The Genius of Archimedes -- 23 Centuries of Influence on Mathematics, Science and Engineering

Starting from Archimedes’ method for calculating the volume of cylindrical wedges, I want to get to describe a method of 18th century for cilindrical groins thought by Girolamo Settimo and Nicolò di Martino. Several mathematicians studied the measurement of wedges, by applying notions of infinitesimal and integral calculus; in particular I examinated Settimo’s

Treatise on cylindrical groins

, where the author solved several problems by means of integrals.

In recent papers we analyzed the historical development of the foundations of the centres of gravity theory during the Renaissance. Using these works as a starting point, we shall briefly present a progression of knowledge with cultural and mathematical Archimedean roots in Torricelli’s mechanics.

We highlight the legacy of Simon Stevin and Gabriel Lamé and show how their work led to some of the most important recent developments in science, ultimately based upon the principles of balance and the act of weighing, virtual or real. These names are also important in the sense of a

unique rational science

and

universal natural shapes

.

In this work we present a twofold educational approach to the reflective properties of surfaces, starting from the historical context of Archimedes “burning mirrors”. The properties of the emerging surface known as “caustic” of a given smoothly shaped mirror are illustrated by an interactive multimedia. An experimental device is also proposed to visualize the geometrical principles underlying the formation of caustics. The proposed didactical trail is intended also to contextualize the figure and work of Archimedes in a perspective tightly linked to modern technology, so to collect young learners’ interest.

In the paper we discuss the three methods that Archimedes employs to deal with the problem of the quadrature of a parabolic segment. We characterize the three approaches as heuristic, mechanical and geometric respectively. We investigate Archimedes’ own attitude towards the three methods, and we conclude with a critical presentation of the prevalent views concerning the matter, which have been expressed in the past by historians of mathematics.

Archimedes practices the heuristic method of analysis and synthesis only in Book II of his

On the Sphere and Cylinder

. This paper has a twofold objective. Firstly, the discussion of his analytical practice through the first problem of Book II, in relation to Pappus’ study of the method of analysis and synthesis in Book VII of his

Mathematical Collection

. The conclusion of this discussion is that Archimedes applies the analytical method in a way, which does not substantially differ from Pappus’ way. Secondly, the discussion about the missing part from the analysis of problem 4 of

On the Sphere and Cylinder

, II, combined with the above conclusion, lead us to advance a conjecture vis-à-vis a lost analytical treatise of Archimedes under the title

Book of Data

.

In this paper I will discuss the position of the Flemish mathematician and engineer Simon Stevin (1546–1620) in the rise of Archimedean mechanics in the Renaissance. Commandino represents the beginning of the Archimedean Renaissance in statics. The next steps were made by Guidobaldo Del Monte and Stevin. Del Monte and Stevin were contemporaries belonging to the generation preceding Galilei (1564–1642). Yet Stevin’s work in mechanics is superior to Del Monte’s. I will discuss the way in which Stevin’s mechanical work, like Del Monte’s, was influenced by the medieval science of weights. For example, the central notion “stalwicht” in Stevin’s work, translated as “apparent weight’ by the editors of Stevin’s

Works

, clearly corresponds to the notion of positional weight (ponderis secundum situm) in the science of weights. I will also argue that while Del Monte remained caught in the conceptual framework of the science of weights the use of the Dutch language helped Stevin in liberating himself from those ideas. For Stevin the use of Dutch was part of his success. Finally I will discuss Stevin’s work on windmills. Not only his original theoretical contributions to statics and hydrostatics but also the unity of theory and practice in Stevin’s work make him in mechanics the first true successor of Archimedes in the Renaissance.

In the paper is discussed the possibility tat Archimedes built and used against the Roman fleet a steam cannon. It is well-known that Archimedes, during the siege of Syracuse, designed and built several war machines to fight against the Romans. Among these war machines, the legend about the large concave mirrors that concentrated the sun rays burning the Roman ships is rather interesting. On this topic are also interesting some drawings by Leonardo Da Vinci where a steam cannon is described and attributed to Archimedes. Starting from passages by ancient Authors (mainly Plutarchos, Petrarca and Da Vinci), the author investigates on the possibility that Archimedes built a steam cannon and used it to hit the Roman ships with incendiary proiectiles.

Starting from a previous model for the shift mechanics of rubber belt variators, this lecture elaborates practical design formulas for the torque and the axial thrust making use of the very close resemblance of the belt path to a linear spiral of Archimedes along a large part of the arc of contact. In addition, as an alternative to the modern calculus tools, it is shown how the drive variables can be equally calculated applying some propositions of Archimedes’ classical treatise

περί ‘ελίκων

(On Spirals).

In the paper are proposed some mechanical systems, all certainly used in the Classic Age, that could be easily adopted to power the siege towers, devices invented by Greek engineers and called Helepolis. These ancient motors are made up by capstans, tread wheels like those used for Greek-Roman cranes and counterweight motors, all installed into the helepolis. The proposed motors are also analyzed from a mechanical point of view in order to examine, at least theoretically, their effectiveness in such applications.

Mathematics forms the common roof for Archimedean spirals on the one side and screw mechanisms on the other side. Moreover, Archimedes was a genius of mechanics and mechanisms and was famous for solving mathematical and mechanical problems. There is also a historical justification for the title of the present paper, because the technical notion “Archimedean water-screw” is well-known to those mechanical engineers who are fond of looking back to the ideas and inventions of some famous protagonists and forerunners in the past and still today want to learn from their successes and failures.

In this paper a relevant contribution of Archimedes is outlined as related to his developments in mechanics with application to mechanism design with a modern vision. He developed theoretical advances that were motivated and applied to practical problems with an enthusiastic behaviour with a modern spirit that can be summarized in his motto ‘Give me a place to stand and I will move the earth’.

From the end of the 16th century to the beginning of the 18th century, Jesuit missionaries introduced Chinese the Western scientific knowledge and technology. In 1612, Archimedean-screw was introduced by Sabbathinus de Ursis and Xu guangqi. Since then, Part of Archimedean mecanical knowledge was transmitted into China. In this paper we will present an account about this transimission. The main points are as the following: in the 17th century, parts of both theoritical were introduced to China; Chinese paid more attention to the practical knowledge then, Theoretical knolwedge was only studied becase that it is the base of making useful devices.

Archimedes is an author who is frequently quoted in Arabic texts in relationship with mathematics and mechanics, including hydraulic devices such as water-clocks. The present study traces transmission paths and evidence for an assessment of the impact of the Archimedean works on the Arabic tradition of mechanics and hydraulics.

Archimedes’s fame is universally more connected to his extra- ordinary inventions and to the legendary events that have been ascribed to him rather than to a deep and real knowledge of the historical personage and of his works. Systems of

levers

and

catapults

,

cochlea

and other mechanical or hydraulic contraptions,

water-clock

,

planetarium

,

heat rays

. Among these and further inventions, real or supposed to be, there is the episode of Hiero’s Crown, Fig. 1. The episode of the apparent fraud goes generally around in two different versions; the first one, which is based on the volumetric comparisons, mentioned by the roman architect Vitruvius, the second one is anonymous, it is related by Priscian and it’s essentially based on the hydrostatic balance. In this paper, we compare and discuss the two reconstructions, both of them to be considered plausible.

Archimedes left to posterity his famous treatise “On Floating Bodies”, which establishes the physical foundations for the floatability and stability of ships and other maritime objects. Yet since this treatise was long lost and also simply ignored by practitioners, it took many centuries before Archimedes’ brilliant insights were actually applied in ship design and ship safety assessment. This article traces the tedious acceptance of Archimedes’ principles of hydrostatics and stability in practical applications. It will document important milestones and explain how this knowledge was passed down through the centuries and ultimately spread into ship design practice.

A recent hypothesis on the giant ship

Syrakosia

, built around 235 B.C under the rule of Jeron II of Syracuse, is presented. In this enterprise Archimedes was a minister or a supervisor of the architect (Archias from Corinth). Comparison is made about opinions on the possibilities Archimedes could have had to plan the ship and to foresee her stability properties. The reconstruction of the actual procedures available to him to evaluate volumes and centres of buoyancy of different solids is studied also with models of the volumes of the ship and of the geometric solids mentioned in the “Floating bodies”, to conclude that he could evaluate the stability properties on a quantitative basis only for the

orthoconoid

, the features of which were exhaustively studied in previous Archimedes’ works. Therefore he must have left to the architect the task of both planning and evaluating empirically the stability of the ship. The last propositions of the first book of the “Floating bodies” suggest that Archimedes may have taken the experience of the

Syrakosia

as a guideline to approach the problem of stability of sectors of spheres, as the ship was launched when the hull reached the floating line and then she was completed afloat. Stability features of these phases appear to be comparable to those of different portions of sectors of the same sphere as presented by Archimedes.

Then, it is thought that Archimedes found the Law of Buoyancy at that moment and proved the theft of the goldsmith. In this paper, the measurement of the overflow volume by golden crown etc. has been tried. At a result, it is proved that the measurement can be done with enough accuracy by using a vessel having enough large opening diameter. Archimedes might have proved the theft of the goldsmith by almost the same method to this measurement. From this result, at “EUREKA” moment, Archimedes did not find the Law of Buoyancy but found the solution of the king Hiero’s problem and specific gravity of things. Also this moment must have been when Archimedes got an inspiration of the idea of the law of Buoyancy.

In this paper the main developments in ship buoyancy, stability and subdivision of ships since the milestone formulation of the basic laws of floatability and stability of floating bodies by Archimedes are reviewed. The continuous progress in the safety of ships as most effective transportation means and the links of the fundamental Archimedean studies to the modern naval architectural approaches to ship stability, design and safety are critically commented.

In a very famous passage dealing with the life of Marcellus, Plutrarch says that Archimedes never wrote a text about mechanics and its practical applications (Plutarch, Life of Marcellus, 17): according to Plutarch, in fact, Archimedes would agreed with the classical Plato’s attack against the knowledge originating from technology and the practice of science because of their vague and inaccurate nature. This paper, focusing on the building of the famous ship “Syrakousia” and its description according to the only existing reference in Atheneus of Naucratis (Deipnosophistae, V, 40–44), is an attempt to rethink Archimedes’ position about mechanical knowledge and the cultural relationship between Syracuse and Alexandria.

A letter preserved in a manuscript of the 16

th

century belonged to Gian Vincenzo Pinelli, shows the high esteem bestowed on Archimedes at that time. An anonymous philologist was working on the critical edition of Pliny the Elder’s

Naturalis Historia

and asked a mathematician for help in amending a doubtful

lectio

of a passage in the 2

nd

book. The philologist’s interpretation is discussed and rejected by the mathematician thanks to the use of Archimedes’ rule for the approximation of

pi

.

Irony of fate! Democritus is the only philosopher mentioned in one of his works by Archimedes, and wrote about mathematical things, while neither Plato nor Aristotle are mentioned by him nor have they written about mathematics, but only witnesses scattered here and there in their writings and very often confused, yet they’re considered Archimedes’ inspirers! (Boyer 1939, Delsedine 1970, Frajese 1974, Gambiano 1992, Reymond 1979). But Archimedes violates the prohibitions of Plato and Aristotle and is inspired by the philosophy of Democritus. It is argued about Archimedes’ sections-weights (toma…-b£rea) and Aristotle’s and Democritus’ indivisible magnitudes (¥toma megšqh). (Luria 1970, Mugler 1970, Ver Ecke 1959, Furley 1967).

This essay is meant to be a

contribution to a philosophical interpretation

of a letter from Archimedes to Eratosthenes.

In this paper our aim is to examine the special interest that some major seventeenth century philosophical figures have shown in the achievements of Archimedean geometry. Given the excellence of the work of Archimedes, it is to his work that philosophers like Descartes, Spinoza or Pascal have been referring in order either to clarify some points of their own philosophies, or to find a sound basis for the modern mathematical conception of nature.

The mutually positive interaction between Science and Technology is first reminded, and the early traces of such a crossfertilisation are sought in Ancient Greece. Subsequently, this phenomenon is examined during the Hellenistic period. Several technical achievements are found to be inspired by scientific knowledge, whereas Technology did offer to Science some practical ideas and, above all, lots of measuring devices. Within this Alexandrian spirit, Archimēdēs was educated and has produced his mathematical and engineering works. Some of his inventions, probably inspired by his own mathematical findings, are mentioned. A more detailed analysis is presented on the scientific bases of the archimēdean planetarium, admired by Cicero. Further on, the innovative views of Archimēdēs are presented on the hybrid demonstration of some geometrical theorems, via both mechanical and theoretical means. Besides, the strange view of Plutarch is critically examined, according to which Archimēdēs considered as “unworthy and vile” any activity related to machines. In conclusion, this assertion is found to be completely unsupported and arbitrary. Finally, the scientific rationality of the design of machines during the Italian Renaissance is mentioned as a confirmation of the validity of the crossfertilisation process.

Cicero’s rediscovery of Archimedes’ tomb shows the interest for the Sicilian scientist in Rome, even if in Italy Archimedes’ geometry was put into practice only by architects and by

Gromatici

, a sort of practical technicians who worked primarily in military and agricultural fields (we have some clear information about their work in a wonderful manuscript of the sixth century now in Wolfenbüttel). Some poets of the classical period were interested in the combination of numbers (like Catullus’ 5 or 7 and Virgil’s

Georgics

2), but they never did open references to Archimedes, for metrical difficulties and embarrassed by his astonishing killing during the Roman occupation of Syracuse. Archimedes’ life and death had an important part on the confluence of eastern and western culture in the third and second centuries B.C., but a good image of the scientist received serious obstacles by the difficulties of his theoretical works (Cicero also didn’t read and understand the mathematical and physical ones) and by his strong and open struggle against the Romans.

The short review of the most considerable Russian researches of engineering and scientific activity of Archimedes is given. Special attention is focused on Ivan Nikolayevich Veselovsky’s original research - one of the largest Russian experts in this area.

According to Richard Westfall (Westfall, 1977) the Scientific Revolution of the seventeenth century was dominated by two themes: the Platonic-pythagorean tradition “which looked on nature in geometric terms” and mechanical philosophy “which conceived of nature as a huge machine”. This paper is an attempt to study the appropriation of Archimedean science in the Scientific Revolution in Western Europe.

This paper is devoted to the questions connected with «Archimedes’ burning mirrors» myth. The scientific disputes, no and con arguments concerning these death mirrors or helio-concentrators are considered. Possibilities of current usage of helio-energetics on the basis of geometrical optics are presented.

The influence of Archimedes on the so-called

theatre of machines

books is reviewed using original manuscripts from the 15

th

to the 19

th

centuries. The evidence shows continuity of knowledge of ancient Greek theory of machines as well as Archimedes principles of statics, hydrostatics and concepts of centers of gravity on the development of machine science.

Archimedes (ca. 287–212 BC) was born in Syracuse, in the Greek colony of Sicily. He studied mathematics probably at the Museum in Alexandria. Archimedes made important contributions to the field of mathematics. Archimedes discovered fundamental theorems concerning the center of gravity of plane geometric shapes and solids. He is the founder of statics and of hydrostatics. Archimedes was both a great engineer and a great inventor, his machines fascinated subsequent writers, and he earned the honorary title “father of experimental science”. Archimedes systematized the design of simple machines and the study of their functions and developed a rigorous theory of levers and the kinematics of the screw. His works contain a set of concrete principles upon which mechanics could be developed as a science using mathematics and reason. His contribution separates engineering science from technology and crafts, often confused for matters arrived at empirically through a process of long evolution. His works have influenced science and engineering from the Byzantine period to the Industrial Revolution and the New Era.

In Akrai, a Syracusan archaic sub-colony, there is a rockrelief in

Intagliatella

’s urban latomy. In this paper, after a trip around the history of studies related to the monument, we propose a new interpretation on the ground of signs, drawn near its position to the town’s entrance, following relevant historical considerations about the figure of Hieron II (306–215 B.C.), the most important king of Hellenistic Syracuse, as well as a friend and protector of Archimedes (287–212 B.C.) who lived and worked in his court.

The data on all known editions of Archimedes’s works in Russian from the middle of XVIII till the middle of XX centuries and the most interesting comments to Archimedes’s works and translations of them are presented.

Steps in elaboration of teaching courses on the history of mechanics and mathematics are illustrated. The courses which are taught in Moscow University and in other universities in Russia, together with their respective textbooks, have always paid a great attention to Archimedes and his original manuscripts. The review of Archimedes’ works that are considered at the lectures on history of mechanics and mathematics in MSU, Department of Mechanics and Mathematics, is given and methodological problems connected with evaluation of Archimedes’ creative ability are discussed.

This article is focused on the way Archimedes’ works and inventions are covered in the academic curricula in Russia. The authors consider different education levels: from the elementary school to the higher technical education. Archimedes’ biography, his inventions and the laws discovered by him are described in a great number of educational literature, both for schoolchildren and for the students of technical universities. Archimedes’ works on hydrostatics, geometry, center of gravity, mechanics of simple machines (levers, pulleys) and the structure of the machines he created are described there.

The aim is to propose, at various levels in secondary schools, Archimedes’ idea for calculating π using the computer as programming tool. In this way, it will be possible to remember the work of one of the greatest geniuses in history and, at the same time, carry out an interdisciplinary project, particularly relevant to the current debate on the Math curriculum.

Archimedes’ genius was derived in no small part from his ability to effortlessly interpret problems in both geometric and mechanical ways. We explore, in a modern context, the application of mechanical reasoning to geometric problem solving. The general form of this inherently Archimedean approach is described and it’s specific use is demonstrated with regard to the problem of finding the geodesics of a surface. Archimedes’ approach to thinking about problems may be his greatest contribution, and in that spirit we present some work related to teaching Archimedes’ ideas at an elementary level. The aim is to cultivate the same sort of creative problem solving employed by Archimedes, in young students with nascent mechanical reasoning skills.

The conquest of Syracuse, which to begin with the Romans expected to lead to a speedy victory, soon turned into a long hard war thanks to Archimede’s extraordinary military defence machines. During the war the scientist was killed - probably by mistake - by a Roman soldier who infringed Marcello’s orders. The author analyses historical sources and outlines some discrepancies that give a totally different reading of the events, a reading more in agreement with Roman politics.