<아르키메데스의 생애>
아르키메데스는 기원전 287년경 시칠리아 섬에 있던 옛 그리스의 도시인 시라큐스(Syracuse)에서 태어났으며 로마가 시라큐스를 정복한 기원전 212년에 죽었다. 그의 아버지는 천문학자였기 때문에 아르키메데스는 어렸을 때부터 천문관측을 배우게 되었다.
그가 코논(Conon),도시테우스(Dositheus),에라토스테네스(Eratosthenes) 등과 교분을 가졌다는 사실로 미루어 보아(앞의 두 사람은 유클리드의 후계자들이고 마니막 사람은 알렉산드리아대학의 사서였는데, 아르키메데스의 많은 연구 결과가 이들과의 편지 속에서 발견되었다), 그는 아마도 이집트에 건너가 수학과 물리학을 가르치는 알렉산드리아의 왕립학교에서 공부했던 것으로 보인다. 학교를 졸업하고 귀국한 아르키메데스는 이론을실용화하고 응용하는 데 천부적인 재능을 발휘해(응용 수학을 경멸하면서도 수학은 단순히 추상적이기만 한 것이 아니라 실용성을 발휘한다는 것을증명해 보이기라도 하듯), 수학자로서뿐 아니라 기술자인 동시에 발명가로도 능력을 발휘했다.
로마의 역사학자들은 아르키메데스에 관한 많은 재미있는 이야기를 전하고 있다. 특히 로마 장군 마르켈루스(Marcellus)가 시라큐스를 공격했을 때 아르키메데스가 시라큐스를 지키기 위하여 고안한 여러 가지 훌륭한 장치에 대한 설명이 있다. 그런 것 중에는 적의 배가 도시 성곽에 가까이 접근했을 때 그 배에 무거운 돌을 떨어뜨릴 수 있는 투석기가 있었는데 그것은 사정거리를 조정할 수도 있고 이동 발사 장치도 갖고 있었다. 또 그는 적의 배를 물에서 끌어올리게 할 수 있는 기중기도 만들었으며 적의 배를 불태우기 위해 커다란 볼록렌즈를 사용했다는 이야기도 있다. 한편 많은 사람들이 달라붙어야 간신히 끌어올릴 수 있는 커다란 배를 그는 복합 도르래 장치를 이용하여 간단히 혼자서 끌어올린 후에 다음과 같이 외쳤다고 한다. "나에게 충분히 긴 지렛대와 서 있을 자리를 다오. 그러면 지구를 움직여 보일 것이다!"
무엇보다도 그에 관한 가장 유명한 일화는 벌거벗은 수학자라는 별명과 "유레카(Eureka)"라는 말이 떠오르는 부력의 법칙과 관련된 이야기일 것이다. 아르키메데스가 부력의 법칙을 발견하게 된 것은, 시라큐스의 통치자인 히에론 2세와의 친분관계 때문이었다. 히에론왕은 우연한 기회에 그 나라에서 가장 솜씨 좋은 금 세공사에게 순금으로 된 왕관을 만들어 오도록 명령했다. 명령을 받은 세공사는 오래지 않아 찬란한 왕관을 가져왔다. 금 세공사가 돌아간 후 히에론왕은 가져온 왕관의 무게를 달아보았다. 그리고 처음 기록해 두었던 순금덩어리의 무게와 비교해 보았다. 무게가 똑같았다. 그러나 얼마 후에 금세공사가 왕에게 받은 금을 전부 사용하지 않고 그 일부를 가로채고 대신 은을 썩어서 왕관을 만들었다는 소문이 나돌았다. 이에 히에론왕은 금관이 정말 순금으로 만들어졌는지를 알아내기 위해 아르키메데스를 불러, 금관을 손상함이 없이 금의 함량을 알아내도록 명령했다. 그 당시 아르키메데스는 지레의 원리를 발표하여 명성을 얻고 있었다. 아르키메데스는 히에론왕으로부터 받아 온 왕관을 두고 고민을 시작하였다. 금이 은보다 무겁다는 사실은 알고 있지만, 금에 은이 섞여 있는지 여부는 어떻게 알아내야 할지, 아르키메데스는 침식도 잊은 채 며칠을 연구실에 틀어박혀 고민했다고 한다. 어느 날 아르키메데스는 우연히 목욕탕에 갔다. 욕조에 가득 찬 물 속에 들어갔을 때 그는 욕조에 몸을 가라앉힌 용적과 같은 양만큼의 물이 넘친다는 사실을 문득 깨달았다. 순간 그는 정신없이 벌떡 일어나 벌거벗은 채로 시라큐스의 말로 "Eureka(발견했다)"라고 소리치면서 자기 집으로 달려갔다고 한다. 그는 왕관과 같은 중량의 금덩이와 은덩이를 각각 만들었다. 그리고 큰 그릇에 물을 가득 채우고 그 속에 은덩어리를 넣자, 은덩어리가 들어간 양만큼의 물이 넘쳐흘렀다. 그리고 다시 은덩어리를 꺼내고 줄어든 만큼의 물을 채운 다음 보충한 물의 양을 측정했다. 따라서 일정한 용적의 물에는 얼마만큼의 은이 해당하는가를 알았다.
아르키메데스는 이다음에 물을 가득 채운 용기에 금덩이를 넣고 넘친 물의 양을 측정했다. 금덩이는 같은 중량의 은덩어리보다도 용적이 적은 만큼 넘친 물의 양도 적다는 것을 알았다. 그래서 아르키메데스는 다시 한 번 용기에 물을 가득 채우고 문제의 왕관을 넣었다. 그랬더니 같은 중량의 금덩이보다는 많은 양의 물이 넘쳤다. 이것으로 금관은 은을 섞어 만들었으며, 금을 많이 빼돌렸었다는 사실이 밝혀졌다. 히에론왕은 여기에 탄복하여 아르키메데스를 극찬하였다. 아르키메데스는 히에론 왕과의 이 사건이 인연이 되어 후일"액체 중에 있는 물체는 그 물체가 밀어낸 액체의 무게만큼 부력을 받는다"라는 유명한 아르키메데스의 원리(부력의 원리)를 발견했다. 수영장에서 무거운 사람을 쉽게 들어올릴 수 있는 것도 이 원리 때문이다.
아르키메데스의 최후는 로마가 시라큐스를 약탈하고 있을 때 맞이하게 된다. 수년 후 시라큐스가 함락되던 날, 그는 죽는 순간까지도 단순한 기술자가 아닌 기하학자로서의 면모를 보여주었다. 그날 아르키메데스는 뜰의 모래 위에 도형을 그리며 기하학의 연구에 몰두하고 있던 중, 다가오는 사람 그림자가 로마 병사인 줄도 모르고 그가 모래판을 짓밟자 "물러서거라, 내 도형이 망가진다."며 호통을 쳤다. 한낱 점령지 시민의 호통소리에 격분한 로마 병사는 그 자리에서 아르키메데스를 죽이고 말았다.
아르키메데스의 명성을 익히 듣고 있던 로마 장군 마르켈루스는 시라큐스를 점령하였을 때 '아르키메데스는 꼭 살려 두라'는 명령을 내렸었다. 아르키메데스가 값진 전리품으로 꼽힐 수 있었기 때문이다. 그러나 그의 명령은 그대로 수행되지 않았다. 아르키메데스를 깊이 존경했던 마르켈루스는 그의 죽음을 애도하여 비록 점령국의 과학자이지만 예우를 갖추었다. "원기둥에 구가 내접한 모양의 묘비를 세워달라"는 아르키메데스의 생전 희망을 받아들여 실현해 준 것이다. 이것은 그가 고심 끝에 발견한 정리(定理)인 "구에 외접하는 원기둥의 부피는 그 구 부피의 1.5배이다"라는 것을 나타낸 것이다.
<아르키메데스의 업적 및실생활 속의 응용>
아르키메데스는 유명한 부력의 원리, 원주율 π 및 그것을 이용한 원의 넓이 구하기, 구의 겉넓이, 구와 원기둥의 부피의 관계를 밝히는등 많은 업적을 남겼다. 실생활에서 응용된 아르키메데스의 원리들에는 다음과 같은 것이 있다.
1) 부력의 원리
① 잠수함
잠수함의 잠수원리는 아르키메데스의 부력의 원리를 이용한 것으로, 한 물체의 일부 또는 전부가 어떤 액체 속에 잠겨 있으면 그 물체에 의해 밀려나온 액체의 중량과 크기가 같고 방향이 반대인 상향력(부력)이 그 물체에 걸리게 된다는 것이다.
즉, 잠수함은 바닷물 속에 잠수되어 있을 때 잠수함의 압력선체(사람이 활동하고 장비가 탑재되는 밀폐된 공간)의 체적만큼 가벼워지는데 이 부력이 잠수함의 무게와 같을 때 잠수함은 바닷물 속에서 뜨거나 가라앉지 않게 된다. 만약 이 상태에서 잠수함이 부력을 더 가지게 된다면 수면으로 떠오르게 될 것이다.
1620년도의 초기 잠수 개념으로는 밀폐된 함정 내에 가죽 격벽을 사용, 안쪽으로 이동시킴으로써 현측에 뚫린 관총구를 통해 물을 유입시켜 배를 무겁게 하여 잠수시키고 나사를 이용, 가죽 격벽을 밀어냄으로써 무게를 줄여서 부력을 가지도록 하였다.
현대 잠수함에서는 주로 공기탱크를 압력선체 외부에 설치하여 공기탱크 내에 물이 채워졌을 때 잠수함의 무게가 부력과 같아지도록 조정하고 수면으로 부상하기 위해 공기탱크 내에 압축공기를 공급하도록 하고 있다.
따라서 잠수는 공기탱크 내의 공기를 밖으로 배출할 수 있도록 공기탱크 상부에 있는 차단밸브를 열어서 이루어지며 이때 자유 충수구로 바닷물이 유입되어 공기탱크를 채우게 된다. 차단밸브의 개폐와 공기탱크 내로의 압축공기 공급은 압력선체 내에서 조작한다. 잠수함의 승선 인원 또는 장비가 추가되어 무거워질 경우 무거워진 만큼의 무게를 보상하기 위한 물탱크를 별도로 압력선체 내에 설치하여 물탱크 내의 물을 함외로 배출하여 무거워진 무게를 조정한다.
② 놀이시설
놀이공원 등에서 볼 수 있는 원형보트는 스릴과 재미를 동시에 즐길 수 있는 놀이시설이다. 이 원형보트도 사실은 아르키메데스의 부력의 원리를 응용하여 만든 것이다. 원형보트는 밑바닥이 평평하고 둥근 모양으로 수로의 폭이 넓으면 물이 느리게 흐르고 수로의 폭이 좁으면 물이 빠르게 흐른다. 보트가 물에 가라앉지 않고 뜨는 것은 바로 물이 물체를 떠받치는 힘인 부력 때문이다. 즉 물이 물체를 떠받치는 힘이 물체가 물에 가하는 힘보다 크기 때문에 원형보트는 절대 뒤집히거나 곤두박질치지 않는다.
2) 지렛대의 원리
아르키메데스가 지구라도 움직여 보이겠다고 주장할 수 있었던 것은 바로 지렛대의 원리를 알았기 때문인데, 이 원리를 응용한 것이 바로 도르레이다. 도르레는 기본적인 기계 요소들 가운데 하나로 바퀴에 기초를 두고 있다. 바퀴에 줄이나 벨트 또는 체인을 걸어 힘의 방향을 바꾸거나 힘의 효력을 확대할 수 있는 것이다.
[도르레의 종류]
· 고정 도르레
고정 도르는레 우물의 두레박에서 쉽게 볼 수 있다. 고정 도르레는 바퀴에 걸린 물건을 끌어올리는 데 이용되는데 힘의 방향을 바꾸어 물건을 쉽게 들어올릴 수 있다는 장점이 있으나, 물건의 무게와 같은 크기의 힘이 들기 때문에 힘의 효과를 크게 하는 작용(힘의 이득)이 없다.
· 움직 도르레
움직 도르레는 물건의 무게와 도르레의 무게를 합한 것의 절반만큼의 힘밖에 들지 않는다. 즉 움직 도르래는 줄을 보통보다 길게 끄는 대신 힘의 효과를 크게 한다.
· 복합도르레
고정 도르레와 움직 도르레의 단점을 보완한 것이 복합 도르레다. 복합 도르레는 고정 도르레와 움직 도르레를 조합하여 힘의 방향을 바꿈과 동시에 힘의 효과를 확대할 수 있다는 이점이 있다.
① 엘리베이터
이러한 도르레를 이용한 것 중 하나가 바로 우리 주위에서 쉽게 찾아볼 수 있는 엘리베이터이다. 대부분의 사람들에게는 물체를 위로 끌어올리는 것보다 아래로 당기는 것이 더 쉽다. 고정 도르레는 운동의 방향을 바꿀 필요가 있는 기계에 쓰이는데 이 중 하나가 엘리베이터이다. 엘리베이터는 엘리베이터의 반대쪽에 평행추가 매달려 있어서 엘리베이터가 올라가면 평행추는 아래로 내려가며, 이 때 도르레의 밧줄을 당기는 데 드는 힘과 물체의 무게는 같다.
② 수원성
주위에서 볼 수 있는 오래된 건물들 역시 지렛대의 원리를 이용한 것이 많은데, 그 대표적인 예로 조선시대 정조대왕 때에 건축된 수원성을 들 수 있다.
수원성은 조선시대 '성곽의 꽃'이라고 불리는데 1794년에 축조하기 시작하여 2년 반이나 걸려 1796년 완성되었다. 당시 수원성의 축조를 맡았던 정약용은 - 그가 아르키메데스의 원리를 서학을 통해 접했는지는 알 수 없지만 - 축성 과정에서 전혀 새로운 차원의 개념을 도입하였다. 우선 작업 과정에서 인부들의 일정한 작업량에 따라 임금을 받을 수 있도록 하여 작업 능률을 올렸고, 자재를 운반하는 새로운 수레와 거중기라는 돌을 들어올리는 첨단기계까지 고안해 냈다. 거중기뿐 아니라 녹로, 유형거 등을 고안하여 40근의 힘으로 25000근의 무게를 움직일 수 있게 된 것이었다. 이러한 기계들은 바로 아르키메데스의 '지렛대의 원리'와 일맥상통하는 것이라고 할 수 있겠다.
3) 원과 구의 원리
아르키메데스는 기하학 중에서도 특히 원, 구에 대하여 많은 고심을 하였다, 그 결과 원주율, 원의 넓이, 구의 겉넓이, 그리고 그가 죽는 순간까지 고심하던 구와 원기둥의 관계 등에 대하여 많은 법칙을 발견할 수 있었다.
이러한 법칙들을 정리하여 가던 중에 그는 지금의 축구공 모양과 비슷한 형태의 다면체는 정이십면체의 꼭지점을 깎아서 만든다는 것을 발견하였다 .정이십면체의 열두 개 꼭지점이 깎여서 열두 개의 정오각형이 되고, 정이십면체의 스무 개의 면은 깎여서 스무 개의 정육각형이 된다. 아르키메데스의 다면체 이론은 크리스탈 등의 결정이론, 바이러스나 생명체 연구, 재료공학, 금속공학, 신소재 연구, 건축 이론 등이나, 예술 작품, 달력 제작, 주사위 등의 놀이기구를 만드는 데에 많이 활용된다.
4)기타
물을 끌어올리는 장치인 '아르키메데스의 스크루펌프'를 발명했고, 마르켈루스가 로마로 가지고 간 2개의 '구'(球)를 만들었다고 여겨진다. 이 구 가운데 하나는 천구(star globe)이고, 다른 하나는 확실하지는 않지만 태양(太陽), 달 그리고 행성들의 움직임을 역학적으로 나타낸 장치이다.
4)기타
물을 끌어올리는 장치인 '아르키메데스의 스크루펌프'를 발명했고, 마르켈루스가 로마로 가지고 간 2개의 '구'(球)를 만들었다고 여겨진다. 이 구 가운데 하나는 천구(star globe)이고, 다른 하나는 확실하지는 않지만 태양(太陽), 달 그리고 행성들의 움직임을 역학적으로 나타낸 장치이다.
역학에 대한 관심은 그의 수학적 사고에 심오한 영향을 끼쳤다. 그는 이론역학과 유체정역학(流體靜力學)에 관한 연구를 집필했을 뿐 아니라 〈역학적인 정리들에 관한 방법 Method Concerning Mechanical Theorems〉에서 새로운 수학정리를 발견하기 위한 발견적 방법으로서 역학적인 논법을 사용했다.
그의 수학 연구는 다른 사람이 새로운 발견을 할 수 있도록 하고자 했다는,역학적인 정리들에 관한 방법에 표현된 그의 희망에도 불구하고 어떤 면에서도 고대에 계속되거나 발전되지 않았다. 회전체 부피를 결정짓는 등 그의 업적들을 확장시키려는 시도가 있었던 8세기 또는 9세기에 몇몇 수학논문들이 아랍어로 번역될 때까지 발전되지 않았다. 중세 초기의 아랍 수학자들에 의한 몇몇 가치 있는 연구는 이러한 아르키메데스의 연구에 의해 자극받았다. 그러나 그의 연구가 이후 수학자들에 미친 가장 큰 영향은 16, 17세기에 이르러 그리스에서 유래된 교과서와 그리스어 교과서 초판이 1544년 바젤에서 인쇄되면서 나타났다. 1558년 페데리코 코만디노에 의해 라틴어로 번역된 많은 아르키메데스의 저서는 그 지식의 전파에 크게 기여했는데, 이것은 요하네스 케플러와 갈릴레오를 비롯한 그당시 가장 유명한 수학자들과 물리학자들의 연구에 반영되어 있다.가장 뛰어난 아르키메데스를 포함한 고대 수학자들의 연구가 재발견되지 않았다면 1550~1650년 유럽에서의 수학의 발전은 생각할 수 없다. 불행히도 역학적인 정리들에 관한 방법 은 아랍과 르네상스의 수학자들에게 알려지지 않아 19세기 후반에야 비로소 재발견되었는데, 만약 르네상스 시대의 수학자들에게 알려졌다면 그들이 새로운 정리들을 발견할 때 자신의 계승자들이 이것을 이용할 것이라는 아르키메데스의 희망이 성취되었을 것이다.
Archimedes of Syracuse (Greek: ?ρχιμ?δη?; c. 287BC ? c. 212BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and the explanation of the principle of the lever. He is credited with designing innovative machines, including siege engines and the screw pump that bears his name. Modern experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors.[1]
Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time.[2][3] He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi.[4] He also defined the spiral bearing his name, formulas for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers.
Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere inscribed within a cylinder. Archimedes had proven that the sphere has two thirds of the volume and surface area of the cylinder (including the bases of the latter), and regarded this as the greatest of his mathematical achievements.
Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c. 530AD by Isidore of Miletus, while commentaries on the works of Archimedes written by Eutocius in the sixth century AD opened them to wider readership for the first time. The relatively few copies of Archimedes' written work that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance,[5] while the discovery in 1906 of previously unknown works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.[6]
This bronze statue of Archimedes is at the Archenhold Observatory in Berlin. It was sculpted by Gerhard Thieme and unveiled in 1972.
Archimedes was born c. 287BC in the seaport city of Syracuse, Sicily, at that time a colony of Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years.[7] In The Sand Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse.[8] A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure.[9] It is unknown, for instance, whether he ever married or had children. During his youth Archimedes may have studied in Alexandria, Egypt, where Conon of Samos and Eratosthenes of Cyrene were contemporaries. He referred to Conon of Samos as his friend, while two of his works (The Method of Mechanical Theorems and the Cattle Problem) have introductions addressed to Eratosthenes.[a]
Archimedes died c. 212BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he not be harmed.[10]
The last words attributed to Archimedes are "Do not disturb my circles" (Greek: μ? μου το?? κ?κλου? τ?ραττε), a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is often given in Latin as "Noli turbare circulos meos," but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch.[10]
The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere and a cylinder of the same height and diameter. Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. In 75BC, 137 years after his death, the Roman orator Cicero was serving as quaestor in Sicily. He had heard stories about the tomb of Archimedes, but none of the locals was able to give him the location. Eventually he found the tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription.[11]
The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome. The account of the siege of Syracuse given by Polybius in his Universal History was written around seventy years after Archimedes' death, and was used subsequently as a source by Plutarch and Livy. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city.[12]
Archimedes may have used his principle of buoyancy to determine whether the golden crown was less dense than solid gold.
The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. According to Vitruvius, a new crown in the shape of a laurel wreath had been made for King Hiero II, and Archimedes was asked to determine whether it was of solid gold, or whether silver had been added by a dishonest goldsmith.[13] Archimedes had to solve the problem without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density. While taking a bath, he noticed that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the volume of the crown. For practical purposes water is incompressible,[14] so the submerged crown would displace an amount of water equal to its own volume. By dividing the weight of the crown by the volume of water displaced, the density of the crown could be obtained. This density would be lower than that of gold if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "Eureka!" (Greek: "ε?ρηκα!," meaning "I have found it!")[15]
The story of the golden crown does not appear in the known works of Archimedes. Moreover, the practicality of the method it describes has been called into question, due to the extreme accuracy with which one would have to measure the water displacement.[16] Archimedes may have instead sought a solution that applied the principle known in hydrostatics as Archimedes' Principle, which he describes in his treatise On Floating Bodies. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.[17] Using this principle, it would have been possible to compare the density of the golden crown to that of solid gold by balancing the crown on a scale with a gold reference sample, then immersing the apparatus in water. If the crown was less dense than gold, it would displace more water due to its larger volume, and thus experience a greater buoyant force than the reference sample. This difference in buoyancy would cause the scale to tip accordingly. Galileo considered it "probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself."[18]
The Archimedes screw can raise water efficiently.
A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer Athenaeus of Naucratis described how King Hieron II commissioned Archimedes to design a huge ship, the Syracusia, which could be used for luxury travel, carrying supplies, and as a naval warship. The Syracusia is said to have been the largest ship built in classical antiquity.[19] According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes screw was purportedly developed in order to remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a low-lying body of water into irrigation canals. The Archimedes screw is still in use today for pumping liquids and granulated solids such as coal and grain. The Archimedes screw described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon.[20][21][22]
Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse.
The 2nd century AD author Lucian wrote that during the Siege of Syracuse (c. 214?212BC), Archimedes destroyed enemy ships with fire. Centuries later, Anthemius of Tralles mentions burning-glasses as Archimedes' weapon.[25] The device, sometimes called the "Archimedes heat ray", was used to focus sunlight onto approaching ships, causing them to catch fire.
This purported weapon has been the subject of ongoing debate about its credibility since the Renaissance. Rene 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.[26] 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 onto a ship. This would have used the principle of the parabolic reflector in a manner similar to a solar furnace.
A test of the Archimedes heat 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 1m). The mirrors were pointed at a plywood mock-up of a Roman warship at a distance of around 160feet (50m). 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 may have aided combustion.[27]
In October 2005 a group of students from the Massachusetts Institute of Technology carried out an experiment with 127 one-foot (30cm) square mirror tiles, focused on a mock-up wooden ship at a range of around 100feet (30m). 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 device was a feasible weapon 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 (570°F).[28]
When MythBusters broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (or failed) because of the length of time and the ideal weather conditions required for combustion to occur. It was also pointed out that since Syracuse faces the sea towards the east, the Roman fleet would have had to attack during the morning for optimal gathering of light by the mirrors. MythBusters also pointed out that conventional weaponry, such as flaming arrows or bolts from a catapult, would have been a far easier way of setting a ship on fire at short distances.[1]
Cicero (106?43BC) mentions Archimedes briefly in his dialogue De re publica, which portrays a fictional conversation taking place in 129BC. After the capture of Syracuse c. 212BC, General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms used as aids in astronomy, which showed the motion of the Sun, Moon and five planets. Cicero mentions similar mechanisms designed by Thales of Miletus and Eudoxus of Cnidus. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius Philus, who described it thus:
Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. Through proof by contradiction (reductio ad absurdum), 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 31?7 (approximately 3.1429) and 310?71 (approximately 3.1408), consistent with its actual value of approximately 3.1416. He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle. In On the Sphere and Cylinder, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. This is the Archimedean property of real numbers.[37]
In Measurement of a Circle, Archimedes gives the value of the square root of 3 as lying between 265?153 (approximately 1.7320261) and 1351?780 (approximately 1.7320512). The actual value is approximately 1.7320508, 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."[38]
As proven by Archimedes, the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure.
In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4?3 times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio 1?4:
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 on the myriad. The word is from the Greek μυρι?? murias, for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion, or 8×1063.[39]
Archimedes is said to have remarked about the lever: Give me a place to stand on, and I will move the Earth.
그의 수학 연구는 다른 사람이 새로운 발견을 할 수 있도록 하고자 했다는,역학적인 정리들에 관한 방법에 표현된 그의 희망에도 불구하고 어떤 면에서도 고대에 계속되거나 발전되지 않았다. 회전체 부피를 결정짓는 등 그의 업적들을 확장시키려는 시도가 있었던 8세기 또는 9세기에 몇몇 수학논문들이 아랍어로 번역될 때까지 발전되지 않았다. 중세 초기의 아랍 수학자들에 의한 몇몇 가치 있는 연구는 이러한 아르키메데스의 연구에 의해 자극받았다. 그러나 그의 연구가 이후 수학자들에 미친 가장 큰 영향은 16, 17세기에 이르러 그리스에서 유래된 교과서와 그리스어 교과서 초판이 1544년 바젤에서 인쇄되면서 나타났다. 1558년 페데리코 코만디노에 의해 라틴어로 번역된 많은 아르키메데스의 저서는 그 지식의 전파에 크게 기여했는데, 이것은 요하네스 케플러와 갈릴레오를 비롯한 그당시 가장 유명한 수학자들과 물리학자들의 연구에 반영되어 있다.가장 뛰어난 아르키메데스를 포함한 고대 수학자들의 연구가 재발견되지 않았다면 1550~1650년 유럽에서의 수학의 발전은 생각할 수 없다. 불행히도 역학적인 정리들에 관한 방법 은 아랍과 르네상스의 수학자들에게 알려지지 않아 19세기 후반에야 비로소 재발견되었는데, 만약 르네상스 시대의 수학자들에게 알려졌다면 그들이 새로운 정리들을 발견할 때 자신의 계승자들이 이것을 이용할 것이라는 아르키메데스의 희망이 성취되었을 것이다.
Archimedes
From Wikipedia, the free encyclopedia
Archimedes Thoughtful by Fetti (1620)
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| Archimedes' Principle, Archimedes' screw, Hydrostatics, Levers, Infinitesimals | |
Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time.[2][3] He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi.[4] He also defined the spiral bearing his name, formulas for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers.
Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere inscribed within a cylinder. Archimedes had proven that the sphere has two thirds of the volume and surface area of the cylinder (including the bases of the latter), and regarded this as the greatest of his mathematical achievements.
Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c. 530AD by Isidore of Miletus, while commentaries on the works of Archimedes written by Eutocius in the sixth century AD opened them to wider readership for the first time. The relatively few copies of Archimedes' written work that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance,[5] while the discovery in 1906 of previously unknown works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.[6]
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Biography
Archimedes died c. 212BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he not be harmed.[10]
The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere and a cylinder of the same height and diameter. Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. In 75BC, 137 years after his death, the Roman orator Cicero was serving as quaestor in Sicily. He had heard stories about the tomb of Archimedes, but none of the locals was able to give him the location. Eventually he found the tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription.[11]
The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome. The account of the siege of Syracuse given by Polybius in his Universal History was written around seventy years after Archimedes' death, and was used subsequently as a source by Plutarch and Livy. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city.[12]
Discoveries and inventions
The Golden Crown
The story of the golden crown does not appear in the known works of Archimedes. Moreover, the practicality of the method it describes has been called into question, due to the extreme accuracy with which one would have to measure the water displacement.[16] Archimedes may have instead sought a solution that applied the principle known in hydrostatics as Archimedes' Principle, which he describes in his treatise On Floating Bodies. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.[17] Using this principle, it would have been possible to compare the density of the golden crown to that of solid gold by balancing the crown on a scale with a gold reference sample, then immersing the apparatus in water. If the crown was less dense than gold, it would displace more water due to its larger volume, and thus experience a greater buoyant force than the reference sample. This difference in buoyancy would cause the scale to tip accordingly. Galileo considered it "probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself."[18]
The Archimedes Screw
The Claw of Archimedes
The Claw of Archimedes is a weapon that he is said to have designed in order to defend the city of Syracuse. Also known as "the ship shaker," the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped onto an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it. There have been modern experiments to test the feasibility of the claw, and in 2005 a television documentary entitled Superweapons of the Ancient World built a version of the claw and concluded that it was a workable device.[23][24]The Archimedes Heat Ray ? myth or reality?
This purported weapon has been the subject of ongoing debate about its credibility since the Renaissance. Rene 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.[26] 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 onto a ship. This would have used the principle of the parabolic reflector in a manner similar to a solar furnace.
A test of the Archimedes heat 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 1m). The mirrors were pointed at a plywood mock-up of a Roman warship at a distance of around 160feet (50m). 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 may have aided combustion.[27]
In October 2005 a group of students from the Massachusetts Institute of Technology carried out an experiment with 127 one-foot (30cm) square mirror tiles, focused on a mock-up wooden ship at a range of around 100feet (30m). 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 device was a feasible weapon 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 (570°F).[28]
When MythBusters broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (or failed) because of the length of time and the ideal weather conditions required for combustion to occur. It was also pointed out that since Syracuse faces the sea towards the east, the Roman fleet would have had to attack during the morning for optimal gathering of light by the mirrors. MythBusters also pointed out that conventional weaponry, such as flaming arrows or bolts from a catapult, would have been a far easier way of setting a ship on fire at short distances.[1]
Other discoveries and inventions
While Archimedes did not invent the lever, he wrote the earliest known rigorous explanation of the principle involved. According to Pappus of Alexandria, his work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth." (Greek: δ?? μοι π? στ? κα? τ?ν γ?ν κιν?σω)[29] Plutarch describes how Archimedes designed block-and-tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that would otherwise have been too heavy to move.[30] Archimedes has also been credited with improving the power and accuracy of the catapult, and with inventing the odometer during the First Punic War. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.[31]Cicero (106?43BC) mentions Archimedes briefly in his dialogue De re publica, which portrays a fictional conversation taking place in 129BC. After the capture of Syracuse c. 212BC, General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms used as aids in astronomy, which showed the motion of the Sun, Moon and five planets. Cicero mentions similar mechanisms designed by Thales of Miletus and Eudoxus of Cnidus. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius Philus, who described it thus:
This is a description of a planetarium or orrery. Pappus of Alexandria stated that Archimedes had written a manuscript (now lost) on the construction of these mechanisms entitled On Sphere-Making. Modern research in this area has been focused on the Antikythera mechanism, another device from classical antiquity that was probably designed for the same purpose. Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing. This was once thought to have been beyond the range of the technology available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient Greeks.[34][35]Hanc sphaeram Gallus cum moveret, fiebat ut soli luna totidem conversionibus in aere illo quot diebus in ipso caelo succederet, ex quo et in caelo sphaera solis fieret eadem illa defectio, et incideret luna tum in eam metam quae esset umbra terrae, cum sol e regione. ? When Gallus moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze contrivance as in the sky itself, from which also in the sky the Sun's globe became to have that same eclipse, and the Moon came then to that position which was its shadow on the Earth, when the Sun was in line.[32][33]
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."[36]
In Measurement of a Circle, Archimedes gives the value of the square root of 3 as lying between 265?153 (approximately 1.7320261) and 1351?780 (approximately 1.7320512). The actual value is approximately 1.7320508, 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."[38]
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 on the myriad. The word is from the Greek μυρι?? murias, for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion, or 8×1063.[39]
Writings
The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse.[40]. The written work of Archimedes has not survived as well as that of Euclid, and seven of his treatises are known to have existed 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.[b] During his lifetime, Archimedes made his work known through correspondence with the mathematicians in Alexandria. The writings of Archimedes were collected by the Byzantine architect Isidore of Miletus (c. 530AD), while commentaries on the works of Archimedes written by Eutocius in the sixth century AD helped to bring his work a wider audience. Archimedes' work was translated into Arabic by Th?bit ibn Qurra (836?901AD), and Latin by Gerard of Cremona (c. 1114?1187AD). 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.[41] 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.[42]Surviving works
- 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, "Magnitudes are in equilibrium at distances reciprocally proportional to their weights."
- Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, parallelograms and parabolas.[43]
- 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 only a rough figure is required.
- 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
- 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 (including its two bases), where r is the radius of the sphere and cylinder. The sphere has a volume and surface area two-thirds that of the cylinder. A sculpted sphere and cylinder were placed 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.
- 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 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.
- 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 could be assembled into the shape of a square. Dr. Netz calculates that the pieces can be made into a square 17,152 ways.[44] The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded.[45] 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 (στ?μαχο?).[46] Ausonius refers to the puzzle as Ostomachion, a Greek compound word formed from the roots of ?στ?ον (osteon, bone) and μ?χη (mach? - fight). The puzzle is also known as the Loculus of Archimedes or Archimedes' Box.[47]
- 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 Wolfenbuttel, Germany in 1773. It is addressed to Eratosthenes and the mathematicians in 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. Amthor[48] in 1880, and the answer is a very large number, approximately 7.760271×10206,544.[49]
- 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, as well as 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 8×1063 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.[50]
- 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.[51]
It has also been claimed that Heron's formula for calculating the area of a triangle from the length of its sides was known to Archimedes.[c] However, the first reliable reference to the formula is given by Heron of Alexandria in the 1st century AD.[52]
Archimedes Palimpsest
Main article: Archimedes PalimpsestThe foremost document containing the work of Archimedes is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg visited Constantinople and examined a 174-page goatskin parchment of prayers written in the 13th century AD. He discovered that it was 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, which was a common practice in the Middle Ages as vellum was expensive. The older works in the palimpsest were identified by scholars as 10th century AD copies of previously unknown treatises by Archimedes.[53] 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 New York.[54] 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.[55]
The treatises in the Archimedes Palimpsest are: On the Equilibrium of Planes, On Spirals, Measurement of a 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, as well as a lunar mountain range, the Montes Archimedes (25.3° N, 4.6° W).[56]The Fields Medal carries a portrait of Archimedes.
The asteroid 3600 Archimedes is named after him.[57]
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).[58]
Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).[59]
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.[60]
A movement for civic engagement targeting universal access to health care in the US state of Oregon has been named the "Archimedes Movement," headed by former Oregon Governor John Kitzhaber.[61]
See also
- 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 (στ?μαχο?).[46] Ausonius refers to the puzzle as Ostomachion, a Greek compound word formed from the roots of ?στ?ον (osteon, bone) and μ?χη (mach? - fight). The puzzle is also known as the Loculus of Archimedes or Archimedes' Box.[47]
- Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, parallelograms and parabolas.[43]
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