Ancient Mathematics

The pseudo-Aristotelian Mechanical Problems is the earliest known ancient Greek text on mechanics, principally concerned with the explanation of a variety of mechanical phenomena using a particular construal of the principle of the lever. In the introduction, the author (thought to be an early Peripatetic) quotes the tragic poet Antiphon to summarise a discussion of the techne-phusis (art-nature) relationship and the status of mechanics as a techne. I argue that this citation of a poet is an Aristotelian cultural signature, intended to guide its reader(s) towards a better understanding of the nature of mechanics as expounded in the Mechanical Problems. By analysing several instances where Aristotle cites Antiphon (as well as other tragic poets) in the Aristotelian corpus, I propose that both the author of the Mechanical Problems and Aristotle use poets for the purpose of persuasion. This is in turn explained by understanding the homologous relationship between mechanics-as-techne (according to the author of the Mechanical Problems) and poetics-as-techne (according to Aristotle) in terms of their shared status as poietike techne (productive art) and claims to universal knowledge. A final facet of the proposed relationship between mechanics and poetry is hypothesised on the grounds of their mimetic nature

Whether the artisan who made the Omphalos at Delphi over 2500 years ago recognized the optical transform properties of its shape or not, its geometrical features are nevertheless those of a space-inverting anamorphoscopic mirror. Specifically, it is similar in shape to a circleinverting anamorphoscope. We speculate that select members of the ancient religious sect at the Temple of Apollo at Delphi realized the symbolism of inverting all of space outside the Omphalos into the image field inside its base, thus making it the virtual centre of the World.

claim that the illustrated technical descriptions in Song Yingxing's late Ming Tiangong kaiwu (Exploitation of the Works of Nature) were for entertainment and not practical instruction. Golas explores who the illustrators were, the economic factors influencing their illustrations, and how their illustrative styles were reflected in the portrayal of technology in the Tiangong kaiwu. The precision in the text was not always matched by the illustrations. By contrast, Bray notes that Dagmar Schä ffer has elsewhere described the Tiangong kaiwu as a moral statement, which had more of an affinity with the congruence of knowledge and action rather than simply representing a technical treatise. Donald Wagner adds to the discussion the fact that hua were usually attributed while tu were usually anonymous. He believes that the Tiangong kaiwu was based on earlier sources, most of which were reliable, but often they were not. Usually the texts and illustrations dealing with iron production, which Wagner focuses on, matched. In some cases, however, the text gave one method while the illustration presented another; in a few cases, the picture had nothing to do with any real methods for producing iron. Here, Wagner and others in the volume perhaps underestimate how tricky the technical drawings used by artisans are to read. Mechanical drawings, as my students keep telling me, are not simply read one way.

In an ancient Egyptian problem of bread distribution from the Rhind mathematical papyrus (dated between 1794 and 1550 B.C.), a procedure of “false position” is used in the calculation of a series of five rations. The algorithm is only partially illustrated in the problem text, and last century's prevailing interpretations suggested a determination of the series by trial and error. The missing part of the computational procedure is reconstructed in this article as an application of the algorithm, exemplified in the preceding section of the papyrus, to calculate an unknown quantity by means of the method of “false position.”

In un antico problema Egizio di distribuzione di pane, nel papiro matematico di Rhind (datato tra il 1794 e il 1550 A.C.), una procedura di “falsa posizione” è utilizzata nel calcolo di una serie di cinque razioni. L'algoritmo è illustrato solo parzialmente nel testo del problema, e le interpretazioni prevalenti nel secolo scorso suggerivano una determinazione della serie per tentativi successivi. La parte mancante dell'algoritmo è ricostruita in questo articolo come applicazione dell'algoritmo, esemplificato nella sezione precedente del papiro, per calcolare una quantità incognita mediante il metodo di “falsa posizione.”

A Roman centuriated cadastre may include other Roman linear
features - such as roads - which are oblique to the square grid,
and appear to ignore it. But initial impressions are deceptive;
there are several cases which reveal clear trigonometrical links.
These relationships are unlikely to have occurred by chance and,
supported by evidence from contemporary documentation, they
indicate that the links were planned. If this is generally so, the
presence of these trigonometrical relationships can suggest that a
centuriated cadastre existed, even if its grid is not immediately
apparent.

As a theme of historical research Diophantus’ work raises two main issues that have been intensely debated among researchers of the period: (i) The proper understanding of Diophantus’ practice; (ii) the recognition of the mathematical tradition to which this practice belongs. The traditional answer to this range of questions – since medieval Islam, through the Renaissance and the Early Modern period, up to the leading historians of mathematics of the 20th century – was that Diophantus’ book is a book on algebra. This traditional approach has been criticized recently by some historians of mathematics who point out the anachronistic methodology that historians in the past often were using in analyzing ancient texts. But, criticizing the methodology by which one defends a historical claim does not mean necessarily that the claim itself is wrong. The paper discusses some crucial issues involved in Diophantus’ problem-solving, thus, giving support to the traditional image about the algebraic character of Diophantus’ work, but put in a totally new framework of ideas.

"The broad reception of Vitruvius in architectural history has especially accounted for the fact that fields of knowledge essential for the understanding of ancient processes of design and planning remain hitherto unconsidered. Although Vitruvius discusses various methods for designing ideal type and modularised architecture the question of mathematical and technical basics for creating a real building is still open, i.e. the practical transformation on the actual building site with all its needs such as architectural surveying and logistics. An as yet widely unsolved problem is which knowledge enabled antique and late antique architects and engineers to provide the rationally comprehensive frame needed to make the theoretical constructions calculatable and plannable buildings.The study of the Hagia Sophia (532-537) and its architects Anthemius of Tralles and Isidore of Miletus leads us to an important source which can fill this gap of knowledge effectively and which proved to be an indispensable basis for understanding ancient architecture in its whole. Late antique sources and primarily the structure of the building itself document that the exceptional achievements of design and planning must be associated with the writings of Heron of Alexandria. From the 1st century AD to the Byzantine period in his name handbooks for engineers of various disciplines were distributed which provided obligatory instruments of calculation with systematically compiled tasks for all groupsof profession engaged in building. Particularly Heron’s scientific discipline of surveying and his treaty on vaults demonstrated to be revised by Isodore, can be assessed as a basis for planning and building. Only if knowing these sources the processes necessary for transforming an ideal plan into a real still existing construction can be reconstructed. TheHagia Sophia therefore is a unique example in which written sources and architectural remains can be analysed and complement each other in a singular way; at the same time it establishes an entirely new model of interpretation for ancient planning praxis."

This paper proposes an updated analysis of the four mathematical problems on the two main fragments of P. Berlin 6619. Photographs, transcription, translation and commentary of the problem texts are included. The analysis focuses in particular on problem 1, on the recto of the large fragment, previously interpreted as the calculation of the sides of two squares. The sentence jr.t HAy.t m wa r nHH (...) indicates that the problem concerns the calculation of two sides of a rectangle.  This interpretation is supported also by architectural and cross-cultural parallels.

Resumen: Examino Acerca del cielo (De caelo) I 2 con el fin de mostrar allí la presencia de la demostración científica. Este desarrollo pretende aportar nueva evidencia en favor de la no discrepancia entre teoría y praxis científica en Aristóteles (Barnes, 1969) y de relativizar la interpretación de que el método real y únicamente usado es la dialéctica (cuyo antecedente se remonta a Owen, 1980). Además, siguiendo la propuesta hermenéutica de Gotthelf (1987) y Detel (1993 y 1997), mostraré de qué modo se utiliza la demostración científica en la prueba de la existencia del cuerpo simple o éter. Ofreceré también una reconstrucción de las pruebas mediante la elaboración de un esquema que abarque el conjunto de las deducciones. Estos últimos desarrollos constituyen el aporte más novedoso del presente artículo.

Models are one of the main instruments in scienti c research. Disciplines have developed a di erent model understanding of the notion, function and purpose. We thus need a systematic approach in order to understand, to build and to use a model. This book gives an insight into the discipline modelling know-how in Kiel and is a rst starting point to develop a general model approach that generalizes and combines for an inter disciplinary use.

Depuis quelques décennies, la question d'une compréhension historiquement correcte de la méthode de Diophante a attiré l'attention des chercheurs. « L'algèbre moderne (c'est-à-dire, post-viètienne) », « la géométrie algébrique », « l'arithmétique », « l'analyse et la synthèse » sont parmi les contextes proposés par certains historiens pour interpréter les procédures résolutoires de Diophante, tandis que la catégorie de « l'algèbre prémoderne » a été proposée récemment par d'autres dans la même finalité. Le but de cet article est d'argumenter contre l'idée de contextualiser le modus operandi de Diophante dans le cadre conceptuel de l'analyse ancienne et d'examiner les quelques exemples, dans les livres conservés des Arithmétiques, qui pourraient être considérés comme liés aux pratiques qui appartiennent au domaine analytique.

In the first argument of Metaphysics Μ.2 against the Platonist introduction of separate mathematical objects, Aristotle purports to show that positing separate geometrical objects to explain geometrical facts generates an ‘absurd accumulation’ of geometrical objects. Interpretations of the argument have varied widely. I distinguish between two types of interpretation, corrective and non-corrective interpretations. Here I defend a new, and more systematic, non-corrective interpretation that takes the argument as a serious and very interesting challenge to the Platonist.

AO 8900, AO 8901, and AO 8902 are three hitherto unpublished Old Babylonian mathematical cuneiform tablets containing multiplication tables. Their physical and textual characteristics suggest that they were produced in the same ancient context. What is remarkable about this small set of tablets is that, unlike most such tablets, two of them have colophons: in AO 8900 we find a month-and-day date, while in AO 8901 we find a damaged year name. These tablets are published here for the first time, together with a discussion of how the information about the year name in AO 8901 fits in what is known about the dating of Old Babylonian mathematics.

In Odes 1.28, Horace deals with one of his favorite topics: death and the appropriate human disposition towards it, by focusing on the Pythagorean mathematician Archytas and his tomb near the sea. The paper tackles the old interpretive difficulty arising from the fact that several of the mathematician’s traits belong rather to Archimedes by arguing that Horace purposefully conflated the two mathematicians to respond to Cicero, who famously portrays himself cleaning Archimedes’ tomb in Tusculans 5.64. By identifying Archimedes with Archytas, Horace accentuates the aura of immortality attributed to Archimedes by Cicero and is able to offer his own contrasting view more forcefully.

For nearly a century there is an ongoing debate about, have the ancient Egyptians known any case of the Pythagorean Theorem and that the triangle 3-4-5 is right-angled? According to the opinions of most scholars, there is no written evidence regarding this dispute. Hence, this paper shows the written evidence in a problem on land survey in the so-called Rhind Mathematical Papyrus-RMP of circa 1550BC, which has been separated by the early scholars into two problems: RMP#53 and RMP#54. The paper shows that RMP#53-54 is one problem on reckoning the dimensions of a triangular plot of land with sides 6-8-10 and its sections, where the triangle's height is a radius of a circular horizon that increases by adding one-tenth of the triangle's hypotenuse. Besides, it shows that the reckonings of RMP#55 are based on the same triangle sketch of RMP#53-54. The paper proves that the ancient Egyptians knew this triangle almost thousand years before the days of Pythagoras. The paper also shows that the ancient Egyptians did not only use the unit fractions but have also used complex fractions.

This paper analyzes the algorithmic structure of geometrical problems in Egyptian papyri of the first half of the second millennium B.C. Processes of transformation of quantities from ‘‘false’’ values into actual values, and conversions from quantities expressed in the abstract system of numbers into metrological quantities, are known in Egyptian mathematics. Three further processes are identified in the present contribution: transformations of ‘‘false’’ dimensions of geometrical objects into true dimensions; transformations of geometrical objects into other geometrical objects; transformations of linear measures of monuments. These processes have relevant implications on the algorithmic structure of the problem texts, resulting in particular in the embedding of sub-algorithms and the creation of parallel structures. More in general, their wide employment in Egyptian mathematics has significant philosophic and cultural implications.

It is well known that Sumerians and Babylonians used a numeration system of base 12 and 60. We still have influence of that system in our nowadays counting of the hours of a day, twelve plus twelve, each hour has 60 minute and each minute 60 seconds. Also the circle has 360 degrees. What is not so well known is that the Sumerians in an earlier period used a ternary system of numeration; the first notice about that system is in Thureau-Dangin (1928). Later Diakonoff (1983) described it in good details and recently, Balke (2010) studied the system and described it with more precision. Still the use of this system and the concept of number involved are open questions. I will answer to those problems making a formalization of the system and showing how it could be related to a cosmogonic design.

Philological uncertainties characterize the analysis of Problem 10 from the Moscow mathematical papyrus, particularly regarding the identification of the object of calculation designated as nb.t. The interpretations previously proposed have differing degrees of plausibility, but none provides decisive evidence. The analysis of a cylindrical measuring vessel from the temple of Karnak, and of a vessel in a three-dimensional model from Beni Hasan, supports the suggestion that the object being calculated is a semi-cylinder.