communication on the calculation of state taxes, customs offices, goods and warehouses, not only the "how" of the "Algho" could be taught in Florence in 1307, but also the term and its pronunciation in the Arab world; the same applies to Alguarismo in Spain: international trade required oral communication and thus phonetically similar terms; Algoa, Alchoa and Al-Ghubar come very close.
United States
135
d
20th century
1970
Dodge
Fihrist Edition
For editorial reasons, includes headings in Fihrist's translation that are not included in the Arabic original; also to Mohammed Ben Mousa al Hwarizmi. The headings are lexical additions of modern times, they suggest to the reader that Musa has always been known as "al Hwarizmi" - which is not the case in the Arabic original, since many scholars from Khorezm are named there.
Lebanon
136
b
20th century
1975
Henry
The singular Syllogisms of Garlandus Composita
Analysis of singular syllogisms in the Dialectica (but confusion: the author is Petrus Abelardus early 12th century, not Garlandus); but still given procedural logic: Fits ghubar-Abacus: Relational sign operations, independent of authorship; Logic test for word syllogisms: Explains how inferences are valid only from the meaning of the word, but can be deceiving (sophisms). Individual sentences (e.g. "der Grammatiker") seem like general ones ("alle Grammatiker"); "all" is often unnecessary. Applies to dialectica logic (early 12th century), useful model for abacus-like computational logic.Burburu mode: Singular identity logic that does not require universal quantifiers. Confirms the validity of an operation solely by the structural position (ex vi terminorum). This is the direct theoretical equivalent of the ġubār hardware: a "token" works through its place in the system, not through its linguistic meaning
England
137
d
20th century
1975
Mazaheri
Les origines persanes de l'arithmétique
the text deals with ghobar numbers several times. It illustrates a recurring historiographical pattern in which theoretical originality is predominantly attributed to the Eastern Islamic (Persian) tradition, while Western Islamic (Maghreb-Andalusian) traditions are primarily presented as transmission-oriented, implementive, or didactic.
Persia
138
c
20th century
1978
Hughes
Algebra Translation
formulates an important contradiction: "One may well wonder if al-Khwarizmi had forgotten that he had written a tract on the decimal system entitled 0/1 Hindu Numerals. 8 There he acknowledged that the decimal system originated with the Hindus; but here in the Liber algebre he credits himself with the discovery." This also speaks against al-Hwarizmi's influence on the Dixit Algorizmi
United States
139
a
20th century
1986
Vernet
Lo que Europa debe al Islam de España
Vernet does two things at the same time: He outlines the algorism family tree with regard to al-hwarizmi, which is still prevalent today, which, as described, contains various demonstrably speculative and false elements (e.g. non-mention that there is no evidence for the book of al-Hwarizmi, indication of fictitious titles for the "lost" book of al-Hwarizmi, which demonstrably comes from Boncompagni. mention of Sacrobosco, but omission of Fibonacci and Villa Dei). He treats "De numero indorum" as if it were the Latin title of a real work by Al-Ḫwārizmī, while at the same time explaining in detail that the content of algorisms corresponds exactly to what was known in Al-Andalus as hisab al-ghubar. Vernet thus explains a West Islamic ghubār practice via an East Islamic book invented by Boncompagni regarding the title, which has not yet been proven. But then he also says: "The consolidation of the Arabic numerals and the positional system took place in our peninsula (al-Andalus)." Vernet's book is explicitly the subject of the "retraction" of Corriente in 1999, who in 1997 explicitly says due to the inconsistencies of this account: The term algorism does not come from al-Hwarizmi. Corriente only retracts strategically. Not in terms of content.
Spain
140
a
20th century
1990
Crossley / Henry
Thus spake al- Kh wārizmī
recognizes many systematic and substantive errors of the Dixit; refers to uncertain old age; Comparisons with algebra reveal inconsistencies; puts forward the thesis that the author of Dixit does not know algorism in the sense of al-Hwarizmi; Reference to the age of the Dixit from the 13th century; in addition, wrong understanding of position; Conclusion: The required use of Dustboard and all other contradictions revealed here will be resolved if al-Hwarizmi ibn Yasamin is assumed to be the author of the texts mentioned in the Dixit instead of al-Hwarizmi.
England
141
d
20th century
1991
Allard
The Arabic Origins and Development of Latin Algorisms in the Twelfth Century
very frequently quoted text that is supposed to confirm al-Hwarizmi eponym; Allard assumes in the article that the Dixit is the oldest document, although it dates from the 13th century; Although Allard describes the material arithmetic practice of algorism, including working with powder, erasure and arithmetic board, he does not take the decisive step of identifying the Latin term "pulveris" with the Arabic technical term "ḥisāb al-ġubār"; Allard also ignores many of Crossley/Henry's findings; although important text often cited to justify the eponym, it cannot resolve any of the inconsistencies listed here in this table; rather, doubts are appropriate, as Eneström has also formulated with regard to Cantor.
France
142
b
20th century
1996
Corriente
Hacia una revisión de los arabismos
Corriente criticizes the eponym despite, or perhaps because of, Allard's publication in several of his own publications; he says that there are no reliable references to al-Hwarizmi, rather the term comes from alguarimso and al-ghubar - as it can be easily reconstructed here in this timeline. He proposes that the RAE replace al-Hwarizmi's thesis with al-ghubar's thesis; It is perhaps the first verifiable, non-convoluted and thus unambiguous statement - interpreted analogously: "The term algorithm does not come from al-Hwarizmi. Stop spreading this nonsense." However, this is precisely for this reason an "attack on the worldwide mainstream" that was still too little documented at the time; Corriente's unequivocal criticism leads to a backlash: Anyone who questions the scientific establishment must expect to be questioned himself!
Spain
143
a
20th century
1997
Corriente
Hacia una revisión de los arabismos
The only scholar who dares to question the al-Hwarizmi eponym unequivocally: "algorisme, alguarisme 'algoritmo': prob. de un bajo lt. *(al)gobarismus, del ár. hisāb alğubār 'cálculo con cifras', y no del nombre del matemático Alxuwarizmī, que no consta en absoluto lo introdujera, ni hay tecnicismo en árabe que lo refleje, a diferencia de aquella expresión." Translated: No! the word algorism does not come from al-Hwarizmi. Nor is there any evidence that he introduced it, nor is there any Arabic technical term to reflect this, as opposed to the expression of hisab al ghubar." He will publicly retract this statement in 1999. In addition, from 2003 onwards, the new editions will be published with the official "al-Hwarizmi" eponym.
Spain
144
a
20th century
1997
Crosby
The Measure of Reality
"The practices of medieval monks!) was to use the numbers 1-9 then the letters A-F. A number written A13D (to base 16) stands for: AX(16} + 1x(16? 4 3x16 + D and has numerical value: 10 x (16)? + 1x(16? + 3x16 + 13, thatis, 41276. Then a cipher with components a, b, c and d (each between 0 and 15) has numerical value: a + bx16 + cx(16)? + dx(16)." He also shows elsewhere that the evaluation of medieval "algoristics" depends entirely on the chosen definition of the term. If algorism in the sense of al-Khwārizmī is understood as written place value calculation with zero, the procedures appear strange and "difficult" and they hardly add any value in the field of trade, among other things. If, however, it is understood in the sense of the Western Arabic ḥisāb al-ghubār – as operative arithmetic in the sand or on the blackboard – the alleged problem disappears.The signs are not "chicken tracks", but visible work steps; and the zero is not a philosophical puzzle, but a functional placeholder.
United States
145
b
20th century
1997
Folkerts
The oldest Latin writing on Indian arithmetic according to al-Hwärizmi
The article impressively shows how highly speculative the title of al-Hwarizmi's "lost" book is constructed or justified. On the one hand, it is confirmed that the fihrist does not name the work on arithmetic. The commentaries that are supposed to be mentioned in Flügel's translation of Fihrist by As-Saidanänl are not even to be found in Dogde's translation of the Fihrist.
Germany
146
b
20th century
1999
Corriente
Diccionario de arabismos y voces afines en Iberorromance (1st edition, 1999)
"The empire strikes back": Only three years later, Corriente publicly retracts his own thesis, which had been put forward shortly before, with reference to the Spanish historian Vernet, who in 1978 supposedly states the reliable original of the Dixit (which, however, essentially refers to the publication of Boncompagni, namely the book with the fantasy title "De numero Indorum"). It is striking that Vernet, of all people, describes in 1978 exactly what Corriente says in terms of content: Ghubar and algorism are largely identical. Corriente gives in despite everything, he withdraws his Ghubar thesis, but at the same time gives explicit references to his previous sources in which he has defended the thesis. True to the motto: "Check for yourself what is more convincing". Conclusion: Anyone who openly opposes a globally institutionalized narrative as a scientist risks his reputation and may even have to genuflect in writing and publicly in order not to be ostracized
Spain
147
b
20th century
1999
Dehaene
The Sense of Numbers
Humans have a preverbal, visual-spatial sense of quantities (more/less, proximity/distance), but no innate exact numbers. Exact numbers are only created by cultural stabilization tools (language, signs, spatial arrangement) and are internalized through practice.al-ġubār (Dust Calculation)s represents such a stabilization tool:(New) numbers are not understood, but made surgically visible (points, lines, places), shifted, deleted and reset. Meaning arises from position in space, not from semantic understanding.Finger arithmetic and al-ġubār also follow the same logic: numbers are not necessarily understood, but shown. Arithmetic is a visible action in space – only later a conceptual activity.
France
148
b
21st century
2011
Høyryp
Tractatus algorismi (translation by Louis Karpinski)
Illuminates an earlier version of Tractatus algorismi by Louis Karpinski (1929). The complete absence of Arabisms in Jacopo da Firenze's algebra suggests that his algebraic practice does not come directly from Arabic texts, but indirectly via a Romance-language environment, especially in the Catalan-Occitan area.
Denmark
149
b
21st century
2001
RAE
DLE (al-ġubār release)
The RAE adopts Corriente's recommendation despite his retraction; Corriente's interpretation of Ghubar is even maintained until 2026, without mentioning al-Hwarizmi; However, the view of the RAE is almost exclusively taken into account or mentioned in the Spanish-speaking world. This "last persistently maintained trace" to the Ghubar past was so important precisely because it was precisely the lead to the al-Ghubar thesis! The example proves the usefulness of thoroughly conservative dictionaries. At the same time, it becomes clear that the RAE and also not Corriente were the founders of the Ghubar thesis, because it was founded comparatively early by Narducci - and Narducci was none other than the secretary of Boncompagni, who in 1857 set the "stone of the algorithm" rolling with his publications. Narducci also managed Boncompagni's estate - there are hardly any works on al-Hwarizmi in it, which can be explained by the fact that Boncompagni himself never advocated the al-Hwarizmi thesis. For him, Fibonacci was the one who received by far the most attention. However, it was only the entry of the RAE that made this old thesis visible in the 21st century. Interesting detail: The entry of the RAE for the older alguarismo, unlike the entry for Algoritmo, corresponds to the al-Hwarizmi eponym; RAE thus (consciously?) keeps both doors open; the more important term, however, is not alguarismo, but algoritmo. Also relevant: The RAE also mentions Corriente's hypothesis of the word "algobarism", which is very close to the result of this table, the "al-ghubar syllogism". Problem: The footnote that Corriente had written in 1996 was not included by the RAE, which misleads the statement that this term exists or that the "quiza" refers to the al-ghubar thesis as a whole and not only to the word algobarism.
Spain
150
a
21st century
2000
Wikipedia / AI
Eponymic popularization
Consolidation of eponyms through AI and Wikipedia as training data; estimates by several AIs that several million entries in online encyclopedias and specialist publications in 300 languages now spread the eponym; AI takes over this as training data without its own source check and thus reinforces the unsecured narrative on purely probabolistic algorithms
Worldwide
151
a
21st century
2001
Folkerts
Medieval Mathematics
Illustrates the evolution of the abacus from Gerbert to Gerland with Arabic numbers. The evolution shown also reflects the evolution of concepts with the Ghubar technique; also the relation of the syllogism to music (Boethius) and thus also the bridge of logic to the Qudrivium is illuminated
Germany
152
a
21st century
2003
King
The Ciphers of the Monks
Many illustrative examples of algorism, which show that arithmetic and algorism were essentially "al-ghubar", among others: "In addition, the said master Jean brought the numbers of the Greeks to England, introduced his confidants to them and explained their meaning to them. These characters are also used to represent letters [numbers]. The most remarkable thing about these characters is that a single character represents any number: op, which does not exist in Latin or in the algorithm. Now we have thought it appropriate to trace these signs in the present work. Take a stick. Draw [vertical] lines from this rod so that each forms a right angle, an acute angle, or an obtuse angle with the rod, as follows." Another example shows that merchants needed two computing systems: one operational on marketplaces, among other things (algorism). Another on accounting.
England
153
a
21st century
2005
Kunitzsch
On the history of the 'Arabic' numerals
It shows something very concrete: Latin scribes of the 10th–13th centuries do not translate numbers uniformly, but sometimes Roman, sometimes Arabic (east / west), sometimes as letters, sometimes mirrored, sometimes in the wrong reading direction, sometimes twice (Roman + Arabic), sometimes as words, sometimes as a circle ("circulus") for zero. The fascinating thing is that the arithmetic itself worked. Because: The numerical value is stable, even though the characters are unstable. But: A star at 23° remains at 23°, regardless of whether you write: XXIII, ٢٣, γκ, "viginti tres" or ⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪⚪The algorism was ud is therefore not a number theory, but a practice of stable placement. Conclusion for algorithms: AI does not "understand" – it reconstructs stable positions in a space of possibilities.
The treatment of the zero is particularly revealing: In Arabic: ṣifr ("emptiness"); In Latin: circulus ("circle") or nichil ("nothing") But: The function remains identical – as a placeholder in the positional logic. Whether you write "0", "Ø", "○" or "nichil" – the carryover works the same. This makes it clear that the value is not in the sign, but in the operation that makes it possible.
Germany
154
b
21st century
2006
Høyryp
Jacopo da Firenze and the beginning of Italian vernacular algebra
Clarified: Jacopo da Firenze (1307) shows an algebraic practice that cannot be derived either from Fibonacci or from the Latin translations of al-Ḫwārizmī or Abū Kāmil. The Arabic origin is certain, but of a different kind: not textual, not terminological, not translation-based. Differences in case order, normalization, and example use point to an independent, practice-oriented Arabic channel of tradition that entered the Romance world without Arabisms, without theory, and without naming authors. Rule-based, deductive computational practices that are algorithmic in the sense of step-by-step rules, but without algebraic symbolism, without theoretical justification, and without author-binding. These practices have an intrinsic consistency and dominated the European approach to algorism until the 19th century.Conclusion: If you want to take the word "algorithm" seriously, you have to stop thinking of it exclusively from the point of view of algebra
Denmark
155
d
21st century
2008
Ambrosetti
La Comparsa dell algebra in india
describes Indian dust calculation in algebraic original form; confirms that no work by al-Hwarizmi has been found to date, but continues to disseminate eponyms on the grounds that its origin was "forgotten" in the Middle Ages or the 16th century. It illustrates how the "forgetting thesis" has manifested itself up to the present day - a methodological approach with which, however, pre-astronautics also wants to prove that extraterrestrials were in South America in the pre-Columbian period - but this was "forgotten" with the demise of these cultures; however, there is no evidence for forgetting the al-Hwarizmi derivation (on the contrary, see Florence 1307, among others)
India
156
a
21st century
2009
Høyryp
Jacopo da Firenze and the beginning of Italian vernacular algebra
Open dispute about authorship, dating and independence of the late medieval abbaco algebra, which makes it clear that the relativization of al-Hwarizmi in Høyryp 2006 is immediately categorically rejected by consensus advocates such as van Egomond. Confirms the existence of algorithmic (rule-based) computational practice in Europe before and alongside algebraic theory.Computational methods were practice-oriented, deductive and rule-driven, without algebraic symbolism or theoretical justification. "Algorithmic" here refers to rules of action, not theory.Algebra appears as a later theorization, not as an origin.Høyrup makes it clear that Louis Karpinski recognized early on that Italian and Anglo-Romance arithmetic traditions could not be derived from al-Ḫwārizmī's Latin translations. Høyrup explains the later neglect of this insight historically, not objectively. The article supports a functional-practical interpretation of algorism as an independent computational culture
Denmark
157
b
21st century
2010
Dehaene
Read!
Reading and writing only work because they dock onto existing neuronal structuresAlgorism was never first writing – it was speaking, pointing and calculating in the common space. Just as children first learn to speak, merchants had to be able to pronounce numbers or understand them phonetically before they could consciously write signs. Similar to the "blackboard" for schoolchildren, the "dustboard" or the al-ghubar principle was ideal for learning aldogisms through operational practice even without much writing knowledge. Therefore, the early texts of algorism were primarily aimed at teaching surgical exercise (Carmen, Tractatus, and Dixit). The dustboard was also a cheap, ubiquitous ephemeral practice and execution medium with which one could learn algorism without having to understand it abstractly and semantically (especially at the beginning).
France
158
d
21st century
2010
hist-math.fr
online encyclopedia
many sites for algorism up to the 19th century. But there is no evidence from the Middle Ages to modern times that clearly refers to al-Hwarizmi. All back-projected representation regarding Dixit without evidence; illustrates how many teaching and learning portals spread the eponym (without ill will)
France
159
b
21st century
2011
Høyryp
Explicit and less explicit algorithmic thinking
Clarified: There were rule-based, deductive computational practices that were algorithmic before algebra existed, without algebraic symbolism, without theoretical justification, but with inner Folgerichtigkeit.In several mathematical cultures – especially those not derived from Greek theory – mathematical practice was based on rules (regula, μέθοδος) and not on theoretical insight in the Greek sense. These sets of rules were later often devalued as merely "empirical". In fact, however, they are algorithmic methods in the sense of straightforward, step-by-step calculations without branches or decision trees. The difference to theoretical mathematics therefore lies not in the structure of the procedures, but exclusively in their subsequent evaluation. Rules could be formulated abstractly and then explained with examples (e.g. in Chinese and Indian texts), or they could be implicitly conveyed by paradigmatic examples followed by the request to "proceed accordingly in similar cases" (as in the late medieval abacus tradition).
Denmark
160
a
21st century
2018
Wikipedia Editorial Team
Algoritmi ou algorizmi
Heated discussion among Wiki editors, which shows how charged the etymology of the term can be: Doubts about the correctness of the al-Hwarizmi thesis lead to resistance: "What, in your opinion, did the word algorithm result???"
Conclusion since Narducci 1883, Eneström 1906 and Corriente 1999: Anyone who doubts the previous etymology of "algorithm" is entering a "hornet's nest".
France
161
d
21st century
2014
Dörflinger
Moritz Cantor's Lectures on the History of Mathematics ̈ from the Perspective of His Critic Gustaf Enestrom
has published a compilation of Eneström's critiques of Cantor's methods and inaccuracy. Interesting above all because Heidelberg administers Cantor's estate - but here is a list of many things that have already been mentioned about Enström and Cantor's questionable methods; it is difficult to assess whether the criticism of Cantor's working methods is shared by the author or not. Unfortunately, it has to be said: The criticism seems justified - but not only of Cantor!
Germany
162
c
21st century
2019
Denkmayr
Diploma Thesis
the connection with the purely theoretical teaching of the contents of the quadrivium explains that the acquisition of higher mathematical knowledge in the early Middle Ages was the privilege of a small educated elite; important for Fibonacci's spread of arithmetic as a merchant -> operative arithmetic vs. monastic theory. Algorism was above all a concept of trade, which indirectly confirms the al-Ghubar thesis; religious content, however, also makes it clear that much in terms of algorism potentially goes back to Gerbert
Austria
163
a
21st century
Computer, Keyboard/ Wikipedia / AI
Emoticons/Smileys
We don't understand smileys. We recognize them anyway. Even with rotation: It's not about the form, but about the invariant structure under transformation. This also applies to AI: in Transformer, ̄\_(ツ)_/ ̄ recognizes not because it "knows what shrugging its shoulders means", but because the token sequence occupies a stable cluster position in a relational embedding space – regardless of rotation, font or language.Conclusion: Even with rotation, distortion or sound shift, the "primal pattern" remains recognizable – not because it "means", but because it is operationally stable. "True" is what proves itself even under transformation. And: Stable systems only show up when they are disturbed, e.g. if instead of 🙂 accidentally writes :-9 on the keyboard (forgetting to press the shift key). If the positional relation of the signs is changed, the meaning is also reversed. Example: (: = 🙂 = positive 🙁 = ): = negative
International
164
a
21st century
Dictionnaire de Citations
historical development of concepts
shows that the al-Hwarizmi in the sense of today's understanding of algorithms is only one of several historically grown readings; "Qu'on peut juger un chiffre en algorisme" (Marot, 16th century)L'algorithme règle les gestes à opérer (Ruyer, 1930)Art de supputer avec justesse et facilité (Encyclopédie)Suite finie et non ambiguë d'opérations (modern usage)— several historical meanings standing side by side at the same time within a single dictionary entry; Conclusion: The term "algorithm" did not etymologically emerge from theory, but from a functional computational practice that was only later interpreted algebraically
