Archimedes

gigatos | January 12, 2022

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Summary

Archimedes of Syracuse (Syracuse, c. 287 BC – Syracuse, 212 BC) was a Sicelian mathematician, physicist and inventor.

Considered as one of the greatest scientists and mathematicians in history, he contributed to the advancement of knowledge in areas ranging from geometry to hydrostatics, from optics to mechanics: he was able to calculate the surface area and volume of the sphere and formulated the laws governing the buoyancy of bodies; in the field of engineering, he discovered and exploited the operating principles of levers and his very name is associated with numerous machines and devices, such as the Archimedes screw, demonstrating his inventive ability; still surrounded by a halo of mystery are instead the war machines that Archimedes would have prepared to defend Syracuse from the Roman siege.

His life is remembered through numerous anecdotes, sometimes of uncertain origin, which have helped to build the figure of the scientist in the collective imagination. It has remained famous over the centuries, for example, the exclamation èureka! (εὕρηκα! – I found it!) attributed to him after the discovery of the principle of buoyancy of bodies that still bears his name.

Historical elements

There are few certain data on his life. All sources agree that he was Syracusan and that he was killed during the Roman sack of Syracuse in 212 BC. There is also the news, handed down by Diodorus Siculus, who has stayed in Egypt and in Alexandria of Egypt has made friends with the mathematician and astronomer Conon of Samos. Most likely it was not really so: the scientist would have wanted to get in touch with the scholars of the time belonging to the school of Alexandria, to whom he sent many of his writings. During this hypothetical stay, Archimedes would have invented the “hydraulic screw”.

The only certain thing is that he was really in contact with Conon (as can be seen from the regret for his death expressed in some works) but may have known in Sicily. He held correspondence with various scientists of Alexandria, including Eratosthenes, to whom he dedicated the treatise The method and Dositeo. A good example that has come to us on the collaboration between the scientist and Alexandrians is the letter of introduction to the treatise On spirals.

According to Plutarch was related to the monarch Hieron II. The thesis is controversial, but is reflected in the close friendship and esteem that, according to other authors, linked them. The date of birth is not certain. It is usually accepted that of 287 BC, on the basis of information, reported by the Byzantine scholar John Tzetzes, who had died at the age of seventy-five years. It is not known, however, if Tzetzes was based on reliable sources now lost or had only tried to quantify the data, reported by various authors, that Archimedes was old at the time of the killing. The hypothesis that he was the son of a Syracusan astronomer named Phidias (otherwise unknown) is based on the reconstruction of a sentence of Archimedes made by the philologist Friedrich Blass, contained in the Arenarius, which in the manuscripts had come corrupted and meaningless. If this hypothesis is correct, it can be assumed that he inherited from his father a love for the exact sciences.

From the preserved works and testimonies we know that he dealt with all the branches of science contemporary to him (arithmetic, plane and solid geometry, mechanics, optics, hydrostatics, astronomy, etc.) and various technological applications.

Polybius, report that during the Second Punic War, at the request of Hieron II, he devoted himself (according to Plutarch with less enthusiasm but according to all three with great success) to the construction of war machines that would help his city to defend itself from the attack of Rome. Plutarch recounts that, against the legions and the powerful fleet of Rome, Syracuse had a few thousand men and the genius of an old man; the machines of Archimedes would have hurled cyclopean boulders and a storm of iron against the sixty imposing quinqueremi of Marcus Claudius Marcellus. He was killed in 212 BC, during the sack of Syracuse. According to tradition, the killer would have been a Roman soldier who, not having recognized, would not follow the order to capture him alive.

Archimedes enjoyed high esteem both in his country, in fact was a reference for King Hieron, both in Alexandria, where he had a correspondence with the most illustrious mathematicians of his time, both among the Romans, so much so that according to legend had been ordered to capture him alive (but was killed). The Roman commander built a tomb in his honor.

The figure of Archimedes fascinated his contemporaries to the point that in the time the biographical events are thickly interlaced to the legends and it is still difficult to distinguish the elements of fiction from the historical reality. To the lack of testimonies is added the fact that Archimedes wrote only works of theoretical and speculative character.

Two famous anecdotes

In the collective imagination Archimedes is inextricably linked to two anecdotes. Vitruvius tells that he would have begun to deal with hydrostatics because the king Hieron II had asked him to determine whether a crown was made of pure gold or using (inside the crown) other metals. He would have discovered how to solve the problem while taking a bath, noting that immersing in the water occurred the rise of its level. The observation would have made him so happy that he would have left the house naked and run through the streets of Syracuse exclaiming “εὕρηκα” (èureka!, I found it!). Had we not been aware of the treatise On Floating Bodies, we could not have deduced the level of Archimedean hydrostatics from the Vitruvian account.

Vitruvius reports that the problem would have been solved by measuring the volumes of the crown and an equal weight of gold by immersing them in a container filled with water and measuring the overflowed water. However, this is an implausible procedure, both because it involves an error too great, and because it has no relation with the hydrostatics developed by Archimedes. According to a more reliable reconstruction, attested in late antiquity, Archimedes had suggested to weigh the crown and a quantity of gold equal in weight both immersed in water. If the crown had been pure gold, the balance would have been in equilibrium. However, since the balance was lowered on the side of the gold, it could be deduced that, since the weights were equal, the crown had undergone a greater hydrostatic upward thrust, and therefore must have had a greater volume, which implied that it must have been manufactured using other metals, since these metals (such as silver) had a lower density than gold.

According to another equally famous anecdote, Archimedes (or Hieron) was able to move a ship thanks to a machine he invented. Exalted by the ability to build machines that could move large weights with small forces, in this or another occasion would have exclaimed: “Give me a point of support and I will raise the Earth. The sentence is reported, with small variations, by various authors, including Pappo of Alexandria

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Death Legends

The legend has handed down to posterity also the last words of Archimedes, addressed to the soldier who was about to kill him: “noli, obsecro, istum disturbare” (don”t ruin, please, this drawing). three different versions of the death of Archimedes.

In the first one he states that a Roman soldier would have ordered Archimedes to follow him to Marcellus; at his refusal the soldier would have killed him.

In the second, a Roman soldier showed up to kill Archimedes and the latter begged him in vain to let him finish the demonstration in which he was engaged.

In the third one, some soldiers would have met Archimedes while he was bringing to Marcellus some scientific instruments, sundials, spheres and squares, in a box; thinking that the box contained gold, the soldiers would have killed him in order to take possession of it.

According to Titus Livius Marcellus, who would have known and appreciated the immense value of the genius of Archimedes and perhaps would have wanted to use it in the service of the Republic, would have been deeply saddened by his death. These authors say that he gave an honorable burial to the scientist. However, this is not reported by Polybius, which is considered a more authoritative source on the siege and sacking of Syracuse.

Cicero tells of having discovered the tomb of Archimedes thanks to a sphere inscribed in a cylinder, which would have been carved in compliance with the will of the scientist.

War devices

Archimedes owes much of his popularity to his contribution to the defense of Syracuse against the Roman siege during the Second Punic War. Polybius, Livy, and Plutarch describe war machines of his invention, including the manus ferrea, a mechanical claw capable of capsizing enemy vessels, and jet weapons perfected by him.

In the 2nd century, the writer Lucian of Samosata reported that during the siege of Syracuse (ca. 214-212 BC), Archimedes destroyed enemy ships with fire. Centuries later, Antemius of Tralles mentions “lenses with fire” as weapons designed by Archimedes. The instrument, called “Archimedes” burning mirrors,” was designed for the purpose of concentrating sunlight on approaching ships, causing them to catch fire.

This hypothetical weapon has been the subject of debate about its veracity since the Renaissance. René Descartes believed it to be false, while modern researchers have attempted to recreate the effect using the only means available to Archimedes. It has been hypothesized that a large array of polished bronze or copper shields had been used as mirrors to focus sunlight onto a ship. This would have used the principle of parabolic reflection in a manner similar to a solar furnace.

An experiment to test Archimedes” burning mirrors was performed in 1973 by Greek scientist Ioannis Sakkas. The experiment took place at the Skaramagas naval base outside Athens. On this occasion, 70 mirrors were used, each with a copper coating and with a size of about 1.5 meters. The mirrors were pointed at a reproduction made of plywood of a Roman warship at a distance of about 50m. When the mirrors focused the sun”s rays accurately, the ship caught fire within seconds. The model had a coating of tar paint that may have aided combustion. Such a coating would have been common on ships of that era.

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Syracuse

Moschione, in a work of which Athenaeus reports extensive excerpts, describes a huge ship wanted by King Hieron II and built by Archia of Corinth The boat, the most impressive of antiquity, was called Syracuse. The name was changed to that of Alexandria when it was sent as a gift to King Ptolemy III of Egypt along with a load of grain, to demonstrate the wealth of the Sicilian city. For this boat, Archimedes adopted an instrument, the cochlea, which allowed to pump water out of the holds, keeping them dry.

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Water Clock

An Arabic manuscript contains a description of an ingenious water clock designed by Archimedes. In the clock, the flow of outgoing water was kept constant by the introduction of a floating valve.

The clock consisted of two tanks, one elevated above the other. The higher one was equipped with a faucet that delivered a constant flow of water into the lower basin.

Above the lower basin was placed a revolving axle to which was coiled a wire to whose ends were tied a small stone and a float.

At the beginning of the day, the bottom tank had to be empty and the line was pulled down so that the float would touch the bottom and the stone would rise to the top.

By opening the faucet, the lower basin would begin to fill by raising the float and lowering the stone.The length of the wire and the flow of water were calibrated so that it was 12 noon when the float was at the height of the stone and 6 p.m. when the stone was at the bottom.

Archimedes faced the problem of maintaining a constant flow from the tap: in fact, emptying the upper tank, the water pressure was reduced and the flow decreased. Then he added, higher than the first two, a third tank that, through a float, filled the second one to keep constant the level and therefore the pressure with which the water came out of the tap.

A merit that today is recognized to Archimedes is also to have been the first to interpret time as a physical quantity that can be analyzed with the mathematical tools used for geometric quantities (for example in the treatise On spirals he represents time intervals with segments and applies to them Euclid”s theory of proportions).

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Mechanical inventions

Athenaeus, tell that Archimedes had designed a machine with which a single man could move a ship with crew and cargo. In Athenaeus the episode is referred to the launching of the Syracuse, while Plutarch speaks of a demonstrative experiment, performed to show the sovereign the possibilities of mechanics. These accounts undoubtedly contain exaggeration, but the fact that Archimedes had developed the mechanical theory that allowed the construction of machines with high mechanical advantage ensures that they were born from a real basis.

According to the testimony of Athenaeus he had invented the mechanism for pumping water, used for the irrigation of cultivated fields, known as the screw of Archimedes.

Technological historian Andre W. Sleeswyk also attributed the odometer, described by Vitruvius, to Archimedes.

The Architronito, described by Leonardo da Vinci, was a steam cannon whose invention dates back to Archimedes of Syracuse around 200 BC. It is thought that the machine was used in the siege of Syracuse in 212 BC and in 49 BC as attested by Julius Caesar during the siege of Marseille.

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The planetarium

One of the most admired achievements of Archimedes in antiquity was the planetarium. The best information on this device are provided by Cicero, who writes that in 212 BC, when Syracuse was sacked by Roman troops, the consul Marcus Claudius Marcellus brought to Rome a device built by Archimedes that reproduced the vault of the sky on a sphere and another that predicted the apparent motion of the sun, moon and planets, equivalent to a modern armillary sphere. Cicero, reporting the impressions of Gaius Sulpicius Gallus, who had been able to observe the extraordinary object, emphasizes how the genius of Archimedes had managed to generate the motions of the planets, so different from each other, starting from a single rotation. It is known thanks to Pappo that Archimedes had described the construction of the planetarium in the lost work On the Construction of the Spheres.

The discovery of the Antikythera machine, a gear device that according to some researches dates back to the second half of the IInd century B.C., demonstrating how elaborate were the mechanisms built to represent the motion of the stars, has rekindled interest in Archimedes” planetarium. A gear that can be identified as belonging to the planetarium of Archimedes was found in July 2006 in Olbia; the studies on the find were presented to the public in December 2008. According to a reconstruction the planetarium, which would have passed to the descendants of the conqueror of Syracuse, could have been lost in the subsoil of Olbia (probable port of call of the journey) before the shipwreck of the ship carrying Marcus Claudius Marcellus (consul 166 BC) in Numidia.

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Pupil diameter measurement

In the Arenarius (book I, chap. 13), after having mentioned a method to proceed to the angular measurement of the Sun using a graduated ruler on which he placed a small cylinder, Archimedes notes that the angle thus formed (vertex in the eye and tangent lines to the edges of the cylinder and of the Sun) does not express a correct measure because the size of the pupil is not yet known. Therefore, by placing a second cylinder of a different color and placing the eye in a position further back than the end of the ruler, we obtain in this way the average diameter of the pupil and, consequently, a more accurate estimate of the diameter of the Sun. The even brief discussion on the subject suggests that Archimedes, rather than referring to the Euclidean writings, in this case also took into account the studies of Herophilus of Chalcedon, who had devoted several writings to the composition of the eye, all entirely lost and known only for the quotations that Galen makes.

Archimedes” scientific achievements can be exposed by first describing the contents of the preserved works and then the evidence about the lost works.

Preserved works

Already in the Bible it was suggested that the ratio of the semicircle to the radius was about 3 and this approximation was universally accepted.

In the short work La misura del cerchio (The measure of the circle), Archimedes first shows that a circle is equivalent to a triangle with base equal in length to the circumference and height equal to the radius. This result is obtained by approximating the circle, from inside and outside, with regular polygons inscribed and circumscribed. With the same procedure Archimedes exposes a method by which he can approximate as much as possible the ratio, which today is indicated by π, between the length of a circle and the diameter of a given circle. The estimates obtained limit this value between 227 (about 3.1429) and 22371 (about 3.1408).

In the work Quadrature of the parabola (which Archimedes dedicated to Dositeo) is calculated the area of a parabola segment, figure bounded by a parabola and a secant line, not necessarily orthogonal to the axis of the parabola, finding that it is worth 43 of the area of the maximum triangle inscribed in it.

It is shown that the maximum inscribed triangle can be obtained by a given procedure. The segment of the secant between the two points of intersection is called the base of the parabola segment. Consider the lines parallel to the axis of the parabola passing through the extremes of the base. Then a third line parallel to the first two and equidistant from them is drawn.

The intersection of the latter line with the parabola determines the third vertex of the triangle. By subtracting the maximum inscribed triangle from the parabola segment, two new parabola segments are obtained, in which two new triangles can be inscribed. The parabola segment is filled with an infinite number of triangles.

The required area is obtained by calculating the areas of the triangles and summing the infinite terms obtained. The final step reduces to summing the geometric series of reason 14:

This is the first known example of the sum of a series. At the beginning of the work, what is now called Archimedes” Axiom is introduced.

Given a parabola segment bounded by the secant AC, inscribe a first maximal triangle ABC.

2 more triangles ADB and BEC are inscribed in the 2 parabola segments AB and BC.

Continue in the same way for the 4 parabola segments AD, DB, BE and EC forming the triangles AFD, DGB, BHE and EIC.

Using the properties of the parabola, we show that the area of triangle ABC is equal to 4 times the area of ADB + BEC and that:ADB+BEC=4(AFD+DGB+BHE+EIC)}

Each step adds 14 of the area of the previous triangle to the area of the triangle.

At this point it is sufficient to show that the polygon that is constructed in this way actually approximates the parabola segment and that the sum of the series of areas of the triangles is equal to 43 of the first triangle.

On the equilibrium of the planes or on the centers of gravity of the planes, work in two books, is the first treatise of statics that has come down to us. Archimedes enunciates a set of postulates on which he based the new science and demonstrates the law of the lever. The postulates also define, implicitly, the concept of barycenter, whose position is determined in the case of different plane geometric figures.

In On Spirals, which is among his major works, Archimedes defines with a kinematic method what today is called Archimedes spiral and obtains two results of great importance. First, he calculates the area of the first turn of the spiral, with a method that anticipates the integration of Riemann. Archimedes” definition of the spiral: a straight line with a fixed end rotates uniformly, on which a point moves with uniform motion: the curve described by this point will be the spiral.

The main results of Della sfera e del cilindro, a work in two books, are that the area of the surface of the sphere is four times the area of its maximum circle and that the volume of the sphere is two-thirds of the volume of the circumscribed cylinder.

According to a tradition handed down by Plutarch and Cicero, Archimedes was so proud of this last achievement that he wanted it to be reproduced as an epitaph on his tomb.

In the work On conoids and spheroids Archimedes defines ellipsoids, paraboloids and hyperboloids of rotation, considers segments obtained by sectioning these figures with planes and calculates their volumes.

On floating bodies is one of the main works of Archimedes, with it is founded the science of hydrostatics. In the first of the two books of the work a postulate is stated from which is deduced as a theorem what today is improperly called the principle of Archimedes. In addition to calculating the positions of static equilibrium of the floats, it is shown that in conditions of equilibrium the water of the oceans takes a spherical shape. Since the time of Parmenides Greek astronomers knew that the Earth had a spherical shape, but here for the first time it is deduced from physical principles.

The second book studies the equilibrium stability of floating paraboloid segments. The problem was chosen for the interest of its applications to naval technology, but the solution also has great mathematical interest. Archimedes studies stability as two parameters vary, a shape parameter and density, and determines threshold values of both parameters that separate stable from unstable configurations. For E.J. Dijksterhuis, these are results “definitely beyond the boundary of classical mathematics.”

In Arenario (see link at the bottom for the Italian translation), addressed to Gelone II, Archimedes proposes to determine the number of grains of sand that could fill the sphere of the fixed stars. The problem arises from the greek system of numeration, which does not allow to express such large numbers. The work, although the simplest from the point of view of mathematical techniques among those of Archimedes, has several reasons of interest. First of all it introduces a new numerical system, which virtually allows to generate numbers however large. The largest number named is the one that today is written 108-1016. The astronomical context then justifies two important digressions. The first relates the heliocentric theory of Aristarchus and is the main source on the subject; the second describes an accurate measurement of the apparent magnitude of the Sun, providing a rare illustration of the ancient experimental method. It should be noted, however, that the challenge to the Aristarchan heliocentric theses is primarily geometric, not astronomical, because even assuming in fact that the cosmos is a sphere with the Earth at its center, Archimedes points out that the center of the sphere has no magnitude and can have no relationship to the surface; Book I, ch. 6.

From the scientific point of view, the demonstrations proposed by Archimedes on the levers, are quite innovative. In fact, the Sicelian scientist adopts a rigorously deductive method based on the mechanics of the equilibrium of solid bodies. To do so, he demonstrates his theses and concepts of equilibrium and barycentre by means of the theory of proportions and with geometrical terms. From these studies was postulated the 1st law of equilibrium of the lever:

Starting from the idea of a balance, composed of a segment and a fulcrum, from which two bodies are hung in equilibrium, it can be said that the weight of the two bodies is directly proportional to the area and volume of the bodies.According to the legend Archimedes would have said: “Give me a lever and I will lift the world” after discovering the second law of levers. By using advantageous levers, in fact, heavy loads can be lifted with a small application force, according to the law:

P:R=bR:bP{displaystyle P:R=b_{R}:b_{P}}

where P{displaystyle P} is the power and R{displaystyle R} is the resistance, and bP{displaystyle b_{P}} and bR{displaystyle b_{R}} are the respective action arms.

The short work The Method on Mechanical Problems, lost at least since the Middle Ages, was first read in the famous palimpsest found by Heiberg in 1906, then lost again, probably stolen by a monk during a manuscript transfer, and found again in 1998. It allows us to penetrate the procedures used by Archimedes in his research. Addressing Eratosthenes, he explains that he used two methods in his work.

Once he had identified the result, to formally prove it he used what was later called the method of exhaustion, of which there are many examples in his other works. However, this method did not provide a key to identify the results. For this purpose Archimedes used a “mechanical method”, based on his statics and the idea of dividing figures into an infinite number of infinitesimal parts. Archimedes considered this method not rigorous but, to the benefit of other mathematicians, he provides examples of its heuristic value in finding areas and volumes; for example, the mechanical method is used to find the area of a parabola segment.

The method also has philosophical connotations as it poses the problem of considering, as a necessary constraint, the application of mathematics to physics. Archimedes used his intuition to obtain immediate and innovative mechanical results, but then undertook to demonstrate them rigorously from a geometric point of view.

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Fragments and testimonies of lost works

The stomachion is a greek puzzle similar to the tangram, to which Archimedes dedicated a work of which two fragments remain, one in Arabic translation, the other contained in the Palimpsest of Archimedes. Analyses carried out in the early 2000”s have allowed to read new portions, which clarify that Archimedes proposed to determine in how many ways the component figures could be assembled in the form of a square. It is a difficult problem in which the combinatorial aspects intertwine with the geometric ones.

The oxen problem consists of two manuscripts that present an epigram in which Archimedes challenges Alexandrian mathematicians to calculate the number of oxen and cows in the Armenti del Sole by solving a system of eight linear equations with two quadratic conditions. This is a diophantine problem expressed in simple terms, but its smallest solution consists of numbers with 206 545 digits.

The question has been faced under a different point of view in 1975 by Keith G. Calkins, subsequently resumed in 2004 by Umberto Bartocci and Maria Cristina Vipera, two mathematicians of the University of Perugia. It is hypothesized that a “small” error of translation of the text of the problem has made “impossible” (some argue that such was the intention of Archimedes) a question that, formulated in a slightly different way, would have been instead addressable with the methods of mathematics of the time.

According to Calogero Savarino, this is not a translation error of the text, but a misinterpretation, or a combination of the two possibilities.

The Book of lemmas has come down through a corrupted Arabic text. It contains a number of geometric lemmas whose interest is diminished by today”s ignorance of the context in which they were used.

Archimedes had written Catoctrica, a treatise, of which we have indirect information, on the reflection of light. Apuleius claims that it was a voluminous work that dealt, among other things, with the magnification obtained with curved mirrors, burning mirrors and the rainbow. According to Olympiodorus the Younger there was also studied the phenomenon of refraction. A scholiosis to the pseudo-Euclidean Catotrics attributes to Archimedes the deduction of the laws of reflection from the principle of reversibility of the optical path; it is logical to think that in this work there was also this result.

In a lost work, of which Pappo gives information, Archimedes had described the construction of thirteen semiregular polyhedra, which are still called Archimedean polyhedra (in modern terminology the Archimedean polyhedra are fifteen because there are also two polyhedra that Archimedes had not considered, those improperly called Archimedean prism and Archimedean antiprism).

The formula of Hero, which expresses the area of a triangle from the sides, is so called because it is contained in the Metrics of Hero of Alexandria, but according to the testimony of al-Biruni the real author would be Archimedes, who would have exposed it in another lost work. The demonstration transmitted by Heron is particularly interesting because a square is squared, a strange procedure in Greek mathematics, since the entity obtained is not representable in three-dimensional space.

Thābit ibn Qurra presents as Book of Archimedes an Arabic text translated by J. Tropfke. Among the theorems contained in this work appears the construction of a regular heptagon, a problem not solvable with a ruler and compasses.

A passage from Hipparchus citing Archimedes” determinations of the solstices, transmitted by Ptolemy, suggests that he had also written works on astronomy. Pappus, Heron, and Simplicius attribute to him various treatises on mechanics, and several titles of works on geometry are transmitted by Arabic authors. The book on the construction of a mechanical water clock, preserved only in Arabic translation and attributed to the pseudo-Archimedes, is actually probably the work of Philo of Byzantium.

The Palimpsest of Archimedes is a medieval parchment codex, containing in the underlying writing some works of the scientist from Syracuse. In 1906, the Danish professor Johan Ludvig Heiberg examining in Constantinople 177 sheets of goatskin parchment, containing prayers of the thirteenth century (the palimpsest), discovered that there were previously writings of Archimedes. According to a very common practice at the time, due to the high cost of parchment, sheets already written were scraped to rewrite other texts, reusing the support. We know the name of the author of the destruction: Johannes Myronas, who finished the rewriting of the prayers on April 14, 1229. The palimpsest spent hundreds of years in a monastery library in Constantinople before being stolen and sold to a private collector in 1920. On October 29, 1998, it was sold at auction by Christie”s in New York to an anonymous buyer for $2 million.

The codex contains seven treatises by Archimedes, including the only surviving copy in Greek (Byzantine) of On Floating Bodies and the only one of the Method of Mechanical Theorems, named in the Suida, which was thought to have been lost forever. The Stomachion has also been identified in the pages, with more precise analysis. The palimpsest was studied at the Walters Art Museum in Baltimore, Maryland, where it underwent a series of modern tests, including the use of ultraviolet and X-rays in order to read the underlying text. Upon completion of the work, Reviel Netz, William Noel, Natalie Tchernetska, and Nigel Wilson published The Archimedes Palimpsest (2011) in two volumes: the first volume is primarily codicological, describing the manuscripts, their histories, the techniques used in their recovery, and the presentation of the texts; the second volume contains, in side-by-side pages, the photographed spread page of the codex with the transcription of the Greek text and the English translation. The pages of the palimpsest are available on the web as photographic images, but are almost impossible to read.

Archimedes” treatises contained in the Palimpsest are: On the Balance of Planes, On Spirals, Measurement of a Circle, On the Sphere and Cylinder, On Floating Bodies, Method of Mechanical Theorems, and Stomachion. The palimpsest still contains two orations of Hyperides (Against Dionda and Against Timander), a commentary on the Categories of Aristotle (probably a part of the commentary Ad Gedalium of Porphyry) and, by unknown authors, a Life of St. Pantaleon, two other texts and a Menaion, a text of the Eastern Church for holidays not dependent on Easter.

Indeed, the compelling story of the palimpsest is only one aspect of the tradition of the corpus of Archimedes” works, that is, the process by which his works have come down to us.

We must begin by noting that even in Antiquity his most advanced texts were not highly regarded, to the point that Eutocius (VI century AD) seems to have known neither the Quadrature of the Parabola nor the Spirals. At the time of Eutocius, in fact, it seems that only the two books of On the Sphere and the Cylinder, the Measure of the Circle and the two books of Equilibrium of the Planes were in circulation. In fact, the Arabs do not seem to have known much more or different from the work of Archimedes, so that in the Latin Middle Ages the only Archimedean text in circulation will be various versions of the Measure of the circle translated from Arabic.

The situation in the Greek world was different: in the ninth century, thanks to Leo the mathematician, at least three codices containing works of Archimedes were set up in Constantinople: codex A, codex ฿ (b ”gothic”) and codex C, the one destined to become a palimpsest in the eleventh century. A and ฿ were found in the second half of the thirteenth century in the library of the papal court of Viterbo: William of Moerbeke used them for his translation of the work of Archimedes made in 1269. William”s translation is today preserved in the ms. Ottob. Lat. 1850 of the Vatican Library where it was discovered by Valentin Rose in 1882. The codex ฿ (which was the only one, besides codex C, to contain the Greek text of the Floats) was lost after 1311. Codex A had a different fate: in the course of the fifteenth century it came into the possession of Cardinal Bessarione, who made a copy of it, now preserved in the Biblioteca Nazionale Marciana in Venice; then of the humanist Giorgio Valla from Piacenza, who published some brief excerpts of Eutocius” commentary in his encyclopedia De expetendis et fugiendis rebus opus, published posthumously in Venice in 1501. Copied several other times, codex A ended up in the possession of Cardinal Rodolfo Pio; sold at his death (1564) it has not been traced since.

However, the numerous copies that remain of it (and in particular the ms. Laurenziano XXVIII,4, which Poliziano had copied for Lorenzo de Medici with absolute fidelity to the ancient ninth-century model) have allowed the great Danish philologist Johan Ludvig Heiberg to reconstruct this important lost codex (Heiberg”s definitive edition of the corpus is dated 1910-15).

The translation carried out in the middle of the fifteenth century by Iacopo da San Cassiano deserves a separate discussion. In the wake of Heiberg, up to this point it was believed that Iacopo had translated using codex A. More recent studies have shown, however, that Iacopo used a model independent of A. His translation thus constitutes a fourth branch of the Archimedean tradition, together with A, ฿, and the palimpsest C.

The work of Archimedes represents one of the high points of the development of science in antiquity. In it, the ability to identify sets of postulates useful to found new theories is combined with the power and originality of the mathematical tools introduced, with a greater interest in the foundations of science and mathematics. Plutarch in fact tells that Archimedes was convinced by King Hieron to devote himself to the more applied aspects and to build machines, mainly of warlike character, to help more concretely the development and security of society. Archimedes dedicated himself to mathematics, physics and engineering, at a time when the divisions between these disciplines were not as clear-cut as today, but in which, however, according to Platonic philosophy, mathematics had to have an abstract character and not applied as in his inventions. The works of Archimedes thus constituted for the first time an important application of the laws of geometry to physics, in particular to statics and hydrostatics.

In antiquity, Archimedes and his inventions were described with wonder and amazement by classical Greek and Latin authors, such as Cicero, Plutarch and Seneca. Thanks to these accounts in the late Middle Ages and early modern era, a great interest moved the research and recovery of the works of Archimedes, transmitted and sometimes lost during the Middle Ages by manuscript. Roman culture was thus mostly impressed by Archimedes” machines rather than by his mathematical and geometrical studies, to the point that the historian of mathematics Carl Benjamin Boyer went so far as to make the more than pungent claim that Cicero”s discovery of Archimedes” tomb was the greatest contribution, perhaps the only one, made to mathematics by the Roman world.

Piero della Francesca, Stevino, Galileo, Kepler, and others up to Newton, studied, resumed and extended in a systematic way the scientific studies of Archimedes, in particular regarding the infinitesimal calculus.

The introduction of the modern scientific method of study and verification of the results obtained was inspired by Galileo to the method with which Archimedes carried out and demonstrated his intuitions. Moreover, the Pisan scientist found the way to apply geometric methods similar to Archimedes” to describe the accelerated motion of falling bodies, finally succeeding in overcoming the description of physics of static bodies developed by the scientist from Syracuse. Galileo himself in his writings called Archimedes “my master”, so much was the veneration for his work and his legacy.

The study of the works of Archimedes, therefore, engaged scholars of the early modern age for a long time and constituted an important stimulus to the development of science as it is understood today. The influence of Archimedes in the last centuries (for example, on the development of a rigorous mathematical analysis) is the object of discordant evaluations by scholars.

Art

In Raphael Sanzio”s famous fresco, The School of Athens, Archimedes is drawn intent on studying geometry. His likeness is by Donato Bramante.

The German poet Schiller wrote the poem Archimedes and the Young Man.

The effigy of Archimedes also appears on stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).

The Italian progressive rock band, Premiata Forneria Marconi within the album States of Imagination has dedicated the last song to the scientist with the title Visions of Archimedes in which the video traces the life and his inventions.

Archimedes is the protagonist of the novel Il matematico che sfidò Roma by Francesco Grasso (Edizioni 0111, Varese, 2014).

Science

On March 14 is celebrated all over the world the pi day, as in Anglo-Saxon countries corresponds to 314. On that day are organized mathematical competitions and also remembered the contributions of Archimedes, who gave the first accurate estimate of pi. In honor of Archimedes both the lunar crater Archimedes and the asteroid 3600 Archimedes were named.

In the Fields medal, the highest honor for mathematicians, on the reverse side of the medal there is a portrait of Archimedes with an inscription attributed to him: Transire suum pectus mundoque potiri, a transliteration of which can be as follows: “Elevate yourself above yourself and conquer the world.

Technology

The Archimede solar car 1.0, a solar powered car, has been designed and built in Sicily.

The Archimede Project has been realized, a solar power plant near Priolo Gargallo that uses a series of mirrors to produce electricity.

Also read, biographies – Theodor Mommsen

Museums and monuments

In Syracuse, a statue was erected in honor of the scientist and the Technopark Archimedes, an area in which inventions were reproduced.

Another statue of Archimedes is at Treptower Park in Berlin.

At Archea Olympia in Greece there is a museum dedicated to Archimedes.

Sources