Editing Archimedes (section)

===Surviving works===
The following are ordered chronologically based on new terminological and historical criteria set by Knorr (1978) and Sato (1986).<ref>{{Cite journal |last=Knorr |first=W. R. |date=1978 |title=Archimedes and the Elements: Proposal for a Revised Chronological Ordering of the Archimedean Corpus |journal=Archive for History of Exact Sciences |volume=19 |issue=3 |pages=211–290 |doi=10.1007/BF00357582 |jstor=41133526}}</ref><ref>{{Cite journal |last=Sato |first=T. |date=1986 |title=A Reconstruction of The Method Proposition 17, and the Development of Archimedes' Thought on Quadrature...Part One |journal=Historia scientiarum: International journal of the History of Science Society of Japan}}</ref>

==== ''Measurement of a Circle'' ====
{{Main|Measurement of a 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 gives an approximation of the value of pi ({{pi}}), showing that it is greater than {{sfrac|223|71}} (3.1408...) and less than {{sfrac|22|7}} (3.1428...).

==== ''The Sand Reckoner'' ====
{{Main|The Sand Reckoner}}
In this treatise, also known as '''''Psammites''''', Archimedes finds a number that is greater than the [[Sand|grains of sand]] needed to fill the universe. This book mentions the [[Heliocentrism|heliocentric]] theory of the [[Solar System]] proposed by [[Aristarchus of Samos]], as well as contemporary ideas about the size of the Earth and the distance between various [[celestial bodies]], and attempts to measure the apparent diameter of the [[Sun]].<ref>{{Cite journal |last=Osborne |first=Catherine |date=1983 |title=Archimedes on the Dimensions of the Cosmos |url=https://www.jstor.org/stable/233105 |journal=Isis |volume=74 |issue=2 |pages=234–242 |doi=10.1086/353246 |jstor=233105 |issn=0021-1753}}</ref><ref name=":7">{{Citation |last1=Rozelot |first1=Jean Pierre |title=A brief history of the solar diameter measurements: a critical quality assessment of the existing data |date=2016 |arxiv=1609.02710 |last2=Kosovichev |first2=Alexander G. |last3=Kilcik |first3=Ali}}</ref> 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{{e|63}} in modern notation. The introductory letter states that Archimedes' father was an astronomer named Phidias. ''The Sand Reckoner'' is the only surviving work in which Archimedes discusses his views on astronomy.<ref>{{cite web |year=2002 |title=English translation of ''The Sand Reckoner'' |publisher=[[University of Waterloo]] |url=http://www.math.uwaterloo.ca/navigation/ideas/reckoner.shtml |archive-date=2002-06-01 |archive-url=https://web.archive.org/web/20020601231141/https://www.math.uwaterloo.ca/navigation/ideas/reckoner.shtml |url-status=dead}} Adapted from {{cite book |last=Newman |first=James R. |title=The World of Mathematics |volume=1 |publisher=Simon & Schuster |location=New York |year=1956}}</ref>

Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as [[Aristarchus of Samos|Aristarchus]]' heliocentric model of the universe, in the ''Sand-Reckoner''.<ref>{{Cite encyclopedia |last1=Toomer |first1=G. J. |last2=Jones |first2=Alexander |date=7 March 2016 |title=Astronomical Instruments |encyclopedia=Oxford Research Encyclopedia of Classics |doi=10.1093/acrefore/9780199381135.013.886 |isbn=9780199381135 |quote="Perhaps the earliest instrument, apart from sundials, of which we have a detailed description is the device constructed by Archimedes for measuring the sun's apparent diameter; this was a rod along which different coloured pegs could be moved."}}</ref> Without the use of either trigonometry or a table of chords, Archimedes determines the Sun's apparent diameter by first describing the procedure and instrument used to make observations (a straight rod with pegs or grooves),<ref>{{Cite journal |last=Evans |first=James |date=1 August 1999 |title=The Material Culture of Greek Astronomy |journal=Journal for the History of Astronomy |volume=30 |issue=3 |pages=238–307 |bibcode=1999JHA....30..237E |doi=10.1177/002182869903000305}}</ref> applying correction factors to these measurements, and finally giving the result in the form of upper and lower bounds to account for observational error.<ref>{{Cite journal |last=Shapiro |first=A. E. |date=1975 |title=Archimedes's measurement of the Sun's apparent diameter. |journal=Journal for the History of Astronomy |volume=6 |issue=2 |pages=75–83 |bibcode=1975JHA.....6...75S |doi=10.1177/002182867500600201}}</ref> 

[[Ptolemy]], quoting Hipparchus, also references Archimedes' [[solstice]] observations in the ''Almagest''. This would make Archimedes the first known Greek to have recorded multiple solstice dates and times in successive years.<ref name="Acerbi2008" />

==== ''On the Equilibrium of Planes'' ====
{{Main|On the Equilibrium of Planes}}
There are two books to ''On the Equilibrium of Planes'': the first contains seven [[Axiom|postulates]] and fifteen [[proposition]]s, while the second book contains ten propositions. In the first book, Archimedes proves the law of the [[lever]],<ref>{{Cite journal |last=Goe |first=G. |date=1972 |title=Archimedes' theory of the lever and Mach's critique |journal=Studies in History and Philosophy of Science Part A |volume=2 |issue=4 |pages=329–345 |doi=10.1016/0039-3681(72)90002-7 |bibcode=1972SHPSA...2..329G}}</ref> which states that:

{{Blockquote|text=[[Magnitude (mathematics)|Magnitudes]] are in equilibrium at distances reciprocally proportional to their weights.}}

Earlier descriptions of the principle of the lever are found in a work by [[Euclid]] and in the ''[[Mechanics (Aristotle)|Mechanical Problems]],'' belonging to the [[Peripatetic school]] of the followers of [[Aristotle]], the authorship of which has been attributed by some to [[Archytas]].<ref name="lever clagett">{{cite book |first=Marshall |last=Clagett |url=https://books.google.com/books?id=mweWMAlf-tEC&q=archytas%20lever&pg=PA72 |title=Greek Science in Antiquity |publisher=Dover Publications |isbn=978-0-486-41973-2 |year=2001}}</ref>

Archimedes uses the principles derived to calculate the areas and [[center of mass|centers of gravity]] of various geometric figures including [[triangle]]s, [[parallelogram]]s and [[parabola]]s.<ref name="works">{{cite book |author=Heath, T.L. |url=https://archive.org/details/worksofarchimede029517mbp |title=The Works of Archimedes |year=1897 |publisher=Cambridge University Press}}</ref>

==== ''Quadrature of the Parabola'' ====
{{Main|Quadrature of the Parabola}}
In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods that the area enclosed by a [[parabola]] and a straight line is 4/3 the area of a [[triangle]] with equal base and height. He achieves this by two different methods: first by applying the [[law of the lever]], and by calculating the value of a [[geometric series]] that sums to infinity with the [[ratio]] 1/4.

==== ''On the Sphere and Cylinder'' ====
{{Main|On the Sphere and Cylinder}}
[[File:Esfera Arquímedes.svg|thumb|A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases|209x209px]]
In this two-volume treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a [[sphere]] and a [[circumscribe]]d [[cylinder (geometry)|cylinder]] of the same height and [[diameter]]. The volume is {{sfrac|4|3}}{{pi}}{{math|''r''}}<sup>3</sup> for the sphere, and 2{{pi}}{{math|''r''}}<sup>3</sup> for the cylinder. The surface area is 4{{pi}}{{math|''r''}}<sup>2</sup> for the sphere, and 6{{pi}}{{math|''r''}}<sup>2</sup> for the cylinder (including its two bases), where {{math|''r''}} is the radius of the sphere and cylinder.

==== ''On Spirals'' ====
{{Main|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 (mathematics)|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 modern [[Polar coordinate system|polar coordinates]] ({{math|''r''}}, {{math|θ}}), it can be described by the equation <math>\, r=a+b\theta</math> with [[real number]]s {{math|a}} and {{math|b}}.

This is an early example of a [[Curve|mechanical curve]] (a curve traced by a moving [[point (geometry)|point]]) considered by a Greek mathematician.

==== ''On Conoids and Spheroids'' ====
{{Main|On Conoids and Spheroids}}
This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of [[cross section (geometry)|sections]] of [[Cone (geometry)|cones]], spheres, and paraboloids.

==== ''On Floating Bodies'' ====
{{Main|On Floating Bodies}}
There are two books of ''On Floating Bodies''. In the first book, Archimedes spells out the law of [[wikt:equilibrium|equilibrium]] of fluids and proves that water will adopt a spherical form around a center of gravity.<ref>{{Cite journal |last=Berggren |first=J. L. |date=1976 |title=Spurious Theorems in Archimedes' Equilibrium of Planes: Book I |journal=Archive for History of Exact Sciences |volume=16 |issue=2 |pages=87–103 |doi=10.1007/BF00349632 |jstor=41133463}}</ref>  

This may have been an attempt at explaining the theory of contemporary Greek astronomers such as [[Eratosthenes]] that the Earth is round.{{cn|date=April 2025}} The fluids described by Archimedes are not {{nowrap|self-gravitating}} since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.{{cn|date=April 2025}}

[[Archimedes' principle]] of buoyancy is given in this work, stated as follows:<ref>{{cite book |last=Netz |first=Reviel |chapter=Archimedes' Liquid Bodies |title=ΣΩΜΑ: Körperkonzepte und körperliche Existenz in der antiken Philosophie und Literatur |year=2017 |pages=287–322 |editor1-first=Thomas |editor1-last=Buchheim |editor2-first=David |editor2-last=Meißner |editor3-first=Nora |editor3-last=Wachsmann |isbn=978-3-7873-2928-1 |place=Hamburg |publisher=Felix Meiner |chapter-url=https://books.google.com/books?id=rQ2KDwAAQBAJ&pg=PA287 |chapter-url-access=limited}}</ref>

<blockquote>Any body wholly or partially immersed in fluid experiences an upthrust equal to, but opposite in direction to, the weight of the fluid displaced.</blockquote>

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.<ref>{{cite book |last=Stein |first=Sherman |chapter=Archimedes and his floating paraboloids |editor1-first=David F. |editor1-last=Hayes |editor2-first=Tatiana |editor2-last=Shubin |title=Mathematical Adventures for Students and Amateurs |publisher=Mathematical Association of America |place=Washington |year=2004 |pages=219–231 |isbn=0-88385-548-8 |chapter-url=https://archive.org/details/mathematicaladve0000unse/page/219 |chapter-url-access=limited}} {{pb}} {{cite journal |last=Rorres |first=Chris |year=2004 |title=Completing Book II of Archimedes's on Floating Bodies |journal=The Mathematical Intelligencer |volume=26 |number=3 |pages=32–42 |doi=10.1007/bf02986750}} {{pb}} {{cite journal |last1=Girstmair |first1=Kurt |last2=Kirchner |first2=Gerhard |title=Towards a completion of Archimedes' treatise on floating bodies |journal=Expositiones Mathematicae |volume=26 |number=3 |year=2008 |pages=219–236 |doi=10.1016/j.exmath.2007.11.002 |doi-access=free}}</ref>

==== ''Ostomachion'' ====
{{Main|Ostomachion}}
[[File:Stomachion.JPG|thumb|''[[Ostomachion]]'' is a [[dissection puzzle]] found in the [[Archimedes Palimpsest]]|200x200px]]
Also known as '''Loculus of Archimedes''' or '''Archimedes' Box''',<ref name=":1" /> 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]]. [[Reviel Netz]] of [[Stanford University]] argued in 2003 that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. Netz calculates that the pieces can be made into a square 17,152 ways.<ref>{{cite news |title=In Archimedes' Puzzle, a New Eureka Moment |author=Kolata, Gina |newspaper=[[The New York Times]] |date=14 December 2003 |url=https://query.nytimes.com/gst/fullpage.html?res=9D00E6DD133CF937A25751C1A9659C8B63&sec=&spon=&pagewanted=all |access-date=23 July 2007}}</ref> The number of arrangements is 536 when solutions that are equivalent by rotation and reflection are excluded.<ref>{{cite web |title=The Loculus of Archimedes, Solved |author=Ed Pegg Jr. |publisher=[[Mathematical Association of America]] |date=17 November 2003 |url=http://www.maa.org/editorial/mathgames/mathgames_11_17_03.html |access-date=18 May 2008}}</ref> The puzzle represents an example of an early problem in [[combinatorics]].

The origin of the puzzle's name is unclear, and it has been suggested that it is taken from the [[Ancient Greek]] word for "throat" or "gullet", ''stomachos'' ({{lang|grc|στόμαχος}}).<ref>{{cite web |first=Chris |last=Rorres |url=http://math.nyu.edu/~crorres/Archimedes/Stomachion/intro.html |title=Archimedes' Stomachion |publisher=Courant Institute of Mathematical Sciences |access-date=14 September 2007}}</ref> [[Ausonius]] calls the puzzle {{Langx|grc|Ostomachion|label=none|italic=yes}}, a Greek compound word formed from the roots of {{Langx|grc|osteon|label=none|italic=yes}} ({{Langx|grc|ὀστέον|label=none|lit=bone}}) and {{Langx|grc|machē|label=none|italic=yes}} ({{Langx|grc|μάχη|label=none|lit=fight}}).<ref name=":1">{{cite web |url=http://www.archimedes-lab.org/latin.html#archimede |title=Graeco Roman Puzzles |publisher=Gianni A. Sarcone and Marie J. Waeber |access-date=9 May 2008}}</ref>

==== The cattle problem ====
{{Main|Archimedes's cattle problem|l1 = Archimedes' cattle problem}}
In this work, addressed to Eratosthenes and the mathematicians in Alexandria, Archimedes challenges them to count the numbers of cattle in the [[The Cattle of Helios|Herd of the Sun]], which involves solving a number of simultaneous [[Diophantine equation]]s. [[Gotthold Ephraim Lessing]] discovered this work in a Greek manuscript consisting of a 44-line poem in the [[Herzog August Library]] in [[Wolfenbüttel]], Germany in 1773. There is a more difficult version of the problem in which some of the answers are required to be [[square number]]s. A. Amthor first solved this version of the problem<ref>Krumbiegel, B. and Amthor, A. ''Das Problema Bovinum des Archimedes'', Historisch-literarische Abteilung der Zeitschrift für Mathematik und Physik 25 (1880) pp. 121–136, 153–171.</ref> in 1880, and the answer is a [[very large number]], approximately 7.760271{{e|206544}}.<ref>{{cite web |first=Keith G |last=Calkins |url=http://www.andrews.edu/~calkins/profess/cattle.htm |title=Archimedes' Problema Bovinum |publisher=[[Andrews University]] |access-date=14 September 2007 |archive-url=https://web.archive.org/web/20071012171254/http://andrews.edu/~calkins/profess/cattle.htm |archive-date=12 October 2007}}</ref>

==== ''The Method of Mechanical Theorems'' ====
{{Main|The Method of Mechanical Theorems}}
As with ''[[Archimedes's cattle problem|The Cattle Problem]]'', ''The Method of Mechanical Theorems'' was written in the form of a letter to [[Eratosthenes]] in [[Alexandria]].

In this work Archimedes uses a novel method, an early form of [[Cavalieri's principle]],<ref name="ArchimedesCalc">{{Cite web |last=Powers |first=J. |date=2020 |title=Did Archimedes do calculus? |url=https://old.maa.org/sites/default/files/images/upload_library/46/HOMSIGMAA/2020-Jeffery%20Powers.pdf |access-date=14 April 2021 |website=maa.org}}; {{Citation |last=Jullien |first=V. |title=Archimedes and Indivisibles |date=2015 |work=Seventeenth-Century Indivisibles Revisited |volume=49 |pages=451–457 |editor-last=J. |editor-first=Vincent |series=Science Networks. Historical Studies |place=Cham |publisher=Springer International Publishing |doi=10.1007/978-3-319-00131-9_18 |isbn=978-3-319-00131-9}}; {{cite web |author1=O'Connor, J.J. |author2=Robertson, E.F. |date=February 1996 |title=A history of calculus |url=https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/ |access-date=7 August 2007 |publisher=[[University of St Andrews]]}}; {{Cite journal |last=Kirfel |first=Christoph |date=2013 |title=A generalisation of Archimedes' method |url=https://www.jstor.org/stable/24496758 |journal=The Mathematical Gazette |volume=97 |issue=538 |pages=43–52 |doi=10.1017/S0025557200005416 |issn=0025-5572 |jstor=24496758}}</ref> to rederive the results from the treatises sent to Dositheus (''Quadrature of the Parabola'', ''On the Sphere and Cylinder'', ''On Spirals'', ''On Conoids and Spheroids'') that he had previously used the [[method of exhaustion]] to prove,{{sfn|Netz|2022|p=131}} using the [[law of the lever]] he applied in ''On the Equilbrium of Planes'' in order to find the [[center of gravity]] of an object first, and reasoning geometrically from there in order to more easily derive the volume of an object.{{sfn|Netz|2022|pp=187-193}} Archimedes states that he used this method to derive the results in the treatises sent to Dositheus before he proved them more rigorously with the method of exhaustion, stating that it is useful to know that a result is true before proving it rigorously, much as [[Eudoxus of Cnidus]] was aided in proving that the volume of a cone is one-third the volume of cylinder by knowing that [[Democritus]] had already asserted it to be true on the argument that this is true by the fact that the pyramid has one-third the rectangular prism of the same base.{{sfn|Netz|2022|p=150-151}}

This treatise was thought lost until the discovery of the [[Archimedes Palimpsest]] in 1906.<ref>{{cite book |last1=Smith |first1=David Eugene |title=Geometrical Solutions Derived from Mechanics: A Treatise of Archimedes |date=1909 |publisher=Open Court Publishing Company |url=https://archive.org/details/geometricalsolu00smitgoog/mode/2up |access-date=4 May 2025 |language=English}}</ref>