History of medieval mathematics

மனித வரலாற்றில் கணக்கறிவு சார்ந்த எண்கள், பின்னங்கள், அளவைகள் பண்பாட்டுப், பொருளியல் செயல்பாடுகளில் பயன்படுத்தப்பட்டுவருகின்றன. இவை இடைக்காலப் பொருளியல் வரலாற்று மீளுருவாக்கதிற்கும் அரசுருவாக்கத்தைப் புரிந்துகொள்ளவும் மிகவும் அவசியமாக உள்ளன. தமிழ் நாட்டில் நிலமேலாண்மைச் செயல்பாடுகள் சங்க காலத்திலிருந்து தொடங்கிவிட்டன எனலாம்.  இடைக்கால அரசுருவாக்கத்தின் போது அளவைகளும், நிலக்கணக்குகளும் மிக முக்கியமான பங்கு வகித்தன.

This article examines al-Ḥāwī fī al-Ḥisāb, a short yet comprehensive treatise by the 14th–century mathematician of the Mamluk Turkish state, Ibn al-Hā‘im, in both mathematical and historical contexts. The work presents the two principal aspects of the science of arithmetic within the classical Islamic mathematical tradition: known (ma‘lūm) and unknown (majhūl) calculations. In the section on known calculation, operations involving integers, fractions, and square roots are addressed in detail. The four basic operations –addition, subtraction, multiplication, and division– are explained through a variety of techniques. Especially noteworthy is the inclusion of nine distinct methods for multiplication, demonstrating the diversity of computational techniques of the time. The division section discusses topics such as prime factorization, divisibility rules, and proportional distribution. Square root operations and approximation techniques are also summarized in a notably concise and striking manner. In the section on unknown calculation, algebraic solutions aimed at determining unknown quantities are offered under the headings of four proportional numbers, the double false position method (ḥisāb al-khata’ayn), and algebra (al-jabr wa al-muqābala). Six fundamental types of equations are defined using classical notation, and the solution algorithms are ex¬plained in detail. The language of the work is clear, concise, and easy to memorize, indicating that it was evidently prepared for instructional purposes. In this respect, al–Ḥāwī is not only a text reflecting the mathematical knowledge of its period but also a didactic source that documents how the science of arithmetic was structured within the classical Islamic educational system.

Ali Kuşçu’nun bilinen kapsamlı tek Arapça matematik kitabı er-Risâletü’l-Muhammediyye fî ʿilmi’l-hisâb, müellifin bizzat kendisi tarafından Fâtih Sultan Mehmed’e ithaf ve hediye edilmiştir. Eldeki nüshalara göre eser, iki asır boyunca çoğaltırlmıştır. Bu da alan uzmanları tarafından okunup, okutulduğu anlamına gelir. İki asır boyunca tedavülde olan bir genel matematik kitabı hakkında mevcut araştırmalar oldukça yetersizdir. Bu eksiklikle müellifin bir polimat olduğu bilgisi birlikte düşünülünce, eser verdiği her bir alandaki araştırmalar, bütünü görmek adına daha fazla önem kazanır. Bundan mülhem bu makalenin konusu Muhammediyye’nin mesâha bölümü olarak seçilmiştir. Zira bu bölüm içeriğiyle müellifin hem astronomi hem de dil yönelimlerine işaret eder. Ancak mesâha bölümünün konumunu belirlemek için öncelikle kitabın tamamı içerisindeki yerini görmek gerekir. Bunun için öncelikle kitabın genel içeriği verilmiş ardından mesâha bölümünün diğer bölümlere göre konumu ve kendi içerisindeki tasnifi ele alınmıştır. Bölümün ayrıntılı içeriği ve bu içeriğin seleflerinin matematik kitaplarının mesâha bölümlerinden farklılaşan özellikleri verilip bunların astronomi ve dil çalışmalarıyla muhtemel ilişkilerine dikkat çekilerek makalenin asli amacı gerçekleştirilmiştir. Neticede de Ali Kuşçu’nun (i) genel matematik kitabı tasnifi, (ii) mesâhayı konumlandırdığı yer, (iii) mesâhanın içeriğindeki ağırlık noktaları, (iv) mesâha kavramlarının tarifleri ve (v) mesâhanın ilişkili olduğu alanlar hususlarında seleflerinden ve çağdaşlarından farklı bir pozisyon aldığı tespit edilmiştir.

An introductory section discusses the utility of the algorithm concept in the historiographic analysis of non-recent mathematics, in particular the sense that can be given to claims that a particular mathematical culture was of algorithmic type. It concludes that the adequacy of this epithet when applied to a mathematical culture does not depend on whether texts used in teaching are built up around paradigmatic examples but on whether the production of rules or algorithms was regarded as a central activity for those whom we would count as "mathematicians" (that is, producers of mathematical knowledge). Three medieval examples of attitudes to algorithms follow. First, Jordanus de Nemore's De numeris datis is shown to develop a method to combine algorithms and deductivity, in an alternative to algebra. Second, Barthélemy de Romans' graphic schemes for organizing the complex algorithms used to solve the sophisticated variants of the problem of the "unknown heritage" are discussed. Third is considered Nicholas Chuquet's dismissal of these schemes and algorithms, in favour of the algebraic tool. An early version of what follows was presented at a workshop dealing with "deductive algorithmic practices in pre-algebraic mathematics". 1 The only word here which does not ask for conceptual clarification is "in"; provisionally, we may pretend to know what "deductive", "mathematics" and "practices" stand for. We may claim, as a mathematician-friend of mine (Bernhelm Booß-Bavnbek, personal communication) once did polemically, that "there was no algebra before Emmy Noether". Or, with Michael Mahoney [1971: 372], be more liberal and accept as algebra that which began in the epoch of Viète and Descartes, while the techniques of an al-Khwa ¯rizmı ¯and a Cardano (etc.) still represent an "algebraic approach" only. Or, finally, we may accept that the technique which al-Khwa ¯rizmı ¯and his successors until the mid-sixteenth century spoke of as "algebra" was algebra. If we take one of the former two of these roads, "pre" in "pre-algebraic" may be taken without trouble in its usual, literal chronological sense, and "prealgebraic" can be take to mean simply "before 1900 CE" or "before 1600 CE". If we choose the third road, the chronological interpretation of the term means "before 800 CE" -and if we accept Sanskrit "algebra" (not to speak of Old Babylonian "algebra") as algebras, we get still earlier limits. This would at best prevent us from looking at anything from the Middle Ages or later, which would not make much sense. Instead, "pre-algebraic" may be interpreted metaphorically, as "not affected by algebraic thinking"; this leaves us the possibility to work, and therefore I shall apply this reading of the term. With this choice it does not matter much whether al-Khwa ¯rizmı ¯etc. made algebra or only had an algebraic approach. On the other hand, it is hard to imagine that anything mathematical which was created in Europe after, say, 1750 CE should not be somehow affected by algebra, given how pervasive use of algebraic symbolism had become by then -even avoiding algebraic reasoning had by then become a deliberate choice, no consequence of ignorance. Regarding algorithms, we may start from the probably earliest paper which tried to apply this concept to non-recent historical material: Donald Knuth's "Ancient Babylonian Algorithms" from [1972]. 2 According to Knuth (p. 672),

Chère (s) lectrice (s).. cher (s) lecteur(s)..
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      Les exercices proposés, dont les solutions détaillées sont présentées immédiatement après les exercices de méthodes de chaque chapitre, sont organisés selon un degré de difficultés progressif, allant du simple au complexe, et du général au particulier.
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NOUREDDINE  HALI

Alī Qūshjī (d. 879/1474), a fifteenth-century astronomer, mathematician, theologian, and linguist, played a key role in the formation of Ottoman scientific life. This role has been analyzed in the history of science and in the history of thought more broadly. Upon Sultan Mehmed II’s invitation, he contributed to the formation of the Fatih Semâniye and Ayasofya Madrasas by transferring the scholarly accumulation of the Samarkand mathematicsastronomy school to Istanbul, and his works were widely used in Ottoman intellectual circles for centuries. Risāla dar ʻIlm-i Hisāb, which is the first of two mathematical works by Alī Qūshjī, was written in Persian, probably in Samarkand, sometime before 861/1456. This work has a remarkable place in the Islamic mathematical tradition given its numerous copies, its widespread distribution in Iran and Anatolia, and its being used as a textbook for centuries. In this work, an analysis based on several copies of the work will be shared, correcting outdated information in the literature and clarifying the place and value of Risāla dar ‘Ilm-i Hisāb in terms of the history of science. As an example of the work’s content, the method of finding the quadratic root of integers will be examined and the proof of the method will be given.

Fâtih dönemi astronomi, matematik, kelam ve dil âlimi Ali Kuşçu’nun (ö. 879/1474) Osmanlılar’da ilmi hayatın teşekkülünde oynadığı kilit rol, bilim ve düşünce tarihi araştırmalarında sıklıkla incelenir. Mensubu bulunduğu Semerkant matematik-astronomi okulunun ilmi birikimini Sultan II. Mehmed’in davetiyle İstanbul’a taşıyarak Fâtih Semâniye ve Ayasofya medreselerinin teşekkülüne katkıda bulunmuş, eserleri Osmanlı entelektüel çevrelerinde yüzyıllarca tedavülde kalarak gündem oluşturmuştur.

Risâle der ʿİlm-i Hisâb Ali Kuşçu’nun telif ettiği iki hesap eserinden Farsça olandır ve 861/1456’dan önce Semerkant’ta telif edilmiştir. Eserin dünya çapında yazma eser kütüphanelerinde şimdiye dek tespit edilmiş dört yüze yakın nüshası bulunur. Bu çalışmada, İran ve Anadolu coğrafyalarında yaygınlığı ve yüzyıllarca okutulması sebebiyle, İslam matematik geleneğinde dikkate değer bir yerde bulunan eserin elde edilen nüshalarına dayanan incelemesi ve telif sürecine dair bulgular paylaşılacaktır. Böylece literatürde yer alan güncelliğini yitirmiş az sayıdaki malumatın tashih edilmesiyle birlikte, Risâle der ʿİlm-i Hisab’ın bilim tarihi açısından durduğu yer ve sahip olduğu değere dair bazı soruların aydınlatılması amaçlanmaktadır. Ayrıca eserin içeriğine bir örnek olarak tam sayıların ikinci dereceden kökünü bulma yöntemi incelenecek ve yöntemin ispatı verilecektir.

During the Islamic period (medieval ages), graphical methods were employed to solve many spherical astronomy problems, a type of descriptive geometry known as the analemma. Analemma method transforms a three-dimensional problem on a sphere into a two-dimensional plane, eliminating the need for trigonometry and mathematical calculations to solve it. One of the practical applications of these methods, discussed in numerous Islamic works, is the determination of meridian and Qibla directions. Zījes were the practical works of astronomy in the Islamic period. Zīj-i Yamīnī, the second oldest known Persian work in the zīj tradition, was composed by Mūḥammad ibn Abī Naṣr Ḥaqāyiqī in Ghazna (present Afghanistan) in the 12th century AD. In Zīj-i Yamīnī, various analemma techniques were collected. One of them, a rare method, is a complex procedure for finding the longitude of the sun (the position of the sun on the celestial sphere along the ecliptic) from the shadow of a gnomon at the desired time and latitude. This analemma method may be based on Ḥabash al-Ḥāsīb's Zīj, The MS Istanbul Yeni Cami 784/2 copy of Ḥabash's Zīj contained many analemma methods. In this article, Zīj-i Yamīnī's analemma construction is explained, and Its accuracy is mathematically demonstrated, also its similarity with Ḥabash's methods is examined.

Traduzione italiana a p. 165. 1. Paulo magistri Dominici, physico Fiorentino, optimo atque doctissimo viro, liber de geometricis transmutationibus Nicolaus de Cusa cardinalis. 2. Etsi veteres magno ingenio praediti sedula indagatione conati sunt milita tunc ab scondita sibi et posteris nota facere perfecerintque utiliter in plerisque maximis atque op timis artibus, non tamen omne desideratum, in quibusque altioribus theoriis attigerunt, praeordinante hoc universorum optimo provisore, ut in nobis vis illa divinae intelligen tiae non torpeat, sed admiratione vehementiori ad ipsa latentia et scitu possibilia feratur. Ardentius quidem in obscuri penetrationem movemur, ut quietius de mentis vigorositate delectemur. Inter ea autem, quae geometricis speculationibus insudantibus hucusque im pedimento fuere, unum ab omnibus, quorum vires ingenii libri, qui ad nostram notitiam deducti sunt, curiosius observant, incognitum remansit, de recti scilicet atque curvi aequa litate aut in invicem transmutatione statuenda, ita ut non minimis, immo paene omnibus huic inquisitioni deditis post immensos labores visum sit viam ad huius notitiam acquiren dam a nobis sublatam, hoc rei impossibilitate agente et natura ipsam tantae oppositionis coincidentiam repellente. Ego vero rei huius difficultatem in parvitate apprehensionis et intermissione diligentiae, non summo acumine vigilantium, ut obscuritas negotii deposcit, potius existimans dato qualicumque otio ad artem novam, qua ad quaesitum pertingerem, me contuli ei diligenter satis ob altiores fines insudando, quousque cunctarum medita tionum mearum subscriptam formula facilem resolutionem efficerem. Quoniam autem in tanta et hactenus ignota arte, a qua non tantum perfectio geometricae transmutationis de pendet, sed et etiam introductio ad altiora ascendendi figuratur, non erat in obscuritate et parvitate ingenioli mei confidendum, recte statui ad te, arbitrum peritissimum atque veritatis zelatorem, confugere et iamdudum probatissimo amico inventionem pandere, ut in statera aequissimi iudicis extimetur. Noli igitur, amice dilectissime, ista, etiam si in maioribus verseris, quasi cruda indigestaque abicere. Lectu enim parva, intellectu vero facillima sunt. Sed quanto me ab annis iuventutis atque adolescentiae nostrae strictiori amicitiae nodo atque cordiali quodam amplexu indesinenter constrinxisti, tanto nunc ac curatius emendationi animum adhibe et in communionem aliorum nisi correctum predire non sinas. 3. Post innumeros paene modos, quibus, semper tamen deficiens, ad institutam artem pervenire contendi, tandem ad principium, quo in De docta ignorantia usus sum, respi ciens via mihi patefacta existit. Exigit autem ars, quam inquiro, praeter ea iam tradita in geometricis versionem curvi in rectum ac recti in curvum. Inter quae cum nulla rationalis proportio cadat, oportet in quadam coincidentia extremorum hoc latere secretum. Quae cum in maximo sit, ut alibi traditur, et maximum sit circulus, qui ignoratur, in minimo, qui est triangulus, inquiri ipsum debere ibi ostenditur. 4. Omnium autem figurarum multorum angulorum, quae polygoniae, et aequalium laterum, quae isopleurae, et eiusdem longitudinis laterum, quae eandem habentes peri pheriam isoperimetrae dicuntur, triangularem minimae capacitatis existere constat. Et cum tanto quaelibet isoperimetra sit capacior, quanto plures angulos habuerit, erit circulus iso perimeter figurarum omnium capacissima. Ad quam per angulorum multiplicationem de veniri nequit, sicut nec in numero ad maximum. Nulla igitur polygonia ad circularem

A manuscript now in Bamberg preserves the only surviving fragment of a thirteenth-century treatise on comets or ‘new stars’, which was written as a letter addressed by an unknown Dominican author to the Master General of his order, John of Vercelli. The present article offers the first discussion of this forgotten work, which was composed in the year after the Great Comet of 1264. Although most of the text has been lost, the inclusion of a geometrical diagram in the manuscript makes it possible to reconstruct a crucial part of its overall argument. The Dominican author was openly critical of the Aristotelian doctrine of comets as atmospheric phenomena and considered the possibility that reliable distance estimates might instead place such objects in the celestial realm. His geometrical investigation of this question is historically significant for containing the earliest known analysis of the effect of cometary distance on its observable parallax, thus anticipating aspects of Johannes Regiomontanus’s seminal '16 Problems' on comets.

At the turn of the first millennium, scientific and philosophical knowledge was far from dormant. Arithmetic, with its diverse calculation techniques and number theory, served as a bridge to philosophy, theology, and the study of the physical world. Even something as simple as a series of multiplication tables could unlock a profound knowledge of both the divine realm and natural phenomena. Such is the case with Abbo of Fleury’s Commentary on the Calculus.
Mathematics and Philosophy at the Turn of the First Millennium sheds light on Abbo’s original philosophical system anchored in two central doctrines, which serve as a compass to navigate it: the theory of unity (henology) and the theory of composition. Yet, the Commentary on the Calculus covers much more. The present study, thus, explores an eclectic range of topics – from water clocks to barleycorns, constellations to human voice, synodic month to the human lifespan, and numbers to God. Abbo’s work is an ambitious attempt to tie together the study of both the visible and invisible realms, what can be measured and what cannot, what can be quantified and what exceeds quantification.

Первый выпуск шестого тома журнала посвящен античной музыке и включает в себя как статьи по истории музыкальной теории и практики, так и ряд переводов античных музыковедческих сочинений. В первом разделе, после общего введения и библиографического очерка, авторы исследуют основные элементы музыкальной теории в Античности и истоки античной идеи гармонии, а также рассматривают эволюцию музыкальной исполнительской практики и роль музыки в воспитании, как они представлены в трактате псевдо-Плутарха О музыке. Во втором разделе в переводе на русский язык публикуются Музыкальные проблемы Аристотелевского корпуса, Деление канона псевдо-Евклида, фрагменты музыкальных сочинений Теофраста, Птолемаиды Киренской и Дидима, выдержка из трактата О музыке Аристида Квинтилиана и анонимный трактат Птолемеева «Музыка», что, вкупе с двумя переводами Руководства по гармонике Никомаха из Герасы в томах 2.2 (2008) и 3.1 (2009) журнала и музыкальными разделами Изложения математических предметов, полезных для изучения Платона Теона Смирнского в томе 3.2 (2009), составляет представительную подборку античных математических сочинений в новых русских переводах. Один из ближайших выпусков журнала будет посвящен естественным наукам в древности. Работы в этот сборник принимаются до конца ноября 2012 г. Приглашаем к сотрудничеству заинтересованных авторов. Сердечно благодарим всех коллег и друзей, принявших участие в наших встречах, и напоминаем авторам, что журнал индексируется The Philosopher's Index и SCOPUS, поэтому присылаемые статьи должны сопровождаться обстоятельными аннотациями и списками ключевых слов на русском и английском языках. Информируем читателей, что все предыдущие выпуски можно найти на собственной странице журнала www.nsu.ru/classics/schole/, а также в составе следующих электронных библиотек: www.elibrary.ru (Научная электронная библиотека) и www.ceeol.com (Central and Eastern European Online Library). Евгений Афонасин Берлин, Германия 25 декабря 2011 г.

Lubb al-Ḥisāb ʿAlī ibn Yūsuf ibn ʿAlī, the author of Lubb al-Ḥisāb (likely from the 12 th AD), drew extensively from various important sources in his work. He has compiled a collection of theoretical and practical arithmetic and geometry topics, incorporating material from the works of prominent mathematicians of the Islamic era. The content of Lubb al-Ḥisāb is linked with the works of mathematicians such as Abū Barzah Ḥāsib (?-910), Muḥammad ibn Mūsā al-Khwārazmī, (c. 780-c. 850), Abū Kāmil al-Ḥāsib al-Miṣrī (?-c. 930), Abū al-Wafā Būzjānī (940-998), Abū Bakr Muḥammad ibn al-Ḥasan al-Karajī (c. 953-c. 1029), ʿAbd al-Qāhir al-Baghdādī (980-1037 AD), Aḥmad ibn Thabāt (1160-1234), and ibn al-Khawwām al-Baghdādī (1245-1325). A significant portion of its geometry and arithmetic topics closely aligns with Ahmad ibn Thabāt's Qunīyat al-Ḥussāb. In addition, the geometry section mirrors the two geometry treatises found in the codex 772 housed in the National Library of France. It appears that the remaining topics of Lubb al-Ḥisāb, not readily found in existing works, are likely derived from important sources from the Islamic period, whose originals may no longer be available. Consequently, investigating the sources used in Lubb al-Ḥisāb can offer valuable insights into the history of science during the Islamic period. Lubb al-Ḥisāb's unique and exquisite Persian manuscript is housed in the Central Library of the University of Tehran (No. 5213). The manuscript consists of 138 folios, but unfortunately, the first page (containing the beginning of the introduction) is missing. This makes it difficult to determine the exact date of its creation. While researchers have differing opinions on the year, it is generally believed to have been written during the 5 th or 6 th centuries AH. Abū al-Qāsim Qurbānī posits that the author, ʿAlī ibn Yūsuf ibn ʿAlī al-Munshī (Mustawfī), was a secretary during this

The translation movement of the twelfth and thirteenth centuries brought a wide range of scientific texts to the attention of Western scholars. 1 Some of these were Greek works, translated directly from Greek or from Arabic translations. A number of original Arabic texts were also translated. This translation movement heavily influenced the evolution of medieval science: Western scholars, such as the Franciscan Roger Bacon, benefitted greatly from the knowledge of these texts. Bacon also benefitted from a cultural environment in which the presence and influence of texts translated from Greek or Arabic had been steadily developing and where the interest in scientific themes was strong. Oxford, where he received part of his education, had seen the magisterium of Robert Grosseteste. Grosseteste, who would become archbishop of Lincoln in 1235, had been the first lector to the Oxford Franciscans between 1229 and 1231. 2 Grosseteste knew Greek and translated Greek texts (including the Nichomachean Ethics and parts of the De caelo), and he had played a pivotal role in developing an interest in sciences in England, especially optics and cosmology. 3 Cecilia Panti has recently shown how that the teaching of Grosseteste and of his successors in Oxford -such as the Franciscan Adam Marsh -was focused on theology and the exegesis of the Holy Writings, and natural philosophy was probably used mostly to illustrate ethical and religious themes; however, the influence of Grosseteste's scientific interests among the English Franciscans remains clear, not only in Bacon's works but also those of Adam of Exeter and the exegetic works of Thomas Docking. 4 Bacon had been educated in Oxford (the dates are uncertain, but probably 1228-1236 or 1234-1242) and had been professor in Paris. He was acquainted with Burnett, 'Humanism and Orientalism'; d' Alverny, 'Translations and Translators' , and Haskins, The Renaissance of the Twelfth Century. Southern, Robert Grosseteste, pp. 74-75. See Gasper and others, this volume. Panti, 'Scienza e teologia' .

Klasik dönem Osmanlı muhasebe matematiği eserleri, ondalık kesirleri ve kök alma işlemlerini ihtiva etmesi bakımından, dönemin diğer genel hesap kitaplarının önüne geçmiş ve ortaçağın doğu ve batı uygarlıklarında sayısal analize yapılan katkıları aralıksız sürdürmüştür. Bunlardan Pîr Mahmud Sıdkı Edirnevî'nin hesap kitabı, yüksek derecelerde hem tam kök ve hem de yaklaşık kök alma metotlarını içermesi itibariyle diğer muhasebe matematiği eserlerinden ayrılmakta, hatta 15. yüzyıl Osmanlı matematiğinin üstün bazı hususlarını Türk dilinde sunmada rehber olmaktadır. Çalışmamızda, kalburüstü bu eserin yazma nüshası ve ikincil diğer kaynaklar yardımıyla, eserdeki üçüncü ve dördüncü dereceden yaklaşık kök alma metotları matematiksel olarak analiz edilerek Edirnevî'nin öncü rolü vurgulanmıştır.

In the texts of the arithmetical-algebraic tradition initiated by al-Karajī (11th c.), the terms of a composite algebraic expression are considered to be positive or negative only in the sense of relational entities. Indeed, within the rules for operating with these algebraic expressions, additives and subtractives represent operands, and tabular methods allow one to compute with subtractives more freely. The strategies adopted for impossible problems further confirm the relational nature of these entities. In my article, I examine these aspects by considering the work of al-Zanjānī (13th c.) and of his predecessors al-Karajī and al-Samaw'al.

Cet article présente une méthode géométrique discrète qui permet de calculer des approximations du nombre Pi et des racines carrées des nombres naturels. Dans un premier temps la méthode de calcul de Pi est présentée, a c c o m p a g n é e d ' e x p l i c a t i o n s g r a p h i q u e s s i m p l e s. Puis, pour la démonstration, on bâtira les calculs d e l ' a rithmétique de base à l ' a i d e d ' u n e méthode géométrique discrète connue en partie dans l' a n t i q u i t é. D a n s le cadre de cette méthode on rappellera les notions de nombre naturel et de rapport. P u i s , g r â c e c e t t e r e p r é s e n t a t i o n e t à l ' a i d e d e r a p p o r t s o n c a l c u l e r a l a r a c i n e c a r r é e d e 2 e t d ' a u t r e s n o m b r e s naturels. On entamera enfin la preuve proprement dite du fait que le calcul exposé au début mène bien vers le nombre Pi. 1 Chercheur indépendant 2 Pour une histoire des rapports et des nombres voir Cousquer [1]. 3 Dans ce cas, un gnomon est une figure géométrique constituée de sous-carrés formant u n e s o r t e d ' é q u e r r e .

Caput sextum. De situ huius cometae, quo ad mundi diametrum, ex ipsius parallaxibus; et an is in Aetherea, an vero Elementari regione extiterit, demonstrative concludere. Paravimus in omnibus antecedentibus viam ad investigandum demonstrandumque id, quod tantopere, tot iam elapsis seculis, ab omnibus pene philosophis, in varias sententias disceptatum est, et a nullo hactenus penitus decisum. Utrum videlicet possible sit, cometas in Aetherea mundi regione, intra orbes coelestes generari, an vero iuxta Peripateticorum placita, omnes infra Lunam, in suprema Aeris regione necessario versentur. Est sane hoc negotium, ut praecipuum inter ea, quae de cometis dici inquirique merito debeant, et sine quo caetera omnia, quae in medium adferuntur, manca existunt, ita etiam omnium difficilimum, et non solum labore, sed etiam magna subtilitate industriaque indiget: adeo ut vulgares astrorum observatores, cum suis puerilibus et ludicris instrumentis, prorsus respuat. Res enim versatur hic circa minima, ex quibus maxima concluduntur, cum illi ut plurimum circa maxima etiam caecutiant et aberrent. Ut ob id non usque adeo mirum sit, tantam esse discrepantiam inter philosophos hac de re, et tam diversas etiam neotericorum ex observationibus erroneis petitas sententias; adeo ut quamplurimi, qui de hac materia aliquid in medium protulerunt, etiam inter eos, qui non vulgares haberi volunt, longissime (quod salvo uniuscuiusque honore dictum volo) a scopo petito aberrarint, ut suo loco singulis satis evidenter demonstrabimus. Neque sane ulterius admiror, tot praestantes astronomos etiam hallucinatos esse circa parallaxin huius cometae indagandam demonstrandamque, cum non pauci ex iis parallaxin sensibilem Stellae Novae attribuerint, adeo ut quidam non dubitarint, Elementarem eam extitisse, pronuntiare. Cum tamen facilime, etiam absque ullo pene instrumento, depraehendi poterat, illam circa verticem aeque ac iuxta horizontem, eandem exquisite a vicinis fixis obtinuisse distantiam, quod fieri nequaquam potuisset, si adeo vicina nobis fuisset, ut Terrae semidiameter sensibilem parallaxin, cui ipsa etiam Luna obnoxia est, induxisset; Verum stella illa revera omnem aspectus diversitatem excludebat, et non aliter quam affixa Sidera, se respectu Terrae revolvebat; ut in priori libro, ubi de hac ex professo egimus, infallibili ratione aliquoties demonstratum reliquimus. Facilitatem autem huius rei observandae pervestigandaeque, peperit tum situs huius stellae semper aspectabilis, eo quod circulum circa Polum magnum quidem, sed cuius pars nulla occideret, motu primi mobilis designaret, neque adeo declivis in minima altitudine fieret, ut vapores circa horizontem, per radium refractum, locum eius aliorsum visui insinuarent, perpetuoque in eodem loco fixa stetit; unde motus proprius nullam in indaganda parallaxi difficultatem causari poterat. At in hoc cometa, quo ad parallaxes enucleandas, maior longe inest laboris perplexitas, et subtiliori opus erit pervestigationis methodo, neque etiam adeo simplici, eo quod is nec in meridiano aspectabilis fuerit, nedum ut non occideret, et motum etiam proprium obtinuerit, eumque non semper aequalem, sed successive se remittentem. Nos tamen certis et diversis rationibus, omnibus his difficultatibus praevenientes, liquido demonstrabimus, hunc cometam minime in Elementari regioni extitisse, sed longe supra Lunae sphaeram in ipso Aethere cursum suum absolvisse; Contra quam Peripatetici, Stagiritae illius authoritati insistentes, hactenus subtilibus suis argumentationibus, nulli tamen experientiae vel demonstrationi certae innixis, nobis persuadere conati sunt. Idque nunc eo audentius contra eos, eorumque asseclas asserere licebit, quod in Nova illa, de qua modo diximus Stella, in ipso Aethere insolitas generationes nonnunquam existere, adeo manifeste apparuit, certoque demonstrabatur, ut qui de hoc amplius haesitare velit, deridendus potius, et tanquam sensu communi carens, a veritatis schola explodendus merito veniat, quam ut responsione dignus censeatur. Cum itaque ratio investigandi parallaxin in hoc cometa, non usque adeo simplex et facilis (ut diximus) existat, qualis in Stella illa Nova sese obtulit, et multae viae alias a mathematicis praestantibus repertae sint ad parallaxium demonstrationem perveniendi; primum quidem ab eximio illo artifice Johanne Regiomontano Franco, edito de hac materia peculiari libello, tum etiam a quibusdam recentioribus non vulgaribus mathematicis: tamen cum nulla earum mihi satisfacere videatur, ad huius cometae parallaxes enucleandas, eo quod maxima pars transitum per meridianum aspectabilem praesupponat, et omnes illae viae, cometae motum nullum alium quam primi mobilis admittant, quae duo in hoc neutiquam locum obtinebant, adde, quod ut plurimum illae rationes, temporis exquisitissimam notitiam requirant, qua in parte quam facile aberrari possit, norunt, qui in hoc pulvere diligentius versati sunt; et ob id illae inductiones ex minimis, quorum parva aberratio, quae vix caveri potest, in maximam crescit deviationem, mihi semper suspectae fuerunt. Idcirco, ut nos in praesenti negotio, omnes a certitudinis scopo abducentes labyrinthos evitemus, et difficultatibus sese ingerentibus oportune occurramus, superatisque errorum scopulis ad veritatis planiciem exoptatam, conscendamus, tribus potissimum modis demonstrabimus, quod cometa hic Elementaris nequaquam extiterit. Primum, et quasi generali ratione, ex ipso ductu et motu, quem toto durationis tempore observavit, circulique tramite et declinatione, quem suo curso designavit. Secundo, particularius idem pervestigabimus ostendemusque ex distantiis a quibusdam peculiaribus fixis sideribus, viae cometae vicinis, quas interlapsis aliquot horis observavimus, cum altior decliviorque ipsius supra horizontem positus conspiceretur. Tertio, ex collatione observationum in semotis Sphaerae inclinationibus, ab aliis mathematicis exquisite deprehensis, et cum nostris, habita ratione interjectae Telluris portionis, diligenter collatis, idem enucleare conabimur. Confidoque his tribus comprobationibus certo convinci posse, cometam hunc supra Lunam, in ipso Aethere locum obtinuisse; quibus tamen, quasi appendicis loco, subjungamus aliqua exempla Regiomontanicae ratiocinationis, quae ex duabus datis altitudinibus et azimuthis, cum intervallo temporis cognito, parallaxin indagare docuit; ne veterum inventa vel ignorasse, vel neglexisse videamur, et ut id, quod prius innuimus, eiusmodi inductiones non ita bene in praxi atque speculatione locum obtinere, manifestum reddatur. Translation pp. 89-92 Chapter 6. To conclude demonstratively on the position of this comet, with respect to the diameter of the World; and whether indeed the comet appeared in the Aether or in fact in the region of the Elements. We have provided in everything that has gone before a way for investigating and demonstrating the question which has, over so many ages, been so much split by almost all philosophers into different opinions, and so far has not been completely settled by anyone: namely, whether it would be possible for comets to be generated in the aethereal region of the World, among the celestial orbs, or whether, instead, according to the views of the Peripatetics, all things under the Moon necessarily dwell in the uppermost region of the Air. Indeed, this issue is chief among those that should deservedly be discussed and inquired into regarding the comet, for without it, all the rest that have been brought forth would remain imperfect, so also it is the most difficult of all, and requires not only labour but also great subtlety and industry, to the extent that the question utterly spurns the common observers of the stars with their puerile and playful tools. For here the matter turns on very small things from which very great ones are concluded, whereas those [common observers] are usually blinded and also go astray regarding very great things. So that because of this, it is not surprising that so far there is such a discrepancy among philosophers on this matter and so many different claims on the basis of erroneous observations of the moderns, such that very many of those who have published something on this matter, even among those who do not want to be regarded as common (because I want what I say to respect everyone's honour), have strayed far from [their] intended target, as in due course, we shall demonstrate sufficiently clearly in the proper place. Nor indeed am I amazed that so many excellent astronomers have also hallucinated in investigating and demonstrating the parallax of this comet, since not a few of them have attributed a detectable parallax to the New Star, to the extent that some people have not hesitated to declare that the star appeared in the [region of the] Elements. But it could be grasped easily, even without any instrument whatsoever, that [the New Star] near the vertex and likewise near the horizon had maintained exactly the same distance from the neighbouring fixed stars, which could never have happened if [the New Star] had been so close to us that the radius of the Earth would have brought about the detectable parallax to which even the Moon is subject; hence that star truly excluded every difference in aspect [i.e. parallax] and carried itself around the Earth, just as a fixed star [does]; as in the former book where I dealt with this explicitly, we have shown demonstrated several times with infallible reasoning. But the position of this always visible star permitted this matter to be easily observed and investigated thoroughly, because the star described with the motion of the Prime Mover a large circle around the pole, no part of which set, nor did its declination come to a sufficiently small altitude for the vapours near the horizon to make its position appear differently to the eye through a refracted ray. And as it (the star) stayed fixed in the same position constantly, no proper motion could present difficulty in detecting...

Except for the fraction 2/3, Egyptian scribes used only unit fractions, i.e. fractions with numerators equal to unity. They expressed all other fractions as a sum of these unit fractions. They used tables such as the Rhind table, providing decompositions of numbers written as 2/n with n an odd integer (from three to 101) into a sum of unit fractions, for which the decomposition procedure remains today mysterious. Here, we propose a simple and unique procedure based on elementary integer summations with which to reproduce the entire Rhind table. We add two prioritized selection criteria and discuss some of the Egyptian scribes' possible preferences, finally resulting in the Rhind table decompositions in 87.8% of cases at the first attempt. All the remaining cases can also be calculated using our procedure, with the same criteria and preferences, but differently prioritised. In addition to understanding how ancient Egyptians computed, this simple procedure improves our understanding of perceptions of numbers in ancient Egypt.

This paper presents a taxonomy of divisibilism, a philosophical perspective advocating for the infinite divisibility of continua. The taxonomy is founded on various conceptualizations of indivisibles, enabling the identification of two types of divisibilism: ‘moderate’ and ‘strong’. The former denies indivisibles as constituent parts of magnitudes, whereas the latter rejects indivisibles as even intrinsic elements (such as limits or junctions) of magnitudes. The paper proceeds to demonstrate how Gregory of Rimini falls into the second category, utilizing geometry and non-entitism as primary tools to dismiss the very possibility of the existence of indivisibles. Within the framework of this ‘strong divisibilism’, the paper also examines Gregory’s interpretation of mathematical items (e.g. points, lines, surfaces) against the backdrop of one of his main sources, that is, Augustine. Lastly, the paper delves into Gregory’s interpretation of the fundamental Aristotelian divisibilist claim, namely, that a continuum is infinitely divisible. It highlights how Gregory’s stance ultimately conflicts with Aristotelian premises, particularly concerning his assertion that the division of all continua is made into infinitely many actual parts.

Öz Osmanlı'da 10 tabanlı sayı sistemi (Hesâb-ı Hindı�) ile birlikte, hesâb-ı zihnı� ve hesâb-ı sittı� nı� adı ile iki farklı sayı sistemi daha kullanılmıştır. Bu sayı sistemlerinden hesâb-ı sittı� nı� adı ile tanımlanan ve genel olarak 60 tabanlı sayı sistemini ifade eden hesap sistemi özellikle astronomik hesaplamalarda tercih edilmiştir. Bununla birlikte ilgili hesap sisteminin trigonometrik hesaplamalara ilişkin matematik kaynaklarında kullanıldığı da bilinmektedir. Bu çalışmada, hesâb-ı sittı� nı� okuyucuya tanıtılarak matematik ya da astronomi araştırmalarında hesâb-ı sittı� nı� ile karşılaşması muhtemel olan bir bilim tarihçisine veya okuyucusuna yardımcı olmak hedeflenmiştir. Ayrıca, araştırmada, daha önce üzerinde geniş çaplı müstakil bir çalışma yapılmamış olan Câbizâde Halil Fa'iz Efendi'nin Fezleketü'l-Hisâb isimli eseri özellikle seçilerek eser incelemesi yolu ile yapılan bilim tarihi araştırmalarına katkı sağlamak amaçlanmıştır. Çalışmada hesâb-ı sittı� nı� tarihsel arka planı ile ele alınmış ve Osmanlı hesap sistemindeki yerine dikkat çekilerek yaygınlığına vurgu yapılmıştır. Sözü edilen amaca uygun şekilde, çalışma boyunca Osmanlı matematiği içerisinde önem arz eden hesap kitaplarında hesâb-ı sittı� nı� sorgulanmıştır.

Manuscripts in Oxford and Erfurt preserve evidence of the earliest known efforts made in Latin Europe to remeasure the eccentricity and maximum equation of the Ptolemaic solar model. The present article analyses and contextualizes this evidence, while also revisiting Ernst Zinner's hypothesis according to which the relevant observations were made by Alard of Diest (fl.1308).

Positional notation and related numerical manipulation techniques of Indian origin were introduced to Europe during the twelfth century through Arabic mediation and vividly described by Fibonacci as modus Indorum, the method of the Indians. This article aims to juxtapose Sanskrit and Latin texts to highlight the connections and differences between matrix and reflection in a complex cultural process of diffusion and assimilation . With reference to positional notation, this contribution examines a conceptual distinction between the graphical notion of position and a logical-classificatory principle hitherto perhaps not fully recognized, although present in the texts of both traditions. The thesis of a consequent diagrammatic conception of positional notation in its origins, as a system of rules relating graphical position and order of magnitude, is also advanced.

Ba~rTz and Thábit b. Qurra translated ioto Arabic {he Arirhmetical lntroduction of Nicomachus, the chapters 00 figurate numbers, based 00 the propenies ilIustrated in rhe table below, were further developed and were introduced in the teaching of marhematics along with other problems of Euclidian number theory. I For the 10 th century, ¡his interest in lhe figurare numbers is confirmed by Ihe mathematical wrilings of al-An!akT 2 and Ibn STna (lbn SIna 1975, 53-61; 1976, 3) and by the encyclopedia of Abii tAbd Alliih al-Khwarizmf (al-KhwiirizmI 1968, 188-91). But lhe cantents of these treatises do nal give us much information 00 early deve[opments in this subject. Also we know that in the central part of the Muslim Empire this tradition was continued for several centuries, as evidenced by the writings of CAbd al-Qahir b. The translation of Habib b. Bahrlz was made (at the end of the 8"' or the begining of the 9'• century) from a Syriae Yersi~n (F. Sezgin 1974, 164-66). For the translation of Thabit b. Qurra, ef. Kutsch 1958. 67-82. 1 Abü'I-Qaslm al-An!akl (ef. F. Sezgin 1974, 30) wrote the KiUib la/srr al-arilhmá!rqr whieh is really a cornmemary on the Arilh!nelical IntroduClion of Nicomachus. Only the third pan has come down to us (Ms. Ankara Saip 5311, ff. 1b-36a).

In this note, we present a 14th-century manuscript which has not been studied before. It contains a complete redaction of the Kitāb al-Istikmāl by the Andalusian mathematician, al-Mu'taman ibn Hū d (11th century), and informs us about the missing pieces of al-Mu'taman's book and about the content of his initial project that had never been completed.

argues that, in the first printed manual on double-entry bookkeeping ('DEB') in 1494, Pacioli presented a novel 'axiomatic' approach that requires a corresponding 'paradigmatic shift' in our view of his contribution. • This paper challenges Sangster's interpretation 25.6.

One of the biggest factor shaping the Islamic civilization is the "science of qh", which determines the individual and social life style of a Muslim. İlm-i ferâiz, which is a subbranch of qh, covers Islamic inheritance law, and ilm-i-muamelat, which is a sub-branch of qh, includes subjects other than the part of qh that regulates worship, such as Islamic debt law, tax and zakat calculations.. Examples of ferâiz problems and transaction calculations are frequently found in the important works of the pre-Ottoman Islamic civilization mathematical literature..

The Kerala School of mathematics, founded by Madhava in Southern India, produced many great works in the area of trigonometry during the fifteenth through eighteenth centuries. This paper focuses on Madhava's derivation of the power series for sine and cosine, as well as a series similar to the well-known Taylor Series. The derivations use many calculus related concepts such as summation, rate of change, and interpolation, which suggests that Indian mathematicians had a solid understanding of the basics of calculus long before it was developed in Europe. Other evidence from Indian mathematics up to this point such as interest in infinite series and the use of a base ten decimal system also suggest that it was possible for calculus to have developed in India almost 300 years before its recognized birth in Europe. The issue of whether or not Indian calculus was transported to Europe and influenced European mathematics is not addressed.

Bu makalede, kendi döneminde Batılılarca da takdir edilmiş olan bir Osmanlı âlimi İsmail Gelenbevî'nin (18. yüzyıl) Hisâbü'l-Küsûr adlı eseri incelenerek, Gelenbevî'nin cebirsel sembolizme yaptığı katkıyı ortaya çıkarmak amaçlanmıştır. Çalışmamızda, İslam klasik geleneğinin son temsilcisi olan Gelenbevî'nin yaşadığı döneme kadar yetişmiş İslam cebircilerinin en önemlilerinin eserleri taranarak, İslam dünyasının oluşturduğu kendine özgü matematiksel sembolizmin Gelenbevî'ye dek kesintisiz devam edip etmediği ve ne ölçüde geliştirildiği ele alınmıştır. Ayrıca, İslam dünyasında ve Avupa'da matematiksel sembolizmin gelişmesi de karşılaştırılmıştır. Bulgularımıza göre, İslam dünyası Avrupa'dan yaklaşık bir asır kadar önce kendine özgü bir matematiksel notasyon sistemi kurmayı başarmış ve bu notasyon sistemi Gelenbevî'ye dek gelişerek devam etmiştir. Endülüslülerin (Batı İslam dünyası) katkısı olarak bilinen, (büyük ihtimalle) Magribli matematikçilerce Osma...

In this article, some terms related to the field of mathematics are discussed in a historical context. The purpose of the article is to study the development and formation of some mathematical terms. It is also discussed some ideas about Egyptian contributions to the development of the terms concerned mathematics and it is written about the scientists who contributed to the development of mathematics in the Middle Ages in the countries of Central Asia and the Middle East.

Bu çalışmada İhvân-ı Safâ'nın Resâil adlı eserinin özellikle "Sayılar (Aritmetik) Risalesi" ile "Hendese (Geometri) Risalesi" esas alınmıştır. Makalede İhvân'ın aritmetik ve geometri ilimlerine bakış açısı, bu ilimleri ele alış biçimi, bu ilimlere verdiği önem ve yararlandığı kaynaklar ortaya konmaya çalışılmıştır. İhvân'ın aritmetik ve geometri risalelerinin içeriği, konuları ve felsefî değeri de etraflıca irdelenmiştir. Ayrıca bu çalışma, özellikle İhvân-ı Safâ'da aritmetik, geometri ve felsefe ilişkilerini incelemektedir.

Full bibliographic details must be given when referring to, or quoting from full items including the author's name, the title of the work, publication details where relevant (place, publisher, date), pagination, and for theses or dissertations the awarding institution, the degree type awarded, and the date of the award.

Al-Samaw'al's work Exposure of the Errors of the Astronomers contains a single numerical table, a highly unusual table of sines. In order to avoid the problem of approximating sin(1 degree), al-Samaw'al redefines the number of degrees in a circle from 360 to 480. In the accompanying chapter, al-Samaw'al criticizes his predecessors for deviating from geometrical precision and explains how his table is to be calculated. However, our analysis demonstrates that al-Samaw'al did not follow his own instructions, simply interpolating his entries from a traditional sine table. Trigonometry was the mathematical heart of ancient Greek and medieval Islamic astronomy. The need to convert geometric models of planetary motion into quantifiable predictions of positions led to the birth of trigonometry and its associated tables, first by the hand of 2 nd century BC scientist Hipparchus of Rhodes (whose table is now lost), and later by Claudius Ptolemy (fl. AD 140) in the Almagest [Toomer, 1984]. The sine function (or, in Greece, the chord) was therefore the mathematical foundation of astronomy. Methods of table computation and levels of precision improved gradually throughout the medieval period. However, the subject of this paper, the 12 thcentury scientist Ibn Ya¬y× al-Samaw'al al-Maghribī, brought a voice of critique to the standard techniques, claiming that a mistake was made in allowing inherently approximative methods to enter the geometric realm of trigonometry. His curious resolution of the matter, to redefine the circle to brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Revistes Catalanes amb Accés Obert

Historians have always seen jabr (restoration) and muqābala (confrontation) as technical terms for specific operations in Arabic algebra. This assumption clashes with the fact that the words were used in a variety of contexts. By examining the different uses of jabr, muqābala, ikmāl (completion), and radd (returning) in the worked-out problems of several medieval mathematics texts, we show that they are really nontechnical words used to name the immediate goals of particular steps. We also find that the phrase aljabr wa'l-muqābala was first used within the solutions of problems to mean al-jabr and/or al-muqābala, and from there it became the name of the art of algebra.

The formation of Islamic culture was accompanied by a process of adoption and integration of the classical scientific tradition. Due to a long dispute over the role of mathematics and astronomy, a new category of disciplines emerged that supported the access of Muslim scholars to mathematics in general and to its applied branches in particular. Selected examples from different fields of applied mathematics (ar. hisdb) demonstrate the extant to which mathematics in the Islamic homelands took root, developed and produced new practical disciplines the Islamic Community could benefit from.