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Archimedes (Greek: Ἀ' χιμήδης) (c. 287 BC – 212 BC) was an ancient Greek mathematician, physicist, engineer, astronomer, and philosopher born in the seaport colony of Syracuse, Magna Graecia, what is now current day Sicily.

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Inttroduction

Many consider him one of the greatest, if not the greatest, mathematicians in antiquity. Carl Friedrich Gauss, himself frequently called the most influential mathematician of all time, modestly claimed that Archimedes was one of the three epoch-making mathematicians (the others being Isaac Newton and Ferdinand Eisenstein). Apart from his fundamental theoretical contributions to math, Archimedes also shaped the fields of physics and practical engineering, and has been called "the greatest scientist ever".

He was a relative of the Hiero monarchy, which was the ruling family of Syracuse (Saracussia), a seaport kingdom. King Hiero II, who was rumored to be Archimedes' uncle, commissioned him to design and fabricate a new class of ships for his navy, which were crucial for the preservation of the ruling class in Syracuse. Hiero II had promised large caches of grain to the Romans in the north in return for peace. Faced with war when unable to present the promised amount, Hiero II commissioned Archimedes to develop a large luxury/supply/war barge in order to serve the changing requirements of his navy. It is rumored that the Archimedes Screw was actually an invention of happenstance, as he needed a tool to remove bilge water. The ship, coined Saracussia, after its nation, may be mythical. There is no record on foundry art, nor any other period pieces depicting its creation. It is solely substaintiated by a description from Plato, who said "it was the grandest equation ever to sail."

He is credited with many inventions and discoveries, some of which we still use today, like his Archimedes screw. He was famous for his compound pulley, a system of pulleys used to lift heavy loads such as ships. He made several war machines for his patron and friend Hiero II. He did a lot of work in geometry, which included finding the surface areas and volumes of solids accurately. The work that has made Archimedes famous is his theory of floating bodies. He laid down the laws of flotation and developed the famous Archimedes principle.

Discoveries and inventions

Archimedes became a very popular figure as a result of his involvement in the defense of Syracuse against the Roman siege in the Second Punic War. He is reputed to have held the Romans at bay with war machines of his own design, to have been able to move a full-size ship complete with crew and cargo by pulling a single rope[1], and to have discovered the principles of density and buoyancy, also known as Archimedes' principle, while taking a bath. The story goes that he then took to the streets naked, being so elated with his discovery that he forgot to dress, crying "Eureka!" ("I have found it!"). He has also been credited with the possible invention of the odometer during the First Punic War. One of his inventions used for military defense of Syracuse against the invading Romans was the claw of Archimedes.

It is said that he prevented one Roman attack on Syracuse by using a large array of mirrors (speculated to have been highly polished shields) to reflect sunlight onto the attacking ships causing them to catch fire. This popular legend, dubbed the "Archimedes death ray", has been tested many times since the Renaissance and often discredited as it seemed the ships would have had to have been virtually motionless and very close to shore for them to ignite, an unlikely scenario during a battle. A group at MIT have performed their own tests and concluded that the mirror weapon was a possibility [2], although later tests of their system showed it to be ineffective in conditions that more closely matched the described siege [3]. The television show Mythbusters also took on the challenge of recreating the weapon and concluded that while it was possible to light a ship on fire, it would have to be stationary at a specified distance during the hottest part of a very bright, hot day, and would require several hundred troops carefully aiming mirrors while under attack. These unlikely conditions combined with the availability of other simpler methods, such as ballistae with flaming bolts, led the team to believe that the heat ray was far too impractical to be used, and probably just a myth.

It can be argued that even if the reflections didn't induce fire, they still could have confused, and temporarily blinded the ship crews, making it hard for them to aim and steer. Making them hot and sweaty before primary battle may have also tired them faster. The effectiveness may have simply been exaggerated. Archimedes also has been credited with improving accuracy, range and power of the catapult.

Archimedes was killed by a Roman soldier during the sack of Syracuse during the Second Punic War, despite orders from the Roman general Marcellus that he was not to be harmed. The Greeks said that he was killed while drawing an equation in the sand; engrossed in his diagram and impatient with being interrupted, he is said to have muttered his famous last words before being slain by an enraged Roman soldier: Μη μου τους κκλους τάαττε ("Don't disturb my circles"). The phrase is often given in Latin as "Noli turbare circulos meos" but there is no direct evidence that Archimedes ever uttered these words. This story was sometimes told to contrast the Greek high-mindedness with Roman ham-handedness; however, it should be noted that Archimedes designed the siege engines that devastated a substantial Roman invasion force, so his death may have been out of retribution [4].

In creativity and insight, Archimedes exceeded any other European mathematician prior to the European Renaissance. In a civilization with an awkward numeral system and a language in which "a myriad" (literally "ten thousand") meant "infinity", he invented a positional numeral system and used it to write numbers up to 1064. He devised a heuristic method based on statistics to do private calculations that we would classify today as integral calculus, but then presented rigorous geometric proofs for his results. To what extent he actually had a correct version of integral calculus is debatable. He proved that the ratio of a circle's circumference to its diameter is the same as the ratio of the circle's area to the square of the radius. He did not call this ratio π but he gave a procedure to approximate it to arbitrary accuracy and gave an approximation of it as between 3 + 10/71 (approximately 3.1408) and 3 + 1/7 (approximately 3.1429). He was the first Greek mathematician to introduce mechanical curves (those traced by a moving point) as legitimate objects of study. He proved that the area enclosed by a parabola and a straight line is 4/3 the area of a triangle with equal base and height. (See the illustration below. The "base" is any secant line, not necessarily orthogonal to the parabola's axis; "the same base" means the same "horizontal" component of the length of the base; "horizontal" means orthogonal to the axis. "Height" means the length of the segment parallel to the axis from the vertex to the base. The vertex must be so placed that the two horizontal distances mentioned in the illustration are equal.)

In the process, he calculated the earliest known example of a geometric progression summed to infinity with the ratio 1/4:

\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3} \; .

If the first term in this series is the area of the triangle in the illustration then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines in the illustration, and so on. Archimedes also gave a quite different proof of nearly the same proposition by a method using infinitesimals.

He proved that the area and volume of the sphere are in the same ratio to the area and volume of a circumscribed straight cylinder, a result he was so proud of that he made it his epitaph.

Archimedes is probably also the first mathematical physicist on record, and the best before Galileo and Newton. He invented the field of statics, enunciated the law of the lever, the law of equilibrium of fluids, and the law of buoyancy. (He famously discovered the latter when he was asked to determine whether a crown had been made of pure gold, or gold adulterated with silver; he realized that the rise in the water level when it was immersed would be equal to the volume of the crown, and the decrease in the weight of the crown would be in proportion; he could then compare those with the values of an equal weight of pure gold). He was the first to identify the concept of center of gravity, and he found the centers of gravity of various geometric figures, assuming uniform density in their interiors, including triangles, paraboloids, and hemispheres. Using only ancient Greek geometry, he also gave the equilibrium positions of floating sections of paraboloids as a function of their height, a feat that would be taxing to a modern physicist using calculus.

Apart from general physics, he was also an astronomer, and Cicero writes that the Roman consul Marcellus brought two devices back to Rome from the ransacked city of Syracuse. One device mapped the sky on a sphere and the other predicted the motions of the sun and the moon and the planets (i.e., an orrery). He credits Thales and Eudoxus for constructing these devices. For some time this was assumed to be a legend of doubtful nature, but the discovery of the Antikythera mechanism has changed the view of this issue, and it is indeed probable that Archimedes possessed and constructed such devices. Pappus of Alexandria writes that Archimedes had written a practical book on the construction of such spheres entitled On Sphere-Making.

Archimedes' works were not widely recognized, even in antiquity. He and his contemporaries probably constitute the peak of Greek mathematical rigour. During the Middle Ages the mathematicians who could understand Archimedes' work were few and far between. Many of his works were lost when the library of Alexandria was burnt (twice) and survived only in Latin or Arabic translations. As a result, his mechanical method was lost until around 1900, after the arithmetization of analysis had been carried out successfully. We can only speculate about the effect that the "method" would have had on the development of calculus had it been known in the 16th and 17th centuries.

Writings by Archimedes

Several of archimededs works were briefly inspected in Constantinople and was published, from photographs, by the Danish philologist Johan Ludvig Heiberg (1854–1928); shortly thereafter it was translated into English by Thomas Heath. His works include:

  • "Equilibrium of Planes"
  • "Spiral Lines"
  • "The Measurement of the Circle"
  • "Sphere and Cylinder"
  • "On Floating Bodies" (only known copy in Greek)
  • "The Method of Mechanical Theorems" (only known copy)
  • "Stomachion" (only known copy)

On the Equilibrium of Planes

2 volumes

This scroll explains the law of the lever and uses it to calculate the areas and centers of gravity of various geometric figures.

On Spirals

In this scroll, Archimedes defines what is now called Archimedes' spiral. This is the first mechanical curve (i.e., traced by a moving point) ever considered by a Greek mathematician.

On the Sphere and The Cylinder

In this scroll Archimedes obtains the result he was most proud of: the relation between the area of a sphere to that of a circumscribed straight cylinder is the same as that of the volume of the sphere to the volume of the cylinder (exactly 2/3).

On Conoids and Spheroids

In this scroll Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.

On Floating Bodies

2 volumes

In the first part of this scroll, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This was probably an attempt at explaining the observation made by Greek astronomers that the Earth is round. Note that his fluids are not self-gravitating: he assumes the existence of a point towards which all things fall and derives the spherical shape. One is led to wonder what he would have done had he struck upon the idea of universal gravitation. In the second part, a veritable tour-de-force, he calculates the equilibrium positions of sections of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, which is reminiscent of the way icebergs float, although Archimedes probably was not thinking of this application.

The Quadrature of the Parabola

In this scroll, Archimedes calculates the area of a segment of a parabola (the figure delimited by a parabola and a secant line not necessarily perpendicular to the axis). The final answer is obtained by triangulating the area and summing the geometric series with ratio 1/4.

Stomachion

This is a Greek puzzle similar to a Tangram. In this scroll, Archimedes calculates the areas of the various pieces. This may be the first reference we have to this game. Recent discoveries indicate that Archimedes was attempting to determine how many ways the strips of paper could be assembled into the shape of a square. This is possibly the first use of combinatorics to solve a problem.

Archimedes' Cattle Problem

Archimedes wrote a letter to the scholars in the Library of Alexandria, who apparently had downplayed the importance of Archimedes' works. In these letters, he dares them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations, some of them quadratic (in the more complicated version). This problem is one of the famous problems solved with the aid of a computer. The solution is a very large number, approximately Template:Sn (See the external links to the Cattle Problem.)

The Sand Reckoner

The Sand Reckoner (Greek: ψαμμιτης - psammites) is probably the most accessible work of Archimedes; in some sense, it is the first research-expository paper. In this work, Archimedes sets himself to challenge the then commonly held belief that the number of grains of sand is too large to count. In this scroll, Archimedes counts the number of grains of sand fitting inside the universe. This book mentions Aristarchus of Samos' theory of the solar system (concluding that "this is impossible"), contemporary ideas about the size of the Earth and the distance between various celestial bodies. From the introductory letter we also learn that Archimedes' father was an astronomer.

Archimedes first has to invent a system of naming large numbers in order to give an upper bound, and he does this by starting with the largest number around at the time, a myriad myriad or one hundred million (a myriad is 10,000). Archimedes' system goes up to

10^{8 \times 10^{16}}

which is a myriad myriad to the myriad myriadth power, all taken to the myriad myriadth power. Another way of describing this number is a one followed by (short scale) eighty quadrillion (8 * 1016) zeroes; compared to this number the otherwise enormous googol, or one followed by one hundred zeroes, seems paltry. This gives a good indication of the notational difficulties encountered by Archimedes, and one can propose that he stopped at this number because he did not devise any new ordinal numbers (larger than 'myriad myriadth') to match his new cardinal numbers. Archimedes also discovered and proved the law of exponents

10a10b = 10a + b

necessary to manipulate powers of 10. Archimedes then sets about estimating an upper bound for the number of grains of sand. Not wanting to be outdone, he counts not only the grains of sand on a beach, but on the entire earth, the earth filled with sand, and then in a universe filled with sand. He then estimates this for the largest model of the universe yet proposed, the heliocentric model of Aristarchus of Samos (in fact, this now lost work is known due to this reference). The reason for this is that a heliocentric model must be much larger if stellar parallax is not clearly measurable. Archimedes proceeds by giving upper bounds for the diameter of the earth, the distance from the earth to the sun, and the diameter of the universe. In order to do this last step, he assumes that the ratio of the diameter of the universe to the diameter of the orbit of the Earth around the Sun, equals the ratio of Earth's solar-orbital diameter to the diameter of the Sun. This simply says that stellar parallax equals solar parallax, and one can interpret this as Archimedes' reason for using this assumption, which is not clearly explained in the text. The resulting estimate is that the radius of the universe is about one light year, which is consistent with current estimates for the radius of the solar system. Archimedes' final estimate gives an upper bound of 1064 for the number of grains of sand in a filled universe.

Archimedes makes some interesting experiments and computations along the way. One experiment estimates the angular size of the sun, as seen from the earth. Archimedes' method is especially interesting as it may be the first known example of experimentation in psychophysics, the branch of psychology dealing with the mechanics of human perception, and whose development is generally attributed to Hermann von Helmholtz (this work of Archimedes is not well known in psychology). In particular, Archimedes takes into account the size and shape of the eye in his experiment measuring the angular diameter of the sun. Another interesting computation accounts for solar parallax, in particular, the differences in distance from the sun, whether taken from the center of the earth or from the surface of the earth at sunrise. Once again, this may be the first known computation dealing with solar parallax.

"The Method"

In this work, which was unknown in the Middle Ages, but the importance of which was realised after its discovery, Archimedes pioneers the use of infinitesimals, showing how breaking up a figure in an infinite number of infinitely small parts could be used to determine its area or volume. The Archimedes Palimpsest[5] is a palimpsest on parchment in the form of a codex which originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors. Archimedes lived in the third century BC, but the copy was made in the 10th century by an anonymous scribe. In the 12th century the codex was unbound and washed, in order that the parchment leaves could be folded in half and reused for a Christian liturgical text. It was a book of nearly 90 pages before being made a palimpsest of 177 pages; the older leaves folded so that each became two leaves of the liturgical book. The erasure was incomplete, and Archimedes' work is now readable using digital processing of ultraviolet, X-ray, and visible light. Archimedes probably considered these methods not mathematically precise, and he used these methods to find at least some of the areas or volumes he sought, and then used the more traditional method of exhaustion to prove them.

What Archimedes did

Although the only mathematical tools at its author's disposal were what we might now consider secondary-school geometry, Archimedes used those methods with rare brilliance, explicitly using infinitesimals to solve problems that would now be treated by integral calculus, which was independently reinvented in the 17th century by Isaac Newton and Gottfried Leibniz. Among those problems were that of calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. Contrary to exaggerations found in some 20th century calculus textbooks, he did not use anything like Riemann sums, either in the work embodied in this palimpsest or in any of his other works. For explicit details of the method used, see how Archimedes used infinitesimals. A problem solved exclusively in the Method is the calculation of the volume of a cylindrical wedge, a result that reappears as theorem XVII (schema XIX) of Kepler's Stereometria.

Some pages of the Method remained unused by the author of the Palimpsest and thus they are still—probably forever—lost. Between them, an announced result concerned the volume of the intersection of two cylinders, a figure that Apostol and Mnatsakian have renamed n = 4 Archimedean globe (and the half of it, n = 4 Archimedean dome), whose volume relates to the n-polygonal pyramid. This is amusing because the collaboration on indivisibles between Galileo and Cavalieri—ranging between years 1626 to around 1635—has as a main argument the hull and pyramid of the n = ∞ dome. So in some sense it is true that the Method is only a theorem behind the modern infinitesimal theory.

In Heiberg's time, much attention was paid to Archimedes' brilliant use of infinitesimals to solve problems about areas, volumes, and centers of gravity. Less attention was given to the Stomachion, a problem treated in the Palimpsest that appears to deal with a children's puzzle. Reviel Netz of Stanford University has shown that Archimedes found that the number of ways to solve the puzzle is 17,152. This is perhaps the most sophisticated work in the field of combinatorics in classical antiquity.

Use of infinitesimals

Archimedes was the first mathematician to make explicit use of infinitesimals. His work with infinitesimals is found in the celebrated Archimedes Palimpsest. The palimpsest embodies Archimedes' account of his "mechanical method", so called because it relies on the concepts of torque exerted on a lever and of center of gravity. Both of those concepts were first introduced by Archimedes. Ironically, Archimedes disbelieved in the existence of infinitesimals, and therefore said explicitly that his arguments fall short of being finished mathematical proofs. The proof of the first proposition in the palimpsest appears below.

The first proposition in the palimpsest

The curve in this figure is a parabola. It the points A and B are on the curve, the line AC is parallel to the axis of the parabola. The line BC is tangent to the parabola. The first proposition states:

The area of the triangle ABC is exactly three times the area bounded by the parabola and the secant line AB.

Proof: Let D be the midpoint of AC. The point D is the fulcrum of a lever, which is the line JB. The points J and B are equidistant from the fulcrum. As Archimedes had shown, the center of gravity of the interior of the triangle is at a point I on the "lever" so located that DI:DB = 1:3. Therefore, it suffices to show that if the whole weight of the interior of the triangle rests at I, and the whole weight of the section of the parabola at J, the lever is in equilibrium. If the whole weight of the triangle rests at I, it exerts the same torque on the lever as if the infinitely small weight of every cross-section EH parallel to the axis of the parabola rests at the point G where it intersects the lever. Therefore, it suffices to show that if the weight of that cross-section rests at G and the weight of the cross-section EF of the section of the parabola rests at J, then the lever is in equilibrium. In other words, it suffices to show that EF:GD = EH:JD. That is equivalent to EF:DG = EH:DB. And that is equivalent to EF:EH = AE:AB. But that is just the equation of the parabola. Q.E.D..

Other propositions in the palimpsest

A series of other propositions of geometry are proved in the palimpsest by similar arguments. Some of them have the location of a center of gravity as the conclusion. One of those states that the center of gravity of the interior of a hemisphere is located 5/8 of the way from the pole to the center of the sphere.

The Palimpsest's modern career

From the 1920s, the manuscript lay unknown in the Paris apartment of a collector of manuscripts and his heirs. In 1998 the ownership of the palimpsest was disputed in federal court in New York in the case of the Greek Orthodox Patriarchate of Jerusalem versus Christie's, Inc. At some time in the distant past, the Archimedes manuscript had lain in the library of Mar Saba, near Jerusalem, a monastery bought by the Patriarchate in 1625. The plaintiff contended that the palimpsest had been stolen from one of its monasteries in the 1920s. Judge Kimba Wood decided in favor of Christie's Auction House on laches grounds, and the palimpsest was bought for $2 million by an anonymous information technology person.

The palimpsest is now at the Walters Art Museum in Baltimore, where conservation continues (as it had suffered considerably from mould). A more accurate edition of the manuscript, including its drawn geometrical figures, is expected, possibly in 2007.

A team of imaging scientists from the Rochester Institute of Technology and Johns Hopkins University has used computer processing of digital images from various spectral bands, including ultraviolet and visible light, to reveal the Archimedes text. Dr. Reviel Netz [6] of Stanford University has been trying to fill in gaps in Heiberg's account with these images.

Four pages that had been painted over with Byzantine-style religious images, which turned out to be 20th-century forgeries intended to increase the value of the prayer book, rendered the underlying text of Archimedes forever illegible, it appeared. Then, in May 2005, highly focused X-rays produced at the Stanford Linear Accelerator Center in Menlo Park, California were used to begin deciphering the parts of the 174-page text that have not yet been revealed. The production of x-ray fluorescence was described by Keith Hodgson, director of SSRL. "Synchrotron light is created when electrons traveling near the speed of light take a curved path around a storage ring—emitting electromagnetic light in X-ray through infrared wavelengths. The resulting light beam has characteristics that make it ideal for revealing the intricate architecture and utility of many kinds of matter—in this case, the previously hidden work of one of the founding fathers of all science." [7].

Quotes

  • ε'ηκα. (Eureka!)
    • Translations: " I have found it!" or "I have got it!"
    • Said to be what he exclaimed as he ran from his bath (without clothes), realizing that by measuring the displacement of water an object produced, compared to its weight, he could measure its density (and thus determine the proportion of gold that was used in making a king's crown).
  • δος μοι που στω και κινω την γην (Dos moi pou sto kai kino taen gaen)
    • Doric Greek: δος μοι π' αν στω και τα γαν κινάσω
    • Translations: "Give me the place to stand, and I shall move the earth." or "Give me a place to stand, and I shall move the world." or "Give me a fulcrum, and I shall move the world."
    • Said to be his assertion in demonstrating the principle of the lever.
  • "Μη μου τους κ'κλους τά'αττε!" (in Greek)
  • "Noli turbare circulos meos."
  • "Noli tangere circulos meos"
    • Translation: "Do not disturb my circles!"
    • Comment: Uttered to a Roman soldier who, despite being given orders not to, killed Archimedes at the conquest of Syracuse.

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