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Archimedes repelled an attack by Roman forces with a burning glass. The device was used to focus sunlight on to the approaching ships, causing them to catch fire. This claim, sometimes called the "Archimedes death ray", has been the subject of ongoing debate about its credibility since the Renaissance. René Descartes rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes.

It has been suggested that a large array of highly polished bronze or copper shields acting as mirrors could have been employed to focus sunlight on to a ship. This would have used the principle of the parabolic reflector in a manner similar to a solar furnace.

In October 2005 a group of students from the Massachusetts Institute of Technology carried out an experiment with 127 one-foot (30 cm) square mirror tiles, focused on a mocked-up wooden ship at a range of around 100 feet (30 m). Flames broke out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes. It was concluded that the weapon was a feasible device under these conditions. The MIT group repeated the experiment for the television show MythBusters, using a wooden fishing boat in San Francisco as the target. Again some charring occurred, along with a small amount of flame. In order to catch fire, wood needs to reach its flash point, which is around 300 degrees Celsius (572 degrees F), and this is hotter than the maximum temperature produced by a domestic oven. When Mythbusters broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (or "failed") due to of the length of time and ideal weather conditions required for combustion to occur.

A similar test of the "Archimedes death ray" was carried out in 1973 by the 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 a size of around five by three feet (1.5 by 1 m). The mirrors were pointed at a plywood mock-up of a Roman warship at a distance of around 160 feet (50 m). When the mirrors were focused accurately, the ship burst into flames within a few seconds. The plywood ship had a coating of tar paint, which is flammable and may have aided combustion.

Mathematics

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch wrote: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life.”

Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. By assuming a proposition to be true and showing that this would lead to a contradiction, he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of π (Pi). He did this by drawing a larger polygon outside a circle and a smaller polygon inside the circle. As the number of sides of the polygon increases, it becomes a more accurate approximation of a circle. When the polygons had 96 sides each, he calculated the lengths of their sides and showed that the value of π lay between 3 + 1/7 (approximately 3.1429) and 3 + 10/71 (approximately 3.1408). He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle.

In The Measurement of a Circle, Archimedes gives the value of the square root of 3 as being more than 265/153 (approximately 1.732) and less than 1351/780 (approximately 1.7320512). The actual value is around 1.7320508076, making this a very accurate estimate. He introduced this result without offering any explanation of the method used to obtain it. This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results."

In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He expressed the solution to the problem as a geometric series that summed to infinity with the ratio 1/4:

${\displaystyle {\begin{smallmatrix}\sum _{n=0}^{\infty }4^{-n}=1+4^{-1}+4^{-2}+4^{-3}+\cdots ={4 \over 3}.\;\end{smallmatrix}}}$

If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and so on. This proof is a variation of the infinite series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.

In The Sand Reckoner, Archimedes set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote: "There are some, King Gelo (Gelo II, son of Hiero II), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited." To solve the problem, Archimedes devised a system of counting based around the myriad. The word is based on the Greek for uncountable, murious, and was also used to denote the number 10,000. He proposed a number system using powers of myriad myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8×10 in modern notation.

Writings

• On the Equilibrium of Planes (two volumes)

The first book is in fifteen propositions with seven postulates, while the second book is in ten propositions. In this work Archimedes explains the Law of the Lever, stating:

Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.

Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, paraboloids, and hemispheres.

• On the Measurement of the Circle

This is a short work consisting of three propositions. It is written in the form of a correspondence with Dositheus of Pelusium, who was a student of Conon of Samos. In Proposition II, Archimedes shows that the value of π (Pi) is greater than 223/71 and less than 22/7. The latter figure was used as an approximation of π throughout the Middle Ages and is still used today when a rough figure is required.

• On Spirals

This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation

${\displaystyle {\begin{smallmatrix}r\ =\ a\ +\ b\theta \end{smallmatrix}}}$

with real numbers a and b. This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician.

• On the Sphere and the Cylinder (two volumes)

In this treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a circumscribed cylinder of the same height and diameter. The volume is 4/3πr³ for the sphere, and 2πr³ for the cylinder; the surface area is 4πr² for the sphere, and 6πr² for the cylinder, where r is the radius. The sphere will have two thirds of the volume and surface area of the cylinder. A carving of this proof was used on the tomb of Archimedes at his request.

• On Conoids and Spheroids

This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.

• On Floating Bodies (two volumes)

In the first part of this treatise, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not self-gravitating, since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.

In the second part, 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, similar to the way that icebergs float. Archimedes' principle of buoyancy is given in the work, stated as follows:

Any body wholly or partially immersed in a fluid experiences an upthrust equal to, but opposite in sense to, the weight of the fluid displaced.

• The Quadrature of the Parabola

A work of 24 propositions addressed to Dositheus. In this treatise Archimedes proves by two methods that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He achieves this by calculating the value of a geometric series that sums to infinity with the ratio 1/4.

• Stomachion

This is a dissection puzzle similar to a Tangram, and the treatise describing it was found in more complete form in the Archimedes Palimpsest. Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. Research published by Dr. Reviel Netz of Stanford University in 2003 argued that Archimedes was attempting to determine how many ways the pieces of paper could be assembled into the shape of a square. The figure given by Dr. Netz is that the pieces can be made into a square in 17,152 ways. The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded. The Stomachion represents an example of an early problem in combinatorics. Stomachion is the Greek word for stomach, στομάχιον; the reason for the name is unknown.

• Archimedes' Cattle Problem

This work was discovered by Gotthold Ephraim Lessing in a Greek manuscript consisting of a poem of 44 lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773. It is addressed to Eratosthenes and the mathematicians at the University of Alexandria. Archimedes challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations. There is a more difficult version of the problem in which some of the answers are required to be square numbers. This version of the problem was first solved by a computer in 1965, and the answer is a very large number, approximately 7.760271×10.

• The Sand Reckoner

In this treatise, Archimedes counts the number of grains of sand that will fit inside the universe. This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos (concluding that "this is impossible"), contemporary ideas about the size of the Earth and the distance between various celestial bodies. By using a system of numbers based on powers of the myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8×10 in modern notation. The introductory letter states that Archimedes' father was an astronomer named Phidias. The Sand Reckoner or Psammites is the only surviving work in which Archimedes discusses his views on astronomy.

• The Method of Mechanical Theorems

This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.

Apocryphal works

Archimedes' Book of Lemmas or Liber Assumptorum is a treatise with fifteen propositions on the nature of circles. The earliest known copy of the text is in Arabic. The scholars T. L. Heath and Marshall Clagett argued that it cannot have been written by Archimedes in its current form, since it quotes Archimedes, suggesting modification by another author. The Lemmas may be based on an earlier work by Archimedes that is now lost.

It has also been claimed by the Arab scholar Abu'l Raihan Muhammed al-Biruni that Heron's formula for calculating the area of a triangle from the length of its sides was known to Archimedes. However, the first reliable reference to the formula is given by Heron of Alexandria in the 1st century  AD.

Archimedes Palimpsest

The written work of Archimedes has not survived as well as that of Euclid, and seven of his treatises are known to exist only through references made to them by other authors. Pappus of Alexandria mentions On Sphere-Making and another work on polyhedra, while Theon of Alexandria quotes a remark about refraction from the now-lost Catoptrica. The writings of Archimedes were collected by the Byzantine architect Isidore of Miletus (c. 530 AD), while translations into Arabic and Latin made during the Middle Ages helped to keep his work alive. Archimedes' work was translated into Arabic by Thābit ibn Qurra (836–901 AD), and Latin by Gerard of Cremona (c. 1114–1187 AD). During the Renaissance, the Editio Princeps (First Edition) was published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin. Around the year 1586 Galileo Galilei invented a hydrostatic balance for weighing metals in air and water after apparently being inspired by the work of Archimedes.

The foremost document containing the work of Archimedes is the Archimedes Palimpsest. A palimpsest is a document written on vellum that has been re-used by scraping off the ink of an older text and writing new text in its place. This was often done during the Middle Ages since animal skin parchments were expensive. In 1906, the Danish professor Johan Ludvig Heiberg realized that a goatskin parchment of prayers written in the 13th century AD also carried an older work written in the 10th century AD, which he identified as previously unknown copies of works by Archimedes. The parchment spent hundreds of years in a monastery library in Constantinople before being sold to a private collector in the 1920s. On October 29, 1998 it was sold at auction to an anonymous buyer for $2 million at Christie's in London. The palimpsest holds seven treatises, including the only surviving copy of On Floating Bodies in the original Greek. It is the only known source of the Method of Mechanical Theorems, referred to by Suidas and thought to have been lost forever. Stomachion was also discovered in the palimpsest, with a more complete analysis of the puzzle than had been found in previous texts. The palimpsest is now stored at the Walters Art Museum in Baltimore, Maryland, where it has been subjected to a range of modern tests including the use of ultraviolet and x-ray light to read the overwritten text.

The treatises in the Archimedes Palimpsest are: On the Equilibrium of Planes, On Spirals, The Measurement of the Circle, On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems and Stomachion.

Legacy

There is a crater on the Moon named Archimedes (29.7° N, 4.0° W) in his honor, and a lunar mountain range, the Montes Archimedes (25.3° N, 4.6° W). The asteroid 3600 Archimedes is named after him.

The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with his proof concerning the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to him which reads in Latin: "Transire suum pectus mundoque potiri" (Rise above oneself and grasp the world).

Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982) and Spain (1963).

The exclamation of Eureka! attributed to Archimedes is the state motto of California. In this instance the word refers to the discovery of gold near Sutter's Mill in 1848 which sparked the California gold rush.

See also

• Archimedes' Axiom
• Archimedes number
• Archimedes Paradox
• Archimedean property
• Archimedean solid
• Archimedes' use of infinitesimals
• Zhang Heng

Notes and references

Notes

a. In the preface to On Spirals addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon's death." Conon of Samos lived c. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works.

b. The treatises by Archimedes known to exist only through references in the works of other authors are: On Sphere-Making and a work on polyhedra mentioned by Pappus of Alexandria; Catoptrica, a work on optics mentioned by Theon of Alexandria; Principles, addressed to Zeuxippus and explaining the number system used in The Sand Reckoner; On Balances and Levers; On Centers of Gravity; On the Calendar. Of the surviving works by Archimedes, T. L. Heath offers the following suggestion as to the order in which they were written: On the Equilibrium of Planes I, The Quadrature of the Parabola, On the Equilibrium of Planes II, On the Sphere and the Cylinder I, II, On Spirals, On Conoids and Spheroids, On Floating Bodies I, II, On the Measurement of a Circle, The Sand Reckoner.

c. Boyer, Carl Benjamin A History of Mathematics (1991) ISBN 0471543977 "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron's formula—k=sqrt(s(s-a)(s-b)(s-c)), where s is the semiperimeter—was known to Archimedes several centuries before Heron lived. Arabic scholars also attribute to Archimedes the 'theorem on the broken chord' Archimedes is reported by the Arabs to have given several proofs of the theorem."

References

1. Hippias, C.2.
2. John Wesley. "A Compendium of Natural Philosophy (1810) Chapter XII, Burning Glasses". Online text at Wesley Center for Applied Theology. Retrieved 2007-09-14.
3. Bonsor, Kevin. "How Wildfires Work". HowStuffWorks. Retrieved 2007-07-23.
4. "Archimedes Death Ray: Testing with MythBusters". MIT. Retrieved 2007-07-23.
5. "Archimedes' Weapon". Time Magazine. November 26, 1973. Retrieved 2007-08-12. {{cite web}}: Check date values in: |date= (help)
6. Plutarch. "Extract from Parallel Lives". fullbooks.com. Retrieved 2007-08-07.
7. Quoted in T. L. Heath, Works of Archimedes, Dover Publications, ISBN 0-486-42084-1.
8. Carroll, Bradley W. "The Sand Reckoner". Weber State University. Retrieved 2007-07-23.
9. Heath,T.L. "The Works of Archimedes (1897). The unabridged work in PDF form (19 MB)". Archive.org. Retrieved 2007-10-14.
10. Kolata, Gina (December 14, 2003). "In Archimedes' Puzzle, a New Eureka Moment". New York Times. Retrieved 2007-07-23. {{cite web}}: Check date values in: |date= (help)
11. Rorres, Chris. "Archimedes' Stomachion". Courant Institute of Mathematical Sciences. Retrieved 2007-09-14.
12. Calkins, Keith G. "Archimedes' Problema Bovinum". Andrews University. Retrieved 2007-09-14.
13. "English translation of The Sand Reckoner". University of Waterloo. Retrieved 2007-07-23.
14. "Archimedes' Book of Lemmas". cut-the-knot. Retrieved 2007-08-07.
15. Wilson, James W. "Problem Solving with Heron's Formula". University of Georgia. Retrieved 2007-09-14.
16. "Editions of Archimedes' Work". Brown University Library. Retrieved 2007-07-23.
17. Van Helden, Al. "The Galileo Project: Hydrostatic Balance". Rice University. Retrieved 2007-09-14.
18. "X-rays reveal Archimedes' secrets". BBC News. August 2, 2006. Retrieved 2007-07-23. {{cite web}}: Check date values in: |date= (help)
19. Friedlander, Jay and Williams, Dave. "Oblique view of Archimedes crater on the Moon". NASA. Retrieved 2007-09-13.{{cite web}}: CS1 maint: multiple names: authors list (link)
20. "Planetary Data System". NASA. Retrieved 2007-09-13.
21. "Fields Medal". International Mathematical Union. Retrieved 2007-07-23.
22. Rorres, Chris. "Stamps of Archimedes". Courant Institute of Mathematical Sciences. Retrieved 2007-08-25.
23. "California Symbols". California State Capitol Museum. Retrieved 2007-09-14.

Further reading

• Boyer, Carl Benjamin (1991). A History of Mathematics. New York: Wiley. ISBN 0-471-54397-7. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)
• Dijksterhuis, E.J. (1987). Archimedes. Princeton University Press, Princeton. ISBN 0-691-08421-1. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help) Republished translation of the 1938 study of Archimedes and his works by an historian of science.
• Gow, Mary (2005). Archimedes: Mathematical Genius of the Ancient World. Enslow Publishers, Inc. ISBN 0-7660-2502-0. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)
• Hasan, Heather (2005). Archimedes: The Father of Mathematics. Rosen Central. ISBN 978-1404207745. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)
• Heath, T.L. (1897). Works of Archimedes. Dover Publications. ISBN 0-486-42084-1. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help) Complete works of Archimedes in English.
• Netz, Reviel and Noel, William (2007). The Archimedes Codex. Orion Publishing Group. ISBN 0-297-64547-1. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)CS1 maint: multiple names: authors list (link)
• Simms, Dennis L. (1995). Archimedes the Engineer. Continuum International Publishing Group Ltd. ISBN 0-720-12284-8. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)
• Stein, Sherman (1999). Archimedes: What Did He Do Besides Cry Eureka?. Mathematical Association of America. ISBN 0-88385-718-9. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)

External links

• Archimedes—The Greek mathematician and his Eureka moments—BBC Radio 4 discussion from In Our Time, broadcast January 25, 2007 (requires RealPlayer)
• The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
• The Mathematical Achievements and Methodologies of Archimedes
• Article examining how Archimedes may have calculated the square root of 3
• PDF scans of Heiberg's edition of the Works of Archimedes, now in the public domain (Text in Classical Greek)
• Photograph of the Sakkas experiment in 1973

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