Papers in English by Marek Otisk
ARCHIWUM HISTORII FILOZOFII I MYŚLI SPOŁECZNEJ • ARCHIVE OF THE HISTORY OF PHILOSOPHY AND SOCIAL THOUGHT, 2023
The Platonic (or more appropriately the Neo-Platonic) doctrine of the elements of matter, their p... more The Platonic (or more appropriately the Neo-Platonic) doctrine of the elements of matter, their properties, and their harmonious interrelations received significant attention during the Middle Ages. One example is the treatise On the Nature of Things (De natura rerum) by Isidore of Seville. In the chapter entitled De partibus mundi, the Bishop of Seville dealt in more detail with the Aristotelian doctrine of the elements (De natura rerum XI, 2–3), but he also introduced the (Neo-)Platonic theory (De natura rerum XI, 1). He supplemented both descriptions of the teachings regarding the elements of matter with a diagram illustrating the theories presented in words. In the case of the drawing to illustrate Plato’s doctrine, his diagram is called figura solida secundum geometricam rationem, and it is very difficult nowadays to determine what this image was intended to look like, what exactly it was to show, what it was supposed to be used for, and what its explanatory value was. The main reason for these uncertainties is the great variety of forms in which the diagram has been preserved; moreover, in many cases it is difficult to uncover the nature of the geometric ratio and harmony, or the spatial character of the diagram in the extant forms. In this paper, the focus is on the diagram figura solida and some of its most frequently preserved forms. The aim of the paper is to show that the widely differing variants of the figura solida can be explained in accordance with period views of solid geometric figures and the nature of the geometric ratio. For geometric solids, the explanation presented by Macrobius in Commentary on the Dream of Scipio is used in this article. Here, plane parallelograms characterise the parallelepipeds, and the three dimensions of solids can be adequately represented by two plane quadrilaterals placed one above the other. Geometric ratios were described by Calcidius (and also, e.g., by Boethius or Martianus Capella), who paid considerable attention to them in his interpretation of Plato’s dialogue Timaeus and also put them in direct reference to the elements and their properties and mutual relations. Using these sources (no doubt familiar to Isidore), this paper offers an interpretation of two often appearing variants of the figura solida in the manuscripts. These two forms both respect Calcidius’ conclusions regarding plane geometric analogies and the justification ofthe application of plane ratios (with one mediator) to solids representing elements (the need for two mediators). Moreover, this interpretation makes it possible to explain the diagram of the figura solida as a circular diagram, which would also make this diagram fully compatible in with the nature of the other diagrams contained in Isidore’s De natura rerum, which is why the name Liber rotarum was adopted for this work in the Middle Ages.
Religions, 2023
The Consolation of Philosophy by Boethius was not only widely read during the Middle Ages, but it... more The Consolation of Philosophy by Boethius was not only widely read during the Middle Ages, but it was also frequently glossed, commented on, and discussed. The ninth poem of the third book, which offers a Platonic image of the creation of the cosmos and the governance over it, had a specific place in the reception of this Boethius’s work. Today we know of numerous debates about the possible interpretations of this poem and its Christian understanding, dating back at least to the 9th century. This paper deals with the commentary on this poem written by Adalbold of Utrecht († 1026). Attention is focused in particular on the role of dialectic in selected passages of this Adalbold’s text and on the inspirational sources of his dialectical knowledge. Specifically, the paper deals with the possibility of definition or description of God (Deus sine nomine), and arguments explaining the appropriateness or inappropriateness of conceptualizing God as the form of the highest good (forma summi boni).
Philosophical Readings, 2022
The paper enquires into the reasons why Boethius altered the passage addressing the definition of... more The paper enquires into the reasons why Boethius altered the passage addressing the definition of number in his loose translation of Introduction to Arithmetic by the Neopythagorean philosopher and mathematician Nicomachus of Gerasa. While Nicomachus's text contains three definitions of number, Boethius lists only two. However, he also pays attention to the definition he omits, even though he does not regard it as a proper definition. In his view it fails to embody the essence of number, and is to be understood as a description of the components constitutive of the substance of number. Although this is a possible explanation of Boethius's dismissal of the definition provided by Nicomachus, the description also occupies an important place in relation to the general characteristic of number, because Nicomachus's definitions fully correspond to the three basic topics which were central to contemporary arithmetic, viz. the science of number: number as discrete quantity, referring to the properties of numbers and their classifications; number as collection of units, leading to the topic of figural numbers; and number as quantity emanating from unit and subsequently returning to it, corresponding with numerical ratios, sequences and their transfers.
Cosmos and History, 2021
The paper is an attempt to interpret a schema that is part of an early medieval fragment known as... more The paper is an attempt to interpret a schema that is part of an early medieval fragment known as the Excerptum de quattuor elementis. In this schema, the individual elements (fire, air, water, earth) are presented as the elements which form the material world using several characterisations typical of the period: as the sum of their natural physical properties in accordance with the Platonic and Aristotelian traditions, as geometric shapes (to put it simply, using the so-called Platonic solids) and as numerical values connected by a 2:1 ratio. The aim of this paper is to propose a possible interpretation of the numerical values and connections found in this schema using contemporary texts and diagrams (especially Calcidius and Isidore of Seville, as well as other sources available at the time), which would include a consistent yet complex characterisation of the elements, their properties and their mutual interconnectedness, as found in the fragmentary text of the Excerptum de quattuor elementis and in the schema itself.
Teorie vědy / Theory of Science, 2020
The paper analyses three preserved reports, depicting Gerbert of Aurillac (also known as: of Reim... more The paper analyses three preserved reports, depicting Gerbert of Aurillac (also known as: of Reims, of Ravenna, of Bobbio, and in 999–1003 as Pope Sylvester II) as a clockmaker. The Benedictine monk William of Malmesbury
(died around 1143) writes about clocks Gerbert made in Reims in The History of the English Kings and describes them as arte mechanica compositum. The Benedictine Arnold Wion (died around 1610) mentions clocks from Ravenna, where Gerbert allegedly constructed a clepsydra, in The Tree of Life. In his Chronicle, Thietmar of Merseburg
(died around 1018) describes a horologium with an observation tube (fistula) from Magdeburg. These three references are analysed from a historical standpoint and especially Williams’s and Thietmar’s short reports are interpreted as possible references to timekeeping devices – the astrolabe and the nocturlabe.
Konštantínove listy/Constantine’s Letters, 2018
Letter on Timekeeping of Gerbert of Aurillac to Brother Adam. This paper focuses on the letter, w... more Letter on Timekeeping of Gerbert of Aurillac to Brother Adam. This paper focuses on the letter, written by Gerbert of Aurillac (Pope Sylvester II) in the late 980s, which was addressed to brother Adam who is otherwise unknown to us from other sources. Gerbert's text deals with the problems of timekeeping and it naturally uses professional astronomical and geographical terminology and concepts which were necessary for timekeeping during this period. Although our contemporary knowledge of brother Adam is lacking, this paper sets out to analyse the letter and to reconstruct the individual fragments of teachings which had to be available to the recipient of the letter (i.e. brother Adam). This paper thus focuses on some topics of the medieval (inspired by antiquity) geocentric view of the organization of the Cosmos together with the basic categories of geographical division of the Earth and, at the same time, the paper is trying to draw attention to the fact that such concepts could have been the part of elementary knowledge of an educated individual by the end of 10th century. Abstrakt: OTISK, Marek. List Gerberta z Aurillacu bratovi Adamovi o meraní času. Štúdia sa venuje listu Gerberta z Aurillacu (neskôr pápež Silvester II.) z konca 80. rokov 10. storočia, ktorý bol adresovaný nám dnes z iných zdrojov neznámemu bratovi Adamovi. Gerbertov list sa venuje problematike merania času a celkom samozrejme užíva odbornú astronomickú i geografickú terminológiou a zároveň odborné koncepcie, ktoré boli potrebné pre určova-nie času v tomto období. Aj keď dnes o bratovi Adamovi nevieme nič, pokúša sa táto štúdia pomocou analýzy Gerbertovho listu rekonštruovať jednotlivé poznatky, ktorými disponoval adresát listu, tj. brat Adam. Táto štúdia sa teda zameriava na určité témy stredovekého (an-tikou inšpirovaného) pohľadu na geocentrické usporiadanie vesmíru, na základné kategórie vtedajšieho geografického členenia Zeme a zároveň sa pokúša upozorniť, že predstavené zna-losti museli byť bežnou výbavou vzdelaného jedinca už na konci 10. storočia. 2 Gerbert's life is not the subject of this paper as well as the famous legend diabolized him since 11th century. Only some elementary information is listed here: he was born before 950 in the region of Auvergne, as a child he entered the monastery in Aurillac, between the years 967 – 970 he studied on the Iberian Peninsula; since 972 he pursued further education and, more importantly, he was a teacher in Reims,
GERBERTVS, 2018
This paper tries to solve the question why did Gerbert of Aurillac in his brief letter to brother... more This paper tries to solve the question why did Gerbert of Aurillac in his brief letter to brother Adam (written in 989) elaborate a table for climate (horologium) where the longest day of the year reaches 18 hours. The standard summaries of climates, available during Gerbert's time, did not mention such climate. Gerbert added this table to his letter because the table of clime with 18 hours solstitial day (similarly like the second added table for the climate of Hellespont) is an exemplary guideline according to which Adam can make his own horologies. Gerbert used this extraordinary climate as a suitable explanatory example due to its mathematical simplicity appropriate demonstrating the astronomical theory of yearly Sun movement.
ARCHIWUM HISTORII FILOZOFII I MYŚLI SPOŁECZNEJ • ARCHIVE OF THE HISTORY OF PHILOSOPHY AND SOCIAL THOUGHT, 2017
The paper deals with early medieval mathematics (mainly arithmetic) and presents mathematical kno... more The paper deals with early medieval mathematics (mainly arithmetic) and presents mathematical knowledge as an important tool for human way to God’s wisdom. The aim of this paper is focused on the definitions of the subject of arithmetic in early medieval texts created between the late 4th and early 7th century. The aim is to highlight the fact that the traditional definitions of a number (i.e., the subject of arithmetic) correspond with the appropriate topics which exist within arithmetic. If a number is characterised as a discrete quantity, it refl ects the classification and typological surveys of the mathematical properties of numbers. If a number is defined as a collection of units, this definition refers to the issue of fi gural numbers, whereas if the number is marked as the quantity that emerges and then returns to the unit, it is possible to detect the themes of numerical sequences and ratios, including their transfers.
Filosofický časopis, 2017
This paper aims to analyse and evaluate the character and role of sense perception in the works o... more This paper aims to analyse and evaluate the character and role of sense perception in the works of Anselm of Canterbury written during the relatively short period of the 1070s and 1080s, namely the Monologion, the Proslogion (including the responses to the objections raised by monk Gaunilo), and De veritate. First, attention is devoted to sense perception in God – whether God possesses this kind of knowledge and whether God can be said to have sensually perceivable characteristics. The subsequent parts examine sense perception in the context of human knowledge on two levels: 1. human sensory knowledge and its role in understanding God (i.e., whether the senses are useful in any way in the struggle to fi nd God) and 2. sensory knowledge and its truthfulness (including sensory illusions). Lastly, an attempt is made to explain why Anselm paid such little attention to sensory perception, even though it seems, according to the analysed texts, that the senses played an important and irreplaceable role in his noetic endeavour.
ARCHIWUM HISTORII FILOZOFII I MYŚLI SPOŁECZNEJ • ARCHIVE OF THE HISTORY OF PHILOSOPHY AND SOCIAL THOUGHT, 2016
The main aim of this paper is an analysis of a dispute between Anselm of Canterbury and Roscelin ... more The main aim of this paper is an analysis of a dispute between Anselm of Canterbury and Roscelin of Compiegne about the interpretation of the Trinity of God. Roscelin’s propositions were condemned at the synod in Soissons in the early 90s of the 11th century, but the first information about his teaching were put down by John, later cardinal-bishop in Tusculum. John wrote a letter to his former teacher Anselm of Canterbury and informed him about Roscellin’s very problematic thesis. Anselm responded with two letters and with the treatise Epistola de incarnatione verbi, a work he re-wrote several times. At the end of this treatise, he mentions (similarly like Roscelin in his later Epistola ad Abaelardum) the key point of the whole controversy: what the Greeks describe as one essence and three substances, the Latins call one substance and three persons; the difference between the Greeks and the Latins is only in words and not in faith. Roscelin and Anselm could read this statement in Augustin’s De trinitate. Anselm probably used also Boethius tract Contra Eutychen et Nestorium and Porphyry’s defi nition of the person (Isagoge 2) to his explanation of the mystery of the Holy Trinity. Roscelin applied Priscian’s assertion (Institutiones grammaticae 2, 18) concerning a noun: each of them (e.g. a person) signifies both a substance and a quality. Roscelin argued that there are no accidents in God; everything in God is a necessary part of the divine essence. Consequently, when we use the noun “person” to refer to God, it must signifies something substantial (and not accidental), in other words the noun “person” signifies the divine substance. If we say that there are three persons in God, together with it we say that there are three divine substances or if the noun “person” is another name for God, than the Father and the Holy Spirit must be incarnated with the Son. It follows that the grammatical and dialectical traditions are the key reasons for Roscelin’s propositions. Inasmuch as Anselm (and Lanfranc) was the most prominent thinkers of the second half of the 11th century who argued for the active use of late ancient grammar (especially Priscian) and logic (Aristotle, Porphyry, Boethius) in questions of doctrine, due to it Roscelin (erroneously) assumed that these teachers would support his theory.
Średniowiecze Polskie i Powszechne, 2015
This paper deals with the abacus of the Latin pre-scholastic Middle Ages. According two descripti... more This paper deals with the abacus of the Latin pre-scholastic Middle Ages. According two descriptions of the abacus written around the year 1000 (one from the third book of the “History” written by Richer of Reims and the other from the first book of the “Liber Abaci” written by Bernelius the younger from Paris) and mainly according nine images of the calculating tool preserved in manuscripts from the end of the 10th to the beginning of the 12th centuries (so called abacus from Echternach and the abacus from Bern – either from the end of 10th century; the abacus from Paris from the beginning of 11th century; abacus from so called pseudo-Boethius’s the “Geometry II” from the half of 11th century; the abacus from Vatican, the abacus from Rouen and the abacus from Paris – all of these from the 11th century, the abacus from Oxford from the beginning of 12th century and so called abacus of Abbo of Fleury) this article presents in detail this arithmetic tool. The emphasis of this paper is given to the reconstruction of the form of this arithmetical tool according surviving images of this instrument as well as mutual comparisons of them together. This text focuses on the detailed description and analysis of the individual parts of the abacus (columns, arcs etc.) and furthermore presents and explains additional mathematical informations emerging in the mentioned images of the abacuses in manuscripts (for example abacistic symbols and names of numerals, markings of the abacus columns and symbols of fractions and relations between them).
GERBERTVS, 2015
The paper deals with two letters written by Gerbert of Aurillac to Constantine of Fleury. In thes... more The paper deals with two letters written by Gerbert of Aurillac to Constantine of Fleury. In these letters Gerbert points out some passages from Boethius's Introduction to Music (II, 10; respectively IV, 2 and II, 21) concerning mathematical operations (multiplication and subtraction) with superparticular ratios i.e. ratios of the type (n+1) : n. The musical harmonies rule the Cosmos and the Celestial Spheres according to Martianus Capella De nuptiis Philologiae et Mercurii; Music is the basis for understanting Astronomy. This paper follows two main aims: philosophical importance of music as liberal art and mathematical basis of the Pythagorean tuning.
GERBERTVS, 2014
This paper deals with comments and glosses to the first chapter of the second book of Boethius's ... more This paper deals with comments and glosses to the first chapter of the second book of Boethius's Introduction to Arithmetic from the last quarter of the 10th century. Those texts were written by Gerbert of Aurillac (Scholium ad Boethii Arithmeticam Institutionem l. II, c. 1), Abbo of Fleury (commentary on the Calculus by Victorius of Aquitaine, the so-called De numero, mensura et pondere), Notker of Liège (De superparticularibus) and by the anonymous author (De arithmetica Boetii). The main aim of this paper is to show that Boethius's statements about the converting numerical sequences to equality from this work could be interpreted minimally in two different ways. This paper discussed also the application of this topic in other liberal arts (like astronomy, music, grammar etc.) and in playing game called rithmomachia, the medieval philosophers' game.
Śląskie Studia Historyczno-Teologiczne, 2014
This paper deals with observing the peripatetic motives and influences of Boethius on the educati... more This paper deals with observing the peripatetic motives and influences of Boethius on the education and thinking of the late 10th and 11th centuries. The connection between Anselm‘s proofs of G od‘s existence from Monologion and Proslogion and so called mensa geometricalis, i.e. the abacus, a counting board used for arithmetical calculations and geometrical demonstrations circa 1,000 A.D., is presented as the entirely natural way of peripatetic interpretation of the intellectual world of Anselm of Canterbury, initiated by Franciscus Salesius Schmitt, through a search for other traces of Aristotelian heritage in the 11th century and in the period around the year 1,000 (primarily under the influence of Boethius‘s texts).
Papers in Czech by Marek Otisk
Filozofia, 2022
Isidore of Seville in On the Nature of Things XI, 1 presents the threefold nature of the four ele... more Isidore of Seville in On the Nature of Things XI, 1 presents the threefold nature of the four elements: fire is acute, subtle and mobile; air is subtle, mobile and obtuse; water is mobile, obtuse and corpulent; earth is obtuse, corpulent and immobile. This (Neo-)Platonic teaching on elements is based on Plato’s dialogue Timaeus. Isidore’s On the Nature of Things was relatively often copied in the Middle Ages. In many manuscripts we can find an illustration to the part XI, 1 of the treatises, usually called the figura solida (according to geometrical ratio), demonstrating the aforementioned qualities of all four elements. The depiction of the figura solida is very different among the surviving manuscripts. The paper focuses on various drawings of the figura solida in the manuscripts. The main aim of this paper is to show how this various version of the figure can be interpreted (especially within the context of Calcidius’ commentary to Timaeus) as an illustration of Platonic teaching on the nature of the elements.
Listy filologické, 2021
The paper deals with the correspondence between Gerbert of Aurillac (from 999 Pope Sylvester II, ... more The paper deals with the correspondence between Gerbert of Aurillac (from 999 Pope Sylvester II, † 1003) and Adalbold (from 1010 Bishop of Utrecht, † 1026/27). Both scholars wrote, in addition to other works, several short texts focused on mathematical problems, including letters addressed to each other. The paper presents, mainly with respect to Gerbert’s letter to Adalbold regarding the area of the equilateral triangle according to geometric and arithmetic rules, the broader context and aims of these mathematical disciplines in the Early Middle Ages. The emphasis is especially on inspiration from ancient and older Early Medieval sources (e.g. Plato, Calcidius, Boethius or Victorius of Aquitaine, the so-called Pseudo-Boethius’ Geometria I and Geometria II), but also other contemporary sources (Abbo of Fleury, Geometria incerti auctoris, etc.) and other texts by Gerbert (primarily Geometria attributed to him) and Adalbold (especially The Commentary to Boethius’ Consolation, III,9). The main goal of the paper is to demonstrate the extremely close relationships between the mathematical and philosophical interpretation of the created world, in accordance with Plato’s legacy.
Filosofický časopis, 2019
Gerbert and Arithmetic: Between the Philosophy of Numbers and the Arithmetic Art
The attitude tow... more Gerbert and Arithmetic: Between the Philosophy of Numbers and the Arithmetic Art
The attitude towards arithmetic in the Middle Ages was closely connected to antiquity’s opinion on the doctrine of the number, in whose framework practical arithmetic (calculating) and theoretical arithmetic (the theory or philosophy of numbers) were differentiated. The latter was traditionally assigned the greater importance in the area of philosophy, but the first had also been, from antiquity, perceived as a means that would lead to an understanding of the nature of numbers and which would cultivate the abstract and philosophical thinking of humans. This study discusses the example of the important, late 10th century thinker Gerbert of Aurillac (Pope Sylvester II) to show how it is possible to use arithmetic for philosophical research. First, the philosophy of numbers is presented, in which Gerbert (in connection with Boethius) reveals an arithmetically based reality, as numbers are also the thoughts of God. Practical arithmetic, for which Gerbert became especially famous (Western Arabic numerals and the abacus), then sharpened the human mind and facilitated the grasping of the nature of numbers.
Pro-Fil, 2019
The paper deals with the dialogue De grammatico written by Anselm of Canterbury. The author of th... more The paper deals with the dialogue De grammatico written by Anselm of Canterbury. The author of the dialogue himself described the work as an introduction to dialectics. In that epoch, the leading role in the given art belonged to Aristotle's Categories. As a result, the article aims to interpret Anselm's dialogue as a commentary to Aristotle's Categories conceived in a pedagogic form. Following Anselm's treatise, the paper analyses particular theses from the Categories (firstly the so-called antepredicamenta, then the categories of substance, quality, and partly also having) and the approach Anselm employed in working with them.
Aither, 2019
The paper deals with the brief treatise De sphaera written by Gerbert of Reims. The text is one o... more The paper deals with the brief treatise De sphaera written by Gerbert of Reims. The text is one of the so-called scientific letters addressed to Gerbert's former pupil and close friend Constantine of Fleury (all these letters were probably written at the end of 70s or at the beginning of 80s of 10th century) and describes a construction and a configuration of an observational hemisphere. The Czech translation of the treatise De sphaera is accompanied by a short presentation of the letter writer and the addressee of the letter, the article also focuses on the content of the Gerbert's letter and analyses a construction, aims and uses of a described hemisphere.
The paper deals with the dialectical dimension of the dialogue De grammatico written by Anselm of... more The paper deals with the dialectical dimension of the dialogue De grammatico written by Anselm of Canterbury during his stay in the Abbey of Le Bec. This dialogue between the teacher and the student addresses the question: How grammaticus is both a substance and a quality? In his work De veritate Anselm described the dialogue De grammatico as an introduction to dialectics. The paper tries to show how this Anselm’s dialogue could serve as an explanatory introduction (a textbook) to this liberal art. It seems that the main source of Anselm’s understanding of dialectics was six Boethius’s books called Commentaries to Cicero’s Topica. Therefore, the article presents basic characteristics of dialectics according to the above mentioned Boethius’s treatise (e.g. dialectics as inveniendi et iudicandi argumenti with the help of definitio, partitio and collectio; the description and components of an argument; the importance of the question in a dialectical disputation etc.), while the text of Anselm’s dialogue De grammatico is confronted with those Boethius’s theses.
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Papers in English by Marek Otisk
(died around 1143) writes about clocks Gerbert made in Reims in The History of the English Kings and describes them as arte mechanica compositum. The Benedictine Arnold Wion (died around 1610) mentions clocks from Ravenna, where Gerbert allegedly constructed a clepsydra, in The Tree of Life. In his Chronicle, Thietmar of Merseburg
(died around 1018) describes a horologium with an observation tube (fistula) from Magdeburg. These three references are analysed from a historical standpoint and especially Williams’s and Thietmar’s short reports are interpreted as possible references to timekeeping devices – the astrolabe and the nocturlabe.
Papers in Czech by Marek Otisk
The attitude towards arithmetic in the Middle Ages was closely connected to antiquity’s opinion on the doctrine of the number, in whose framework practical arithmetic (calculating) and theoretical arithmetic (the theory or philosophy of numbers) were differentiated. The latter was traditionally assigned the greater importance in the area of philosophy, but the first had also been, from antiquity, perceived as a means that would lead to an understanding of the nature of numbers and which would cultivate the abstract and philosophical thinking of humans. This study discusses the example of the important, late 10th century thinker Gerbert of Aurillac (Pope Sylvester II) to show how it is possible to use arithmetic for philosophical research. First, the philosophy of numbers is presented, in which Gerbert (in connection with Boethius) reveals an arithmetically based reality, as numbers are also the thoughts of God. Practical arithmetic, for which Gerbert became especially famous (Western Arabic numerals and the abacus), then sharpened the human mind and facilitated the grasping of the nature of numbers.
(died around 1143) writes about clocks Gerbert made in Reims in The History of the English Kings and describes them as arte mechanica compositum. The Benedictine Arnold Wion (died around 1610) mentions clocks from Ravenna, where Gerbert allegedly constructed a clepsydra, in The Tree of Life. In his Chronicle, Thietmar of Merseburg
(died around 1018) describes a horologium with an observation tube (fistula) from Magdeburg. These three references are analysed from a historical standpoint and especially Williams’s and Thietmar’s short reports are interpreted as possible references to timekeeping devices – the astrolabe and the nocturlabe.
The attitude towards arithmetic in the Middle Ages was closely connected to antiquity’s opinion on the doctrine of the number, in whose framework practical arithmetic (calculating) and theoretical arithmetic (the theory or philosophy of numbers) were differentiated. The latter was traditionally assigned the greater importance in the area of philosophy, but the first had also been, from antiquity, perceived as a means that would lead to an understanding of the nature of numbers and which would cultivate the abstract and philosophical thinking of humans. This study discusses the example of the important, late 10th century thinker Gerbert of Aurillac (Pope Sylvester II) to show how it is possible to use arithmetic for philosophical research. First, the philosophy of numbers is presented, in which Gerbert (in connection with Boethius) reveals an arithmetically based reality, as numbers are also the thoughts of God. Practical arithmetic, for which Gerbert became especially famous (Western Arabic numerals and the abacus), then sharpened the human mind and facilitated the grasping of the nature of numbers.
article mainly wants to focus on the correlation between those rules and
the subject of interest of the medieval intellectuals in the field of arithmetic (especially in the context of Boethius’s Introduction to Arithmetic).
The timekeeping was motivated for example by the religious obligations (the date of Easter
or exact times of all daily prayers in the monasteries) and due to it many of the Early Medieval intellectuals
paid close attention to an astronomical context of the timekeeping. This article points
out the definitions of basic timekeeping categories (year, month, week, hour etc.) and their astronomical
and geographical bases according to pre-scholastic thinkers.
Gerbert of Reims holds a firm position in history of philosophy and scientific thinking. Measured by today’s categories he can be noted as an important astronomer, logician, rhetorician, philosopher or mathematician who was also active in music theory and praxis, geometry as well as in area of practical and theoretical arithmetic. The last mentioned activity is subject of this book bringing the Latin original and Czech translation of seven Gerbert’s letters dedicated to mathematics.
First five of them were addressed to Gerbert’s friend, co-operator and perhaps pupil Constantine of Fleury. All letters to Constantine were written by the end of the 70-ties or at the beginning of the 80-ties of the 10th century, therefore during the time when Gerbert worked as teacher in Reims. Letter 1 is reaction to the period debate on conversion of three-membered numerical sequences arranged according to a certain ratio in the three same sums. Gerbert, following Boethius’s Introduction to arithmetic, deals with a conversion of superparticular numbers arranged in ratio 5:4 to a parity, i.e. ratio 1:1 and vigorously delimits the non-systematic method that was probably commonly used, however without respecting the nature of numbers, metaphysical hierarchy of relationships between the numerical ratios and ignoring the Boethius’s rules of conversions. Numerous and clear rejection of alternative approach of the ratios conversion is not even reduced by mathematical final correctness of the unrecognized method.
Letters 2 and 3 represent Gerbert’s commentaries on various text passages of Boethius’s Introduction to music. Boethius’s work is always quoted at first then Gerbert explains the arithmetic base of the quoted postulate. In Letter 2 Gerbert targets multiplication of ratios and classification of resulting products according to relative nature of numbers. Letter 3 describes and by using specific example clarifies a process of subtraction of smaller superparticular ratio from immediately subsequently bigger superparticular ratio. The resulting difference was found smaller then half of the subtracted ratio (subtrahend) as the double quantity of the result is a semitone smaller then subtracted ratio.
While the first three Gerbert’s letters are dedicated to theoretical arithmetic, Letters 4 and 5 represent an accompanying text of an abacist tract. Their subject is introduction to practical arithmetic, i.e. computing. Both abacist letters have very similar structures: Gerbert praises Constantine for his interest in study; and more he calls him consolation of his struggle and reason for writing the abacist treatises. Then Gerbert delimits himself towards other period scholars who only mention the abacus computations, however Gerbert criticises them for their disinterest in scientific texts of older authorities as well as for inability to understand neither the basic principles of abacus numeration nor methods of arithmetic computations.
Another two Gerbert’s mathematic letters were written in later period of Gerbert’s life (Letter 6 in year 988, Letter 7 probably in year 999). Both are part of a correspondence between Gerbert and Remigius of Trier (Letter 6) and Adelbold of Utrecht (Letter 7). First of the two letters deals in its mathematic part with divisible numbers, i.e. with way of measuring one number by another one. This topic is also linked to abacus numeration as well as to disintegration of numbers according to superparticular ratios. Letter 7 in general manner as well as with help of the specific examples explains different method of expressing the surface area of equilateral triangle using geometric and mathematic skill.
Individual letters are accompanied by commentaries closely introducing mainly the historical context of the individual problem and mathematic method presented in Gerbert’s letters.
The book opens an introductory study in order to ease orientation in the topics targeted by Gerbert’s mathematic letters as well as to closely present Boethius’s Introduction to arithmetic, the theoretical base of arithmetic knowledge initial for Gerbert’s texts (especially a character of number as such, character of numbers in relation to other numbers, i.e. above all number measuring or numeral sequences and ratios, further long also figural numbers and their discovering and depicting). Early medieval abacus computing tradition is also briefly presented including the design and structure of this unique mathematic device.
The introduction to theoretical and practical arithmetic is preceded by condensed description of Gerbert’s personality. The author of the letters is mainly presented as an important scholar of his period who was also very active in field of diplomacy. The book doesn’t miss out so called Gerbert’s legend reflecting already during the medieval period the personality of the great intellectual and mathematician.