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own as psammites archimedes finds a number that is greater than the grains of sand needed to fill the universe this book mentions the 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 61 62 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 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 63 archimedes discusses astronomical measurements of the earth sun and moon as well as aristarchus heliocentric model of the universe in the sand reckoner 64 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 65 applying correction factors to these measurements and finally giving the result in the form of upper and lower bounds to account for observational error 66 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 4 on the equilibrium of planes main article on the equilibrium of planes there are two books to on the equilibrium of planes the first contains seven postulates and fifteen propositions while the second book contains ten propositions in the first book archimedes proves the law of the lever 67 which states that 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 mechanical problems belonging to the peripatetic school of the followers of aristotle the authorship of which has been attributed by some to archytas 68 archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles parallelograms and parabolas 69 quadrature of the parabola main article 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 70 on the sphere and cylinder main article on the sphere and cylinder a sphere has 2 3 the volume and surface area of its circumscribing cylinder including its bases 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 circumscribed cylinder of the same height and diameter the volume is 4 3 π r 3 for the sphere and 2 π r 3 for the cylinder the surface area is 4 π r 2 for the sphere and 6 π r 2 for the cylinder including its two bases where r is the radius of the sphere and cylinder 71 72 on spirals main article on spirals this work of 28 propositions is also addressed to dositheus the treatise defines what is now called the archimedean spiral 73 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 modern polar coordinates r θ it can be described by the equation r a b θ displaystyle r a b theta with real numbers a and b 74 this is an early example of a mechanical curve a curve traced by a moving point considered by a greek mathematician on conoids and spheroids main article 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 75 76 on floating bodies main article on floating bodies there are two books of on floating bodies in the first book archimedes spells out the law of equilibrium of fluids and proves that water will adopt a spherical form around a center of gravity 77 archimedes principle of buoyancy is given in this work stated as follows 78 any body wholly or partially immersed in fluid experiences an upthrust equal to but opposite in direction to the weight of the fluid displaced 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 79 ostomachion main article ostomachion ostomachion is a dissection puzzle found in the archimedes palimpsest also known as loculus of archimedes or archimedes box 80 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 81 the number of arrangements is 536 when solutions that are equivalent by rotation and reflection are excluded 82 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 στόμαχος 83 ausonius calls the puzzle ostomachion a greek compound word formed from the roots of osteon ὀστέον bone and machē μάχη fight 80 cattle problem main article 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 herd of the sun which involves solving a number of simultaneous diophantine equations 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 numbers a amthor first solved this version of the problem 84 in 1880 and the answer is a very large number approximately 7 760271 10 206 544 55 the method of mechanical theorems main article the method of mechanical theorems as with 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 85 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 86 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 87 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 88 this treatise was thought lost until the discovery of the archimedes palimpsest in 1906 89 apocryphal works main article pseudo archimedes archimedes book of lemmas or liber assumptorum is a treatise with 15 propositions on the nature of circles the earliest known copy of the text is in arabic t l heath argued that it cannot have been written by archimedes in its current form since it quotes archimedes by name suggesting the lemmas were compiled by a later author he thought its ideas like the arbelos were archimedean in origin 90 tannery supposes that if such a work once existed it had already been lost by the time of pappus who put similar lemmas in the collection calling them ancient and from various works 91 92 other questionable attributions to archimedes work include the latin poem carmen de ponderibus et mensuris 4th or 5th century which describes the use of a hydrostatic balance to solve the problem of the crown and the 12th century text mappae clavicula which contains instructions on how to perform assaying of metals by calculating their specific gravities 93 94 lost works many written works by archimedes have not survived or are only extant in heavily edited fragments 95 pappus of alexandria mentions on sphere making as well as a work on semiregular polyhedra and another work on spirals while theon of alexandria quotes a remark about refraction from the now lost catoptrica principles addressed to zeuxippus explained the number system used in the sand reckoner there are also on balances on centers of gravity 95 scholars in the medieval islamic world also attribute to archimedes a formula for calculating the area of a triangle from the length of its sides which today is known as heron s formula due to its first known appearance in the work of heron of alexandria in the 1st century ad and may have been proven in a lost work of archimedes that is no longer extant 96 archimedes palimpsest main article archimedes palimpsest in 1906 the archimedes palimpsest revealed works by archimedes thought to have been lost in 1906 the danish professor johan ludvig heiberg visited constantinople to examine a 174 page goatskin parchment of prayers written in the 13th century after reading a short transcription published seven years earlier in 1899 by papadopoulos kerameus 97 98 he confirmed that it was indeed a palimpsest a document with text that had been written over an erased older work palimpsests were created by scraping the ink from existing works and reusing them a common practice in the middle ages as vellum was expensive the older works in the palimpsest were identified by scholars as 10th century copies of previously lost treatises by archimedes 97 99 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 treatises in the archimedes palimpsest include on the equilibrium of planes on spirals measurement of a circle on the sphere and cylinder on floating bodies the method of mechanical theorems stomachion speeches by the 4th century bc politician hypereides a commentary on aristotle s categories other works the parchment spent hundreds of years in a monastery library in constantinople before being sold to a private collector in the 1920s on 29 october 1998 it was sold at auction to an anonymous buyer for a total of 2 2 million 100 101 the palimpsest was stored at the walters art museum in baltimore maryland where it was subjected to a range of modern tests including the use of ultraviolet and x ray light to read the overwritten text 102 it has since returned to its anonymous owner 103 104 legacy bronze statue of archimedes in berlin further information list of things named after archimedes sometimes called the father of mathematics 105 and mathematical physics 106 historians of science and mathematics almost universally agree that archimedes was the finest mathematician from antiquity 107 classical antiquity the reputation that archimedes had for mechanical inventions in classical antiquity is well documented 108 athenaeus 109 recounts in his deipnosophistae how archimedes launched the largest known ship in antiquity the syracusia while apuleius 110 talks about his work in catoptrics 111 plutarch 112 had claimed that archimedes disdained mechanics and focused primarily on pure geometry but this is generally considered to be a mischaracterization by modern scholarship fabricated to bolster plutarch s own platonist values rather than to an accurate presentation of archimedes 113 middle ages this section needs expansion you can help by making an edit request adding missing information april 2025 archimedes work was translated into arabic by thābit ibn qurra and into latin via arabic by gerard of cremona direct greek to latin translations were later done by william of moerbeke and iacobus cremonensis 114 115 renaissance and early modern europe 1612 drawing of a now lost bronze coin depicting archimedes 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 116 which were an influential source of ideas for scientists during the renaissance and again in the 17th century 117 118 leonardo da vinci repeatedly expressed admiration for archimedes and attributed his invention architonnerre to archimedes 119 120 121 galileo galilei called him superhuman and my master 122 123 while christiaan huygens said i think archimedes is comparable to no one consciously emulating him in his early work 124 gottfried wilhelm leibniz said he who understands archimedes and apollonius will admire less the achievements of the foremost men of later times 125 italian numismatist and archaeologist filippo paruta and leonardo agostini reported on a bronze coin in sicily with the portrait of archimedes on the obverse and a cylinder and sphere with the monogram armd in latin on the reverse 126 although the coin is now lost and its date is not precisely known ivo schneider described the reverse as a sphere resting on a base probably a rough image of one of the planetaria created by archimedes and suggested it might have been minted in rome for marcellus who according to ancient reports brought two spheres of archimedes with him to rome 127 in modern mathematics the fields medal carries a portrait of archimedes carl friedrich gauss s heroes were archimedes and newton 128 and moritz cantor who studied under gauss in the university of göttingen reported that he once remarked in conversation that there had been only three epoch making mathematicians archimedes newton and eisenstein 129 likewise alfred north whitehead said that in the year 1500 europe knew less than archimedes who died in the year 212 bc 130 the historian of mat...