This is the second volume of the first fully-fledged English translation of the works of Archimedes - antiquity's greatest scientist and one of the most important scientific figures in history. It covers On Spirals and is based on a reconsideration of the Greek text and diagrams, now made possible through new discoveries from the Archimedes Palimpsest. On Spirals is one of Archimedes' most dazzling geometrical tours de force, suggesting a manner of 'squaring the circle' and, along the (...) way, introducing the attractive geometrical object of the spiral. The form of argument, no less than the results themselves, is striking, and Reviel Netz contributes extensive and insightful comments that focus on Archimedes' scientific style, making this volume indispensable for scholars of classics and the history of science, and of great interest for the scientists and mathematicians of today. (shrink)

This paper deals with the ethical issues concerning living organ donor transplantation in the context of Immanuel Kant’s Ethical Theory. It primarily aims to refute the common perception about Kant’s categorical opposition to organ transplantation as violative to his concept of duty of self-preservation (transplantation as a form of mutilation or partial suicide). In this paper we will argue that: 1) Kantian concept of mutilation, or partial self-murder, should be perceived within the context of Kant’s prohibition of killing oneself; 2) (...) Kant does not prohibit all forms of organ donation as contrary to duty of self-preservation (i.e., duty to maintain quality of life and donation as violative to the strict prohibition against mutilation). Organ donation could hardly be considered as mutilation or contrary to duty of self-preservation if the following conditions are satisfied: 1) the organ to be donated and the surgical procedure involved, would not pose grave risk (e.g. physical disability and death) to the donor; and 2) the moral personality of the donor would not be at risk by donating his organ. (shrink)

Defining the mechanism of evolution is a controversial issue that, until now, divides the scientific community. Some have argued in the strictest Darwinian terms that evolution’s primary mechanism is necessity—“survival of the fittest”. Other evolutionists followed in the footsteps of Jacques Monod, the French biologist, who argued for a mixture of random chance and necessity. Teilhard de Chardin, it is widely believed, took Monod one step further by asserting that evolution is the fundamental motion of the entire universe, an ascent (...) along a privileged and necessary pathway toward consciousness—thus, evolution was guided chance and necessity. However, if evolution is being guided, what is doing the guiding? And where, ultimately, is it going? His bold answers brought Teilhard to the heart of a widely perceived scientific, as well as religious, heresy. A heresy that was effectively silenced, and soon would re-emerge as the world began witnessing exponential advancements in Science and Technology (specifically, on computing, nano-technology, robotics and genetic engineering). Almost half a century after the publication of Phenomenology of Man, many futurist thinkers have began noticing that the super-fast acceleration in the passage of time for evolution is moving in a very different direction than that for the Universe from which it emerges. This paper puts forward the thesis that the philosophical underpinning of a “human-sponsored variant of evolution” (i.e. evolution towards convergence of biological and non-biological intelligence) finds support and meaning within Teilhard de Chardin’s theory of evolution (i.e. evolution towards consciousness). It specifically covers 1) the implications of advancing technologies in human evolution and consciousness within the context of Teilhard’s theory of evolution; 2) how, after homo sapiens silently emerged around 500,000 years ago (with larger brains, particularly in the area of the highly convoluted cortex responsible for rational thought), and after they develop computing, the story of evolution has progressed exponentially paving the way for the possibility of turning Teilhard’s controversial ideas (such as the Noosphere) more than a poetic image; and 3) how the grandest creations of evolution—consciousness and intelligence—provide for the very tool that may allow homo sapiens to take over the course and direction of their own evolution—without necessarily shedding their desire to search for spiritual truth in a secular universe. (shrink)

Cassius Dio's fragmentary Roman History 15 contains an account of Archimedes’ role in defending Syracuse during the Roman siege of 213–212 b.c., incorporating a legendary tale about a solar reflector Archimedes constructed to burn Roman warships, and including details of his death when the city fell. The textual basis of this famous episode depends on two derivative twelfth-century works: Zonaras’ Epitome of Histories (9.4–5) and Tzetzes’ Chiliades (2.35). After clarifying the present state of enquiry, this paper introduces two (...) new witnesses, overlooked by editors of Dio and extensive scholarship on Archimedes, and assesses their value for reconstructing Dio's text. Comparative analysis of corresponding Dio-derived material in Tzetzes’ Carmina Iliaca and Hypomnema in S. Luciam, especially verbal correspondences with Zonaras’ Epitome, demonstrates that they are independent and, sometimes, superior witnesses to Dio's wording and content, reflecting Tzetzes’ selective use of the Roman History in different verse and prose compositions over several decades. The study considers editorial implications for this section of Dio's work and general characteristics of Tzetzes’ writings as repositories of testimonia and fragments. (shrink)

Archimedes of Syracuse (ca. 287–212 B.C.) was, by reputation at least, the most accomplished mechanic of antiquity. He was also perhaps the first to formulate what I call the mechanical challenge as the general goal of mechanics—to move any weight, however large, with any power, however small—and the first apparently to have found its general solution. From this insight he allegedly boasted that, given a place to stand, he could move the earth (the motto to this chapter), and he (...) was also said to have devised numerous ingenious mechanical instruments and weapons of war. His contribution to theoretical mechanics, on the other hand, is usually taken to be his rigorous, formal proof of the law of the equilibrium of the balance—that two weights are in equilibrium on a balance when their magnitudes are inversely as their distances from the pivot. It has generally been inferred that the law of equilibrium was in fact his theoretical solution to the mechanical challenge, and that his main purpose in proving this law was to establish on sound foundations the science of mechanics—the science of designing machines to accomplish great works with small powers. This may be true, although there is no direct evidence that he actually applied the law of equilibrium to the design of machines; Hero of Alexandria was perhaps the first to do this explicitly. Rather, what one finds in Archimedes’ mechanical works is an entirely different purpose. Instead of proving the law of equilibrium in order to apply it to practical machines, he seems to have proved it in order to apply it to purely geometrical problems. By identifying the pivot of the balance with the common centre of gravity of two weights, he extended the law of equilibrium into the more general law of centres of gravity—that all the parts of a magnitude, taken together in balanced pairs in accordance with the law of equilibrium, act as though at a single point called the centre of gravity of that magnitude. And with the law of centres of gravity, he devised a mechanical method in geometry to determine the areas, volumes, and centres of gravity of curvilinear surfaces and bodies. How he did this is the subject of this chapter. (shrink)

This marvelous volume takes the reader on a journey across the centuries as it explores eponymous physical laws—from Archimedes' Law of Buoyancy and Kepler's Laws of Planetary Motion to Heisenberg's Uncertainty Principle and Hubble's Law of Cosmic Expansion—whose ramifications have profoundly altered our everyday lives and our understanding of the universe.

SummaryWith the rediscovery by J. L. Heiberg of the manuscript of Archimedes' Method and its publication by him in 1907, a suspicion held for centuries by mathematicians and mathematical historians was at last corroborated by direct evidence: Archimedes had indeed applied in his discovery of geometric theorems a heuristic technique of analysis, although the highly formal synthetic proofs given in their demonstration concealed this fact. Disappointingly little effort has since been made to search for other indications of growth (...) in the composition of Archimedes' treatises. In particular, no one has attempted to exploit the relative chronology determined by J. Torelli in 1792 for a major part of the corpus. Despite the potential of even this partial ordering for a deeper understanding of Archimedes' work, no subsequent edition, translation or commentary has retained it as the organizing principle. In the present reexamination of this question of Archimedean chronology, I will (1) show how tenuous are the foundations of the orderings now in standard usage; (2) present a chronological order which differs from the standard accepted sequence in certain important respects, notably, in the position of the Dimension of the Circle and the Method; and (3) distinguish an “early group” of studies from the “mature group”, employing as my criterion the predominantly Euclidean character of the techniques used in the former. I believe that the results of my inquiry have broader implications, as in reinterpreting as Archimedean certain contributions preserved by such later authors as Hero and Pappus. (shrink)

As part of the laboratory activities designed for the students of the second year of the Mathematical High School by the research group in Mathematics Education of the Department of Mathematics of the University of Salerno, the theme “From Archimedes and the oxen problem to Pell’s equation” has been designed. The course has not yet been delivered to students and in this paper the didactic path to be developed in the classroom will be described. The students initially will retrace (...) the various solution approaches proposed over the centuries. Subsequently, with a heuristic approach, they will tackle the resolution of the proposed problem by exploiting properties related to the theory of numbers. With the help of some schedules, provided by the teachers-tutors, they will conjecture and argue by answering questions aimed at stimulating didactic reflection. This workshop path involves processes such as exploring and conjecturing through the articulation of different ways of working (individual and in groups), manual and digital. (shrink)

This article describes the mystery of a long lost codex of Archimedes that resurfaced briefly at the turn of the last century by Johan Ludwig Heiberg. Long enough for the Danish historian of mathematics Heiberg to identify, photograph and eventually transcribe “The Method” and several other works by Archimedes of considerable mathematical interest. In 1879 Heiberg completed his dissertation, Quaestiones Archimedeae, devoted to Archimedes’ life, works, and transmission of his texts.Este artículo describe el misterio de un códice (...) de Arquímedes perdido hace mucho tiempo que reapareció brevemente a principios del siglo pasado de la mano de Johan Ludwig Heiberg. Tiempo suficiente para que el historiador danés de las matemáticas Heiberg pudiese identificar, fotografiar y, finalmente, transcribir “El Método” y varias otras obras de Arquímedes de interés matemático considerable. En 1879 Heiberg completó su tesis doctoral, Quaestiones Archimedeae, dedicado a la vida de Arquímedes, las obras, y la transmisión de sus textos. (shrink)

Published in 1880–1, this three-volume edition of the extant works of the Greek mathematician Archimedes of Syracuse was edited by the Danish philologist and historian Johan Ludvig Heiberg, whose Quaestiones Archimedeae is also reissued in this series. He compiled this edition from a Florentine codex, which he compared with other extant sources. Volume 1 contains On the Sphere and the Cylinder, On the Measurement of a Circle and On Conoids and Spheroids. Volume 2 contains On Spirals, On the Equilibrium (...) of Planes, The Sand Reckoner, The Quadrature of the Parabola, On Floating Bodies, the Liber Assumptorum, the cattle problem and fragments. Volume 3 contains the editor's own Latin prolegomena, the commentaries on Archimedes by Eutocius of Ascalon and indexes. The texts are given in Greek with parallel Latin translation, notes and introductory material. (shrink)

Archimedes was the greatest scientist of antiquity and one of the greatest of all time. This book is Volume I of the first authoritative translation of his works into English. It is also the first publication of a major ancient Greek mathematician to include a critical edition of the diagrams and the first translation into English of Eutocius' ancient commentary on Archimedes. Furthermore, it is the first work to offer recent evidence based on the Archimedes Palimpsest, the (...) major source for Archimedes, lost between 1915 and 1998. A commentary on the translated text studies the cognitive practice assumed in writing and reading the work, and it is Reviel Netz's aim to recover the original function of the text as an act of communication. Particular attention is paid to the aesthetic dimension of Archimedes' writings. Taken as a whole, the commentary offers a groundbreaking approach to the study of mathematical texts. (shrink)

The content of Stevin’s hydrostatics was original and became a cornerstone of the advances in hydrostatics that were to follow it. Stevin grasped the fact that the force on a solid surface in contact with water is independent of the orientation of that surface and depends only on its depth beneath the uppermost surface of the water. Some of Stevin’s proofs were highly ingenious. However, his postulates, the most significant of which acknowledged that the depth to which a vessel sinks (...) is proportional to the weight it carries, were insufficient to yield the content of his theory as deductive consequences. In a way that he did not make explicit, Stevin inserted into his theory features of the behavior of water with which he was familiar as a hydraulic engineer but which were not licensed by his postulates. For instance, he assumed, rather than proved or explained, that water presses horizontally against a vertical surface, a phenomenon with which Stevin was familiar through his dealings with lock gates. It was only during the course of the seventeenth century that the facts about hydrostatics identified by Stevin were explained and adequately theorized. Those developments required the recognition that experiment needs to be read as supplying evidence for hydrostatics rather than as being mere applications of it, as Stevin had assumed. (shrink)

"Why did Archimedes jump from his bath and run naked through the streets shouting 'Eureka!'? What is a quantum and where does it leap? Do you know your id from your ego? Does God play dice? his books answers all those questions and more, taking the revolutionary and perplexing ideas of Western thought and extracting their essence. From Greek philosophy to contemporary economics, physics and architecture, A Little Knowledge covers some of the most often heard but least understood theories (...) and explains them simply and entertainingly, including: Existentialism The Oedipus Complex Original Sin 'I think, therefore I am' Fuzzy Logic The Big Bang 'Religion is the Opium of the People'. (shrink)

A review of Huw Price's Time's Arrow and Archimedes' Point.

There have been two attempts in the history of human speculation to estimate the number of particles in the universe. The first was that of Archimedes of Syracuse about 216 B. C., and the second that of Sir Arthur Eddington nearly two thousand years later. What is surprising is that they both arrive at the same number. This is the number obtained by multiplying ten by itself seventy-nine times.

This paper discusses four diagrams in the Archimedes Palimpsest, a manuscript that provides among other texts the only extant witness to Archimedes’ Method. My study of the two diagrams for Method 12 aims to open up discussions about the following two questions. First, I want to question the assumed relationship between diagram and geometric configuration. Rather than a representation-represented relation, I argue that the two diagrams for Method 12 have a stronger independence from the geometric configuration they are (...) related to. Their connection with the theorem rather lies in their role in shaping the way in which the proof is written. Secondly, I want to examine the connection between geometric intuition and figure. Geometric intuition is the recognition of certain properties or relations in a geometric configuration not through deduction and is held to be parallel to, if not dependent on, the empirical observation of a physical figure. Method 12 provides a curious case where an intuitive fact is not only unrepresented, but even hindered by the diagrams, and the proof assumes that the reader has come to realise that fact, which is most easily achieved when the diagrams are ignored. (shrink)

_Sunto_ Nella proposizione 3 della κύκλου μέτρησις Archimede costruisce due numeri razionali che approssimano la misura del rapporto tra la circonferenza e il suo diametro, oggi noto come π. In questo lavoro vogliamo porre l’accento su due aspetti del trattato di Archimede: il primo riguarda le scelte dei valori razionali approssimati per eccesso e per difetto di √3 utilizzati all’inizio della procedura di calcolo, che risultano essere i migliori approssimanti in base alla moderna teoria delle frazioni continue, i cui termini (...) si possono ricavare dall’antico ανθυφαίρεσις euclideo; il secondo riguarda alcune stime utilizzate da Archimede, apparentemente non giustificate e sub-ottime, che invece sono frutto di un preciso procedimento logico da noi ricostruito in termini di un algoritmo. Di tale studio è stata proposta una trasposizione didattica che parte dal testo antico e giunge alla produzione di una traduzione commentata in cui lo studente è protagonista del processo di scoperta. _Abstract_ In proposition 3 of the κύκλου μέτρησις Archimedes constructs two rational numbers that approximate the measure of the ratio between the circumference and its diameter, today known as π. In this work we emphasize two aspects of Archimedes' proof. The first one concerns the choices of the rational values approximated by excess and defect of √3 used at the beginning of the calculation procedure. These values turn out to be the best approximants based on the modern theory of continued fractions, whose terms can be obtained from the ancient Euclidean ανθυφαίρεσις. The second one concerns some estimates used by Archimedes, apparently unjustified and sub-optimal, which instead are the result of a precise logical procedure that we recontruct in terms of an algorithm. We also propose an educational transposition of this study that starts from the ancient text and leads to the production of a commented translation in which the student is the protagonist of the discovery process. (shrink)

Using Riemann’s Rearrangement Theorem, Øystein Linnebo (2020) argues that, if it were possible to apply an infinite positive weight and an infinite negative weight to a working scale, the resulting net weight could end up being any real number, depending on the procedure by which these weights are applied. Appealing to the First Postulate of Archimedes’ treatise on balance, I argue instead that the scale would always read 0 kg. Along the way, we stop to consider an infinitely jittery (...) flea, an infinitely protracted border conflict, and an infinitely electric glass rod. (shrink)